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Simulation of charge motion in ball mills. Part 2: numerical simulations
International Journal of Mineral Processing, 40 (1994) 187-197
187
Elsevier Science B.V., Amsterdam
Simulation of charge motion in ball mills. Part 2: numerical simulations B.K. Mishra and Raj K. Rajamai Comminution Center, University of Utah, Salt Lake City, UT 84112, USA (Received 28 April 1993; accepted after revision 30 June 1993)
ABSTRACT The formulation and verification of the discrete element model for the ball charge motion in a ball mill are described in Part I. Here, the model is used to simulate the charge motion in an industrial size mill. Many unobservable facts about charge motion are revealed by this simulation approach: It is shown that larger balls segregate to the center at high speeds and to the shell at lower speeds. The frequency of collisions in a 4.75 m diameter mill mostly lie within one joule. The friction between the ball charge and the mill shell can increase the power draft. The center of the ball mass shifts in distance as much as 4% of the mill diameter during a complete rotation. Finally, the distribution of collision energy and the spatial locations of high- and low-energy collisions are shown. While many of the simulated results may never be verified experimentally, these results are closer to actual values since the simulation is based on sound principles of physics. Above all, the simulation provides collision frequency information which is the key to mill design and optimization.
INTRODUCTION
In conventional studies on ball mills, two basic experimental measuremerits are made to asses the performance of the mill. These data involve the power draw of the mill and the product size distribution of the particulate material. Based on this information one can only speculate as to what really goes on inside the mill. It is easy to study the motion of the charge inside the mill in a laboratory; however, it is difficult to establish how a particular mode of charge motion would relate to the distribution of forces inside the mill. Also, motion analysis becomes difficult when mill diameter increases. Therefore, numerical analysis of such problems becomes natural. With a numerical scheme it is easy to perform parametric studies without excessive labor and cost. Also, multiple simulation runs with varying liner geometries can be easily made, and relevant information from intermediate simulation runs can be obtained. These advantages favor numerical simulation over physical experiments especially when the simulation is based on sound principles of physics. Experimental measurements of the impact forces operating inside the mill have been done in two ways: imbedding strain gauges and integrated circuitry 0301-7516/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved.
SSDI 0301-7516 (93)E0036-K
188
B.K. MISHRAANDR.K RAJAMAI
in the ball or, alternatively, imbedding sensors in the mill wall. Dunn and Martin (1978) inserted accelerometers inside a ball to measure and record m a x i m u m decelerations of the ball in an operating ball mill. This way they were able to predict the stresses imposed on the mill liner due to falling balls. Vermeulen et al. (1984) inserted piezo-electric sensors in some of the bolts used to clamp the liner block to the mill shell; these sensors recorded impact forces. Using load sensors on the mill shell, Yashima and Hashimoto ( 1988 ) obtained data related to the frequency of impact of balls corresponding to a given fall-height. Liddell and Moys (1988) studied the effect of mill speed and filling on the position of the toe and shoulder of the ball charge and hence the torque. They used electrical conductivity probes inserted in the liner to locate the angular position of the toe and the shoulder. This work pointed out the deficiency in some of the torque formulae; it was concluded that the angle of repose of the charge depended on operating conditions. Furthermore, it is too simplistic to assume that the ball load lies below the chord drawn from the toe to the shoulder. Rolfet al. ( 1982 ) have developed a new measuring technique for the direct recording of the energy distribution within the ball charge. This technique uses deflection of springs embedded in a ball to measure force. When the impact force exceeds the pretension force of the spring, an electrical impulse is generated and sent to a digital counter. These counts are read by means of a computer after the experiment. Rolf and Theudchai (1984) used these instrumented balls to measure the energy distribution in the ball mill. High-impact collisions are necessary to achieve fine grinding. However, such collisions result in extensive wear of liners. Mclvor ( 1983 ) studied----on similar lines proposed by Davis--the trajectory of the outer row of balls in a mill fitted with lifters of different face angles. He found, among other things, that the particle trajectories are highly sensitive to lifter face angle. Vermeulen and Howat ( 1988 ) analyzed cinematographic films to assess the effect of rectangular lifters on the motion of grinding elements en masse. They found, in contradiction to McIvor, that the lifter bar height has little effect on the motion of the rods. However, the lifter bars increase the intensity of impactive interaction of the rods. Information gathered by experimental means is still inadequate, primarily because the instrumentation itself is not well-developed. Therefore, as of date, neither experiments nor models have been able to provide reasonable estimates of the energy dissipated in grinding mills. The purpose of this paper is to establish that discrete element modeling is capable of handling multibocly collisions and, hence, accurately establish the motion of balls inside the mill which in turn drives the calculation for the power draw of the mill.
