A Direct Approach of Modeling Batch Grinding

Int. J. Miner. Process. 64 (2002) 181 – 200 www.elsevier.com/locate/ijminpro

A direct approach of modeling batch grinding in ball mills using population balance principles and impact energy distribution Amlan Datta, Raj K. Rajamani* Department of Metallurgical Engineering, University of Utah, 135 South 1460 East Rm 412, Salt Lake City, UT 84112 0114, USA Received 29 May 2000; received in revised form 5 February 2001; accepted 3 April 2001

Abstract The design and scale-up of ball mills are important issues in the mineral processing industry. Incomplete knowledge about the mechanics of charge motion often forces researchers to rely on phenomenological modeling to formulate scale-up procedures. These models predict the behavior of large industrial-scale mills using the data obtained in small laboratory-scale mills. However, the differences in charge motion in plant-scale and lab-scale mills introduce significant inaccuracies in the predictions. In this article, a batch-grinding model using the impact energy distribution of the mill is explained. The distribution of impact energy is obtained from the simulation of the charge motion using the discrete element method. The model is also verified with experimental data, and the strengths and the weaknesses of the model in its current form have been identified. It is anticipated that a model, which uses information about impact-energy distribution will overcome some of the difficulties faced by the phenomenological models. D 2002 Elsevier Science B.V. All rights reserved. Keywords: batch grinding models; ball mill scale-up; impact-energy distribution; discrete element method

*

Corresponding author. Tel.: +1-801-581-6386; fax: +1-801-581-4937. E-mail address: rajamani@mines.utah.edu (R.K. Rajamani).

0301-7516/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 ( 0 1 ) 0 0 0 4 4 - 8

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A. Datta, R.K. Rajamani / Int. J. Miner. Process. 64 (2002) 181–200

1. Introduction Comminution is a process step for a wide range of industries including cement, ceramics, pharmaceutics, paper, pigments, and minerals. Many industrial surveys have established that a significant portion of the total cost of metal production is expended in comminution processes. The grinding operation in a ball mill is a capital- and energyintensive process. Hence, a marginal improvement in the efficiency of mill operation will be of immense economic benefit to the industry. A typical scale-up procedure for designing large industrial-scale mills consists of several steps (Herbst and Fuerstenau, 1980). First, laboratory experiments in smaller size mills are conducted under identical operating conditions to obtain the breakage properties of a particular ore. Then, these properties are scaled to larger mills using suitable mathematical models. In the end, the mill dimensions are computed from the feed and the estimated product size distributions. However, the fundamental drawback to this approach is that the motion of the charge in a small mill and that in a large mill are significantly different. The large mill runs at a much lower absolute speed than the small mill, even though the percent critical mill speed value is the same. For example, at 60% critical speed, the charge is very clearly divided into cascading and cataracting zones in the case of a large mill, while for the same speed only a cataracting zone exists in a small mill. Moreover, the laboratory-scale mill uses a ball size distribution with a smaller top ball size, which also alters the breakage regime in these two mills. The earliest scale-up model for prediction and design of the performance of an industrial-scale ball mill was formulated by Bond (1952, 1960), a procedure that evolved from the classical energy-size reduction principle (Austin, 1973). Several criticisms of Bond’s model are found in the literature (Austin et al., 1984; Gumtz and Fuerstenau, 1970). First of all, the entire size distribution of feed and product is characterized by a single parameter called 80% passing size. Secondly, all the grinding sub-processes are lumped in a single work index term, and also, the information about ball size distribution and lifter design is absent in the scale-up procedure. Some of the deficiencies of Bond’s scale-up procedure were overcome in a grinding model developed using the population balance principles (Herbst, 1979). The evolution of size distribution in the mill is described by the following equation: i1 X d½Hmi ðtÞ ¼ Si Hmi ðtÞ þ bij Sj Hmj ðtÞ dt j¼1

ð1Þ

where mi is the mass fraction of a particular size class i, Si is the selection function or fractional breakage rate of size class i, bij is the breakage distribution function of the size class, and H is the mass hold-up of the mill. This phenomenological model has the required kinetic parameters, which is an improvement over the Bond model. In scale-up procedures utilizing the population balance model (PBM), the selection and breakage functions are determined in a small laboratory mill. The laboratory experiments are done with nearly identical feed materials and operating conditions. Then, these parameters are scaled for bigger industrial mills. It has been shown experimentally that the breakage function does not depend upon the grinding environment and can be normalized


