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Load behavior and mill power
Int. J. Miner. Process. 69 (2003) 11 – 28 www.elsevier.com/locate/ijminpro
Load behavior and mill power H. Dong *, M.H. Moys School of Process and Materials Engineering, University of the Witwatersrand, Private Bag 3, Johannesburg, WITS, 2050, South Africa Received 1 December 2001; received in revised form 21 April 2002; accepted 22 April 2002
Abstract Load orientation is conventionally described by using shoulder and toe positions and a chord connecting the shoulder and toe position with its slope equivalent to the angle of repose of mill load. This model suffers a marked discrepancy with the true load orientation at low ball filling or high speed of rotation. One reason for this discrepancy is that the load in a mill will dilate and change shape substantially at high speed. It means that the distribution of grinding elements in the mill chamber does not match the above description at all. So some torque-arm power models based on load behavior failed to predict power draw with a satisfactory accuracy at high speed. In this paper, experimental position density plots (PDPs) are used to describe load behavior in a tumbling mill. A PDP is a digital, visual and statistical representation of the charge distribution in a mill. It is constructed by superimposing a number of independent digitized images of a mill load at steady state. Not only the parameters about load behavior, such as dynamic angle of repose, shoulder angle, and toe angle, are determined from PDP, power drawn by the entire load and en masse load can be calculated as well by use of the torque-arm model directly. D 2003 Elsevier Science B.V. All rights reserved. Keywords: comminution; grinding; mill; power
1. Introduction Mill power drawn is a function of mill size (L and D), the loading ( J), and operating conditions such as mill speed N, slurry/void ratio, slurry density, etc. However, these are not all the variables having influence on mill power. As systematically summarized by Rose and Sullivan (1958), apart from the aforementioned variables, which can be clearly
*
Corresponding author. Fax: +27-11-4031471. E-mail addresses: [email protected] (H. Dong), [email protected] (M.H. Moys).
0301-7516/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 ( 0 2 ) 0 0 0 4 6 - 7
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defined and often included in power equations, physical properties of materials in a mill (hardness, density, size distribution, coefficient of restitution, and coefficient of friction), lifter design (shape and number), slurry rheological attributes and slurry hold up that change as a mill works in grate discharge mode or overflow discharge mode, influence the power required to drive a mill to some extent. Owing to the great number of variables, different treatments of the major variables lead to various power equations. It is impossible to do justice to the enormous amount of work in this area. Briefly speaking(Austin et al., 1984), there are two main approaches in the derivation of power equations. One approach (Davis, 1919; Rose and Sullivan, 1958; Morrell, 1993) calculates the paths of balls tumbling in the mill and integrates the energy required to raise the balls over all possible paths. The other approach (Hogg and Fuerstenau, 1972; Moys, 1990) uses the torque-arm model to get the power equations. Davis (op. cit) analyzed the force balance between the centrifugal force and gravitational force acting on a ball in a tumbling mill. Then he found an equilibrium surface where the balls commence their parabolic motion and established his power equation. One obvious deficiency of this model is that the frictional force between balls that produces frictional heat was not considered. Rose and Sullivan (op. cit) carried out a more comprehensive force balance analysis among the gravity force, centrifugal force and frictional force and derived a power formula using dimensional analysis. With the understanding of the load behavior deepened by measurements of load behavior in a glass-sided laboratory mill, Morrell proposed a model based on the observation of toe and shoulder position and angular velocity gradient of successive layers in the charge. His model was validated with a broad database of the power draws of industrial ball mills, SAG mills and AG mills and an excellent agreement was observed. The torque-arm model uses a chord connecting the toe and shoulder of the charge to delineate the charge shape. When the information about toe and shoulder positions is not available, the angle of repose or dynamic angle of repose is used as the slope of the chord to define the profile of the charge. However, the free rolling surface of the cascading charge, at high speeds in particular, is far from a straight line. So the torque-arm model does not work well at high mill speeds as it does at low speeds. Moys modified the torque model by treating the whole charge as two parts: centrifuged part and non-centrifuged part. The centrifuged part draws no power and its thickness varies with mill speed, lifter configuration, and slurry rheology. The mill power was then determined by use of Bond’s power equation according to the behavior of non-centrifuged part that was described by effective speed, filling and diameter of the mill. This treatment seems to describe fairly accurately the behavior of mill power over a wide range of operating conditions. The basic form of the torque-arm model for mill power is 2p N ðrpmÞT ðNmÞ 60
ð1Þ
T ¼ Force ðN Þ Torque-arm ðmÞ ¼ Mgxcog
ð2Þ
P ðwattsÞ ¼
All the torque-arm models attempted to relate torque to mill design, load properties, and operating conditions. In this paper, a direct application of the above formula to calculate
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the power drawn by a laboratory mill of an internal diameter of 0.55 m is reported. The torque is calculated from a position density plot (PDP, defined below), from which the center of gravity (COG) of the load is statistically determined. PDPs visually reflect the influences of mill speed, filling, and lifter configuration on load behavior and mill power. An excellent agreement is observed by the comparison of calculated power against measured power.
2. Experimental apparatus The experimental mill has an internal diameter of 0.55 m and 0.0235 m in length fitted with twelve 0.022 0.022 m square lifter bars. The load is composed of chrome steel balls of diameter 0.02225 m. Only one layer of balls in the axial direction can be housed to ensure that all the balls can be observed at any time. So it is termed 2D mill. A transparent end plate is used to allow the load to be photographed. The 2D mill, with its driving shaft connected with a load beam measuring the torque when the mill is running, has its speed adjusted from 0% up to 200% of critical speed. A digital camera is used to take pictures of the load in the 2D mill with a digital stroboscope as the light source.
