A Review of Computer Simulation of Tumbling Mills

www.elsevier.com/locate/ijminpro

Int. J. Miner. Process. 71 (2003) 95–112

A review of computer simulation of tumbling mills

by the discrete element method

Part II—Practical applications

B.K. Mishra

Department of Materials and Metallurgical Engineering, Indian Institute of Technology, Kanpur, India

Received 23 October 2001; received in revised form 6 March 2003; accepted 7 March 2003

Abstract

The potential of the discrete element method (DEM) for the design and optimization of tumbling

mills is unequivocally accepted by the mineral engineering community. The challenge is to

effectively use the simulation tool to improve industrial practice. There are several areas of

application in the analysis of tumbling mills where DEM is most effective. These include analysis of

charge motion for improved plant operation, power draw prediction, liner and lifter design and

microscale modeling for calculation of size distribution. First, it is established that charge motion in

ball and SAG mills can be computed with ease using DEM. The simulation results in the case of the

ball mill are verified by comparing snapshots of charge motion. Furthermore, it is shown that power

draw of ball as well as SAG mills can be predicted within 10%. Finally, it is demonstrated that in the

near future direct simulation of the entire comminution process by DEM will become possible by

using the impact energy spectra and the breakage distribution data.

D 2003 Elsevier Science B.V. All rights reserved.

Keywords: tumbling mill; motion analysis; power draw; DEM

1. Introduction

Grinding is an essential step in the mineral processing industries. It has been the subject

of research for almost 50 years. Over these years, a number of comprehensive review

papers have been written to summarize the state of knowledge relating to the grinding

theory and practice (see Austin, 1997; King, 1993). In spite of its economic importance

0301-7516/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0301-7516(03)00031-0

E-mail address: [email protected] (B.K. Mishra).

B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–11296

and years of research, grinding in practice suffers from cyclic and surging behavior of the

charge, erratic product quality, high circulating ratio, and unplanned shutdowns. There is

no formal methodology for designing comminution circuits. For example, many of the

leading copper processing plants use different circuits; there is no established technique to

decide the configuration and shape of lifter bars of tumbling mills; varying L/D ratio of

mills for similar milling practice. Over the years, use of population balance has helped in

many applications with mill optimization, scale-up and design. However, this approach

does not allow analysis of elementary processes involved in grinding of particles where

impact geometries and other local environmental factors are very important. Lack of

understanding of these elementary processes makes the designing approach a bit

empirical. It is in this regard, the discrete element method (DEM) has been making a

significant contribution. As a numerical tool, it has so far offered a qualitative under-

standing of the effects of different design and operational variables of the mill on the

dynamic state of the charge. With the advancement in comminution theory coupled with

availability of computing power, soon it will be possible to make quantitative predictions

using DEM.

The application of DEM to understand qualitative as well as quantitative aspects of

tumbling mill operation is reviewed here. In a qualitative sense we will analyze how DEM

allows prediction of charge behavior in tumbling mills. Power draft of the mill turns out to

be a natural offshoot of the numerical exercise as it is intimately related to the charge

motion. We will show how this information can be effectively put into practice in order to

monitor and improve the plant operation. Finally, we will touch upon various ideas that

evolved over the years to make quantitative predictions of size distribution by evolving a

microscale comminution model based on impact energy spectra and single-particle

breakage characteristics.

2. Charge motion analysis

The motion of grinding media and the energy distribution have a profound influence on

the comminution of particles in tumbling mills. The dynamic charge has been charac-

terized according to its profile. For example, when the rotating grinding chamber transfers

energy to the grinding media, the charge inside the mill may assume the so-called

cascading and cataracting types of motion. The operation of the mill is dependent largely

on the prevailing motion characteristic of the charge, subject to various operating and

design conditions of the mill. The earliest analysis of ball motion in tumbling mills dates

back to early 1900, when Davis (1919) calculated trajectories of a single ball based on

simple force balance. Rose and Sullivan (1958), while reviewing the work relating to

charge motion, emphasized the need to consider the frictional factor, which was neglected

up to that point. Up until 1990, numerical analysis of charge motion was limited to single

ball trajectory calculations. However, several interesting research efforts were made by

experimental means (Rogovin and Herbst, 1989; Vermeulen and Howatt, 1988) that

primarily attempted to show the effect of various design and operational parameters on

charge motion. Direct evidence of charge motion was limited to only laboratory scale

mills. Later when numerical tools, particularly the discrete element method were

B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 97

introduced (Cundall and Strack, 1979), it was adapted initially by Mishra (1991) and

subsequently by many others, to track en masse motion of the charge in large diameter

tumbling mills.