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with respect to the size (Broadbent and Callcott, 1956; Herbst and Fuerstenau, 1980; Herbst et al., 1981). So the size-normalized values obtained for a particular ore in a laboratory-scale mill can be used for an industrial mill. Herbst and Rajamani (1982) applied the specific selection function hypothesis (Herbst and Fuerstenau, 1973; Herbst et al., 1973) for scaling up the selection function. The selection function is proportional to the mass-specific power input to the mill. The constant of proportionality is known as the ‘‘specific selection function’’. Therefore, knowing the required mill capacity and the power consumption, one can calculate the selection function of the industrial mill. However, the variation of grinding regimes as influenced by the mill diameter, the ball sizes, and the lifter configuration was not included in the model. Austin et al. (1984) proposed a more elaborate model where each individual design and operating parameter including mill diameter, rotational speed, ball size distribution, and powder filling were accounted for while computing the selection functions for the industrial mill. However, this method requires calculation of several empirical parameters. The problem with all the methodologies is that the mill is treated as a black box. In other words, instead of incorporating the mode of energy expenditure in the mill, PBM links the feed and the product size distribution via a series of model parameters. The effect of ball size distribution on grinding has not been clearly interpreted. Breakage regime and the mixing efficiency are dependent on the mill size, ball size distribution, lifter configurations, and other parameters. The model envisaged here is an approach wherein the numerous collisions occurring in the mill are modeled first, and the results are coupled with breakage of particles as they are caught in these collisions. As a result, one obtains the evolution of size distribution in the mill. It is anticipated that this technique eventually will lead to a procedure where the capacity of a mill can be directly estimated without any intermediate scale-up procedure.

2. Grinding model based on impact energy Previously, several researchers have proposed the concept of energy-based breakage rate and breakage distribution functions and have attempted to derive these functions from the collision patterns inside a mill (Narayanan, 1987; Cho, 1987; Hofler and Herbst, 1990; King and Bourgeois, 1993; Morrell and Man, 1997). Some of these models did not have adequate information about impact patterns in the mill, and in other cases, the models were too complex and required a considerable amount of parameter estimation. In order to reduce the difficulties encountered in the previous efforts, a simple and direct model is formulated here that requires minimal parameter estimation. In practice, all the operating parameters, such as ball size distribution, absolute value of mill speed, and lifter configurations, differ between small and large mills. These parameters directly affect charge motion and, hence, the distribution of collision energy. Therefore, the collision energy distribution is the most fundamental phenomenon that should be the basis of any grinding model of a ball mill. During the tumbling action in a

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ball mill, it is assumed that each collision nips a certain amount of material of a particular size class and breaks some material in that class. The broken material is redistributed in the lower sizes, and this distribution depends on the energy of that collision. The breakage due to the tumbling action, as shown in Fig. 1, is assumed to be equivalent to subjecting

Fig. 1. Grinding phenomenon in a ball mill as interpreted in terms of collision energy and frequency.


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individual layers of particles to a series of impacts of various energy levels at an identical rate. Thus, the grinding action is idealized as each of the ball-to-ball collisions capturing a specific mass of particles. It is assumed that collisions of energy ek are generated in the tumbling charge at a rate of kk collisions per second and that each collision of energy ek nips mj,k grams of particles in size interval j. The breakage function based on the energy, bij,k, is defined as the fraction of that mj,k grams of broken particle from size j that reports to a smaller size i. For all size classes of particles and all collisions associated with various levels of energy, a population balance equation 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

