Discrete element modelling of the influence of lifters on powerdraw of tumbling mills q

N. Djordjevic

Julius Kruttschnitt Mineral Research Centre, University of Queensland, Brisbane, Australia

Received 11 December 2002; accepted 13 January 2003

Abstract

Crushing and grinding are the most energy intensive part of the mineral recovery process. A major part of rock size reduction

occurs in tumbling mills. Empirical models for the power draw of tumbling mills do not consider the effect of lifters. Discrete element

modelling was used to investigate the effect of lifter condition on the power draw of tumbling mill. Results obtained with PFC3D

code show that lifter condition will have a significant influence on the power draw and on the mode of energy consumption in the

mill. Relatively high lifters will consume less power than low lifters, under otherwise identical conditions. The fraction of the power

that will be consumed as friction will increase as the height of the lifters decreases. This will result in less power being used for high

intensity comminution caused by the impacts. The fraction of the power that will be used to overcome frictional resistance is de-

termined by the material�s coefficient of friction. Based on the modelled results, it appears that the effective coefficient of friction for

in situ mill is close to 0.1.

� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Comminution; SAG milling; Modelling

1. Introduction

Lifters are an essential element in the tumbling action

of a grinding mill, whose purpose is to prevent slippage

of the mill charge. Such slippage consumes energy

without substantial breakage, and the mill may not lift

the charge sufficiently high to promote impact breakage.Hence, the size, shape and number of lifters will have a

strong impact on the tumbling action of the media

(King, 2001). The power required to operate large mills

is typically 5–10 MW. Hence, optimisation of the power

consumption will have a significant impact on the

overall economic performance and environmental im-

pact of the mineral processing plant.

Empirical models for the power draws of the tum-bling mills do not consider the effect of lifters (Morrell,

1996; Napier-Munn et al., 1996; King, 2001). Input

parameters are size of the mill, rotational speed and

parameters related to the rock and steel balls. The issues

of the lifter height and number are not considered.

The computing power of modern desktop computers

presents an opportunity to use a discrete element mod-

elling (DEM) for mill modelling (Mishra and Rajamani,

1994), in this case Particle Flow Code 3D (Itasca Inc.,

1999). Power draw calculated using PFC3D refers to the

total power draw accumulated for each rock and steel

ball in the mill. In order to compare modelled powerwith measured power, it is necessary to add an estimate

of the power required to rotate the empty mill.

Cleary (2001) used DEM to investigate the effect of

lifter pattern of the specific power draw of the 2D mill.

He concluded that substantial influences of the lifter

pattern occur only for relatively high rotational velocities

of the mill (greater than 80% of critical). He noticed that

by reducing the number of lifters by half, the specificpower draw (kW/tonne) of the mill could be increased

substantially. This means that more efficient grinding can

be achieved by reducing the number of lifters (from 24 to

12 in his 2D model of the 5 m mill) (Cleary, 2001).

2. Discrete element modelling using PFC3D code

PFC3D code models the behaviour of particles,

which may be enclosed within a finite volume by the

qPresented at Minerals Engineering �02, Perth, Australia, September

2002.

E-mail address: n.djordjevic@uq.edu.au (N. Djordjevic).

0892-6875/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0892-6875(03)00019-0

Minerals Engineering 16 (2003) 331–336This article is also available online at:

www.elsevier.com/locate/mineng

non-deformable walls. The code keeps a record of in-

dividual particles and updates any contact with other

particles or walls. Each calculation step includes appli-

cation of the laws of motion to a particle, a force–dis-

placement law to each contact, and constant updating of

wall position (Cundall and Strack, 1979).

The PFC3D modelling is based on the assumptionthat the individual particles (balls) are treated as stiff

bodies. At contacts, rigid particles can overlap. The

magnitude of the overlap is related to the contact force.

Overlaps are small relative to the size of the particles.

Individual particles can also be bonded to form clusters

to mimic rock shape and strength.

During contact, the behaviour of a material is simu-

lated using a linear contact model. The contact forcevector between two balls or ball and wall is composed of

a normal and shear components. The normal contact

force vector is calculated using the formula:

Fn ¼ KnUnni

where Kn is normal stiffness at the contact; Un is the

relative contact displacement in the normal direction; niis unit normal vector.

