A Review of Computer Simulation of Tumbling Mills

Int. J. Miner. Process. 71 (2003) 95 – 112 www.elsevier.com/locate/ijminpro

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

E-mail address: [email protected] (B.K. Mishra). 0301-7516/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-7516(03)00031-0

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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 understanding 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 characterized 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


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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.

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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 15cm-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.


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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, keeping the 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

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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) 32 lifters, 8.95 rpm and face angle of 30j; (c) 32 lifters, 10.28 rpm and face angle of 7.5j.


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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 Mb rg sina

ð1Þ

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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 Semiautogenous 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


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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)

Grinding mill

Diameter

Length

Chuquicamata Kanowna Belle McCoy Ellimon Mount Isa Cyprus Baghdad Leeudoorn Henderson Forrestania Nickel Vaal Reefs Fimiston Fimiston Amandelbult STP Ghana Bibiani Ghana KIOCL, India

9.60 7.35 6.40 5.34 9.75 9.76 5.00 8.53 3.62 4.85 10.80 10.80 4.27 6.15 5.49 9.6

4.57 2.85 3.36 1.78 4.85 3.96 11.00 4.26 5.62 9.15 5.65 5.65 4.27 7.6 8.7 4.27

Speed (rpm)

Percent filling

Power (kW)

Ball/rock

Installed

Calculated

10.2 10.9 13.8 13.6 10.7 10.3 16.7 10.9 15.6 17.2 9.3 9.9 15.6 10.4 13.54 10.4

12:13 11:19 12:16 12:13 5:22 7:20 8:17 10:15 00:35 13:12 13:8.6 13:12.2 12:13 15:15 15:15 12:15

5890 2134 2150 596 5700 4073 2660 – 4000 5222 550 3000 9255 10 374 1250 3800 3350 4800

5406 1652 1714 516 4798 4166 2848 3547 430 2593 8766 9432 776 3303 3038 4665

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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 20j and 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


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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.

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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 emphasizing 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.


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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 N N X i1 X Mj ðtÞ dMi ðtÞ Mi ðtÞ X ¼ þ : kk mi;k kk mj;k bij;k dt H H k¼1 k¼1 j¼1

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.

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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.


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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, microscale 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 environment. 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.

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A Review of Computer Simulation of Tumbling Mills

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