FUNDAMENTAL DISCRETE ELEMENT CHARGE ... - Science Direct

Minerals Engineering, Vol. 11, No. 12, pp. 1161-1178, 1998

Pergamon 0892-6875(98)00103--4

© 1998 Elsevier Science IAd All rights re~rved 0892-6875/98/$- see front matter

FUNDAMENTAL DISCRETE ELEMENT CHARGE MOTION MODEL VALIDATION

P. RADZISZEWSKY *° and S. MORRELL* § D6partement des sciences appliqu~es, Universit~ du Qu6bec en Abitibi-T~miscamingue, • ~5, blvd de l'Universit~, Rouyn-Noranda, Qu6bec J9X 5FA, Canada E-mail: Peter.Radziszewski @uqat.uquebec.ea t Julius Kruttschnitt Mineral Research Centre, Isles Road, Indooroopilly, Brisbane, Queensland 4068, Australia • On sabbatical leave at the JKMRC (Received 27 April 1998; accepted 6 July 1998)

ABSTRACT

With improving and faster computers, research into understanding the complex interactions of mineral processes is now a practical proposition for aiding design and optimization. In comminution research, recent trends have been made to describe internal dynamics of mills and relate them to such things as power draw [1-3]. In this work, a charge motion model is described that takes advantage of the empirical approach of Morrell while being anchored in the fundamentals of discrete element models such as those of lnoue and Mishra/Rajamani. In this present work, a fundamental discrete element ball charge motion model is defined with the help of an arbitrary discretization scheme of the charge. The model is then calibrated and validated using two ball mill data sets with simu,~ated power being compared with observed power. Further, this technique permits ct~rge motion to be calculated while determining a frequency distribution of impacts for any given mill be it grate or overflow discharge. © 1998 Elsevier Science Ltd. All rights reserved Keywords Grinding; modelfing INTRODUCTION With improving and faster computers, research into understanding the complex interactions of mineral processes is now a practical proposition for aiding design and optimization. In comminution research, recent trends have been made to describe internal dynamics of mills. Here one can cite the recent works of Morrell [1], Inoue [2], Mishra and Rajamani [3] and Radziszewski [4,5]. In Morrell's work [1], the determination of charge power draft was determined using an empirical charge motion model developed from experimental data and observation of charge motion. In particular, the Dmodel as programmed on a PC gives good results with a mean relative error of < 0.2 % and a standard deviation of 3.3 % for the cases tested [1]. In the case of Inoue [2] and Mishra/Rajamanis' work [3], they have developed discrete element models of charge motion simulating the motion of each in(fividual ball in Presented at Comminuaon '98, Brisbane, Australia, February 1998

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P. Radziszewskyand S. Morrell

the mill charge. These models allow for the determination of charge motion and power draft calculation as a function of the physical and operating parameters. They also allow the determination of the frequency distribution of media collision energies, a necessary prerequisite for the determination of breakage products and media/liner wear [5,6,7]. Each of these efforts present certain advantages and disadvantages. In this paper, a charge motion model is described that takes advantage of the empirical approach of Morrell while being anchored in the fundamentals of the discrete element approach as adopted by Inoue and Mishra/Rajamani. The original model is described elsewhere [4,5,8,9]; here it will be calibrated and validated as a discrete element model for charge motion.

FUNDAMENTAL ASPECTS OF BALL MILL CHARGE MOTION Particle breakage, as well as media and liner wear, occur through the impact and/or sliding motion of the media in a mill. Both of these actions continually nip and break ore while simultaneously impacting and abrading ball and liner surfaces. These events, breakage and wear, are the final ones in a chain of energy transformations which commence with the rotation of the mill. A typical charge motion profile (Figure 1) shows four zones that can be characterized by the type of action produced there, namely: the flight zone where balls in flight follow a parabolic path under the action of gravity; the crushing zone where balls in flight re-enter the charge and crush rock particles at relatively high energies; the grinding zone where ball layers slide over one another grinding the material trapped between them; and finally, the tumbling zone where balls roll over one another and break material in low energy breakage.

~NE OF" ~LLING BALL

IT

~

I"t~O q~,r ,|m,

V

Fig. 1 Typical ball charge motion profile. As a ball charge is a system of interdependent and interactive elements all working together to achieve charge motion, it is necessary to decompose this system into its constituent elements and describe the interactions between them. Essentially, the elements of this system are the balls, the liner (riflers) and the mill shell. The interactions between these elements occur cyclically as can be described using the four quadrants of a circle (I, H, HI, IV).

Fundamental discrete element charge

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Quadrant I relationships Here only one type of interaction needs to be described, that being ball flight as defined using flight trajectory relationships under gravity, those being:

x:v~At+Xoi

(1)

v~=v~ y= -~g A tZ+v~At+y~ v~=-gAt +vyoi where: Xoi, Yoi, Vxoi, Vyoi are the coordinates and velocities of ball i at time t; xi, Yi, Vxi, Vyi are the coordinates and velocities of ball i after time At; g is the acceleration due to gravity. Quadrant H relationships

Examining quadr~at II of the charge profile, one can define two relationships describing phenomena in this region: slurry pool[ effect on ball trajectories and foot or toe stability.

Slurry pool Ball mills have one of two discharge mechanisms: grate or overflow. In the case of grate discharge, no slurry pool will form unless flow rates are excessive and therefore do not affect ball trajectories. However, this is not the case for overflow mills where a slurry pool is intrinsic to this type of mill discharge. As such, a ball entering a slurry pool experiences a deceleration due to the viscous effects of the slurry as well as buoyancy forces. Viscous effects can be described using the drag coefficient of spheres as a function of ball velocity, diameter and slurry viscosity, while buoyancy can be expressed as a function of slurry density and ball volume. Taking these factors into account, ball trajectories through a slurry pool can be defined as follows:

x,i = vnAt +xo:lanAt z va ffi v~+a~At y = -1QAt~" +v .At+y .+la ~AtZ+ 1 Pst~trgAt: 2"

~

o~ 2 y,

2p,m ~

v~ = - 8 A t +vyot+a~At + P~ gAt Pmml where:the deceleration due to drag in the x direction is defined as:

(2)

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P. Radziszewskyand S. Morreil

2

3Cdpswryv2 4Pmetaf/b~

_

%

mb~

(3)

Pslurry, Praetal are slurry and ball metal densities [kg/m3], A is the cross-sectional area of the ball [m2], and the drag coefficient for a sphere as a function of Reynolds number Re is approximated using:

(4)

C d=O.2021e4.7297R, "°'2m similarly the deceleration can be defined in the y-direction.

Foot stability The criterion for ball displacement at the charge foot is a moment calculation (see Figure 2) which describes the stability of a given ball at the charge foot giving:

unstable mb~Zl-mbc'qoJZ21)O: stable 2 f