The incremental shear force is calculated using the

formula:

DFs ¼ �Ks DUs

where Ks is the shear stiffness at contact; DUs is the in-cremental shear displacement at contact.

PFC3D also includes a slip model. The slip model is

defined by the friction coefficient at the contact, where

the active relevant friction coefficient is taken to be the

minimum friction coefficient of the two contacting en-

tities. Each contact is checked for slip conditions, by

calculating the maximum allowable shear contact force:

Fs ðmaxÞ ¼ labsðFnÞwhere l is the friction coefficient; Fn is the normal force.

The energy state of the entire set of particles can be

examined by recording various forms of energy. Fric-

tional work is defined as the total cumulative energy

dissipated by frictional sliding at all contacts. Intensity

of deformation can be assessed using strain energy,

which is defined as the total strain energy stored at all

contacts assuming a linear contact-stiffness model.

The PFC3D model of the mill is composed of anumber of walls which represent liner and lifters as well

as balls which represents mill charge. The power of the

mill is calculated for each instant of time by summing

products of moments that are applied to the mill liner

and lifters and rotational velocity of the mill.

A critical aspect of DEM is selection of material

parameters. Parameters such as material stiffness, coef-

ficient of friction, and damping ratio may affect not justthe value of the power draw but also computational

time. Considering that realistic full 3D modelling of the

tumbling mills may require many thousands of particles,

the ability to produce a modelling result within a rea-

sonable time is of large practical significance.

Strain energy at contact can be defined as:

E ¼ 0:5ðF 2n =Kn þ F 2

s =KsÞwhere E is the strain energy stored at the contact of two

colliding objects; Fn, Fs are the magnitudes of the normal

and shear component of the contact force; Kn, Ks are the

normal and shear contact stiffness.

The contact stiffness is computed assuming that the

stiffness�s of two contacting entities (ball and wall) act in

series:

Kn ¼ KbKw=ðKb þ KwÞwhere Kb, Kw are the normal stiffness of the ball and

wall; Kn is the normal contact stiffness.In the case of perpendicular impact, shear component

can be ignored. Contact force is described with formula:

Fn ¼ KnUn

where Kn is the normal contact stiffness; Un is total

normal displacement.

By combining these equations, for the case of normal

impact, the strain energy can be described with the

formula:

E ¼ KnU 2n

Considering that normal displacement at contact is

inversely proportional to the contact stiffness, it is likely

that an increase of stiffness of the ball will not have a

dramatic influence on the amount of strain energy.Within a mill, all energies eventually need to come from

the external power source. Consequently, induced strain

energy is directly proportional to the power. In order to

preserve the accuracy of the computation, sampling time

was chosen to always be 20% of the minimum stable

critical time step for the particular modelling case.

Previous work (Djordjevic, 2002), has showed that a

full sized mill can be reliably represented by a verticalslice of sufficient thickness. In absence of the fluid in-

duced pressure gradient, the motion of the particles

within the PFC3D model is essentially constrained

within a relatively narrow disk, with the main motion

occurring in the vertical plane, while mixing of the

particles along the length of mill is relatively insignifi-

cant. The previous results also showed that in terms of

power draw, the PFC3D model power is consistent withthe power draw predicted using empirical models.

Hence, as a standard case we used one meter thick slices

for a mill with a diameter of 5 m.

3. Influence of lifters height on power

The effect of the lifter height is computed in terms ofenergy distribution that occurs within the mill. Some

332 N. Djordjevic / Minerals Engineering 16 (2003) 331–336

fraction of the power required to rotate the mill will be

consumed in friction between balls and between balls

and the mill shell. We postulate that the fraction of the

input power that will be consumed in friction will in-

crease with a decrease of the lifter height. With increases

in power consumption as frictional energy, the influence

of rock abrasion will increase. This will result in lessenergy being consumed in high-energy impact breakage.

Consequently, the overall rate of size reduction within

the mill will decrease, although the product may become

finer.

