a review of computer simulation of tumbling mills
TRANSCRIPT
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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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–11298
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 99
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112100
Fig. 3. Charge motion inside a planetary mill; R =� 1 (Mishra, 1995).
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 101
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
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112102
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 103
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
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112104
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 105
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112106
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 107
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112108
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 109
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.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112110
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.
References
Acharya, A., 2000. A distinct element approach to ball mill mechanics. Commun. Numer. Methods Eng. 16,
743–753.
Agrawala, S., Rajamani, R.K., Songfack, P., Mishra, B.K., 1997. Mechanics of media motion in tumbling mills
3D discrete element method. Miner. Eng. 10 (2), 215–227.
Austin, L.G., 1997. Concepts in process design of mills. In: Komar Kawatra, S. (Ed.), Comminution Practices.
SME Publications, Littleton, CO, pp. 339–348. Chapter 40.
Bhimji, D., Thornton, C., Adam, M., 2001. DEM batch granulation simulations with a rotating drum. Proceed-
ings of Particle Technology UK Forum 3, University of Birmingham, UK, 12–13 July 2001.
Bond, F.C., 1961. Crushing and grinding calculations. Allis-Chalmers. Publication No. 07R9235C, 1961.
Bourgeois, F., King, R.P., Herbst, J.A., 1992. In: Kawatra, S.K. (Ed.), Low Impact-Energy Single-Particle
Fracture. Comminution—Theory and Practice. SME, pp. 99–108. Chap. 8.
Bwalya, M.W., Moys, M.H., Hinde, A.L., 2001. The use of discrete element method and fracture mechanics to
improve grinding rate predictions. Miner. Eng. 14 (6), 565–573.
Cho, K., 1987. Breakage Mechanism in Size Reduction. PhD Thesis. Department of Metallurgical Engineering,
University of Utah.
Cleary, P.W., 1998. Predicting charge motion, power draw, segregation, wear and particle breakage in ball mills
using discrete element methods. Miner. Eng. 11 (11), 1061–1080.
Cleary, P.W., 2000. DEM simulation of industrial particle flows: case studies of dragline excavators, mixing in
tumblers and centrifugal mills. Powder Technol. 109, 83–104.
Cleary, P.W., 2001. Modelling communition devices using DEM. Int. J. Numer. Anal. Methods Geomech. 25,
83–105.
Cleary, P.W., Hoyer, D., 2000. Centrifugal mill charge motion and power draw: comparison of DEM predictions
with experiment. Int. J. Miner. Process. 59, 131–148.
Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29,
47–65.
Datta, A., Rajamani, R.K., April 2002. A direct approach of modeling batch grinding in ball mills using
population balance principles and impact energy. Int. J. Miner. Process. 64 (4), 181–200.
Datta, A., Mishra, B.K., Rajamani, R.K., 1999. Analysis of power draw in ball mill by discrete element method.
Can. Metall. Q. 38 (16), 130–138.
Davis, E.W., 1919. Fine crushing in ball mills. Trans. AIME 61, 250–296.
Dong, D.J., Moys, M.H., 2002. Assessment of discrete element method for one ball bouncing in grinding mill.
Int. J. Miner. Process. (in press).
Dunn, D.J., Martin, R.C., 1978. Measurement of impact forces in ball mills. Mining, 384–388.
Fuerstenau, D.W., Kapur, P.C., Velamakanni, B., 1990. A multi-torque model for the effects of dispersants and
slurry viscosity on ball milling. Int. J. Miner. Process. 28, 81–98.
Guerrero, P.K., Arbiter, N., 1960. Tumbling mill power at cataracting speeds. AIME Trans. 217, 73–87.
Harris, C.C., Schonock, E.M., Arbiter, N., 1985. Grinding mill power consumption. Miner. Process. Technol.
Rev. 1, 297–345.
Herbst, J.A., 1997. Response to the population balance model challenge. In: Komar Kawatra, S. (Ed.), Commi-
nution Practices. SME Publications, Littleton, CO, pp. 47–53. Chapter 7.
Hofler, A., Herbst, J.A., 1990. Ball mill modeling through micro-scale fragmentation studies: fully monitored
particle bed comminution versus particle impact tests. Proc. 7th European Symp. on Comminution, Ljubljana,
pp. 1–17.
Hogg, R., Fuerstenau, D.W., 1972. Power relationships for tumbling mills. AIME Trans. 252, 418–423.
Hoyer, D.I., 1999. Discrete element method for fine grinding scale-up in Hicom mills. Powder Technol. 105 (1),
250–256.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112 111
Inoue, T., Okaya, K., 1996. Grinding mechanisms of centrifugal mills—a batch ball mill simulator. Int. J. Miner.
Process. 44 (45), 425–435.
Jones, S.M., 2001. Autogenous and semiautogenous mill update. In: Barratt, D., Allan, M.J., Mular, A.L. (Eds.),
International Autogenous and Semiautogenous Grinding Technology, pp. I-362– I-372.
Kano, J., Naoki, C., Saito, F., 1997. A method for simulating the three dimensional motion of balls under the
presence of a powder sample in tumbling ball mill. Adv. Powder Technol. 8 (1), 39–51.
King, R.P., 1993. Comminution research—a success story that has not yet ended. Proc. of XVIII Intl. Mineral
Processing Congress, Sydney, Australia, 1993.
King, R.P., Bourgeois, F., 1993. A new conceptual model for ball milling. Proc. 18th International Mineral
Processing Congress, Sydney, pp. 81–86.