SIMULATION OF CHARGE MOTION IN BALL MILLS. NUMERICAL SIMULATIONS
189
MOTION ANALYSIS
Segregation effects Motion of charge is intimately related to the operating speed and the liner profile. It is easy to visualize the speed effect on the overall motion of the ball charge. However, with speed another motion phenomenon becomes apparent that deals with preferential motion of balls of different size. This is the segregation effect--a known physical problem involving charge motion whereby balls of a particular size move to the mill shell while others form the core of the charge, resulting in poor grinding. Generally, ball mills are operated with a distribution of balls according to their mass. Therefore, the mill operates at any time with different sizes of balls. The difference in the sizes and the mass of the balls coupled with the different kinds of forces that are operative inside the mill cause the balls to segregate. The results of the simulation done in a 4.3 m diameter mill fitted with eight rectangular lifter bars are shown in Figs. 1 and 2. In these figures "snapshots" taken at different speeds are shown. The mill is operated with three different
Fig. 1. Segregationof balls in a 4.3 m diameter mill; mill speed is 60% of critical.
190
B.K. MISHRA AND R.K. RAJAMAI
Fig. 2. Segregation ofbaUs in 38 cm diameter mill; mill speed is 80% of critical.
sizes of balls: 4.0, 2.5 and 1.25 cm. As shown in the figure, the balls of smaller size clearly segregate to the interior of the charge at about 60% of the critical speed. At about 80% of critical speed, s m ~ e r balls ~ n to centrifuge. The replication of the segregation behavior-- a complex phenomenon - - is a strong claim for the validity of the computer code developed here from a discrete element approach.
Effect of friction The coefficient of friction between the rolling layers of ball charge has a significant effect on the power draw (Lidddl and Moys, 1988). This effect is observed by studying the overall motion of the charge by varying the coefficient of friction at the contacts. In mills with no lifter bars, the motion of the charge strongly depends on the friction at the wall. However, it is interesting to observe the ball motion in a mill fitted with lifter bars: a 4.75 m ( 16 ft) diameter mill operating with approximately 45% ball load, using eight different ball sizes, is simulated. At 70% of critical speed the change in the profile of the load is shown in Fig. 3. The snap-shots in this ease correspond to the
SIMULATIONOF CHARGE MOTION IN BALL MILLS. NUMERICAL SIMULATIONS
191
Fig. 3. Variation in charge profile due to change in the coefficient of friction at 70% of critical speed. The charge profile left and right correspond to coefficient of friction of 0.7 and 0.2, respectively.
profile of the load at the end of rive revolutions. It is seen that, when a low coefficient of friction at the walls is used, balls tend to flow down the surface of the charge, and a toe begins to form. As the friction at the wall increases, cataracting motion is observed. A comparison of the power draw shows that for a coefficient of friction of 0.7 this mill would draw about i.5 times more power than in the case where it is 0.2.
Dilation of charge It is known that the profile of the charge inside the mill does vary with mill speed and filling. However, it is seen here that even for a given speed and rifling the charge profile changes as the mill rotates. This is illustrated in Fig. 4 where the variation of the center of gravity of the charge in a 55 cm diameter mill operating with 40% ball filling is shown. It is seen from this figure that the center of gravity is not a fixed point in time and space but shifts as the mill rotates. Also, for counterclockwise motion of the mill, the center of gravity shifts upward and to the left as the mill rotation was increased from 60% to 80% of the critical speed. Fig. 4 can be translated to show the variation in the number of contacts between balls as the mill rotates. The shift in the center of gravity of the ball charge is due to en-masse motion and local dilation or contraction of the charge. Therefore, Fig. 4 indicates that, as the mill rotates, the ball charge may
192
B.K. MISHRA
AND R.K
RA,IAMAI
24 Mill Diameter= 55cm
Ball Load=40% o
o
25
o
o t.)
~
22
°o
o
o
~
Speed 60%
Critical ° "~80%
>L
÷ +
21
÷
÷
34
.(
÷÷÷ ÷
+ +
÷
35
56
X-distance, cms
Fig. 4. Variation in the center of mass of the ball charge in a 55 cm diameter mill during motion.
dilate locally, resulting in a pore structure. This local behavior certainly affects the rate of breakage and transport of the particulate material. DISTRIBUTION OF ENERGY AND FALL-HEIGHT
The impact of grinding media inside a ball mill is the most difficult of all parameters to measure or predict. While the liberation of valuable minerals is achieved by impacts of balls on particles, at the same time these impacts cause the liners to wear, deform, and eventually crack. Therefore, it is essential to quantify the forces acting inside a mill so that the material used and its design can be improved. The individual collisions are modeled by a pair of spring and dashpot. Therefore, it is possible to calculate the energy distribution of collisions in the ball charge. More importantly, the difficulties associated in the experimental measurements (Rolf et al., 1982 ) in finding the energy expended in oblique impacts (oblique in particular) are easily overcome.