DEM allows numerical simulation of the dynamic interaction of the tumbling mill liner

with the milling media contained in it. Unlike single-particle analysis, DEM allows

calculation of the trajectories of individual entities in the entire grinding charge as they

move in the mill and collide with one another and the mill shell. The calculations are based

on the fundamental laws of motion and take into account the exact geometry, dimensions,

and material property of each individual steel ball, chunk of rock, mill liner, and lifter. In

essence, the DEM approach to the tumbling mill problem rests in understanding the

dynamics of the charge which is the genesis of the force field that is responsible for

particle breakage, wear of balls and liner walls, etc.

Almost all the researchers who have taken up DEM to solve mineral engineering

problems have analyzed the motion of the charge in tumbling mills. Currently, there are

several concerted research efforts directed towards understanding charge dynamics in

tumbling mills; notable among them are

Cleary (1998, 2000, 2001)

Inoue and Okaya (1996)

Kano et al. (1997)

Mishra and Rajamani (1990, 1992), Rajamani et al. (1999, 2000a,b)

Powell and Nurick (1996)

Radziszewski (1999)

Van Nierop et al. (2001), Bwalya et al. (2001)

Zhang and Whiten (1996, 1998)

Several others have been working outside the comminution area and notable amongst them

are Acharya (2000), Misra and Cheung (1999), and Bhimji et al. (2001).

The study of media mechanics becomes challenging when the mill diameter is in the

range of 4–6 m. These mills have not operated efficiently when the diameter is

increased beyond 5 m. One of the major problems is the governing scale-up procedure

that is typically used for designing. Any improvement in the scale-up and design

demands a better understanding of the overall milling process. For this reason, extensive

research has been done on single-particle breakage (Datta and Rajamani, 2002; Tavares

and King, 1998; Bourgeois et al., 1992; Hofler and Herbst, 1990; Narayanan, 1987; to

name a few), media motion (Rogovin and Herbst, 1989; Vermeulen and Howatt, 1988;

Tarasiewicz and Radziszewski, 1989; Mishra and Rajamani, 1992; Cleary, 2001), and

measurement of forces inside the mill (Moys and Skorupa, 1993; Dunn and Martin,

1978; Rolf and Vongluekiet, 1984) to correctly identify the microscopic processes

responsible for grinding in tumbling mills. It is believed that the best way to tackle the

scale-up problem is to get a comprehensive and accurate model for the dynamic motion

of the balls within the mill. It turns out that it can be easily done by means of DEM.

What follows here in this paper is a systemic analysis of charge motion in a ball mill

that results in the distribution of impact energy, which in turn can be used in an

improved grinding model that will hopefully eliminate the scale-up problem.


DEM has been used in many simulation studies to analyze the motion of the charge

in a wide range of tumbling mills. It went through several stages of rigorous validation

process before being accepted as a viable numerical tool for motion analysis. To

illustrate, we compare typical experimental data obtained from a 90-cm diameter and 15-

cm-long batch mill fitted with eight 4� 4-cm lifters with DEM simulation results. Fig. 1

shows the comparison, where the similarity between the experimental and predicted

charge profile is evident. Extensive validation of experimental data can be found in

Venugopal and Rajamani (2001) and Dong and Moys (in press). A more rigorous

validation would be to predict the velocity, acceleration, and force on a ball. These

quantities are difficult to measure but Agrawala et al. (1997) have attempted to do so,

albeit with limited success.

Fig. 1. Comparison of charge motion in a 90-cm diameter mill.