Influence of the lifter height on the power draw of the

mill and amount of power that will be consumed in the

form of frictional energy has been investigated. Fric-

tional forces will be developed at contacts of balls andballs and mill liner. Frictional energy is converted into

frictional power by dividing accumulated energy by

time. Within the PFC3D code, frictional work dissi-

pated by frictional sliding at all contacts (ball/ball or

ball/wall) is defined as:

Ef ¼X

Fis DUis

where Ef is the cumulative frictional work or energy; Fisis the average shear force at the specific contact for the

current time step; DUis is the average increment of slip

displacement at the contact for the current time step;P

is the summation for all contacts for the current time

step.

Power (Pf ) required for frictional work in a certain

time interval (DT ) is given by:

Pf ¼ Ef=DT

The modelled mill was 5 m diameter, with 20 lifters

with height of 0.2, 0.1 and 0.05 m. The width of the

lifters remained constant (0.1 m). Angular rotational

velocity of the mill was kept constant, at 70% (or 1.38

rad/s) of its critical speed. Relatively low values of mill

fill factor of 5% and 10% were used in order to reduce

the number of balls, which makes it possible to complete

modelling within a reasonable time using a standarddesktop personal computer.

The properties of balls are given in Table 1.

The heights of the lifters were chosen to equal 2, 1 or

0.5 ball diameters, or high, medium and low lifters. Low

lifters correspond to the case where due to severe wear

and tear, lifters were reduced to 25% of their initial

height. The pattern of ball motion and their velocity

vectors for three types of lifters for constant speed and

fill factor is shown in Figs. 1–3.

High lifters are able to more efficiently lift a larger

numbers of balls. This reduces the thickness of the mill

charge in the base of the mill. Due to the slightly longer

contact time with the mill shell, balls detach later from

the mill. After free fall, they tend to intercept the millshell at a higher position. Clearly, the number of balls

that are involved in tumbling action per unit of time is

larger with high lifters.

The highly variable net-power draw time history is

shown in Fig. 4, smoothed time history in Fig. 5. This

net-power draw refers to the 1 m thick slice of the 5 m

mill. Hence, for any comparison with real mills, it is

Table 1

Properties of balls used for the modelling

Diameter 0.1 m

Density 2500 kg/m3

Normal stiffness 0.1 MN/m

Shear stiffness 0.1 MN/m

Coefficient of friction 0.3 (0.1)

Viscous damping coefficient 0.3

Coefficient of restitution 0.38

Fig. 1. Ball motion pattern and velocity vectors for the mill with high

lifters.

Fig. 2. Ball motion pattern and velocity vectors for the mill with

medium lifters.

Fig. 3. Ball motion pattern and velocity vectors for the mill with low

lifters.

N. Djordjevic / Minerals Engineering 16 (2003) 331–336 333

necessary to multiply the values by the length of the

equivalent 5 m mill and add power for effect of slurry

and no-load power.Beside net-power, frictional power was modelled.

Frictional power is defined as the ratio of cumulative

frictional energy at all contacts within the mill divided

by cumulative time. Frictional energy is only a fraction

of the net-power consumed by the mill (Fig. 6). Ac-

cording to the empirical models (Morrell, 1996; King,

2001), the power required to overcome frictional resis-tance, losses due to vibration and heat within the mill is

about 20% of the total net-power draw of the mill.

The results show that frictional power is also influ-

enced by the size of the lifters. As lifters height decrease,

Fig. 4. Unsmoothed time record of the modelled mill power draw.

Fig. 5. Smoothed time history of the net-power draw.

334 N. Djordjevic / Minerals Engineering 16 (2003) 331–336

the amount of power which is consumed to overcome

friction increases. For constant coefficient of friction

(0.3), and fill factor (5%), the influence of the lifter

height on the net-power draw and frictional power draw

are given in the Fig. 7.The ratio of the frictional-power to net-power clearly

increases with decrease in lifter height. The modelled

ratios are much higher than 20%. This could be due to a

relatively high coefficient of friction (0.3) used in this

modelling, or due to the low filling factor. In the case of

the real mills, the presence of the slurry will act as a

lubricant, reducing the effective coefficient of friction.