Distribution of energy The distribution of the energy inside the mill is often related to the rate of breakage of the particulate material. Simulation results show that the distribution of energy inside the mill changes with mill speed, filling, ball size distribution, and the shape of the lifter bars. While the overall effect of these parameters is being studied by means of statistical analysis (Mishra and Ra-
SIMULATIONOF CHARGEMOTION IN BALLMILLS.NUMERICALSIMULATIONS
193
10 4 80% crit.
I0 3
5 10 2
o
101
10 ° 0
10
20
30
40
50
60
70
Meen Energy (J)
Fig. 5. Frequency distribution o f impacts in 4.75 m d i a m e t e r mill.
jamani, 1992b), the individual effects of some parameters are shown here. A 4.75 m ( 16 ft) diameter mill is simulated to study the distribution of energy inside the mill. This mill has 60 equally spaced wavy lifters and contains about 45% ball load. The ball sizes correspond to eight size classes, and the mass distribution follows the equilibrium ball size distribution with a top size ball diameter of 12.5 era. For two different speeds, viz. 60 and 80 percent of the critical speed, the distribution of energy is plotted in Fig. 5. Fig. 5 is a frequency distribution diagram of impact frequency versus impact energy where the specific number of impacts is the total number of impacts per unit of time. This bar diagram shows that most of the collisions are between 0 to 1 joule for both speeds and only the relative positions of the bars change with speed. In other words, the total number of collisions in one revolution may vary with speed, however, most of the collisions during one revolution will still be within one joule. Apart from mill speed, filling, and diameter, it is known that lifter bar shape and viscosity of the charge may affect the energy distribution inside the mill. In smaller mills (25 to 55 cm), it is observed that all the collisions are of the order of millijoules. As the diameter is increased, the collision energy also increases. The high energy collisions are mostly due to the impactive pressure of the charge. A ball that is caught at the bottom of the toe during its ascent experiences the load of the entire charge. If contact is preserved during its ascent then the collision energy at that contact will be very high. The other source of high-energy collisions is the cataracting balls. It is shown (Mishra
J 94
B.K. MISHRA AND R.K. RAJAMAI
and Rajamani, 1992a) that in large mills the impact energy may go as high as 200 joules, at impact forces of the order of 1000 newtons, but these impacts constitute only 10 to 20% of the total number of impacts. The power supplied to the mill is used to sustain the charge in its dynamic position, which in turn does work on the particles. Energy is wasted inside the mill in unwanted collisions. To show the energy map inside the mill, a contour plot of the dissipated energy in various parts of the mill is prepared. To do this, the entire mill (4.75 m diameter) is divided into 35 X 35 square grids. The location of the impact and the corresponding energy are mapped into these grids. Thus at the end of the simulation, the total energy supplied is shown as distributed within the mill. The result of one such simulation is I
H~LI
I
Speed
[
I
=
I
i
I
[
I
I
80% C r ~ t ,
I
] i
l
I
f
l
I
I
Fig. 6. Contour plots of the energy inside a 4.75 m d i a m e t e r mill. C o n t o u r f r o m 0 to 1020. C o n t o u r i n t e r v a l 60. x i n t e r v a l = 0.49 m , y i n t e r v a l = 0 . 4 9 m .
SIMULATIONOF CHARGE MOTION IN BALLMILLS. NUMERICAL SIMULATIONS
19 5
shown in Fig. 6 for a mill that rotates counterclockwise. The x- and y-axes of this plot correspond to the diameter of the mill. It can be noted from this figure that the supplied energy is distributed in accordance with the motion of the charge. Most of the energy is accumulated in the toe region. Thus active grinding takes place in this area. There are pockets that may constitute more than half the area of the mill where no impacts take place.