The simulation tool can also be used to optimize the performance of an operating plant

by analyzing the motion of the charge. Optimization is achieved by adjusting the mill

speed and changing the shape and configuration of the lifter bars. The test mill is a 36-ft

(10.75-m) diameter SAG mill. For the sake of simplicity and better interpretation of

numerical results, three-dimensional snapshots are avoided in favor of two-dimensional

ones. Fig. 2 shows three snapshots where the leading face angle and the number of lifters

were varied. Fig. 2a shows the charge motion inside the mill fitted with 64 lifters (face

angle 7.5j) operating at 8.95 rpm (70% of critical speed). The motion is both cascading

and cataracting. When the speed was increased to 80% critical speed, balls began to strike

at the 9 o’clock position, causing high impact forces on the shell. It was observed that the

gap between the two lifters serves to scoop the balls to the 12 o’clock position and then

release them. This feature of charge motion remained, although to a lesser extent, even

when the face angle was increased to 30j. In both cases, much of the power was wasted in

ball-on-liner impact.

It has been observed that in most instances, SAG mill liner breakage is typically due to

continuous ball-on-liner impact. Clearly, a way of preventing liner breakage in a SAG mill

is simply to avoid situations that allow continuous, direct ball-on-liner impact. To this end,

we compare the numerical results for the same 36-ft diameter mill where the number of

lifters was reduced from 64 to 32 and the face angle increased from 7.5j to 30j, keepingthe mill speed the same as before. It was observed that the charge motion changed

significantly as evident in Fig. 2b. The motion is predominantly cascading. Whenever

balls do cataract, they fall on the belly of the charge. Now it appears that the speed of the

mill could be increased to allow more balls to cataract to the extent that they fall on the toe

of the charge instead of its belly (Fig. 2c). This modification is highly desirable as it has

been determined that by increasing the speed the mill draws about 15% more power than

the first case (Fig. 2a), and as evident from the snapshots it reduces the risk of liner failure.

Such a 32-lifter design could be even further improved by choosing a greater lifter height

to achieve a desired capacity.

While most investigators have used DEM for large-scale tumbling mill analysis, there

are several others who have successfully utilized the technique to analyze charge motion in

technically more sophisticated mills. Mishra (1995) applied DEM for analysis of charge

motion in planetary mill. A typical snapshot of the charge motion in a planetary mill as

predicted by 2D DEM is shown in Fig. 3. It is a 10-cm diameter mill that is connected to a

gyration shaft of 60 cm. The mill was loaded with 400 balls of 3-mm diameter to obtain a

mill filling of approximately 50%. Here the snapshots were taken at equal intervals of time

representing one complete revolution. It is seen from the figure that the charge within the

mill is displaced in the direction of the mill rotation (counterclockwise), which is the most

common feature of charge dynamics in this type of mill.

As long as the type of contacts between colliding bodies and the corresponding

mathematical models are known, DEM in principle can be used for the analysis of any

type of grinding mills. Several researchers have applied the DEM technique more

rigorously to study centrifugal mills (Inoue and Okaya, 1996; Cleary and Hoyer, 2000;

Cleary, 2000). Rajamani et al. (2000a,b) have shown how DEM can be applied to analyze

charge motion in vibration mills, and at the same time Hoyer (1999) has applied DEM to

analyze charge motion in Hicom mills. In short, it appears DEM has great potential to

Fig. 2. Charge motion inside a 10.75-m diameter SAG mill: (a) 64 lifters, 8.95 rpm and face angle of 7.5j; (b) 32lifters, 8.95 rpm and face angle of 30j; (c) 32 lifters, 10.28 rpm and face angle of 7.5j.


Fig. 3. Charge motion inside a planetary mill; R =� 1 (Mishra, 1995).


extend its range of applicability beyond tumbling mills to more complicated and

sophisticated mills that are used in the industry for ultrafine grinding.

3. Power draw analysis

The power draft and grinding efficiency of tumbling mills depend solely on the motion

of the grinding charge and the ensuing ball collisions that utilize the input power to cause

particle breakage. Power draw analysis was first attempted by Davis (1919) by considering

the dynamics of the ball charge through individual ball paths and velocities. In the

subsequent analyses, the profile of the cascading charge began to figure in the mill power

prediction. The main task was to locate the center of gravity of the cascading charge so

that the mill power could be calculated by a torque-arm formula. Empirical correlations

sprang up to calculate the mill power draft from design and operating parameters (Bond,

1961; Hogg and Fuerstenau, 1972; Guerrero and Arbiter, 1960; Harris et al., 1985; Moys,

