.Powder Technology 109 2000 105112www.elsevier.comrlocaterpowtec
Discrete element analysis of tumbling millsR.K. Rajamani a,), B.K. Mishra b, R. Venugopal a, A. Datta a
a Department of Metallurgical Engineering, Uniersity of Utah, Salt Lake City, UT 84112, USAb Department of Materials Science and Metallurgy, Indian Institute of Technology, Kanpur, UP 208016, India
Accepted 21 September 1999
The tumbling mill simulation problem has been intractable due to the multibody collision events involving thousands of balls androcks. The harsh environment in the mill prevents any intrusion with measuring sensors. Simpler physical models with some empiricism
.could not describe the charge motion within the mill adequately. The discrete element method DEM provides a fitting solution to this .problem. In this paper, the formulation of both two- and three-dimensional 2D and 3D, respectively DEM algorithms for the tumbling
mill is presented. Experimental validation of the models is also presented. q 2000 Elsevier Science S.A. All rights reserved.
Keywords: Discrete element method; Tumbling mills; Modeling
The mining industry routinely uses ball mills and semi-autogenous mills for processing ore bodies. A typical minesite processes 80,000 tonsrday of ore employing twosemi-autogenous mills and four-ball mills. Since the en-
w xergy consumption is 25 kW hrton of ore 1 for grindingrock particles to sieve mesh sizes, a substantial portion ofthe cost of ore processing is expended in tumbling mills.
A tumbling mill is a cylindrical drum fitted with conicalend plates on both sides. The drum is filled with steel ballsoccupying about 30% of the volume. The internal shell ofthe drum is fitted with rectangular bars, called lifters, tohelp carry the ball and rock charge as the drum rotates.Mineral ore or rocks are fed on one end of the drum anddischarged through the other. As the drum rotates, the ballsand rocks tumble, which leads to grinding of the ore. The
.discrete element method DEM is ideally suited for theanalysis of charge motion in such a slowly rotating cylin-drical device.
The energy efficiency of tumbling mills can be exam-ined by directly looking at the motion of rocks and steelballs inside the mill. The make-up of the charge and thelifter bars attached to the inside of the mill shell can be
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designed particularly to maximize power drawn by themill, and hence capacity. At the same time, the unneces-sary collisions of steel balls against the mill shell can beavoided. Furthermore, the cascading charge flow can bealtered in such a way as to maximize grinding efficiency.
The environment inside a tumbling mill of 4-m diame-ter, considered small in the industry, is so harsh that any ofthe sensors available has a very slim chance of survivingfor a reasonable duration of time. The only informationavailable from an operating mill is the instantaneous powerdraft. It has long been believed that maximizing powerdraft implies maximizing mill capacity. Unfortunately, theimpact of steel balls directly on the shell plates has notbeen understood sufficiently for lack of sensors. However,as numerical methods advanced, such as the DEM, thesimulation of charge motion in tumbling mills began to
w xemerge 2 . In the last 8 years, the DEM for simulation oftumbling mills has advanced sufficiently that it has be-
w xcome a very practical tool 3 in the mining industry.This manuscript gives an overview of the DEM as
applied to the tumbling mill problem. Both the two- and .three-dimensional 2D and 3D, respectively models are
described, followed by experimental validation.
2. Discrete element method
The DEM refers to a numerical scheme that allowsfinite rotations and displacements of rigid bodies, where
0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. .PII: S0032-5910 99 00230-2
( )R.K. Rajamani et al.rPowder Technology 109 2000 105112106
complete loss of contacts and formation of new contactsbetween bodies can occur as the calculation cycle pro-
w xgresses. DEM was first proposed by Cundall and Strack 4to model the behavior of soil particles subjected to dy-namic loading conditions. Since its inception this tech-nique has been adapted to model a variety of physicalsystems as an alternative to the continuum mechanics
w xapproach 58 . Here, a brief description of the 2D DEMis given that is particularly useful to model motion of largesteel balls inside tumbling mills.
In the DEM the grinding media are modeled as discs intwo dimensions, where a given disc moves according tothe forces acting on it. When a contact between two discsis detected, the collision is modeled by a pair of spring-and-dashpot, one in the normal direction and one in thetangential direction. In other words, the discs are allowedto overlap at the boundaries according to a contact model.The acceleration of the body is computed from the netforce, which is then integrated for velocity and displace-ment.