Distribution offall-heights The average vertical distance through which the colliding bodies travel is a good indication of the intensity of the forces that operate inside the mill. This vertical distance is caged the fall-height. Fig. 7 compares the fall-heights at two different speeds, in a 4.75 m ( 16 ft) diameter mill. One important aspect of ball motion is now apparent from this figure: At lower speeds average fall height is small, whereas at high speeds balls may fall from more than three quarters of the diameter of the mill. The fag height by itself is misleading in determining the severity of impact; it should be analyzed with the corresponding velocity. Nevertheless, it is a good indicator of the motion of the charge. Having studied the fall-height distribf~tion, control of the impactive forces may be exercised by adjusting the mass of the grinding media. However, Rose and Sullivan (1958) pointed out that the efficiency of impact is impaired when the balls are excessively large in diameter. Therefore, the mass of the 10 4
~
80% crit.]
10 3
~
10 2
0
(,~ 101
10° 0
60
120
1BO
240
300
360
Fall-Height (Cm) Fig. 7. Frequency distribution of impacts in a 4.75 m diameter mill.
196
B,K. MISHRA AND R.K. RAJAMAI
balls and their diameter have to be judiciously adjusted so that the grinding efficiency is optimized. Further insight to this problem can be obtained by studying the fragmentation and axial flow properties of the ore. CONCLUSIONS
The numerical calculations presented here are helpful in studying the events that take place inside the mill. Such information is otherwise difficult to obtain by experimental means. Most of the trends observed here pertain to large mills; the results of the simulations can aid in mill design and operation. The following observations are made from motion analysis in ball mills: The ball charge inside the mill exhibits different modes of motion depending on the operating factors. At different mill speeds a graded ball charge segregate according to the size. Large balls tend to move to the center of the charge at high speeds and smaller balls move to the center at low speeds. Charge level inside the mill also changes during one revolution, indicating dilation and compaction of the ball bed. - At any given speed, the charge profile depends on the frictional forces developed along the sliding surfaces. At a high coefficient of friction (0.7 and above), more layers of balls tend to cataract. The distribution of energy and fall-heights inside the mill points to one important feature of charge motion: Even in large mills of 4.75 m ( 16 ft) diameter, the energies involved in most of the collisions are within 1 joule. This is surprising, because a 12 cm diameter ball when falling from a distance of half the diameter of the mill should have an impact energy around 150 joules. The fact that such high-intensity collisions are few in number suggests that most of the ball charge moves in a cascading manner. Some of the high-energy impacts recorded in these simulations could be due to the impactive charge pressure and may not be due to the cataracting action of the charge. More simulations need to be done to delineate this effect. -
-
ACKNOWLEDGEMENT
The authors would like to thank the Utah Super Computing Institute, which is funded by the State of Utah and the IBM Corporation, for their support in this endeavor. This research has been supported by the Department of Interior's Mineral Institute program administered by the Bureau of Mines through the Generic Mineral Technology Center for Comminution under grant number G1175149.
SIMULATION OF CHARGE MOTION IN BALL MILLS. NUMERICAL SIMULATIONS
197
REFERENCES Dunn, J. and Martin, R., 1978. Measurement of impact forces in ball mills. Min. Eng.: 30:384388. Liddell, K.S. and Moys, M.H., 1988. The effects of mill speed and filling on the behavior of the load in a rotary grinding mill. J. S. Afr. Inst. Min. Metall., 88: 49-57. McIvor, R.E., 1983. Effect of speed and liner configuration on ball mill performance. Min. Eng., 7: 617-622. Mishra, B.K. and Rajamani, R.K., 1992a. Analysis of media motion in industrial mills. In: S.K. Kawatra (Editor), Comminution - - Theory and Practice. Society for Mining Metallurgy and Exploration, Littleton, CO, pp. 427-440. Mishra, B.K. and Rajamani, R.K., 1992b. Analysis of power draw in tumbling mills. 93-217: Preprint of paper presented at SME-AIME Annual Meeting, Reno, NV. Rolf, L. and Theudchai, V., 1984. Measurement of energy distributions in ball mills. Ger. Chem. Eng., 7: 287-292. Rolf, L., Vongluekiet, T. and Uygun, M., 1982. Stress energy in ball and vibration mills. Bergbau: 311-318. Rose, H.E. and Sullivan, R.M.E., 1958. Ball, Tube and Rod Mills. Chemical Publishing Co., New York, NY. Vermeulen, L., Fine, M. and Schakowski, F., 1984. Physical information from the inside of a rotary mill. J. S. Aft. Inst. Min. Metall., 84: 247-253. Vermeulen, L.A. and Howat, D.D., 1988. Effect of lifter bars on the motion of en-masse grinding media in milling. Int. J. Miner. Process., 24: 143-159. Yashima, S. and Hashimoto, H., 1988. Measurement of kinetic energy of grinding media in tumbling ball mills. In: E. Forssberg (Editor), XVI International Mineral Processing Congress, Stockholm. Elsevier Amsterdam, pp. 299-309.