1993). In all the power draw correlations, mill diameter and mill speed figure in the

expressions, presumably in lieu of impact energy produced in the ball mass. Nevertheless,

all the correlations were based on the torque-arm principle where the charge is considered

as a single mass. The torque necessary to maintain the offset in the center of gravity from

the rest position is

T ¼ Mbrgsina ð1Þ


where T is the torque, Mb is the mass of the balls, rg is the distance from the mill center to

the center of gravity of the load, and a is the angle of repose of the ball charge. For mill

speed of N rpm, the power draft is given by

P ¼ 2pTN : ð2Þ

The torque-arm approach to determine power had many shortcomings. Several factors

such as shifts in the center of gravity and mill-operating conditions such as mill-speed, ball

load, etc., were not incorporated. Keeping that in mind, the next important modification

was proposed by Fuerstenau et al. (1990). They incorporated the cascading and cataracting

portion of charge separately in their model. Powell and Nurick (1996) also considered the

ball mass that are in free flight in his model. Recently, Morrell (1992) and Morrell and

Man (1997) developed models that are similar to many of the earlier works on this subject

but have much wider applicability and accuracy.

Despite these improvements over the years with regard to power prediction, one

wonders why even today different manufacturers provide widely differing power estimates

for identical mills. The reason lies in a lack of detailed information about the mill charge

motion, thereby precluding an accurate steady state prediction of power draw. The discrete

element method on the other hand allows the balls to cascade, interpenetrate between

layers while cascading, and also cataract (Datta et al., 1999; Cleary, 1998). The balls can

bounce off the mill shell and lifters, and moreover, balls of different sizes can collide with

each other at oblique angles. Here the collision of a ball with the mill wall or another ball

is modeled and the energy consumed in each of the thousands of collisions is summed to

arrive at power draft. For these reasons, this is a powerful technique for computing power

draft. Since the internal geometry of the mill shell is explicitly taken into account in the

calculation of ball charge motion, the method allows prediction of power draft for

variations in lifter designs as well as variations in ball size distribution. In the following

paragraphs, we show how well DEM predicts power for both ball mills and SAG mills of

varying size.

To illustrate the accuracy of power draw prediction, a DEM-based computer program,

Millsoftn (1999), is used. It was originally developed to understand the motion of the

charge in ball mills under various operating and design conditions. It has been extensively

used for predicting power draft of ball mills over a wide range of diameters. Fig. 4 shows

the predictive capability of the DEM model (Datta et al., 1999) where the power draw

comparison is made for different diameter mills in the range of 0.25–4.8 m. The best-fit

straight lines in log-scale have a slope of 2.5 and 2.3 for laboratory- and industrial-scale

mills, respectively, which are close to the Bond exponent of 2.5. Such predictions verify

that the total energy loss summed over all the individual collisions is an accurate indication

of power integrated over a specified time.

To predict the power draw of SAG mills the design and operating data for a variety of

mills were taken from the Proceedings of the International Autogenous and Semi-

autogenous Grinding Technology (SAG 96) (Mular et al., 1996) held at Vancouver,

Canada in 1996. The data lacked certain information, which for the simulation purpose

were judiciously assumed and an attempt was made to show that power predictions by

DEM agree with plant observations in a practical sense. For the simulation purpose, a

Fig. 4. Comparisons of power draw for ball mills of different diameters.


linear contact model with proven parameters for the spring and dashpots were used.

However, more realistic parameters can be obtained and implemented in a nonlinear DEM

model as suggested by Mishra and Murty (2001), for better estimates of power. Fifteen

different mine sites employing mills from as small as 3.62 m diameter to as large as 10.8 m

diameters are listed in Table 1. In the cases of Chuquicamata, Kanowna Belle, Mount Isa,

Table 1

A comparison power prediction with industrial semi-autogenous grinding mill data

Mine site Dimension (m) Speed (rpm) Percent filling Power (kW)

