Discrete Particle Simulation of Particle Flow in a Stirred Mill: Effect of

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Discrete Particle Simulation of Particle Flow in a Stirred Mill: Effect of Mill Properties and Geometry C. T. Jayasundara, R. Y. Yang, and A. B. Yu* Laboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia ABSTRACT: Particle flow in a stirred mill was modeled using discrete element method, focusing on the effect of mill properties and stirrer configurations, such as particle-wall friction, the size of disc holes, distance between stirrers, and stirrer shape, on the flow properties of grinding media. The flow properties were analyzed in terms of velocity field, porosity distribution, collision frequency, collision energy, impact energy, and power draw. The results indicate that although particle-wall sliding friction coefficient affects the energy transfer from discs to particles, too high a sliding friction may lead to a decrease in energy efficiency. The distance between discs significantly affects the circulation of grinding media between discs. Among the different stirrer types considered, energy transfer is more effective when disc holes are present. Pin stirrer shows increased grinding rates which also results in relatively high power consumption. Although different collision environments exist with different stirrer types, it is shown that the grinding rate can be determined by the first-order kinetics where the rate constant is dependent on the impact energy, for a given material. Grinding efficiency has been compared for different grinding materials under different operating conditions. The results suggest that selection of stirrer geometry also depends on the feed size and the type of material to be ground. Discussion has also been made of the usefulness of particle scale information in the design and control of stirred mills of different types.

1. INTRODUCTION IsaMill is one of the most commonly used stirred mills for fine and ultrafine grinding in industry.1 Basically this stirred mill consists of a stationary drum, a rotating shaft running along the axis of the drum, and discs with holes for stirring, which are evenly spaced along the shaft. Compared with the conventional grinding mills such as ball mill, it can significantly reduce the energy consumption, the media cost, and the capital cost of fine grinding.1,2 In the past, there have been a considerable number of studies related to IsaMill and other stirred mills.311 Most of these studies are based on experimental observation and the findings led to the formulation of various empirical relations which are difficult to generalize. To understand the grinding mechanism, fundamental understanding of the particle flow in a mill is required. However, such flow information is very difficult to obtain using the conventional experimental techniques. Alternatively, numerical models based on the discrete element method (DEM) can produce information about the forces acting on and the trajectory of individual particles. DEM models have been used increasingly in the past decade and have shown to be an effective way to understand particle flow in different types of mills at a particle scale.1219 DEM has been further developed and used to study many particulate systems as recently reviewed by Zhu et al.20,21 We recently performed DEM modeling of the flow of grinding media in a high-speed horizontal stirred mill.22,23 The model was validated by experiments, and the microdynamic properties relating to flow and force structures were analyzed based on the simulation results. The effects of the properties of grinding media, such as particle size, density, and roughness were also investigated.24 Numerical modeling performed under different mill speeds and media fill levels showed that the grinding r 2011 American Chemical Society

performance can be described by the first-order kinetics where the rate constant has a well-defined correlation with the impact energy.25 More recently, DEM has been coupled with Computational Fluid Dynamics (CFD) to simulate particle-slurry flow in stirred mills.26 Slurry properties such as slurry density and viscosity were found to significantly affect the flow of grinding media. In practice, different types of discs have been used for different grinding materials. Mill properties such as different types of discs, disc sliding friction, and space between discs may affect the mill performance significantly. The optimum selection of these parameters may have to be based on the results from many tests, which is experimentally very costly. Numerical modeling, often coupled with supportive physical modeling, provides an alternative method to achieve this goal. To date, however, such study has not been made on the performance of stirred mills under different mill properties. In this paper, we carry out a DEM study to investigate the effects of mill properties such as particledisc sliding friction, and mill geometry, i.e., the distance between discs, the size of disc holes, and the stirrer geometry on the flow behavior in a model stirred mill relevant to IsaMill. The use of the resulting particle scale information in the evaluation of grinding performance is also discussed.