ð2Þ

The term Mi(t)/H in Eq. (2) represents the instantaneous mass fraction of size class i in the mill. This term acts as an ‘‘effectiveness factor’’ for the collision. It implies that the total number of collisions of an energy level is distributed to all the size classes in proportion to their instantaneous mass fraction. This idea is indirectly in accordance with the principle of first-order breakage kinetics. The model equation incorporates three input terms: the impact energy spectra, broken mass in the particle bed at a given impact energy, and the energy-based breakage function. The first one is obtained from the simulation of ball charge motion, and the other two are experimentally determined from drop-ball tests. The implicit assumptions of the model are the following. (1) Breakage of particles due to abrasion is negligible. Impact is the only mode of energy transfer from the grinding media to the particles. In other words, shearing of particles during ball-to-ball or ball-to-wall collision is absent. (2) Each collision takes part in grinding. This means each collision nips some material regardless of the location of the collision. This is a valid assumption, if the fractional mill filling of the media interstices is more than 1.0. (3) The content of the mill is perfectly mixed, i.e., the size distribution of the particle bed caught between two colliding balls is the same as the prevailing distribution in the entire mill. This assumption is valid for dry grinding where internal classification is not as prevalent as in wet grinding. (4) There is no progressive damage leading to breakage due to low-energy impacts, i.e., breakage due to repeated collisions at low energy is not accounted for in the model. (5) The linearity assumption is also implied here, i.e., the broken-mass and breakage functions remain the same as the size distribution changes in the mill. It has been established by earlier research (Schonert, 1979) that the breakage characteristics of a particle depend upon the size distribution of particles around it. Nevertheless, this assumption simplifies the effort needed to conduct the drop-ball experiments. The alternative is to conduct drop-ball tests for all possible grinding environments, which would be too time-consuming. (6) In deriving both the breakage parameters by drop-ball tests, it is assumed that the particle bed that is nipped between two colliding bodies is made up of four layers of material. There is a significant difference in the breakage characteristics when a single particle, a monolayer, and a multilayer of particles are stressed in a collision. Therefore, this assumption implies an average grinding environment in the mill.

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3. Obtaining impact energy spectra by the discrete element method The impact energy spectrum is obtained from numerical simulation of the ball charge motion, which is based upon the discrete element method (DEM) (Cundall and Strack, 1979). This technique computes the finite displacement and rotation of bodies isolated by a distinct and undeformable boundary, as these bodies undergo collision and translation. It is a suitable choice for a ball mill, where the balls and the mill shell are treated as distinct elements. There are a few studies on the measurement impact forces in the ball mass that have been reported in the literature (Dunn and Martin, 1978; Rolf and Vongluekiet, 1984; Vermeulen et al., 1984; Yashima et al., 1988; Van Nierop and Moys, 1996). However, it is very difficult to develop suitable instrumentation to measure the impact energy distribution in the harsh environment that prevails inside a mill. Therefore, numerical simulation of the ball charge motion appears to be a better tool for this purpose. Morrell (1992) modeled the cascading ball charge as a series of concentric ball layers, each one slipping against the adjacent layer of balls. However, this approach does not predict the impact distributions. Powell and Nurick (1996a,b), on the other hand, modeled the trajectory of one ball carried upward and released by the lifter, which gives only the impact energy due to cataracting motion. The most suitable simulation is the DEM simulation (Mishra and Rajamani, 1992, 1994; Rajamani et al., 2000; Kano et al., 1997; Cleary, 1998), which accounts for the geometry of the lifters and which allows individual balls to take their own free trajectory depending on their position in the mill. In the two-dimensional discrete element simulation scheme, a ball is represented as a disc with a mass equal to that of the sphere, and the circular mill shell including the lifters is represented by a series of straight lines (walls). The DEM models the ball-to-ball collision with a linear spring and dashpot. The spring provides the repulsive force and the dashpot dissipates a portion of the relative kinetic energy. During collision, the balls are allowed a virtual overlap D x, and the normal vn and tangential vt relative velocities determine the collision forces. The normal force is given by: Fn ¼ kn D x þ Cn vn

ð3Þ

where kn is the normal spring constant and Cn is the normal damping coefficient. The balls rebounding after collision follow a free-flight trajectory as per Newton’s law until the next collision. Likewise, a pair of spring and dashpot is used in the tangential direction. The two quantities of interest within the scope of the current work are the mill power draft and the impact energy distribution. At each collision, a fraction of the supplied energy is consumed, which is modeled by the dashpot. Thus, the addition of the product of normal and shear force on the dashpot and respective overlap (subscripts n and s denote normal and shear direction, respectively) gives the energy lost at that contact, which is given by: E¼

X X j t

k

k Cn Avn A2 Dt þ Cs Avs A2 Dt

ð4Þ


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where the energy loss term is summed up over all the collisions (k) and for all the time steps (t). The energy associated with each of the collisions is summed over a few revolution of the mill, which leads to the impact energy distribution and power draft. Lacking any measurement technique for recording energy in individual collision inside the mill, we rely on the accuracy of the simulation’s power draft prediction. A detailed investigation of the capability of the DEM simulation for predicting the power draft of ball mills of different sizes is reported elsewhere (Datta et al., 1999). The agreement between the measured and predicted power draft implies that the impact energy distribution computed with this simulation scheme is a reasonable approximation of the true spectrum prevailing in the mill.