Grinding mill Diameter Length Ball/rock Installed Calculated

Chuquicamata 9.60 4.57 10.2 12:13 5890 5406

Kanowna Belle 7.35 2.85 10.9 11:19 2134 1652

McCoy 6.40 3.36 13.8 12:16 2150 1714

Ellimon 5.34 1.78 13.6 12:13 596 516

Mount Isa 9.75 4.85 10.7 5:22 5700 4798

Cyprus Baghdad 9.76 3.96 10.3 7:20 4073 4166

Leeudoorn 5.00 11.00 16.7 8:17 2660–4000 2848

Henderson 8.53 4.26 10.9 10:15 5222 3547

Forrestania Nickel 3.62 5.62 15.6 00:35 550 430

Vaal Reefs 4.85 9.15 17.2 13:12 3000 2593

Fimiston 10.80 5.65 9.3 13:8.6 9255 8766

Fimiston 10.80 5.65 9.9 13:12.2 10374 9432

Amandelbult 4.27 4.27 15.6 12:13 1250 776

STP Ghana 6.15 7.6 10.4 15:15 3800 3303

Bibiani Ghana 5.49 8.7 13.54 15:15 3350 3038

KIOCL, India 9.6 4.27 10.4 12:15 4800 4665


Leeurdon, Fimiston, STP, Bibiani, and KIOCL where power draft was reported, the

predictions are as close as one can expect, given the lack of measured design and operating

data. In all other SAG mill operations, the predicted power is between 70% and 80% of

installed power.

4. Shell lifter design

In tumbling mills, the charge motion and the power draw are largely a consequence of

type and configuration of shell lifters for a given mill speed and filling. In recent years,

with increasing mill size whose power draw has reached over 20 MW, the use of large face

angle shell lifter especially in combination with wider lifter spacing has been used. The

claimed benefits are reduced ball on liner impact leading to increased liner life and less

ball breakage, and improved mill performance. These developments, i.e. the designing of

lifters are largely due to the predictive capability of DEM that allows direct visualization

of the charge as a function of various operating and design parameters.

The design of lifter-bar should be such that the mill is able to draw adequate power to

achieve maximum throughput and at the same time minimize liner damage. The predicted

power draw by DEM takes into account the changes in lifter design in terms of its

geometry and arrangement. Analysing the data available in the Proceedings of the

International Autogenous and Semi-autogenous Grinding Technology (see Jones, 2001),

held at Vancouver, Canada in 2001 it seems there is a trend towards replacing the top-hat

lifters with widely spaced trapezoidal lifters. We analyzed some selected plant data

available in the proceedings using DEM. These data pertain to Cadia Mines, Australia,

Collahuasi Mines, Chile, Alumbrera Mines, Argentina, and Barrick Goldstrike, Nevada,

USA. In all these cases, it has been found that by reducing the number of lifters and

increasing the face angle the desired charge motion for better grinding and peak power

could be obtained.

A typical simulation result of a 5.89� 7.6-m SAG mill is presented in Fig. 5. This mill

operates at 10.4 rpm and uses 10% ball load at a total filling level of 30%. The mill is fitted

with 40 lifters of top-hat type with an 11j face angle. It draws on an average 2481 kW of

power. The mill is driven by 3800 kW motor drive system. With a view to improve the

power draw and capacity, the charge motion inside the mill was computed using Millsoftn

and the result of simulation is shown in Fig. 5a. As observed from the snapshot, this type

of lifter arrangement causes packing between lifters. By increasing the face angle to 20jand increasing the mill speed to 12.5 rpm, it was found that the power draw increased by

500 kW. The overall charge motion under the modified design is shown in Fig. 5b which

gives a much better charge profile that gives a net power draw of 2908 kW.

DEM has been playing a crucial role in decision making as to the size, shape, and

configuration of steel as well as rubber lifters. In case of rubber lifters, simply the material

properties are changed to effect a change in charge motion. It has been observed through

DEM simulations that the trajectories of balls where rubber lifters are used lie below that

of steel lifters under identical conditions. This alone turns out to be a practical piece of

information that can be used to decide the operating parameters of the mill. For more

accurate simulations for the comparison of trajectories and overall charge motion, contact

Fig. 5. Charge motion in a 5.89� 7.6-m SAG mill fitted with 40 top hat type lifters. (a) Snapshots of charge

motion at 10.4 rpm for 11j face angle lifters and (b) charge motion at 12.5 rpm for 20j face angle lifters.


models incorporating material properties must be considered (see Mishra and Thornton,

2002).

5. Liner wear

Liner wear is a very complex phenomenon because it results from several complicated

processes that occur simultaneously. The wear rate is influenced by the liner hardness and

design, size distribution of the charge, mill speed, ore abrasion index, forces on media and

liners, and the extent of corrosion. The cost of liner and media wear in grinding compares

well with the cost of electric energy consumed. The economics of mill liner is even more

important in large diameter semi-autogenous mills because of the high expense of shut

down time for relining. Independent studies have been devoted to milling efficiency and

liner wear, although it is well known that the two are closely related. Recently,

Radziszewski (2002) used DEM to compute the impact energy associated with the

collisions inside the mill to estimate wear. This is a step in the right direction that has

lot of potential.