2. DEM MODEL AND SIMULATION CONDITIONS 2.1. Simulation Method. The DEM model used in this work has been detailed in our previous work,22 thus it is only briefly Received: August 25, 2011 Accepted: November 28, 2011 Revised: November 23, 2011 Published: November 28, 2011 1050

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described here. In DEM simulations, each particle possesses translational motion and rotational motion, which can be described by Newton’s second law of motion, given by dvi ¼ dt

∑

dω i ¼ dt

∑

mi

Fnij þ Ftij þ mi g

ð1Þ

following two equations: pffiffiffi pffiffiffipffiffiffiffiffi 2 îj n îj Fnij ¼ E R̅ ξn 3=2 γn E R̅ ξn vij 3 n 3 and

h 3=2 i Ftij ¼ sgnðξs ÞμjFnij j 1 1 min ξs , ξs, max =ξs, max

and Ii

R i Ftij μr Ri jFnij jω i

ð2Þ

where vi, ωi, and Ii are, respectively, the translational velocity, angular velocity, and moment of inertia of particle i, Ri is a vector running from the center of the particle to the contact point with its magnitude equal to particle radius Ri. Fnij and Ftij represent, respectively, the normal contact force and the tangential contact force imposed on particle i by particle j. In connection with our previous studies,23,24 they are determined according to the

Figure 1. Lab model of the stirred mill.

ð3Þ

ð4Þ where E = Y/(1σ ~ ), and Y and σ ~ are, respectively, Young’s modulus and Poisson’s ratio; ξn is the overlap between particles i and j; n îj is a unit vector running from the center of particle j to the center of particle i; R = RiRj/(Ri + Rj). The normal damping constant, γn, is the material property directly linked to the coefficient of restitution e. ξs and ξs,max are, respectively, the total and maximum tangential displacements of particles during contact. The model mill in this work consists of a fixed chamber, a rotating shaft, and three stirrers (Figure 1). Different types of stirrers were used as shown in Figure 2c, namely disc without holes (0H), disc with three holes (3H), disc with five holes (5H), and pin type stirrer. In the simulations, the mill material properties, the gap between stirrers, and the stirrer shape were varied to examine their effects on the flow. Note that the slurry flow (including ground materials) was not considered in this work focused on the flow of grinding media. Table 1 lists the base values and their varying ranges of the parameters used in the simulation. Note that Young’s modulus in the simulations is much smaller than that of real glass beads (∼100 GPa). As the time step in DEM is inversely proportional to the hardness, a smaller Young’s modulus can reduce simulation time considerably. Previous studies have confirmed the simulation results are independent of Young’s modulus as long as the maximum 2

Figure 2. Geometry of the model stirred mill: (a) sectional front view; (b) sectional end view; and (c) different stirrer geometries of 0H, 3H, 5H and PIN (from left to right, dimensions in mm). 1051

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Industrial & Engineering Chemistry Research overlap is less than 23% of particle diameter. The current value ensures that the maximum overlap is less than 2% of particle diameter and thus has insignificant effect on the final results.27,28 Unless otherwise specified, the so-called base values set a standard condition to study the effect of a variable. A simulation starts with a packing process in which the shaft and the discs rest and particles are fed into the mill to form a stable packed bed. The shaft and the discs then rotate at a given speed to agitate the particles. The values and varying ranges of key parameters in the simulations are listed in Table 1. The values Table 1. Physical and Operational Parameters Used in the Present Study parameter

base value

number of particles, N

41,000

particle diameter, d (mm)

3

particle density, F (kgm3)

2.3 103

2

Young’s modulus, Y (Nm )

varying range

1.0 107

Poisson’s ratio, σ ~

0.29

particle/particle sliding friction coefficient,

0.2

μs,pp particle/disc sliding friction coefficient, μs,pd 0.2 0.01 rolling friction coefficient, μr

0.1 1.0

restitution coefficient, e

0.68

rotation speed, Ω (rpm)

1200

4001200

mill loading, J (%)

80

60

distance between discs, ld (mm)

30

1730

disc hole diameter, dn (mm)

18

1224

stirrer

5H

0H, 3H, 5H, PIN

time step, (s)

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and varying ranges of key parameters in the simulations are listed in Table 1. When the effect of one parameter is investigated, only its value is changed while other parameters keep their base values. All the results are analyzed for four complete revolutions of the disc, when the system reaches the macroscopically steady state. The steady state is here determined when the torque on the mill shaft just fluctuates around a constant. With the current configuration consisting of 41 000 particles, it takes about 40 CPU hours to simulate 1 s of real time on an Intel 2.8 GHz Xeon PC. Also note that in the following discussion, while the snapshots are taken from the YY0 (radial plane) and XX0 (axial plane) regions in Figure 2, the statistical distributions and mean values are calculated based on the total flow (i.e., all the particles) in the mill. Based on the simulation results, the particle flow is analyzed in terms of flow velocity, collision energy Ce, collision frequency Cf, and impact energy Ei. These variables have been demonstrated to be useful in characterizing particle flow in a stirred mill.2325 Here collision energy is defined as the kinetic energy in the normal direction of two colliding particles 1/2mν2n,ij, where m is the mass of a particle and vn,ij is the normal component of the relative velocity of the two particles. Collision frequency is defined as the number of collisions per particle per second. Both local and total impact energies are used in the following discussion. Local impact energy is the summation of the energies of all the collisions in a particular region or cell at a unit time, while the total impact energy Ei (= NCfCe) is the summation of the energy of all the collisions in the whole system of N particles divided by time.