4. Experimental work A drop-ball apparatus was used in this work to determine experimentally the brokenmass (mi,k) and the energy-based breakage function (bij,k). The apparatus used in this work is similar to the ultra-fast load cell (Hofler and Herbst, 1990; Bourgeois et al. 1992). The key factors in this method are the drop-ball diameter, particle bed configuration, and anvil geometry. A certain amount of material corresponding to four layers of particle bed was placed on top of the anvil in a thin paper cup. Then, a ball of desired size was dropped on that bed from a desired height. After breakage, the broken mass was size analyzed for the distribution of progeny particles. For each collision energy, the breakage functions were determined by averaging the results from 40 to 50 test samples. Hence, experimental error was minimized to a large extent. Broken-mass data was obtained from about 5 to 10 test samples for each collision energy. Small sieves of diameter 7.6 cm (3 in.) were used in size analysis to minimize material loss when handling the small amounts of sample. Batch grinding tests were performed in three mills (diameters 25.4, 38.1, and 90.0 cm) using limestone as feed material. In addition, the data of Siddique (1977) on the same 25.4-cm mill were also examined. In all these experiments, the power draft was calculated from experimentally measured torque on the drive shaft. Table 1 shows the operating conditions maintained in these experiments. Limestone from two different sources, referred to as limestone-A and limestone-B, was used. Breakage parameters were obtained separately for these two types of material. Two exploratory batch-grinding tests were conducted using both materials under the conditions reported by Siddique (1977) in the 25.4-cm diameter mill. The mill product size distributions over various grinding times obtained from Limestone-A were close to those reported by Siddique. Hence, the breakage parameter from that particular limestone was used while examining Siddique’s data.

5. Results and discussions The charge motion of the experimental mills was simulated by the DEM simulation code using the parameters presented in Table 1. As seen in Table 1, the predicted mill powers are in good agreement with the measured mill power data. The computed impact

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D L (cm)

Feed size (mm)

Ball load (%)

% Critical mill speed

Lifter configurations

Ball size distribution (size in cm wt.%)

Measured mill power (W)

Predicted mill power (W)

25.4 29.2 38.1 29.2 90.0 14.0 25.4 29.2 (Siddique, 1977) 38.1 29.2 (Siddique, 1977)

2.36 1.7 2.36 1.7 9.5 6.35 1.7 natural feed 1.7 natural feed

35 35 20 50

65 65 40 60

8, Rectangular (1.9 0.5 cm) 10, Rectangular (2.5 0.9 cm) 8, Square (4.0 4.0 cm) 8, Rectangular (1.9 0.5 cm)

60.0 165.0 270 82

58.0 155.0 281 71

50

60

10, Rectangular (2.5 0.9 cm)

Monosize (5.08 – 100%) Monosize (5.08 – 100%) Monosize (5.08 – 100%) Multisize (3.81 – 53%, 2.54 – 30%, 1.91 – 12%, and 1.27 – 5%) Multisize (3.81 – 53%, 2.54 – 30%, 1.91 – 12%, and 1.27 – 5%)

209

207


Table 1 Experimental conditions and mill power draft of batch grinding tests


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energy distributions are presented in Fig. 2. Generally, the total number of impacts in the 25.4-cm mill is less than that in the other two mills. Further, the maximum energy of impact in this mill is only 0.475 J. The impacts in mid- and high-energy intervals are greater in number inside the 90.0-cm mill than in the 38.1-cm mill. Moreover, in the lower energy range, the numbers of collisions are very close for the 90- and 38.1-cm mills. Even though the 90.0-cm mill exhibits only a cascading charge, compared with the 38.1-cm mill, which exhibits both cascading and cataracting charge, the number of low-energy collisions is the same in both the mills. In recording the impact spectrum from simulation, an energy interval of 0.01 J was used to increase the accuracy of predictions made with the impact-energy-based population balance model. First, to parameterize the model, a study of the breakage behavior of limestone-A and limestone-B was undertaken. The general trend of variation of broken mass with impact energy is shown in Figs. 3 and 4. The broken mass is defined as the amount of material passing through the bottom screen, which brackets the monosize feed material. The broken-mass versus impact-energy data was fitted with a logarithmic function for interpolation purposes. In the lower energy range, the broken mass increases sharply with impact energy, which is not the case at higher energy. At the same level of impact energy, the broken mass of particles of bigger size is more than that of smaller particles, and this is attributed to the greater density of internal flaws in bigger particles. However, this trend is reversed in extremely low-energy collisions, as evident in Fig. 4. At such low impact

Fig. 2. Impact energy distribution in ball mill of different diameter. Conditions are described in Table 1.