One of the major contributions of DEM is that it allows isolating individual collisions.

Since the precise location of each impact is known accurately, it is now possible to

determine the contribution of each collision event to wear. An attempt is made just to show

how the energy gets distributed on the liner which in turn gets translated to wear. Here we

use Millsoftn to monitor the collisions that take place on a rectangular lifter segment of an

11.89-m diameter mill. The mill operates at 70% of critical speed and rotates in

counterclockwise manner. The individual collisions on the lifter segment are cumulated

over three revolutions of the mill and a qualitative assessment of initial wear is made. Fig.

6 shows the expected initial wear pattern on the four wall segments that comprise the lifter.

Fig. 6. Wear pattern on a rectangular lifter comprising four wall segments. Mill rotates counterclockwise. The

wear pattern is depicted above the wall segment.


It is observed from the figure that the two leading edges of the lifter are more prone to

wear and the corners are even more severely affected. Interestingly, the wall segment that

is parallel to the liner is experiencing relatively higher degree of wear at the middle

compared to the edges.

Other issues that are relevant here are (a) how does liner wear affects the mill power

and mill capacity, and (b) how does liner design and operating conditions affect media and

liner wear. All these issues can be addressed by DEM simulation of mills with worn out

lifters and with the different set of parameters to induce different extent of wear. An

extensive research is underway to use DEM predict wear of liners and lifters in SAG mills.

6. Microscale modeling

There have been arguments and counter arguments about the validity of the population

balance model in predicting particle size distribution in the context of grinding (Herbst,

1997). Microscale modeling potentially offers an alternative to population balance. We

will elaborate the salient features of this modeling approach while particularly emphasiz-

ing the role of DEM. In the milling context, microscale modeling involves combining the

data relating to single-particle breakage and impact energy to compute the particle size

distribution. In simple terms, single-particle breakage data are obtained by dropping a ball

of a given size from various heights onto a bed of monosize particles resting on an anvil

(see Narayanan, 1987; Cho, 1987; Hofler and Herbst, 1990; King and Bourgeois, 1993;

Morrell and Man, 1997; Datta and Rajamani, 2002). The impact energy spectra can be

obtained through DEM simulations. The model involves combining these two specific

pieces of information in a framework that predicts size distribution similar to population

balance models.


A conceptually simple modeling approach was originally introduced by Rajamani et al.

(1993) and followed later by Mishra and Rajamani (1994) and most recently by Datta and

Rajamani (2002). It works within the premise that milling involves impacts of various

energies. Grinding occurs when these impacts are imparted to particles that are positioned

for breakage. Thus, given the impact energy spectra and the corresponding breakage

behavior of particles it is possible to make a direct calculation of the resulting size

distribution.

According to the above approach, an accounting of the number of impacts (k) and their

corresponding energies is needed. Then for each specific impact energy e, the size

distribution of the daughter fragments is determined. It is assumed that collisions of

energy e are generated in the tumbling charge at a rate of kk collisions per second and that

each collision of energy e nips m grams of particles in size interval j. The breakage

function based on the energy, bij,k, is defined as the fraction of particles of mj,k from size

class j that reports to size class i. For all size classes of particles and all collisions

associated with various levels of energy, a population balance equivalent for a batch

grinding mill can be written as

dMiðtÞdt

¼ �XN

k¼1

kkmi;kMiðtÞH

þXN

k¼1

Xi�1

j¼1

kkmj;kbij;kMjðtÞH

: ð3Þ

The term Mi(t)/H in Eq. (3) represents the instantaneous mass fraction of size class i in the

mill. Clearly, the model is meaningless without the knowledge of impact energy spectra.

Lacking any reliable measurement technique for recording energy in individual collisions

inside the mill, we rely on the DEM’s capability to provide this data accurately. During the

simulation of the mill, the energy associated with each of the collisions is determined,

which leads to the impact energy distribution.