187
Elsevier Science B.V., Amsterdam
Simulation of charge motion in ball mills. Part 2: numerical simulations B.K. Mishra and Raj K. Rajamai Comminution Center, University of Utah, Salt Lake City, UT 84112, USA (Received 28 April 1993; accepted after revision 30 June 1993)
ABSTRACT The formulation and verification of the discrete element model for the ball charge motion in a ball mill are described in Part I. Here, the model is used to simulate the charge motion in an industrial size mill. Many unobservable facts about charge motion are revealed by this simulation approach: It is shown that larger balls segregate to the center at high speeds and to the shell at lower speeds. The frequency of collisions in a 4.75 m diameter mill mostly lie within one joule. The friction between the ball charge and the mill shell can increase the power draft. The center of the ball mass shifts in distance as much as 4% of the mill diameter during a complete rotation. Finally, the distribution of collision energy and the spatial locations of high- and low-energy collisions are shown. While many of the simulated results may never be verified experimentally, these results are closer to actual values since the simulation is based on sound principles of physics. Above all, the simulation provides collision frequency information which is the key to mill design and optimization.
INTRODUCTION
In conventional studies on ball mills, two basic experimental measuremerits are made to asses the performance of the mill. These data involve the power draw of the mill and the product size distribution of the particulate material. Based on this information one can only speculate as to what really goes on inside the mill. It is easy to study the motion of the charge inside the mill in a laboratory; however, it is difficult to establish how a particular mode of charge motion would relate to the distribution of forces inside the mill. Also, motion analysis becomes difficult when mill diameter increases. Therefore, numerical analysis of such problems becomes natural. With a numerical scheme it is easy to perform parametric studies without excessive labor and cost. Also, multiple simulation runs with varying liner geometries can be easily made, and relevant information from intermediate simulation runs can be obtained. These advantages favor numerical simulation over physical experiments especially when the simulation is based on sound principles of physics. Experimental measurements of the impact forces operating inside the mill have been done in two ways: imbedding strain gauges and integrated circuitry 0301-7516/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved.
SSDI 0301-7516 (93)E0036-K
188
B.K. MISHRAANDR.K RAJAMAI
in the ball or, alternatively, imbedding sensors in the mill wall. Dunn and Martin (1978) inserted accelerometers inside a ball to measure and record m a x i m u m decelerations of the ball in an operating ball mill. This way they were able to predict the stresses imposed on the mill liner due to falling balls. Vermeulen et al. (1984) inserted piezo-electric sensors in some of the bolts used to clamp the liner block to the mill shell; these sensors recorded impact forces. Using load sensors on the mill shell, Yashima and Hashimoto ( 1988 ) obtained data related to the frequency of impact of balls corresponding to a given fall-height. Liddell and Moys (1988) studied the effect of mill speed and filling on the position of the toe and shoulder of the ball charge and hence the torque. They used electrical conductivity probes inserted in the liner to locate the angular position of the toe and the shoulder. This work pointed out the deficiency in some of the torque formulae; it was concluded that the angle of repose of the charge depended on operating conditions. Furthermore, it is too simplistic to assume that the ball load lies below the chord drawn from the toe to the shoulder. Rolfet al. ( 1982 ) have developed a new measuring technique for the direct recording of the energy distribution within the ball charge. This technique uses deflection of springs embedded in a ball to measure force. When the impact force exceeds the pretension force of the spring, an electrical impulse is generated and sent to a digital counter. These counts are read by means of a computer after the experiment. Rolf and Theudchai (1984) used these instrumented balls to measure the energy distribution in the ball mill. High-impact collisions are necessary to achieve fine grinding. However, such collisions result in extensive wear of liners. Mclvor ( 1983 ) studied----on similar lines proposed by Davis--the trajectory of the outer row of balls in a mill fitted with lifters of different face angles. He found, among other things, that the particle trajectories are highly sensitive to lifter face angle. Vermeulen and Howat ( 1988 ) analyzed cinematographic films to assess the effect of rectangular lifters on the motion of grinding elements en masse. They found, in contradiction to McIvor, that the lifter bar height has little effect on the motion of the rods. However, the lifter bars increase the intensity of impactive interaction of the rods. Information gathered by experimental means is still inadequate, primarily because the instrumentation itself is not well-developed. Therefore, as of date, neither experiments nor models have been able to provide reasonable estimates of the energy dissipated in grinding mills. The purpose of this paper is to establish that discrete element modeling is capable of handling multibocly collisions and, hence, accurately establish the motion of balls inside the mill which in turn drives the calculation for the power draw of the mill.