3. Construction and validation of PDP A PDP is a plot constructed by analyzing tens of photographs of the charge in the mill operating at steady state to statistically quantify the charge distribution within the mill. Each digital picture is then analyzed with an image analysis package to read the position of every point that represents the center of a ball in the mill. A key issue to get the true position of a ball center from an image is to calibrate image distortions. The calibration process has been discussed in our previous publication (Dong and Moys, 2001). After calibration, the maximum position shift between the true position and image position is less than 1.2 mm. This means that the maximum relative measurement error is 0.22% relative to the mill diameter. A sample digitized frame is shown in Fig. 1(a) where there are 160 points representing the centers of 160 balls in the mill at 80% of critical speed. All the photos for one PDP were taken at a rate of two frames per second (fps) to make sure that every single frame is an independent observation. Thirty digitized frames are superimposed together to build a PDP as shown in Fig. 1(b) where there are 4800 points. This ensures that a load behavior over a time of 15 s (corresponding to about 12 revolutions at 80% of critical speed) is obtained. Fig. 1(c) is a picture taken simultaneously under the same operating conditions except that the exposure time of the camera was set at 2 s and the flashing rate of the strobe 60 fps. In other words, there are 120 frames during one and half revolution of the mill. It can be seen from Fig. 1(b) and (c) that: (1) balls in the tumbling mill are moving in layers (at least four layers can be clearly distinguished); (2) the outermost layer is raised much higher by the square lifters and landed on the bare shell at nine o’clock position; (3) the inner layers also experience a brief period of cataracting that is attributed to the lifting
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Fig. 1. Load behavior at J = 30% and N = 80% critical speed. (a) A sample digitized frame of mill; (b) digital PDP J30N80 of 30 frames; (c) a photograph of the mill with 120 exposures spread over 2 s.
effect of the outer layer of balls. It is safe to say, by comparing Fig. 1(b) and (c), that the PDP is an excellent digital visual representation of charge distribution within a mill.
4. Results The load behavior of the 2D mill at a wide range of speed of rotation and filling level has been investigated systematically and is presented in the form of PDPs in Fig. 2. Each PDP is labeled with its major operating conditions as a double check. For example, the PDP with a label of J30N40 can be found at row N = 40% of critical speed and column J = 30% of load volume. Every PDP is made up of more than 30 frames to
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ensure its representativity. Analysis of these PDPs will be made in the following sections.
5. Analysis 5.1. Free rolling surface It has been long established that the charge in a tumbling mill moves in two modes: cascading and cataracting. It is of practical significance to differentiate them on the PDPs. The profiles of en masse charge are abstracted into four types from all PDPs in Fig. 2: linear, bilinear, trilinear-chair, and trilinear-saddle, which are illustrated in Fig. 3. These four types of profiles can be modeled with one general trilinear form of two points and two slopes. If the two points converge to one point and the two slopes are identical, the trilinear model is reduced to the linear form (Fig. 3(a)); if the two points converge to one and the two slopes are different, the trilinear model is reduced to the bilinear form (Fig. 3(b)); if the two points are separate and the two slopes are all positive, the chair-type of profile can be expected (Fig. 3(c)); if the two points are separate and the slope for the left line is negative and that for the right line positive, the saddle profile applies (Fig. 3(d)). We use the trilinear model to express all free surfaces in Fig. 2. The two endpoints determine the middle section and the two slopes together with their corresponding endpoints define the other two segments. For example, as for N = 70% and J = 40%, the free surface drawn from the trilinear model is shown in Fig. 4. The free rolling surface is not fixed due to the vibration of the mill and the surging of the load. It can be precisely determined from every single frame of the charge in the mill. However, the workload would be doubled and the number of points grouped into en masse charge will change from frame to frame in so doing. This will cause difficulty in the following step of processing of data. So the trilinear surface is determined by visual examination on the PDP. Linear free surface is observed at low speeds and high filling levels as in J40N10 and J40N20 of Fig. 2 where the balls roll down from shoulder to toe. The slopes of these linear surfaces can be used to derive the dynamic angle of repose. It is computed, from the first three PDPs at low speeds of J40, to be 29.5j for balls of 22 mm in diameter. Continuing to increase the speed of rotation and keeping J = 40%, the free surface changes from bilinear through chair-type to saddle-type. At high speeds of rotation, the balls are thrown far onto the down-moving side of the mill and consolidate at the toe region forming one wing of the saddle. When the filling levels are low, the free surface appears to be chair-type at low speeds and saddle-type at high speeds. No linear profile can be formed even at low speed of rotation because there are not enough balls in the en masse region to support the balls falling down from the top of the charge. 5.2. Equilibrium surface Apart from the free rolling surface, another feature of interest is the so-called equilibrium surface that can be observed in Fig. 4. Equilibrium surface refers to a plane where all the forces acting on an element are in equilibrium. Then the element leaves the
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Fig. 2. (Part 1 of 2) Position density plots of the mill for J = 20 – 40% and N = 10 – 50% of critical speed. (Part 2 of 2) Position density plots of the mill for J = 20 – 40% and N = 60 – 100% of critical speed.
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Fig. 2 (continued ).
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Fig. 3. Profiles of free rolling surface of en masse load.
circular path and is ejected into the mill chamber. This surface was initially described by Davis with a circular arc, called ‘‘Davis circle’’, which is also included in Fig. 4. The center of Davis circle is g/2x2 above the mill center that is also the radius of Davis circle Rd. g g 447:28 Rd ¼ ¼ ¼ 0:269m ð3Þ ¼ 2x2 2 2pN 2 N2 60
where x is angular velocity of the mill and N mill speed in rpm, N = 40.74 rpm. A modification on Davis’ circular equilibrium surface was made by Rose and Sullivan (op. cit) taking frictional force into account as well as gravitational force and centrifugal force and arrived at an expression of an equiangular spiral. However, the parameters relating to mill speed (N), filling level ( J), and coefficient of friction (l) were not clearly defined in their expression. Some equiangular spirals obtained from their monograph at different coefficients of friction are superimposed onto Fig. 4 too. Comparison of the equilibrium surfaces of Davis circle and equiangular spirals with PDPs in Figs. 2 and 4 show that both of them do not predict well. Furthermore, variables related to slurry properties were not included in both models. At low mill speed, the equilibrium surface and the free rolling surface converge to one because the speed of particles after projection is low and particles begin to roll down immediately after they leave the equilibrium surface. 5.3. Charge in flight and en masse It should be noted that the behavior of the outermost layer is very different from that of the inner layers. The reason for this is that balls in this layer are subjected to the strong lifting action of the 12 square lifter bars. Balls in inner layers experience lifting action only
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Fig. 4. The trilinear free rolling surface for PDP J40N70 and corresponding Davis circle and equiangular spirals for l = 0.25, l = 0.5, and l = 1 (redrawn after Rose and Sullivan).