To illustrate how the model works, we show a typical calculation procedure to

determine the particle size distribution in a 90-cm diameter mill. As evident from the

foregoing, the model requires three types of input data: (i) the impact energy spectra, (ii)

broken mass in the particle bed at a given impact energy, and (iii) the energy-based

breakage function. The impact energy spectra can be obtained by the DEM simulation. In

the DEM, since we track individual collisions to monitor the en masse motion of charge, it

is a matter of extending the bookkeeping practice to store the associated energies of each

impact. Thus at any given time, the number of impacts in various energy ranges of interest

known. Now we show the results of simulation for a 90� 14-cm diameter mill that was

fitted with eight square lifters and used monosize balls of 5.08-cm diameter. The measured

power draw of the mill was on average 270 W. Fig. 7 shows the impact energy distribution

of the mill that was operated at 18 rpm under identical condition as the experiment. On this

plot, the collisions are spread among a very wide range of energy levels. It is a natural

characteristic of tumbling mills that is desirable because the material being ground in the

mill is of a wide size distribution and probably variable strength. The shape of the impact

energy distribution diagram changes with mill speed, lifter bar shape and configuration,

ball size distribution, and total ball load. Thus, the information contains the effect of both

the design and the operational parameters of the mill.

Fig. 7. Impact energy spectra of 90-cm diameter mill at 18 rpm and 20% ball load.


Next, the single-particle breakage data specific to the material and milling condition

was obtained from the literature (Datta and Rajamani, 2002). They carried out drop ball

experiments on limestone in the size range of 9.5� 6.35-mm using a steel ball of 5.08-cm

Fig. 8. Predicted and measured product size distributions: 90.0-cm mill, 20% ball load, 18 rpm, 5.08-cm balls.


diameter. Their breakage data allowed computation of broken mass in the particle bed at a

given impact energy and of the energy-based breakage function. Putting all these pieces of

information together, the size distribution of the particles after 0.5 and 4.0 min of grinding

was calculated for a monosize feed of 9.5� 6.35 mm. The result of this simulation is

compared with the experimental data carried out under identical conditions. Fig. 8 shows

that the predictions agree with the experimental result within the limits of experimental

error. Thus, the DEM combined with the single-particle breakage approach allows the

detailed physics of the process to be incorporated in the modeling of the breakage process,

which eliminates a lot of inherent empiricism in earlier approaches. Nevertheless, micro-

scale modeling has a long way to go before its range of applicability is increased.

7. Conclusions

In the last decade, with the availability of computer power and advanced numerical

tools such as DEM, substantial progress has been made in understanding and quantifying

various theoretical as well as practical aspects of grinding in tumbling mills. Although still

in the process of development, DEM research is ready to be applied in the industry for

design, monitoring, and control of tumbling mill circuits. The designing can be done by

analyzing the en masse motion of the charge using a discrete element code such as

Millsoftn. For example, the configuration and shape of lifters in the tumbling mill for

better capacity utilization can be decided using systematic simulation studies.

DEM also opens up avenues to devise ways to operate the plant in a manner that is

conducive to the most favorable mode of breakage of particles within the mill environ-

ment. This can be done by using a soft sensor via DEM. These include torque variations,

vibrational analysis, etc.

The development of the past demands further research. Some of the key areas that must

be targeted include:

� First and foremost, the need to improve the DEM structure to bring it to the PC

platform so that it can be widely used. For example, mills consisting of a million

particles and a thousand balls should not pose any problem. In principle this is not

unrealistic, given the fact that DEM is quite amenable to parallel solution schemes.

Thus, more research effort is required to develop efficient parallelized codes. Most

researchers using an explicit time stepping scheme solve the equilibrium equations in

DEM. Without compromising stability and accuracy, one should look at implicit

integration scheme that allow for larger time steps. Work is also needed to integrate

improved contact detection schemes into the DEM code since contact detection and

resolution of contact forces are one of the most computationally demanding

components of the numerical scheme.� Extending the success of the DEM application to the tumbling mill to study granulation

and agglomeration problems in similar devices. Much remains to be explored in this

area with the application of DEM.� Research activities in the area of understanding the breakage and fragmentation

behavior of particles and particle agglomerates by applying the DEM technique.


Overall, there is a significant improvement in the understanding of the tumbling mill by

the use of the discrete element method that makes the literature much richer than it was in

the early 1990s.

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Documents

A Review of Computer Simulation of Tumbling Mills