SIMULATION OF CHARGE MOTION IN BALL MILLS. NUMERICAL SIMULATIONS
189
MOTION ANALYSIS
Segregation effects Motion of charge is intimately related to the operating speed and the liner profile. It is easy to visualize the speed effect on the overall motion of the ball charge. However, with speed another motion phenomenon becomes apparent that deals with preferential motion of balls of different size. This is the segregation effect--a known physical problem involving charge motion whereby balls of a particular size move to the mill shell while others form the core of the charge, resulting in poor grinding. Generally, ball mills are operated with a distribution of balls according to their mass. Therefore, the mill operates at any time with different sizes of balls. The difference in the sizes and the mass of the balls coupled with the different kinds of forces that are operative inside the mill cause the balls to segregate. The results of the simulation done in a 4.3 m diameter mill fitted with eight rectangular lifter bars are shown in Figs. 1 and 2. In these figures "snapshots" taken at different speeds are shown. The mill is operated with three different
Fig. 1. Segregationof balls in a 4.3 m diameter mill; mill speed is 60% of critical.
190
B.K. MISHRA AND R.K. RAJAMAI
Fig. 2. Segregation ofbaUs in 38 cm diameter mill; mill speed is 80% of critical.
sizes of balls: 4.0, 2.5 and 1.25 cm. As shown in the figure, the balls of smaller size clearly segregate to the interior of the charge at about 60% of the critical speed. At about 80% of critical speed, s m ~ e r balls ~ n to centrifuge. The replication of the segregation behavior-- a complex phenomenon - - is a strong claim for the validity of the computer code developed here from a discrete element approach.
Effect of friction The coefficient of friction between the rolling layers of ball charge has a significant effect on the power draw (Lidddl and Moys, 1988). This effect is observed by studying the overall motion of the charge by varying the coefficient of friction at the contacts. In mills with no lifter bars, the motion of the charge strongly depends on the friction at the wall. However, it is interesting to observe the ball motion in a mill fitted with lifter bars: a 4.75 m ( 16 ft) diameter mill operating with approximately 45% ball load, using eight different ball sizes, is simulated. At 70% of critical speed the change in the profile of the load is shown in Fig. 3. The snap-shots in this ease correspond to the
SIMULATIONOF CHARGE MOTION IN BALL MILLS. NUMERICAL SIMULATIONS
191
Fig. 3. Variation in charge profile due to change in the coefficient of friction at 70% of critical speed. The charge profile left and right correspond to coefficient of friction of 0.7 and 0.2, respectively.
profile of the load at the end of rive revolutions. It is seen that, when a low coefficient of friction at the walls is used, balls tend to flow down the surface of the charge, and a toe begins to form. As the friction at the wall increases, cataracting motion is observed. A comparison of the power draw shows that for a coefficient of friction of 0.7 this mill would draw about i.5 times more power than in the case where it is 0.2.
Dilation of charge It is known that the profile of the charge inside the mill does vary with mill speed and filling. However, it is seen here that even for a given speed and rifling the charge profile changes as the mill rotates. This is illustrated in Fig. 4 where the variation of the center of gravity of the charge in a 55 cm diameter mill operating with 40% ball filling is shown. It is seen from this figure that the center of gravity is not a fixed point in time and space but shifts as the mill rotates. Also, for counterclockwise motion of the mill, the center of gravity shifts upward and to the left as the mill rotation was increased from 60% to 80% of the critical speed. Fig. 4 can be translated to show the variation in the number of contacts between balls as the mill rotates. The shift in the center of gravity of the ball charge is due to en-masse motion and local dilation or contraction of the charge. Therefore, Fig. 4 indicates that, as the mill rotates, the ball charge may
192
B.K. MISHRA
AND R.K
RA,IAMAI
24 Mill Diameter= 55cm
Ball Load=40% o
o
25
o
o t.)
~
22
°o
o
o
~
Speed 60%
Critical ° "~80%
>L
÷ +
21
÷
÷
34
.(
÷÷÷ ÷
+ +
÷
35
56
X-distance, cms
Fig. 4. Variation in the center of mass of the ball charge in a 55 cm diameter mill during motion.
dilate locally, resulting in a pore structure. This local behavior certainly affects the rate of breakage and transport of the particulate material. DISTRIBUTION OF ENERGY AND FALL-HEIGHT
The impact of grinding media inside a ball mill is the most difficult of all parameters to measure or predict. While the liberation of valuable minerals is achieved by impacts of balls on particles, at the same time these impacts cause the liners to wear, deform, and eventually crack. Therefore, it is essential to quantify the forces acting inside a mill so that the material used and its design can be improved. The individual collisions are modeled by a pair of spring and dashpot. Therefore, it is possible to calculate the energy distribution of collisions in the ball charge. More importantly, the difficulties associated in the experimental measurements (Rolf et al., 1982 ) in finding the energy expended in oblique impacts (oblique in particular) are easily overcome.