Mishra, B.K., 1991. Study of media mechanics in tumbling mills. PhD Thesis. Department of Metallurgical
Engineering, University of Utah.
Mishra, B.K., 1995. Charge dynamics in planetary mill. KONA Powder Part. 13, 151–158.
Mishra, B.K., Murty, C.V.R., 2001. On the determination of contact parameters for the realistic DEM simulations
of ball mills. Powder Technol. 115, 290–297.
Mishra, B.K., Rajamani, R.K., 1990. Numerical simulation of charge motion in ball mills. Proceedings of the 7th
European Conference on Comminution, Ljubljan, Yugoslavia, pp. 555–570.
Mishra, B.K., Rajamani, R.K., 1992. The discrete element method for the simulation of ball mills. Appl. Math.
Model 16, 598–604.
Mishra, B.K., Rajamani, R.K., 1994. Simulation of charge motion in ball mills: Part 1. Experimental verifica-
tions. Int. J. Miner. Process. 40, 171–186.
Mishra, B.K., Thornton, C., 2002. An improved contact model for ball mill simulation by the discrete element
method. Adv. Powder Technol. 13 (1), 25–41.
Misra, A., Cheung, J., 1999. Particle motion and energy distribution in tumbling mills. Powder Technol. 105,
222–227.
Morrell, S., 1992. Prediction of grinding mill power. Trans. Inst. Min. Metall. Section C: Miner. Process. Extr.
Metall. 101, 25–32.
Morrell, S., Man, Y.T., 1997. Using modeling and simulation for the design of full scale ball mill circuits. Miner.
Eng. 12 (10), 1311–1327.
Moys, M.H., 1993. A model for mill power as affected by mill speed, load volume and liner design. J. S. Afr.
Inst. Min. Metall. 93 (6), 135–141.
Moys, M.H., Skorupa, J., 1993. Measurement of the forces exerted by the load on a liner in a ball mill, as a
function of liner profile, load volume and mill speed. Int. J. Miner. Process. 37, 239–256.
Mular, A., Barratt, D., Knight, D. (Eds.), 1996. Proceedings of an International Conference on Autogenous and
Semiautogenous Grinding Technology Held in Vancouver, B.C., Canada, October 6–9, pp. 1–390.
Narayanan, S.S., 1987. Modeling the performance of industrial ball mills using single particle breakage data. Int.
J. Miner. Process. 20, 211–228.
Powell, M.S., Nurick, G.N., 1996. A study of charge motion in rotary mills. Parts 1, 2, and 3, Miner. Eng. Vol. 9,
No. 3, 259–268; 343–350; Vol. 9, No. 4, 399–418.
Radziszewski, P., 1999. Comparing three DEM charge motion models. Miner. Eng. 12 (12), 1501–1520.
Radziszewski, P., 2002. Exploring total media wear. Miner. Eng. 15, 1073–1087.
Rajamani, R.K., Agrawala, S., Mishra, B.K., 1993. Mill scaleup: ball collision frequency and collision energy
density in laboratory and plant-scale mills. Proc. of XVIII Intl. Mineral Processing Congress, Sydney,
Australia, pp. 103–109.
Rajamani, R.K., Mishra, B.K., Songfack, P., Venugopal, R., 1999. MILLSOFT—a simulation software for
tumbling mills design and troubleshooting. Min. Eng., Dec., 41–47.
Rajamani, R.K., Songfack, P., Mishra, B.K., 2000a. Impact energy spectra of tumbling mill. Powder Technol.
109, 105–112.
Rajamani, R.K., Mishra, B.K., Venugopal, R., Datta, A., 2000b. Discrete element analysis of tumbling mills.
Powder Technol. 109 (1–3), 105–112.
Rogovin, Z., Herbst, J.A., 1989. Charge motion in a semi-autogenous grinding mill. Miner. Metall. Process. 6 (1),
18–23.
Rolf, L., Vongluekiet, T., 1984. Measurement of energy distribution in ball mills. Ger. Chem. Eng. 7, 287–292.
B.K. Mishra / Int. J. Miner. Process. 71 (2003) 95–112112
Rose, H.E., Sullivan, R.M.E., 1958. ATreatise on the Internal Mechanics of Ball, Tube, and Rod Mills. Constable
and Company, London.
Tarasiewicz, S., Radziszewski, P., 1989. Ball mill simulation: Part I—a kinetic model of ball charge motion.
Trans. Soc. Comput. Simul. 6 (2), 75–88.
Tavares, L.M., King, R.P., 1998. Single-particle fracture under impact loading. Int. J. Miner. Process. 54 (1),
1–28.
Van Nierop, M.A., Glover, G., Hinde, A.L., Moys, M.H., 2001. A discrete element method investigation of the
charge motion and power draw of an experimental two-dimensional mill. Int. J. Miner. Process. 59, 131–148.
Venugopal, R., Rajamani, R.K., 2001. 3D simulation of charge motion in tumbling mills by the discrete element
method. Powder Technol. 115, 157–166.
Vermeulen, L.A., Howatt, D.D., 1988. Effect of lifter bars on the motion of en-masse grinding midia in milling.
Int. J. Miner. Process. 24 (1–2), 143–159.
Zhang, D., Whiten, W.J., 1996. The calculation of contact forces between particles using spring and damping
models. Powder Technol. 88, 59–64.
Zhang, D., Whiten, W.J., 1998. An efficient calculation method for particle motion in discrete element simu-
lations. Powder Technol. 98, 223–230.