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Fig. 3. Variation of broken mass with particle size and impact energy. Material = limestone-A; drop-ball diameter = 5.08 cm; particle bed configuration = four layers.

force, the applied stress does not exceed the critical fracture stress considering the high cross-sectional area of the bigger particle. The critical stress of a large particle is lower than that of a small particle due to the higher density of internal flaws. The general trends of the breakage distribution function are presented in Figs. 5 and 6 for the two limestones. They exhibit a finer distribution with the increase in the energy of the impact. It was established by earlier research (Hofler and Herbst, 1990) that highenergy impacts are low in efficiency. A significant portion of the energy of impact is wasted in the collision between the anvil and the drop-ball, as well as in displacing the particles in the bed. Therefore, as we increase the impact energy, the fineness of the product increases steeply in the beginning. As shown in Figs. 5 and 6, the extent of fineness diminishes in the higher energy range. It has been often found that the breakage functions used in the population balance models are normalizable with respect to the parent size interval (Herbst et al., 1973; Siddique, 1977). The normalized size is defined as the ratio of the bottom size of each interval to that of the parent size class. This particular behavior of broken particle fragments is extremely convenient for modeling breakage phenomena, since a progeny distribution obtained from one size class of particle is applicable for other size classes also. In the drop-ball tests reported here, two different size classes exhibit the same progeny distribution, as shown in Fig. 7, for identical impact energy when plotted against the


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Fig. 4. Variation of broken mass with particle size and impact energy. Material = limestone-B; drop-ball diameter = 5.08 cm.; particle bed configuration = four layers.

Fig. 5. Breakage functions for different impact energies. Material = limestone-A, 2.36 1.7 mm (four layers); drop-ball diameter = 5.08 cm.

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Fig. 6. Breakage functions for different impact energies. Material = limestone-B, 9.5 6.35 mm (four layers); drop-ball diameter = 5.08 cm.

normalized size. However, several previous research initiatives (Austin et al., 1984; Cho, 1987) demonstrated that an abnormal nature of the progeny distribution was evident in

Fig. 7. Breakage function normalized with respect to parent size. Material = limestone-A; particle bed configuration = four layers; drop-ball diameter = 5.08 cm.


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some grinding experiments using coarser mill feed. Thus, it is anticipated that there is a dependence of the breakage distribution function on feed size and ball size distribution. Tavares (1997) also confirmed the dependence of breakage distribution on particle size, by single-particle breakage experiments. Likewise, as seen in Fig. 8, the coarse particle exhibited a non-normalizable breakage function. The size distribution of progenies is coarser for the first two size intervals compared to the smaller particle sizes for the same energy input. Therefore, whenever large particles were used as the feed material, two different sets of breakage functions were used for the large and small size classes in the computation of the mill product size distribution. While the model is formulated in precise detail regarding the collision energy and the size of particles nipped in a specified collision, in the actual milling operation inevitably the situation is much different. Hence, a multiplication factor of 0.8 was found necessary in the appearance and disappearance terms in the right-hand side of Eq. (2), particularly for the predictions for the 38.1- and 90.0-cm mills. This possibly indicates that the efficiency of grinding reduces when there is greater amount of fine material in the mill. However, it is difficult to quantify this effect and incorporate it explicitly in the model. Nevertheless, this simple correction parameter gave reasonably good predictions of all the experimental results.

Fig. 8. Breakage function normalized with respect to size. Material = limestone-B; particle bed-configuration = four layers; drop-ball diameter = 5.08 cm.