3. DYNAMICS OF GRINDING MEDIA 3.1. Effect of Stirrer Sliding Friction. In practice, stirrers are coated with steel, ceramic, or rubber liners. Choice of liner materials depends on a number of factors such as material cost, installation,

Figure 3. Snapshots showing particle flow pattern (top) and porosity distribution and velocity vectors (bottom) for different particle-stirrer sliding friction coefficients: (a) μs,pd = 0.1; and (b) μs pd = 1.0 when J = 80% and Ω = 1200 rpm. 1052



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Figure 4. (a) Mean collision frequency and collision energy; and (b) impact energy and input power for different particle/disc sliding friction coefficients when J = 80% and Ω = 1200 rpm.

wear, and other factors including noise and vibration reduction. Depending on the type of the material, particlewall or particle stirrer sliding friction coefficient varies. To reflect this effect, the stirrer sliding friction in this work is changed so that the particle stirrer sliding friction coefficient μs,pd varies from 0.1 to 1.0. Figure 3 shows the snapshots of particle flow pattern and velocity and porosity distributions at different μs,pd. With a small μs, pd (= 0.1), particles rotate from the lower region (about 4 o’clock position) to the upper region (around 11 o’clock). Those particles in the upper region reach high velocities and then accelerate toward the mill drum where they collide with the upper part of the drum. This leaves the flow with a low porosity in the bottom part and high porosity in the upper part. The region of high porosity is caused by the unconfined movement of particles. With a large μs,pd (= 1.0) and therefore more efficient energy transfer from stirrer to particles, particles are more densely packed and form a solid centrifugally driven layer around the entire perimeter of the mill, creating a large void in the center. Particle flow also becomes faster but the velocity gradient is reduced along the radial plane. Figure 4 shows the variation of mean collision frequency, collision energy, impact energy, and input energy. Although collision frequency increases with sliding friction as shown in Figure 4a, the rate of increase slows down when μs,pd increases. An increase of μs,pd from 0.1 to 0.6 will decrease the average Ce. A further increase in μs,pd up to 1.0 does not change the distribution significantly. As a result, when μs,pd increases from 0.1 to 0.6, average collision energy decreases (Figure 4a) and further increase of μs,pd shows a flat variation in average collision energy. This suggests that stirrers with very small (μs,pd < 0.6) friction coefficient should increase the collision energy. However, as stirrers with larger friction have higher collision frequency compared with those with smaller friction as shown in Figure 4a, particles in the mill with rougher stirrers can have a larger number of collisions with low collision energy. Figure 4b shows the variation of impact energy and input power. When the particle disc sliding friction is high, the impact energy increment is not significant. But the power draw still increases with the sliding friction, because more input energy is required to overcome the friction between particles and stirrers. Therefore, too high a particlestirrer friction coefficient may simply consume more energy with insignificant improvement in grinding. 3.2. Effect of Distance between Stirrers. As shown in Figure 5, a mill is filled with grinding media between discs giving a few grinding chambers. A circulation flow of media can be generated in each chamber, which can generate high density

Figure 5. Discs arrangement: (a) ld = 17 mm, (b) ld = 30 mm.

energy regions that would contribute to particle breakage.29 Therefore, the proper selection of distance between discs is another important parameter in the IsaMill design. To examine its effect, distances between discs have been changed by changing the number of discs while keeping the length of the mill unchanged (Figure 5). Note that the largest distance between discs was set to 30 mm which corresponds to the distance used in the scaled down 4-L laboratory mill.29 Figure 6 shows that, by increasing the disc distance to 30 mm, more particles stay in the center of the mill. This is because when the distance increases, the proportion of particles near a disc is reduced. The energy transferred from the discs cannot be propagated effectively to particles far away from discs and therefore particles do not gain enough kinetic energy to move along the mill. For smaller disc distances, particles can gain higher kinetic energy which in turn increases the centrifugal force. As a result, particles tend to move toward the mill drum, creating empty space in the middle. If the flow is viewed from the axial plane as shown in Figure 7, the stirrer distance has a major influence on the flow circulation between stirrer discs. The particle flow with a disc distance of 17 mm does not show detectable recirculation between the discs, while flow with a disc distance of 30 mm reveals a circulating flow between the stirrers (Figure 7b). If slurry is introduced to the system, the intensity of circulation flow can be further promoted.4,26 Figure 8a shows the variations of collision frequency and collision energy with stirrer distance. If the stirrer distance increases, particle disc interaction reduces. As a result, energy transfer from discs to particles is limited, which in turn decreases the mobility of particles. When particle mobility reduces, the number of collisions among particles is also reduced, resulting in decreased collision frequency. The relative velocity between particles increases, however, because some particles obtain higher velocities than those with less mobility. Therefore, slightly increased collision energy can be observed. If the stirrer distance is reduced, more particles come into contact 1053