from balls in their outer adjacent layers, which is much weaker than that from the lifters. So the outermost balls are lifted higher than the inner balls. It is worth noting that the lifting height of balls in the outermost layer is only a function of the speed of rotation. On the contrary, the lifting height of balls in inner layers is related not only to the speed of rotation but to dynamic pressure in a mill and coefficient of friction as well. With this phenomenon in mind, a ball in the outer layer is considered as belonging to the en masse regime if RzRm Hl
ð4Þ
or if it is below the trilinear free surface. Otherwise, it is considered as in flight. In doing so, charge in the en masse regime for J40N70 is illustrated in Fig. 5. The mass of charge in en masse regime is 81% of the total charge by counting all the points within the en masse area. For J40N80 and J40N90, the masses of charge in the en masse regime are 82.3% and 81.6% of the total charge, respectively. This result is in excellent agreement with the
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Fig. 5. The en masse load of J40N70.
experimental observation on a 4.88 9.14 m (16 30 ft) rod mill by Vermeulen et al. (1984) which stated that 80.7% of the total charge was en masse when the mill operated at N = 82% and J = 50% with 32 lines of lifter bars. In general, square lifter bars have a more marked lifting action than other types of linings, so the amount of material in flight at normal operating speed would be no more than 20%. At very low speed, say 20% of critical speed, only about 5% of the total charge is in flight. With speed increasing, the amount of charge in flight increases as well. 5.4. Arrangement of balls in en masse regime It is obvious, from Fig. 2, that balls move in circular bands in the en masse area of a rotary mill. At low speeds or high charge level, four to six bands are clearly observed. On the contrary, when the speed is high and charge level is low, most balls are centrifuged onto the shell of the mill. No band is expected to appear except the centrifuged layer. It is of practical significance to know the distance between two adjacent bands. The digital PDPs make it possible to calculate the average radius of the circular bands from the position of every point. Take the PDP J40N70 for example, the points of the outer four bands in the IV quadrant of the anti-clockwise tumbling mill are plotted in Fig. 6 and the average radii for each band are also marked in Fig. 6.
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Fig. 6. Bands of load in the fourth quadrant and their average radii for PDP J40N70.
Fig. 7. The two extreme packing patterns of balls in en masse region of a mill and the distances between two layers. (a) Close packed; (b) packed in discrete layers.
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Two extreme patterns in which balls are packed are illustrated in Fig. 7. The distance between two layers of close-packed balls is 0.01927 m (1.732rb) as in Fig. 7(a) and 0.02225 m (2rb) for discrete layers as in Fig. 7(b). It is seen that the distances between any two adjacent layers of balls in the mill shown in Fig. 6 fall in between the two extreme cases. This means that slippage between layers may occur, which lead to abrasion breakage by frictional work. The frictional work is a function of frictional force and velocity gradient at the interface of two layers. The dynamic coefficient of friction, ld, can be computed from the aforedetermined dynamic angle of repose; ld = tan 29.5j = 0.57, which is a major factor influencing frictional work between layers of balls.
6. Power draw based on load behavior Mill power is related to load behavior in some way. Some power models use shoulder position, toe position, or angle of repose as a description of load behavior to derive their power equations. As can be seen from Fig. 2, a chord connecting the shoulder and toe of the load does not precisely express the configuration of load in a mill at the whole range of speed. That is why models based on shoulder and toe angles do not work well at high speed and need several correction factors to allow for this inaccuracy. Position density plot is a precise statistical representation of load behavior of a rotary mill. It allows the center of gravity of the load in a mill to be computed, which can be used to calculate the power drawn by the mill through Eqs. (1) and (2). 6.1. Instantaneous power from individual frame of load behavior The centers of gravity of load for all 32 frames composing PDP J30N70 in Fig. 2 are illustrated in Fig. 8. Even if the mill runs at steady state, the center of gravity of the mill load is not fixed at a point. It migrates within a 2 1.8 cm box. It can be expected that the center of gravity of the load in a bigger industrial mill will move in a wider range. The average center of mass is marked in Fig. 8 with a plus sign. Power drawn by the mill corresponding to every frame of PDP J30N70 using Eqs. (1) and (2) was calculated; the average power is 23.93 W with a standard deviation of 1.48 W. The variation of power drawn by every frame suggests that the torque to drive the mill will fluctuate with time. In general, measured power is only an average over a specific time interval. The average power can also be calculated from the average center of gravity of the PDP. 6.2. Power drawn by individual layers of load It has been demonstrated that balls in the mill are arranged in concentric layers in the en masse region. The mass of load in each layer and the power drawn by them are shown in Fig. 9(a), which was calculated from PDP J30N70. The layers of balls are grouped according to their distances from the center of the mill. Each layer has a thickness of one ball diameter, 0.02225 m, as illustrated in Fig. 9(b) by the dotted circles. The outermost layer, layer 1 in Fig. 9(a), includes 19.6% of total load by mass and consumes 30.7% of
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Fig. 8. Center of gravity for the whole load for all frames composing PDP J30N70. The origin is at the mill rotation center.
power input. Then the power consumed by the inner layers decreases as the radius of layer decreases. Layer 6 is the inner circular core of load in the mill enclosing 22.5% of total load but drawing only 10.6% of total power input. The sum of power drawn by all six layers equals to power drawn by the whole mill. As shown in Fig. 9(b), the layered power is calculated from all the balls falling into each circular band with the assumption that every band moves at the same rotational speed as the mill shell. Morrell (1993) also proposed a power model based on layered power, called ‘‘D-model’’. The major difference is that he assumed that the loads in free flight and down-going (cascading) neither draw power from the mill drive nor return energy to the mill. So his torque-based power is only part of energy balancebased power. However, as will be demonstrated later, the power drawn by the whole load is a better estimation than that by the active load only. As a matter of fact, the balls in flight do not draw power and they generally return power to the mill when they impact on the descending wall of the mill or on the toe of the load. So if they are not taken into account in power estimation, the power will be over-estimated at high speed in particular where there are about 20% of load in flight as shown in Section 5.3.