Distribution of energy The distribution of the energy inside the mill is often related to the rate of breakage of the particulate material. Simulation results show that the distribution of energy inside the mill changes with mill speed, filling, ball size distribution, and the shape of the lifter bars. While the overall effect of these parameters is being studied by means of statistical analysis (Mishra and Ra-
SIMULATIONOF CHARGEMOTION IN BALLMILLS.NUMERICALSIMULATIONS
193
10 4 80% crit.
I0 3
5 10 2
o
101
10 ° 0
10
20
30
40
50
60
70
Meen Energy (J)
Fig. 5. Frequency distribution o f impacts in 4.75 m d i a m e t e r mill.
jamani, 1992b), the individual effects of some parameters are shown here. A 4.75 m ( 16 ft) diameter mill is simulated to study the distribution of energy inside the mill. This mill has 60 equally spaced wavy lifters and contains about 45% ball load. The ball sizes correspond to eight size classes, and the mass distribution follows the equilibrium ball size distribution with a top size ball diameter of 12.5 era. For two different speeds, viz. 60 and 80 percent of the critical speed, the distribution of energy is plotted in Fig. 5. Fig. 5 is a frequency distribution diagram of impact frequency versus impact energy where the specific number of impacts is the total number of impacts per unit of time. This bar diagram shows that most of the collisions are between 0 to 1 joule for both speeds and only the relative positions of the bars change with speed. In other words, the total number of collisions in one revolution may vary with speed, however, most of the collisions during one revolution will still be within one joule. Apart from mill speed, filling, and diameter, it is known that lifter bar shape and viscosity of the charge may affect the energy distribution inside the mill. In smaller mills (25 to 55 cm), it is observed that all the collisions are of the order of millijoules. As the diameter is increased, the collision energy also increases. The high energy collisions are mostly due to the impactive pressure of the charge. A ball that is caught at the bottom of the toe during its ascent experiences the load of the entire charge. If contact is preserved during its ascent then the collision energy at that contact will be very high. The other source of high-energy collisions is the cataracting balls. It is shown (Mishra
J 94
B.K. MISHRA AND R.K. RAJAMAI
and Rajamani, 1992a) that in large mills the impact energy may go as high as 200 joules, at impact forces of the order of 1000 newtons, but these impacts constitute only 10 to 20% of the total number of impacts. The power supplied to the mill is used to sustain the charge in its dynamic position, which in turn does work on the particles. Energy is wasted inside the mill in unwanted collisions. To show the energy map inside the mill, a contour plot of the dissipated energy in various parts of the mill is prepared. To do this, the entire mill (4.75 m diameter) is divided into 35 X 35 square grids. The location of the impact and the corresponding energy are mapped into these grids. Thus at the end of the simulation, the total energy supplied is shown as distributed within the mill. The result of one such simulation is I
H~LI
I
Speed
[
I
=
I
i
I
[
I
I
80% C r ~ t ,
I
] i
l
I
f
l
I
I
Fig. 6. Contour plots of the energy inside a 4.75 m d i a m e t e r mill. C o n t o u r f r o m 0 to 1020. C o n t o u r i n t e r v a l 60. x i n t e r v a l = 0.49 m , y i n t e r v a l = 0 . 4 9 m .
SIMULATIONOF CHARGE MOTION IN BALLMILLS. NUMERICAL SIMULATIONS
19 5
shown in Fig. 6 for a mill that rotates counterclockwise. The x- and y-axes of this plot correspond to the diameter of the mill. It can be noted from this figure that the supplied energy is distributed in accordance with the motion of the charge. Most of the energy is accumulated in the toe region. Thus active grinding takes place in this area. There are pockets that may constitute more than half the area of the mill where no impacts take place.