The influence of a low coefficient of friction (0.1) on thepower draws is presented in Fig. 8. The ratio of the

frictional power and net-power draw is now close to 0.2,

which corresponds favourably with empirical models.

For the extreme case when the coefficient of friction is

nil there will be no frictional losses within the mill.

However, this does not result in overall net-power draw

being lower. Hence, it appears based on this modelling

that the coefficient of friction will influence how theapplied power will be consumed within the mill (friction

or impacts), but it will not have a significant effect on the

overall power draw of the mill. This confirms the results

obtained earlier by Cleary (2001).

Another contributing factor to the relatively high

values of the frictional power is low fill factor (5%).

Empirical models are developed with typical fill factors

in the range of 35–45%. An increased fill factor will re-duce the relative amount of power required to overcome

frictional resistance within the charge, and at the con-

tacts of balls and mill shell. Results show that when the

Fig. 7. Influence of the lifter height on the power draw. Fig. 8. Influence of the balls coefficient of friction on the power draw.

Fig. 6. Smoothed time history of the power consumed by friction.

N. Djordjevic / Minerals Engineering 16 (2003) 331–336 335

fill factor is increased from 5% to 10%, the fraction of

power consumed as friction decreases from 0.4 to 0.34

for the case of high lifters. A similar trend is also ob-served for the case of medium high and small lifters (Fig.

9). Although increased fill factor may reduce the relative

significance of the frictional energy, it is clear that the

major influence is the coefficient of friction. In the real

mills effective coefficient of friction, due to effect of

water, is probably quite low.

4. Conclusions

DEM results obtained with the PFC3D code show

that the lifter condition will have a significant influence

on the power draw and on the mode of energy con-

sumption with the mill. Mill with relatively high lifters

will consume less power than mill with low lifters, under

otherwise identical conditions. The fraction of the powerthat will be consumed as friction will increase as the

height of the lifters decreases. This means that less

power is available for high intensity comminution

caused by impacts. The fraction of the power that must

be used to overcome frictional resistance is determined

by the material�s coefficient of friction. Based on the

modelled results, it appears that effective coefficient of

friction for real mill is close to 0.1. Slurry acts as a lu-

bricant, reducing the coefficient of friction of dry feed or

steel balls. When higher value for coefficient of friction is

used in modelling (0.3), the fraction of net-power con-

sumed for friction is much higher than that which em-pirical models suggests (0.2).

Acknowledgement

The author wishes to thank Dr. Robert Morrison for

the useful discussions and helpful comments.

References

Morrell, S., 1996. Power draw of wet tumbling mills and its

relationship to charge dynamics. Part 2: an empirical approach to

modelling of mill power draw. Trans. Inst. Mining Metall. (Section

C: Mineral Process. Ext. Metall.) 105, C54–C62.

Napier-Munn, T.J., Morrell, S., Morrison, R.D., Kojovic, T. Mineral

Comminution Circuits. Their Operation and Optimisation. Julius

Kruttschnitt Mineral Research Centre, Brisbane, 1996.

King, R.P., 2001. Modelling and Simulation of Mineral Processing

Systems. Butterworth-Heinemann.

Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for

granular assemblies. Geotechnique 29, 47.

Cleary, P.W., 2001. Charge behaviour and power consumption in ball

mills: sensitivity to mill operating conditions, liner geometry and

charge composition. Int. J. Mineral Process. 63, 79–114.

Djordjevic, N., 2002. Discrete element modelling of power draw of

tumbling mills. In: Proceedings of IMM/AusIMM, submitted for

publication.

Mishra, B.K., Rajamani, R.K., 1994. Simulation of charge motion in

ball mills: Part 1. Experimental verifications. Int. J. Mineral

Process. 40, 171–186.

PFC3D Particle Flow Code in 3 Dimensions, 1999. Itasca Consulting

Group Inc., Minneapolis, MN.

Fig. 9. Influence of lifter height and fill factor (5% and 10%) on the

fraction of power consumed by friction.

336 N. Djordjevic / Minerals Engineering 16 (2003) 331–336