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Fig. 9. Balls in layers in PDP J30N70. (a) Mass of balls in each layer and power drawn by each layer; (b) layers of balls in PDPJ30N70.
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6.3. Power drawn by the whole load and en masse load The trilinear model for the free rolling surface discussed in Section 5.1 divides the entire load in a tumbling mill into two portions: the en masse load and the cataracting load. The power drawn by the entire load and that by the en masse load calculated from PDPs in Fig. 2, as well as the measured value are presented in Fig. 10. The power based on the entire load is an average of all frames constituting the corresponding PDPs. A standard deviation is also plotted with each calculated power. It is evident that the standard deviations increase as the speed of rotation of the mill increases. It implies that (1) the center of gravity of the charge in the mill migrates within a wider area at high speeds than it does at low speeds; (2) the PDP method for mill power works better at low speeds than it does at high speeds. Power computed from the en masse load is only a single value because the center of gravity of en masse load is determined from all points in the en masse area (refer to Fig. 5). Comparing the computed power from PDPs against the experimentally measured power reveals that the power computed from PDPs of all load is a better measurement of mill power than that from the en masse load. All the measured powers are within the range of powers of PDP all load plus/minus one standard deviation with 95% confidence. The reader should note that this is not a surprising conclusion. The model for torque expressed in Eq. (2) is only correct if the load is regarded as a single consolidated mass, the center of gravity of which is displaced to a distance xcog from the center of the mill by the forces exerted on it by the shell of the mill. In fact the load consists of balls which are moving in relation to each other, tumbling down the surface of the load, then rapidly accelerated in the toe of the load in the opposite direction into the en masse part of the load and are lifted towards the shoulder again. Some of these balls are then ejected into free space by the lifters; the latter balls exert no force on the mill. The other balls (especially those in the toe) exert forces, which exceed their weight because they are being accelerated
Fig. 10. Calculated and measured power at different speeds and ball filling levels.
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rapidly. The net effect is that the torque is best estimated with reference to all balls in the mill, independent of whether they are in contact with the mill liners or not. This observation is of theoretical and practical significance. It makes it possible to use PDP to control and optimize grinding mills because mill power and load behavior (toe position in particular) can be obtained from PDP, which are prime parameters used in grinding circuit control. However, if PDP en masse, not all grinding elements, is used to calculate mill power, the arbitrarily selected trilinear free rolling surface will yield a mill power with substantial uncertainty. Fig. 11 shows the center of gravity for the entire load from 10% to 100% of critical speed. The origin of Cartesian coordinate system is set at the rotation center of the mill. The lowest point for every filling level is for 10% of critical speed and the highest point for 100% of critical speed. The traces of center of gravity for the entire load are quite smooth. As the speed increases, the center of gravity moves upwards and inwards. That is to say, the load is lifted up along the mill shell. At 50 –60% of critical speed, the center of gravity of the whole load at any filling levels begins to move towards the center of the mill. This means that some balls were projected into the center of the mill. In the meantime, the torque to drive the mill should also peak at 50– 60% of critical speed where the xcoordinate of COG reaches its maximal value according to Eq. (2). So the trace of the
Fig. 11. Centers of gravity for all balls at different speeds and filling levels. Lowest point for 10% critical speed and highest point for 100%.
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center of gravity is a good indicator for load behavior and can be used to optimize operating conditions or even to develop a universal power equation.
7. Conclusions A digital visual representation of load behavior, called position density plot (PDP), is employed to systematically investigate the distribution of balls in a tumbling mill over a wide range of speeds of rotation and ball filling levels. A digital PDP makes it possible to determine the traditional parameters describing load behavior such as angle of repose, shoulder and toe positions, to count the number of balls in flight and en masse, and even to calculate the power drawn by the mill through the torque-arm formula. The free rolling surface of load in a mill is defined with a trilinear model. At low speeds and high ball filling fraction, the dynamic angle of repose of 22 mm balls is evaluated to be 29.5j that is equivalent to 0.57 of a dynamic coefficient of friction of ball to ball. The trilinear free rolling surface separates the whole charge into cataracting balls and cascading balls. At normal operating speed (70% to 80% critical speed), about 80% load is en masse. Balls in the en masse region are neither close packed nor in discrete layers. Slippage between adjacent layers may occur. Mill power can be calculated from the PDP using all grinding elements including those in flight. List of symbols g acceleration due to gravity Hl height of lifter bars, 0.022 m J fractional ball filling, % M the mass of mill load, kg N speed of mill rotation in rpm or percentage of critical speed P mill power in watts Pi power drawn by layer i R, Ri distance from ball center to mill center, i, layer of balls rb radius of ball, 0.02225 m Rd radius of Davis circle, m Rm radius of the 2D mill, 0.55 m T torque, Nm xcog the horizontal distance between the center of the mill and the center of gravity of the load ld dynamic coefficient of friction x angular velocity of mill, rad/s
References Austin, L.G., Klimpel, R.R., Luckie, P.T., 1984. Process Engineering of Size Reduction: Ball Mill. SME, New York, 560 pp.
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Davis, E.W., 1919. Fine crushing in ball mills. Transactions of American Institute of Mining and Metallurgy 61, 250 – 296. Dong, H., Moys, M.H., 2001. A technique to measure velocities of a ball moving in a tumbling mill and its applications. Minerals Engineering 14 (8), 841 – 850. Hogg, R., Fuerstenau, D.W., 1972. Power relationships for tumbling mills. Transaction of SME/AIME 252, 418 – 423. Morrell, S., 1993. The prediction of power draw in wet tumbling mills. PhD Thesis, University of Queensland, 228 pp. Moys, M.H., 1990. A model for mill power as affected by mill speed, load volume, and liner design. In: Schonert, K. (Ed.), Preprints of the 7th European Symposium on Comminution. University of Yugoslavia, Ljubljana, pp. 595 – 607. Rose, H.E., Sullivan, R.M.E., 1958. A Treatise on the Internal Mechanics of Ball, Tube, and Rod Mills. Constable and Company, London, 258 pp. Vermeulen, L.A., Ohlson De Fine, M.J., Schakowski, F., 1984. Physical information from the inside of a rotary mill. Journal of South Africa Institute of Mining and Metallurgy 84 (8), 247 – 253.