Distribution offall-heights The average vertical distance through which the colliding bodies travel is a good indication of the intensity of the forces that operate inside the mill. This vertical distance is caged the fall-height. Fig. 7 compares the fall-heights at two different speeds, in a 4.75 m ( 16 ft) diameter mill. One important aspect of ball motion is now apparent from this figure: At lower speeds average fall height is small, whereas at high speeds balls may fall from more than three quarters of the diameter of the mill. The fag height by itself is misleading in determining the severity of impact; it should be analyzed with the corresponding velocity. Nevertheless, it is a good indicator of the motion of the charge. Having studied the fall-height distribf~tion, control of the impactive forces may be exercised by adjusting the mass of the grinding media. However, Rose and Sullivan (1958) pointed out that the efficiency of impact is impaired when the balls are excessively large in diameter. Therefore, the mass of the 10 4
~
80% crit.]
10 3
~
10 2
0
(,~ 101
10° 0
60
120
1BO
240
300
360
Fall-Height (Cm) Fig. 7. Frequency distribution of impacts in a 4.75 m diameter mill.
196
B,K. MISHRA AND R.K. RAJAMAI
balls and their diameter have to be judiciously adjusted so that the grinding efficiency is optimized. Further insight to this problem can be obtained by studying the fragmentation and axial flow properties of the ore. CONCLUSIONS
The numerical calculations presented here are helpful in studying the events that take place inside the mill. Such information is otherwise difficult to obtain by experimental means. Most of the trends observed here pertain to large mills; the results of the simulations can aid in mill design and operation. The following observations are made from motion analysis in ball mills: The ball charge inside the mill exhibits different modes of motion depending on the operating factors. At different mill speeds a graded ball charge segregate according to the size. Large balls tend to move to the center of the charge at high speeds and smaller balls move to the center at low speeds. Charge level inside the mill also changes during one revolution, indicating dilation and compaction of the ball bed. - At any given speed, the charge profile depends on the frictional forces developed along the sliding surfaces. At a high coefficient of friction (0.7 and above), more layers of balls tend to cataract. The distribution of energy and fall-heights inside the mill points to one important feature of charge motion: Even in large mills of 4.75 m ( 16 ft) diameter, the energies involved in most of the collisions are within 1 joule. This is surprising, because a 12 cm diameter ball when falling from a distance of half the diameter of the mill should have an impact energy around 150 joules. The fact that such high-intensity collisions are few in number suggests that most of the ball charge moves in a cascading manner. Some of the high-energy impacts recorded in these simulations could be due to the impactive charge pressure and may not be due to the cataracting action of the charge. More simulations need to be done to delineate this effect. -
-
ACKNOWLEDGEMENT
The authors would like to thank the Utah Super Computing Institute, which is funded by the State of Utah and the IBM Corporation, for their support in this endeavor. This research has been supported by the Department of Interior's Mineral Institute program administered by the Bureau of Mines through the Generic Mineral Technology Center for Comminution under grant number G1175149.
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REFERENCES Dunn, J. and Martin, R., 1978. Measurement of impact forces in ball mills. Min. Eng.: 30:384388. Liddell, K.S. and Moys, M.H., 1988. The effects of mill speed and filling on the behavior of the load in a rotary grinding mill. J. S. Afr. Inst. Min. Metall., 88: 49-57. McIvor, R.E., 1983. Effect of speed and liner configuration on ball mill performance. Min. Eng., 7: 617-622. Mishra, B.K. and Rajamani, R.K., 1992a. Analysis of media motion in industrial mills. In: S.K. Kawatra (Editor), Comminution - - Theory and Practice. Society for Mining Metallurgy and Exploration, Littleton, CO, pp. 427-440. Mishra, B.K. and Rajamani, R.K., 1992b. Analysis of power draw in tumbling mills. 93-217: Preprint of paper presented at SME-AIME Annual Meeting, Reno, NV. Rolf, L. and Theudchai, V., 1984. Measurement of energy distributions in ball mills. Ger. Chem. Eng., 7: 287-292. Rolf, L., Vongluekiet, T. and Uygun, M., 1982. Stress energy in ball and vibration mills. Bergbau: 311-318. Rose, H.E. and Sullivan, R.M.E., 1958. Ball, Tube and Rod Mills. Chemical Publishing Co., New York, NY. Vermeulen, L., Fine, M. and Schakowski, F., 1984. Physical information from the inside of a rotary mill. J. S. Aft. Inst. Min. Metall., 84: 247-253. Vermeulen, L.A. and Howat, D.D., 1988. Effect of lifter bars on the motion of en-masse grinding media in milling. Int. J. Miner. Process., 24: 143-159. Yashima, S. and Hashimoto, H., 1988. Measurement of kinetic energy of grinding media in tumbling ball mills. In: E. Forssberg (Editor), XVI International Mineral Processing Congress, Stockholm. Elsevier Amsterdam, pp. 299-309.