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The logarithmic functions shown as solid lines in Figs. 3 and 4 between the amount of broken mass and impact energy were used for interpolation. Broken-mass values of size classes at which drop-ball tests were not conducted were calculated by interpolation. The energy-dependent broken-mass values were determined only for the first five size intervals due to the difficulty in executing a drop-ball test for extremely fine sizes of particles. Therefore, the mill product size distribution was also predicted for the first five size classes only. Some dependence of broken mass on the drop-ball diameter was observed in this study. Therefore, for the grinding experiments with multisize balls in the charge, a combined broken-mass data in proportion to the mass fraction of balls of different size was used for prediction. This technique was previously used by Austin et al. (1976) to modify the selection function in the PBM equation in proportion to mass fractions of media sizes. The breakage function data for the complete range of energy values were also determined by interpolation, since it was not possible to do drop-ball tests for extremely low values of impact energy; the required data were obtained by extrapolating the available values. Breakage functions were determined from the parent size class and applied to other sizes assuming that they are normalizable with respect to size. There was no variation of breakage function due to the size of drop-ball; hence, the breakage function obtained from the top-size of the ball charge was used as input for grinding simulations with multisize ball charge. The predicted results are in good agreement with the measured data for 25.4- and 38.1cm mills (Figs. 9 and 10), and the model is able to predict the size distribution with reasonable accuracy even for longer grinding times. The predicted values of median size (d50) and the 80% passing size (d80) shown in Figs. 11 and 12 are quite close for these grinding cases. However, the predicted results for the 90-cm mill shown in Fig. 13 were in general not as good as the other two mills. The high particle-to-ball diameter ratio in this

Fig. 9. Predicted and measured product size distributions: 25.4-cm mill, 35% ball load, 55 rpm speed, 100% 5.08-cm balls.


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Fig. 10. Predicted and measured product size distributions: 38.1-cm mill, 35% ball load, 41 rpm speed, 100% 5.08-cm regular steel balls.

mill is the reason for the poor prediction. The anomalous nature of the breakage of coarse particles is well documented in the literature (Austin et al., 1984). When the ball diameter is relatively larger compared to the particle size, then the particles fill up high individual interstitial volumes. Hence, the chance of breakage due to abrasion is greater. Nevertheless, in its current state the model is quite useful to estimate the mill performance in terms of the median size (d50), as shown in Fig. 14.

Fig. 11. Predicted and measured d50 and d80 of the mill product: 25.4-cm mill, 35% ball load, 55 rpm speed, 100% 5.08-cm balls.

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Fig. 12. Predicted and measured d50 and d80 of the mill products: 38.1-cm mill, 35% ball load, 41 rpm speed, 100% 5.08-cm regular steel balls.

The model was also tested on grinding experiments where multi-size feed and balls constituted the mill charge. The operating conditions are listed in the last two rows of Table 1. A natural feed of 10 mesh (1.7 mm) was charged in the mill in both tests. The ball charge was made up of a mixture of 3.81-, 2.54-, 1.91-, and 1.27-cm diameter balls. The predictions are shown in Figs. 15 and 16. Although the grinding media was made up of four size classes of ball, breakage data for only 3.81- and 2.54-cm balls were available. Hence, the mass of balls in the 1.91- and 1.27-cm diameter classes was equally distributed

Fig. 13. Predicted and measured product size distributions: 90.0-cm mill, 20% ball load, 18 rpm speed, 5.08-cm balls (100%).


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Fig. 14. Predicted and measured d50 of the mill product: 90-cm mill, 20% ball load, 18 rpm speed, 100% 5.08-cm balls.

into the 3.81- and 2.54-cm classes while simulating the charge motion of these two mills. Even with this approximation, the predictions were close to the measured product size distribution. It is anticipated that, with further modification in the simulation and prediction scheme, the model will become more accurate in the estimation of mill product size distributions. For example, the probability of a collision nipping some particles has been assumed to be equal irrespective of the location. This assumption will certainly introduce an error in the

Fig. 15. Predicted and measured product size distributions: 25.4-cm diameter mill, 50% ball load, 55-rpm mill speed, 3.81-cm top-size ball.

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Fig. 16. Predicted and measured product size distributions: 38.1-cm diameter mill, 50% ball load, 43 rpm mill speed, 3.81-cm top-size ball.

prediction and lead to overestimation of the fineness of the product size distribution. Further work will divide the collisions in terms of their spatial distribution and a probability factor incorporated in the model. Furthermore, the nipped particle bed was assumed to be made up of four layers of material. More realistically, the particle bed between two colliding balls will be composed of something in between a monolayer and several layers of material. In order to avoid this shortcoming of the model, the distance traveled by two balls in the mill during the course of a collision should be taken into account. This distance will be a fairly good indication of the number of layers of material caught in the collisions. However, the complexity of the prediction scheme will increase if these modifications are incorporated.