Load behavior and mill power H. Dong *, M.H. Moys School of Process and Materials Engineering, University of the Witwatersrand, Private Bag 3, Johannesburg, WITS, 2050, South Africa Received 1 December 2001; received in revised form 21 April 2002; accepted 22 April 2002
Abstract Load orientation is conventionally described by using shoulder and toe positions and a chord connecting the shoulder and toe position with its slope equivalent to the angle of repose of mill load. This model suffers a marked discrepancy with the true load orientation at low ball filling or high speed of rotation. One reason for this discrepancy is that the load in a mill will dilate and change shape substantially at high speed. It means that the distribution of grinding elements in the mill chamber does not match the above description at all. So some torque-arm power models based on load behavior failed to predict power draw with a satisfactory accuracy at high speed. In this paper, experimental position density plots (PDPs) are used to describe load behavior in a tumbling mill. A PDP is a digital, visual and statistical representation of the charge distribution in a mill. It is constructed by superimposing a number of independent digitized images of a mill load at steady state. Not only the parameters about load behavior, such as dynamic angle of repose, shoulder angle, and toe angle, are determined from PDP, power drawn by the entire load and en masse load can be calculated as well by use of the torque-arm model directly. D 2003 Elsevier Science B.V. All rights reserved. Keywords: comminution; grinding; mill; power
1. Introduction Mill power drawn is a function of mill size (L and D), the loading ( J), and operating conditions such as mill speed N, slurry/void ratio, slurry density, etc. However, these are not all the variables having influence on mill power. As systematically summarized by Rose and Sullivan (1958), apart from the aforementioned variables, which can be clearly
*
Corresponding author. Fax: +27-11-4031471. E-mail addresses: [email protected] (H. Dong), [email protected] (M.H. Moys).
0301-7516/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 ( 0 2 ) 0 0 0 4 6 - 7
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defined and often included in power equations, physical properties of materials in a mill (hardness, density, size distribution, coefficient of restitution, and coefficient of friction), lifter design (shape and number), slurry rheological attributes and slurry hold up that change as a mill works in grate discharge mode or overflow discharge mode, influence the power required to drive a mill to some extent. Owing to the great number of variables, different treatments of the major variables lead to various power equations. It is impossible to do justice to the enormous amount of work in this area. Briefly speaking(Austin et al., 1984), there are two main approaches in the derivation of power equations. One approach (Davis, 1919; Rose and Sullivan, 1958; Morrell, 1993) calculates the paths of balls tumbling in the mill and integrates the energy required to raise the balls over all possible paths. The other approach (Hogg and Fuerstenau, 1972; Moys, 1990) uses the torque-arm model to get the power equations. Davis (op. cit) analyzed the force balance between the centrifugal force and gravitational force acting on a ball in a tumbling mill. Then he found an equilibrium surface where the balls commence their parabolic motion and established his power equation. One obvious deficiency of this model is that the frictional force between balls that produces frictional heat was not considered. Rose and Sullivan (op. cit) carried out a more comprehensive force balance analysis among the gravity force, centrifugal force and frictional force and derived a power formula using dimensional analysis. With the understanding of the load behavior deepened by measurements of load behavior in a glass-sided laboratory mill, Morrell proposed a model based on the observation of toe and shoulder position and angular velocity gradient of successive layers in the charge. His model was validated with a broad database of the power draws of industrial ball mills, SAG mills and AG mills and an excellent agreement was observed. The torque-arm model uses a chord connecting the toe and shoulder of the charge to delineate the charge shape. When the information about toe and shoulder positions is not available, the angle of repose or dynamic angle of repose is used as the slope of the chord to define the profile of the charge. However, the free rolling surface of the cascading charge, at high speeds in particular, is far from a straight line. So the torque-arm model does not work well at high mill speeds as it does at low speeds. Moys modified the torque model by treating the whole charge as two parts: centrifuged part and non-centrifuged part. The centrifuged part draws no power and its thickness varies with mill speed, lifter configuration, and slurry rheology. The mill power was then determined by use of Bond’s power equation according to the behavior of non-centrifuged part that was described by effective speed, filling and diameter of the mill. This treatment seems to describe fairly accurately the behavior of mill power over a wide range of operating conditions. The basic form of the torque-arm model for mill power is 2p N ðrpmÞT ðNmÞ 60
ð1Þ
T ¼ Force ðN Þ Torque-arm ðmÞ ¼ Mgxcog
ð2Þ
P ðwattsÞ ¼
All the torque-arm models attempted to relate torque to mill design, load properties, and operating conditions. In this paper, a direct application of the above formula to calculate
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the power drawn by a laboratory mill of an internal diameter of 0.55 m is reported. The torque is calculated from a position density plot (PDP, defined below), from which the center of gravity (COG) of the load is statistically determined. PDPs visually reflect the influences of mill speed, filling, and lifter configuration on load behavior and mill power. An excellent agreement is observed by the comparison of calculated power against measured power.
2. Experimental apparatus The experimental mill has an internal diameter of 0.55 m and 0.0235 m in length fitted with twelve 0.022 0.022 m square lifter bars. The load is composed of chrome steel balls of diameter 0.02225 m. Only one layer of balls in the axial direction can be housed to ensure that all the balls can be observed at any time. So it is termed 2D mill. A transparent end plate is used to allow the load to be photographed. The 2D mill, with its driving shaft connected with a load beam measuring the torque when the mill is running, has its speed adjusted from 0% up to 200% of critical speed. A digital camera is used to take pictures of the load in the 2D mill with a digital stroboscope as the light source.