Referring to Fig. 1, every disc is identified separately,virtual overlap is allowed at each contact point, and thecalculation is done for every disc in turn. The relativevelocity of the disc i with respect to the discs in contact .discs k, l, m, n, o, p, and q is first determined. Theserelative velocities for every contact of disc i are resolved
in the normal along the line drawn through the centers of
Fig. 1. Schematic representation of an assembly of discs and the spring-and-dashpot model in the normal and tangential directions at a contacttangential.
.a pair of discs in contact and tangential direction and theforce calculation is then done for each contact as follows:D F sk D tn n nD F sk D ts s s 1 .Dd sc n n nDd sc s s swhere, in an incremental time D t, D F and D F are then sincremental forces due to the springs, Dd and Dd are then sincremental forces due to the dashpots, and are then srelative velocities, and k and c are the spring stiffness anddashpot constant, respectively. Then, the contact forcesand other body forces acting on the disc are vectoriallysummed to determine the net out-of-balance force actingon it. The acceleration of disc i of mass m is given by:i
1x s F i x
1y s F 2 .i y
1u s M ,i 0I0 i
where x and y are the acceleration in the x and y directions respectively, u is the angular acceleration, I is0 i
the moment of inertia of the disc i, and M is the total0moment acting on the disc.
In light of the spring-and-dashpot model of collision,the tangential force due to the dashpot is limited by themaximum that can exists at the contact, which is given by:F smF , 3 .s ,max nwhere m is the coefficient of friction and F is the normalnforce at the contact. If the absolute value of the tangentialforce in the spring-and-dashpot exceeds F , then slip iss,maxpresumed to occur. In this situation, during the computa-tion, the dashpot in the tangential direction is omitted andthe F value is used instead.s,max
Since the model deals with individual contacts, it isnecessary to get realistic values of the disc-to-disc anddisc-to-wall contact properties. These parameters are mate-rial stiffness, coefficient of restitution, and coefficient offriction. Material stiffness property correctly establishesthe forces generated in the spring. The coefficient ofrestitution is a measure of the damping property of thematerial, and hence it figures in the dashpot constant of thematerial, which in turn establishes the forces developed inthe dashpot.
A method for measuring material stiffness is with anw xapparatus called an ultra-fast load cell 9 . It consists of a
long vertically mounted steel rod in which fast-respondingstrain gauges are imbedded. When a steel ball is made tostrike the rod, the strain gauges record the primary waveand reflected waves. From these recordings, the force vs.displacement curve can be determined. Simply, the slope
( )R.K. Rajamani et al.rPowder Technology 109 2000 105112 107
of this curve is the required material stiffness. In thisapparatus, a bed of ore particles can be interposed betweenthe striking ball and the rod, and hence an effectivestiffness for the tumbling mill application can be found
w xreadily 2 .The tangential stiffness is determined from a theoretical
standpoint. For a Poisson ratio below 0.5, it is known fromHertzian contact theory for spheres that the tangentialstiffness k may vary from 2r3 to 1 of the normalsstiffness k . Hence, k is taken as 2r3 of the value of k .n s n
The value of the damping constant is estimated from thecoefficient of restitution. Since the latter is the measure of
w xthe energy loss upon collision, Corkum 10 showed thatfor a given coefficient of restitution e the damping con-stant l is given by:c
2 ln e k m .( nl sy , 4 .c 2 2(ln e qp . .where msm sm r m qm and m and m are the1 2 1 2 1 2
masses of the colliding discs. If m is very large in1comparison to m , as in the case of the tumbling mill2walls, then m is just the mass of the disc.
A simple explicit leapfrog integration algorithm alge-braically equivalent to the popular Verlet scheme wasfound to be sufficient for the tumbling mill simulation
.problem. A typical acceleration quantity in Eq. 2 , such asx, is integrated from t to t to calculate the ny1r2 nq1r2velocity as follows:
x s x q x D t . 5 . . . . nq1r2 ny1r2i i iThe value of the velocity calculated for time t isnq1r2used to compute displacement:
x s x q x D t . 6 . . . .nq1r2 nq1r2i i in .The other two acceleration quantities in Eq. 2 were
integrated in a similar manner.The stability of the numerical algorithm is important in
assuring accurate results. Since the integration schemeadopted is a central difference scheme, the numericalstability depends on the time step chosen. A stability
w xanalysis 4 leads to a critical value given by:
D t-2 mrk , 7 .( nwhere m is the smallest mass of discs present in thesystem. Failure to ensure this condition resulted in unreal-istic motion of the discs.