6. Conclusion A direct simulation approach for the modeling of the evolution of particle size in a tumbling mill is described. This approach treats the grinding process as a multitude of collisions of various energy values in which a bed of particles is caught and broken. The breakage process in a collision is idealized as a bed of four layers of particles being nipped in an impact of a specified energy. Schemes for the calculation of impact energy and measurement of breakage process in a collision are described. Such an idealization obviously implies severely restrictive assumptions. Despite the assumptions, the model predictions are satisfactory, which suggests that this approach is worth further development. For example, all of the collisions are taken as normal collision forces that nip a bed of particles, whereas, in reality, numerous collisions occur in oblique directions. This results in shearing of the particle bed, and shearing in turn produces finer fragments. A shear cell apparatus may be useful for characterizing this type of breakage of the particle bed. With


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these suggested modifications in the mill charge simulation as well as the experimental technique, it is anticipated that the model predictions will be improved further. Acknowledgements This research has been supported under Grant Number G1115149 from the United States Department of Interior, administered by the US Bureau of Mines through the Generic Mineral Technology Center of Comminution. References Austin, L.G., 1973. A commentary on the Kick, Bond and Rittinger Laws of grinding. Powder Technol. 7, 315 – 318. Austin, L.G., Shoji, K., Luckie, P.T., 1976. The effect of ball size on mill performance. Powder Technol. 14, 71 – 79. Austin, L.G., Klimpel, R.R., Luckie, P.T., 1984. Process Engineering of Size Reduction: Ball Milling. SME-AIME, New York. Bond, F.C., 1952. The third theory of comminution. Trans. AIME 193, 484 – 494. Bond, F.C., 1960. Crushing and grinding calculation. Br. Chem. Eng. 6, 378 – 391. 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 Publications, Littleton, Colorado, USA, pp. 99 – 108 Chap. 8. Broadbent, S.R., Callcott, T.G., 1956. A matrix analysis of processes involving particle assembly. Philos. Trans. R. Soc. London, Ser. A 249, 99 – 123. 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 and wear in ball mills using discrete element methods. Miner. Eng. 11 (11), 1061 – 1080. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assembly. Geotechnique 29, 47 – 65. 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. Dunn, D.J., Martin, R.G., 1978. Measurement of impact forces in ball mills. Mining Engineering 4, 384 – 388. Gumtz, G.D., Fuerstenau, D.W., 1970. Simulation of locked-cycle grinding. Trans. SME/AIME 247, 330 – 335. Herbst, J.A., 1979. In: Sohn, H.Y., Wadsworth, M.E. (Eds.), Rate Processes of Multiparticle Metallurgical Systems, Rate Processes of Extractive Metallurgy, Plenum Press, pp. 61 – 74. Herbst, J.A., Fuerstenau, D.W., 1973. Mathematical simulation of dry ball mill using specific power information. Trans., vol. 254. SME Publications, Littleton, Colorado, USA, pp. 343 – 348. Herbst, J.A., Fuerstenau, D.W., 1980. Scale-up procedures for continuous grinding mill design using population balance models. Int. J. Miner. Process. 7, 1 – 31. Herbst, J.A., Rajamani, R.K., 1982. In: Mular, A.L., Jergensen, G.V. (Eds.), Developing a Simulator for Ball Mill Scale-Up: A Case Study, Design and Installation of Comminution Circuit. AIME, New York, pp. 325 – 345. Herbst, J.A., Grandy, G.A., Fuerstenau, D.W., 1973. Population balance models for the design of continuous grinding mills. Proc. 10th International Mineral Processing Congress, pp. 27 – 37. Herbst, J.A., Siddique, M., Rajamani, K., Sanchez, E., 1981. Population balance approach to ball mill scale-up: bench and pilot scale investigations. Trans. SME/AIME 272, 1945 – 1954. 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. Kano, J., Chujo, N., Saito, F., 1997. A method for simulating the three-dimensional motion of balls under the presence of a powder sample in a tumbling ball mill. Adv. Powder Technol. 8 (1), 39 – 51.

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A Direct Approach of Modeling Batch Grinding

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