3. Construction and validation of PDP A PDP is a plot constructed by analyzing tens of photographs of the charge in the mill operating at steady state to statistically quantify the charge distribution within the mill. Each digital picture is then analyzed with an image analysis package to read the position of every point that represents the center of a ball in the mill. A key issue to get the true position of a ball center from an image is to calibrate image distortions. The calibration process has been discussed in our previous publication (Dong and Moys, 2001). After calibration, the maximum position shift between the true position and image position is less than 1.2 mm. This means that the maximum relative measurement error is 0.22% relative to the mill diameter. A sample digitized frame is shown in Fig. 1(a) where there are 160 points representing the centers of 160 balls in the mill at 80% of critical speed. All the photos for one PDP were taken at a rate of two frames per second (fps) to make sure that every single frame is an independent observation. Thirty digitized frames are superimposed together to build a PDP as shown in Fig. 1(b) where there are 4800 points. This ensures that a load behavior over a time of 15 s (corresponding to about 12 revolutions at 80% of critical speed) is obtained. Fig. 1(c) is a picture taken simultaneously under the same operating conditions except that the exposure time of the camera was set at 2 s and the flashing rate of the strobe 60 fps. In other words, there are 120 frames during one and half revolution of the mill. It can be seen from Fig. 1(b) and (c) that: (1) balls in the tumbling mill are moving in layers (at least four layers can be clearly distinguished); (2) the outermost layer is raised much higher by the square lifters and landed on the bare shell at nine o’clock position; (3) the inner layers also experience a brief period of cataracting that is attributed to the lifting
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Fig. 1. Load behavior at J = 30% and N = 80% critical speed. (a) A sample digitized frame of mill; (b) digital PDP J30N80 of 30 frames; (c) a photograph of the mill with 120 exposures spread over 2 s.
effect of the outer layer of balls. It is safe to say, by comparing Fig. 1(b) and (c), that the PDP is an excellent digital visual representation of charge distribution within a mill.
4. Results The load behavior of the 2D mill at a wide range of speed of rotation and filling level has been investigated systematically and is presented in the form of PDPs in Fig. 2. Each PDP is labeled with its major operating conditions as a double check. For example, the PDP with a label of J30N40 can be found at row N = 40% of critical speed and column J = 30% of load volume. Every PDP is made up of more than 30 frames to
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ensure its representativity. Analysis of these PDPs will be made in the following sections.
5. Analysis 5.1. Free rolling surface It has been long established that the charge in a tumbling mill moves in two modes: cascading and cataracting. It is of practical significance to differentiate them on the PDPs. The profiles of en masse charge are abstracted into four types from all PDPs in Fig. 2: linear, bilinear, trilinear-chair, and trilinear-saddle, which are illustrated in Fig. 3. These four types of profiles can be modeled with one general trilinear form of two points and two slopes. If the two points converge to one point and the two slopes are identical, the trilinear model is reduced to the linear form (Fig. 3(a)); if the two points converge to one and the two slopes are different, the trilinear model is reduced to the bilinear form (Fig. 3(b)); if the two points are separate and the two slopes are all positive, the chair-type of profile can be expected (Fig. 3(c)); if the two points are separate and the slope for the left line is negative and that for the right line positive, the saddle profile applies (Fig. 3(d)). We use the trilinear model to express all free surfaces in Fig. 2. The two endpoints determine the middle section and the two slopes together with their corresponding endpoints define the other two segments. For example, as for N = 70% and J = 40%, the free surface drawn from the trilinear model is shown in Fig. 4. The free rolling surface is not fixed due to the vibration of the mill and the surging of the load. It can be precisely determined from every single frame of the charge in the mill. However, the workload would be doubled and the number of points grouped into en masse charge will change from frame to frame in so doing. This will cause difficulty in the following step of processing of data. So the trilinear surface is determined by visual examination on the PDP. Linear free surface is observed at low speeds and high filling levels as in J40N10 and J40N20 of Fig. 2 where the balls roll down from shoulder to toe. The slopes of these linear surfaces can be used to derive the dynamic angle of repose. It is computed, from the first three PDPs at low speeds of J40, to be 29.5j for balls of 22 mm in diameter. Continuing to increase the speed of rotation and keeping J = 40%, the free surface changes from bilinear through chair-type to saddle-type. At high speeds of rotation, the balls are thrown far onto the down-moving side of the mill and consolidate at the toe region forming one wing of the saddle. When the filling levels are low, the free surface appears to be chair-type at low speeds and saddle-type at high speeds. No linear profile can be formed even at low speed of rotation because there are not enough balls in the en masse region to support the balls falling down from the top of the charge. 5.2. Equilibrium surface Apart from the free rolling surface, another feature of interest is the so-called equilibrium surface that can be observed in Fig. 4. Equilibrium surface refers to a plane where all the forces acting on an element are in equilibrium. Then the element leaves the
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Fig. 2. (Part 1 of 2) Position density plots of the mill for J = 20 – 40% and N = 10 – 50% of critical speed. (Part 2 of 2) Position density plots of the mill for J = 20 – 40% and N = 60 – 100% of critical speed.
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Fig. 2 (continued ).
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Fig. 3. Profiles of free rolling surface of en masse load.
circular path and is ejected into the mill chamber. This surface was initially described by Davis with a circular arc, called ‘‘Davis circle’’, which is also included in Fig. 4. The center of Davis circle is g/2x2 above the mill center that is also the radius of Davis circle Rd. g g 447:28 Rd ¼ ¼ ¼ 0:269m ð3Þ ¼ 2x2 2 2pN 2 N2 60
where x is angular velocity of the mill and N mill speed in rpm, N = 40.74 rpm. A modification on Davis’ circular equilibrium surface was made by Rose and Sullivan (op. cit) taking frictional force into account as well as gravitational force and centrifugal force and arrived at an expression of an equiangular spiral. However, the parameters relating to mill speed (N), filling level ( J), and coefficient of friction (l) were not clearly defined in their expression. Some equiangular spirals obtained from their monograph at different coefficients of friction are superimposed onto Fig. 4 too. Comparison of the equilibrium surfaces of Davis circle and equiangular spirals with PDPs in Figs. 2 and 4 show that both of them do not predict well. Furthermore, variables related to slurry properties were not included in both models. At low mill speed, the equilibrium surface and the free rolling surface converge to one because the speed of particles after projection is low and particles begin to roll down immediately after they leave the equilibrium surface. 5.3. Charge in flight and en masse It should be noted that the behavior of the outermost layer is very different from that of the inner layers. The reason for this is that balls in this layer are subjected to the strong lifting action of the 12 square lifter bars. Balls in inner layers experience lifting action only
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Fig. 4. The trilinear free rolling surface for PDP J40N70 and corresponding Davis circle and equiangular spirals for l = 0.25, l = 0.5, and l = 1 (redrawn after Rose and Sullivan).