There are a number of contact models of varying com-plexity. However, in the context of colliding metal spheres
w xthe relevant model is the Hertz solution 11 for the normalforce between elastic spheres that increases as the 3r2
.power of the relative approach or overlap of the spheres.On the other hand, measurements of forcedisplacement of
w xmetal spheres 12 exhibit a mildly non-linear contactbehavior that can be approximated to a linear contactmodel. The linear model used in this tumbling mill prob-
lem is supported by the fact that the simulation resultswere verified experimentally in a variety of laboratory-sizeand industrial-scale tumbling mills.
2.1. Boxing scheme and contact search
The computation of the net out-of-balance force on adisc requires the evaluation of the forces exerted on thedisc at all its contacts. Therefore, it becomes essential tokeep track of all the elements that are in contact with agiven ball at every step in time. This procedure is referredto as a contact search. Regardless of the shape of elementsinvolved, simulation of N interacting particles with DEM
.involves an N Ny1 r2-pair of contacts search problem.However, the time spent on searching can be reduced to .O N by dividing the entire working area into squares. In
the DEM literature, this procedure is referred to as boxingw x4 . The dimension of a cell is set at the maximumdiameter of a disc. A disc is supposed to be contained inall those boxes where the corners of its circumscribingsquare have an entry. In the case of a line elementcorresponding to the mill walls, all the boxes throughwhich the line passes are identified and used for contact
detection. Once the box list the list of all elements in a.given box is generated, only those elements that have
entries into boxes associated with a given disc are assumedto be in potential contact with it. Actual disc-to-disccontact is calculated from the coordinates of the disccenters. Once the contact is determined, then the amountof overlap is found which in turn is used in the contactmodel to compute the contact force. After the force calcu-lation and integration of the equation of motion, the posi-tions of the discs are updated, and accordingly, the neigh-
Fig. 2. Typical snapshot of charge motion in an 11-m-diameter semi-au-togenous mill.
( )R.K. Rajamani et al.rPowder Technology 109 2000 105112108
Table 1Simulation parameters for mill power analysisParameter Steelsteel Steelrubber
. .Normal stiffness 400,000 Nrm 234,000 Nrm . .Tangential stiffness 300,000 Nrm 176,000 Nrm
Coefficient of restitution 0.6 0.45Coefficient of friction 0.5 0.9
bor list is updated. Maintaining the neighbor list reducesthe searching effort.
2.2. Energy input
The two quantities of interest within the scope of thecurrent work are mill power draft and the impact energyspectra. Mill power draft is readily obtained from thesimulation of ball charge motion. The power supplied tothe mill is expended to maintain the ball charge in motion.While doing this work, energy is also lost in friction andcollisions. At each collision, a part of the total energy islost, which is mimicked by the dashpot. Thus, the additionof the product of normal and tangential force on thedashpot and respective displacement gives the energy lostat that contact. The energy lost in two dimensions isexpressed as:
Es F D tqF D t . 8 . normal normal tangential tangentialt k
Here, F is the dissipative force and is the velocity. Asshown in the above expression, the energy loss term is
.summed up over all the collisions k and for all the time .steps t . During this calculation, the energy associated
with each of the collisions is maintained in a record, whichat the end constitutes the total energy loss. A frequencyplot of the number of collisions vs. the collision energy isknown as an impact energy spectrum and is a valuable aid
w xin the practical study of tumbling mills 13 .
2.3. Computer implementation
The algorithm described above has been implementedin a computer code called Millsofte. This 2D code gener-
ates the circular mill shell fitted with lifter bars up to amill diameter of 13 m. Then, it generates a distribution ofdiscs of various sizes until the mill is filled to a prescribedlevel. First, the code allows the discs to settle on thebottom of the mill shell, after which the shell begins torotate. A typical semi-autogenous mill of diameter 12 mrequires 7500 discs of sizes 25 to 150 mm in the simula-tion. The simulation time for two full revolutions of themill, i.e., 24 s of real time, is 10 h on a 450-MHz Pentiumprocessor with Windows 95 operating system. The codestores the instantaneous position, velocity, and force infor-mation for all entities in the simulation. A typical snapshotof the charge motion in an 11-m-diameter tumbling mill isshown in Fig. 2. Fig. 2 shows a number of discs in
.cataracting parabolic trajectory from 2 oclock positionmotion and a greater number of discs in cascading discs