from balls in their outer adjacent layers, which is much weaker than that from the lifters. So the outermost balls are lifted higher than the inner balls. It is worth noting that the lifting height of balls in the outermost layer is only a function of the speed of rotation. On the contrary, the lifting height of balls in inner layers is related not only to the speed of rotation but to dynamic pressure in a mill and coefficient of friction as well. With this phenomenon in mind, a ball in the outer layer is considered as belonging to the en masse regime if RzRm Hl
ð4Þ
or if it is below the trilinear free surface. Otherwise, it is considered as in flight. In doing so, charge in the en masse regime for J40N70 is illustrated in Fig. 5. The mass of charge in en masse regime is 81% of the total charge by counting all the points within the en masse area. For J40N80 and J40N90, the masses of charge in the en masse regime are 82.3% and 81.6% of the total charge, respectively. This result is in excellent agreement with the
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Fig. 5. The en masse load of J40N70.
experimental observation on a 4.88 9.14 m (16 30 ft) rod mill by Vermeulen et al. (1984) which stated that 80.7% of the total charge was en masse when the mill operated at N = 82% and J = 50% with 32 lines of lifter bars. In general, square lifter bars have a more marked lifting action than other types of linings, so the amount of material in flight at normal operating speed would be no more than 20%. At very low speed, say 20% of critical speed, only about 5% of the total charge is in flight. With speed increasing, the amount of charge in flight increases as well. 5.4. Arrangement of balls in en masse regime It is obvious, from Fig. 2, that balls move in circular bands in the en masse area of a rotary mill. At low speeds or high charge level, four to six bands are clearly observed. On the contrary, when the speed is high and charge level is low, most balls are centrifuged onto the shell of the mill. No band is expected to appear except the centrifuged layer. It is of practical significance to know the distance between two adjacent bands. The digital PDPs make it possible to calculate the average radius of the circular bands from the position of every point. Take the PDP J40N70 for example, the points of the outer four bands in the IV quadrant of the anti-clockwise tumbling mill are plotted in Fig. 6 and the average radii for each band are also marked in Fig. 6.
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Fig. 6. Bands of load in the fourth quadrant and their average radii for PDP J40N70.
Fig. 7. The two extreme packing patterns of balls in en masse region of a mill and the distances between two layers. (a) Close packed; (b) packed in discrete layers.
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Two extreme patterns in which balls are packed are illustrated in Fig. 7. The distance between two layers of close-packed balls is 0.01927 m (1.732rb) as in Fig. 7(a) and 0.02225 m (2rb) for discrete layers as in Fig. 7(b). It is seen that the distances between any two adjacent layers of balls in the mill shown in Fig. 6 fall in between the two extreme cases. This means that slippage between layers may occur, which lead to abrasion breakage by frictional work. The frictional work is a function of frictional force and velocity gradient at the interface of two layers. The dynamic coefficient of friction, ld, can be computed from the aforedetermined dynamic angle of repose; ld = tan 29.5j = 0.57, which is a major factor influencing frictional work between layers of balls.
6. Power draw based on load behavior Mill power is related to load behavior in some way. Some power models use shoulder position, toe position, or angle of repose as a description of load behavior to derive their power equations. As can be seen from Fig. 2, a chord connecting the shoulder and toe of the load does not precisely express the configuration of load in a mill at the whole range of speed. That is why models based on shoulder and toe angles do not work well at high speed and need several correction factors to allow for this inaccuracy. Position density plot is a precise statistical representation of load behavior of a rotary mill. It allows the center of gravity of the load in a mill to be computed, which can be used to calculate the power drawn by the mill through Eqs. (1) and (2). 6.1. Instantaneous power from individual frame of load behavior The centers of gravity of load for all 32 frames composing PDP J30N70 in Fig. 2 are illustrated in Fig. 8. Even if the mill runs at steady state, the center of gravity of the mill load is not fixed at a point. It migrates within a 2 1.8 cm box. It can be expected that the center of gravity of the load in a bigger industrial mill will move in a wider range. The average center of mass is marked in Fig. 8 with a plus sign. Power drawn by the mill corresponding to every frame of PDP J30N70 using Eqs. (1) and (2) was calculated; the average power is 23.93 W with a standard deviation of 1.48 W. The variation of power drawn by every frame suggests that the torque to drive the mill will fluctuate with time. In general, measured power is only an average over a specific time interval. The average power can also be calculated from the average center of gravity of the PDP. 6.2. Power drawn by individual layers of load It has been demonstrated that balls in the mill are arranged in concentric layers in the en masse region. The mass of load in each layer and the power drawn by them are shown in Fig. 9(a), which was calculated from PDP J30N70. The layers of balls are grouped according to their distances from the center of the mill. Each layer has a thickness of one ball diameter, 0.02225 m, as illustrated in Fig. 9(b) by the dotted circles. The outermost layer, layer 1 in Fig. 9(a), includes 19.6% of total load by mass and consumes 30.7% of
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Fig. 8. Center of gravity for the whole load for all frames composing PDP J30N70. The origin is at the mill rotation center.
power input. Then the power consumed by the inner layers decreases as the radius of layer decreases. Layer 6 is the inner circular core of load in the mill enclosing 22.5% of total load but drawing only 10.6% of total power input. The sum of power drawn by all six layers equals to power drawn by the whole mill. As shown in Fig. 9(b), the layered power is calculated from all the balls falling into each circular band with the assumption that every band moves at the same rotational speed as the mill shell. Morrell (1993) also proposed a power model based on layered power, called ‘‘D-model’’. The major difference is that he assumed that the loads in free flight and down-going (cascading) neither draw power from the mill drive nor return energy to the mill. So his torque-based power is only part of energy balancebased power. However, as will be demonstrated later, the power drawn by the whole load is a better estimation than that by the active load only. As a matter of fact, the balls in flight do not draw power and they generally return power to the mill when they impact on the descending wall of the mill or on the toe of the load. So if they are not taken into account in power estimation, the power will be over-estimated at high speed in particular where there are about 20% of load in flight as shown in Section 5.3.
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Fig. 9. Balls in layers in PDP J30N70. (a) Mass of balls in each layer and power drawn by each layer; (b) layers of balls in PDPJ30N70.
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6.3. Power drawn by the whole load and en masse load The trilinear model for the free rolling surface discussed in Section 5.1 divides the entire load in a tumbling mill into two portions: the en masse load and the cataracting load. The power drawn by the entire load and that by the en masse load calculated from PDPs in Fig. 2, as well as the measured value are presented in Fig. 10. The power based on the entire load is an average of all frames constituting the corresponding PDPs. A standard deviation is also plotted with each calculated power. It is evident that the standard deviations increase as the speed of rotation of the mill increases. It implies that (1) the center of gravity of the charge in the mill migrates within a wider area at high speeds than it does at low speeds; (2) the PDP method for mill power works better at low speeds than it does at high speeds. Power computed from the en masse load is only a single value because the center of gravity of en masse load is determined from all points in the en masse area (refer to Fig. 5). Comparing the computed power from PDPs against the experimentally measured power reveals that the power computed from PDPs of all load is a better measurement of mill power than that from the en masse load. All the measured powers are within the range of powers of PDP all load plus/minus one standard deviation with 95% confidence. The reader should note that this is not a surprising conclusion. The model for torque expressed in Eq. (2) is only correct if the load is regarded as a single consolidated mass, the center of gravity of which is displaced to a distance xcog from the center of the mill by the forces exerted on it by the shell of the mill. In fact the load consists of balls which are moving in relation to each other, tumbling down the surface of the load, then rapidly accelerated in the toe of the load in the opposite direction into the en masse part of the load and are lifted towards the shoulder again. Some of these balls are then ejected into free space by the lifters; the latter balls exert no force on the mill. The other balls (especially those in the toe) exert forces, which exceed their weight because they are being accelerated
Fig. 10. Calculated and measured power at different speeds and ball filling levels.
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rapidly. The net effect is that the torque is best estimated with reference to all balls in the mill, independent of whether they are in contact with the mill liners or not. This observation is of theoretical and practical significance. It makes it possible to use PDP to control and optimize grinding mills because mill power and load behavior (toe position in particular) can be obtained from PDP, which are prime parameters used in grinding circuit control. However, if PDP en masse, not all grinding elements, is used to calculate mill power, the arbitrarily selected trilinear free rolling surface will yield a mill power with substantial uncertainty. Fig. 11 shows the center of gravity for the entire load from 10% to 100% of critical speed. The origin of Cartesian coordinate system is set at the rotation center of the mill. The lowest point for every filling level is for 10% of critical speed and the highest point for 100% of critical speed. The traces of center of gravity for the entire load are quite smooth. As the speed increases, the center of gravity moves upwards and inwards. That is to say, the load is lifted up along the mill shell. At 50 –60% of critical speed, the center of gravity of the whole load at any filling levels begins to move towards the center of the mill. This means that some balls were projected into the center of the mill. In the meantime, the torque to drive the mill should also peak at 50– 60% of critical speed where the xcoordinate of COG reaches its maximal value according to Eq. (2). So the trace of the
Fig. 11. Centers of gravity for all balls at different speeds and filling levels. Lowest point for 10% critical speed and highest point for 100%.
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center of gravity is a good indicator for load behavior and can be used to optimize operating conditions or even to develop a universal power equation.
7. Conclusions A digital visual representation of load behavior, called position density plot (PDP), is employed to systematically investigate the distribution of balls in a tumbling mill over a wide range of speeds of rotation and ball filling levels. A digital PDP makes it possible to determine the traditional parameters describing load behavior such as angle of repose, shoulder and toe positions, to count the number of balls in flight and en masse, and even to calculate the power drawn by the mill through the torque-arm formula. The free rolling surface of load in a mill is defined with a trilinear model. At low speeds and high ball filling fraction, the dynamic angle of repose of 22 mm balls is evaluated to be 29.5j that is equivalent to 0.57 of a dynamic coefficient of friction of ball to ball. The trilinear free rolling surface separates the whole charge into cataracting balls and cascading balls. At normal operating speed (70% to 80% critical speed), about 80% load is en masse. Balls in the en masse region are neither close packed nor in discrete layers. Slippage between adjacent layers may occur. Mill power can be calculated from the PDP using all grinding elements including those in flight. List of symbols g acceleration due to gravity Hl height of lifter bars, 0.022 m J fractional ball filling, % M the mass of mill load, kg N speed of mill rotation in rpm or percentage of critical speed P mill power in watts Pi power drawn by layer i R, Ri distance from ball center to mill center, i, layer of balls rb radius of ball, 0.02225 m Rd radius of Davis circle, m Rm radius of the 2D mill, 0.55 m T torque, Nm xcog the horizontal distance between the center of the mill and the center of gravity of the load ld dynamic coefficient of friction x angular velocity of mill, rad/s
References Austin, L.G., Klimpel, R.R., Luckie, P.T., 1984. Process Engineering of Size Reduction: Ball Mill. SME, New York, 560 pp.
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Davis, E.W., 1919. Fine crushing in ball mills. Transactions of American Institute of Mining and Metallurgy 61, 250 – 296. Dong, H., Moys, M.H., 2001. A technique to measure velocities of a ball moving in a tumbling mill and its applications. Minerals Engineering 14 (8), 841 – 850. Hogg, R., Fuerstenau, D.W., 1972. Power relationships for tumbling mills. Transaction of SME/AIME 252, 418 – 423. Morrell, S., 1993. The prediction of power draw in wet tumbling mills. PhD Thesis, University of Queensland, 228 pp. Moys, M.H., 1990. A model for mill power as affected by mill speed, load volume, and liner design. In: Schonert, K. (Ed.), Preprints of the 7th European Symposium on Comminution. University of Yugoslavia, Ljubljana, pp. 595 – 607. Rose, H.E., Sullivan, R.M.E., 1958. A Treatise on the Internal Mechanics of Ball, Tube, and Rod Mills. Constable and Company, London, 258 pp. Vermeulen, L.A., Ohlson De Fine, M.J., Schakowski, F., 1984. Physical information from the inside of a rotary mill. Journal of South Africa Institute of Mining and Metallurgy 84 (8), 247 – 253.