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Figure 6. Snapshots of particle flow (top) and spatial distributions of velocities and porosity (bottom) for different disc distances: (a) ld = 17 mm, and (b) ld = 30 mm when J = 80% and Ω = 1200 rpm.

Figure 7. Spatial distributions of velocities and porosity at section XX0 for different stirrer distances: (a) ld = 17 mm, and (b) ld = 30 mm when J = 80% and Ω = 1200 rpm.

with the disc. As a result, more energy is transferred from discs to particles, which increases the particle kinetic energy. With increased kinetic energy, particles move toward the mill drum where they are closely packed. When particles are closely packed, they undergo a rapid series of collisions with their neighbors, leading to a high collision frequency. Collision energy slightly decreases because most of the particles are in motion with a similar magnitude of velocity which decreases the relative velocities among particles. Figure 8b shows the variation of Pin and Ei with stirrer distance. When the stirrer distance reduces, as a result of more energy being transferred to particles, Pin and Ei will be increased. However, the rate of increase in Pin is much higher than that of Ei. The results suggest that when the distance decreases, a relatively high input energy is consumed by the mill.

3.3. Effect of the Size of Holes. In stirred mills, energy is transferred from stirrers to particles mainly in two ways: through particlestirrer friction and through normal impact between particles and disc holes. The effect of particlestirrer friction has been discussed earlier. Here we quantify the effect of the size of disc holes on particle flow. The diameter of holes dh was varied from 12 to 24 mm and Figure 9 shows particle flow, and spatial velocity and porosity distributions for different sized holes. The characteristics of the two distributions are very similar, but small holes result in a lower velocity. The porosity distribution also indicates that, with small holes, more particles are in the bottom part of the mill. By increasing the hole size, high-porosity regions are more evident in the mill center. This is because the energy transferred from disc holes to particles is the primary mechanism 1054



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Figure 8. (a) Mean collision frequency and collision energy; and (b) impact energy and power draw for different ld when J = 80% and Ω = 1200 rpm.

Figure 9. Snapshots of particle flow (top), and spatial distribution of porosity and velocity vectors (bottom) for disc holes of different sizes: (a) dh = 12 mm, and (b) dh = 24 mm when J = 80% and Ω = 1200 rpm.

as the holes act as lifters for particles. With large holes, more particles are captured by the rotating holes, and therefore gain more kinetic energy. Figure 10a shows the mean collision frequency and collision energy with hole diameter. Both Cf and Ce increase linearly with increasing hole size as a result of high energy transfer to particles. Figure 10b shows that impact energy and power draw increase almost linearly with dh. Obviously, at large dh, more particles are captured and lifted and therefore more power is required to maintain the motion of particles. 3.4. Effect of Stirrer Geometry. In stirred mills, energy transfer takes place primarily through the stirrers. It has been observed that stirrers of different types can be used for different processes and product size control. As shown in Figure 2, here we have examined four types of stirrers, namely a disc without holes (0H), with three holes (3H), with five holes (5H), and a pin type stirrer (PIN). Figures 11 and 12 show particle flow, velocity field, and porosity

distribution for these stirrers. Note that when the particledisc sliding friction coefficient is too low, energy transfer from 0H disc to particles is too low to agitate particles. Therefore, for all the stirrers, a relatively high sliding friction coefficient (μs,pd = 0.5) was used in this analysis. For the 0H stirrer, quite a low velocity profile can be observed and particles stay in the bottom part of the mill, because of the absence of normal impacts between particles and disc hole. As the agitation is caused only by the friction between the particles and stirrers, less energy is transferred to the particles. For the 3H and 5H stirrers, energy transfer to particles is more effective when holes are present. The velocity profile has a maximum near disc holes with the 5H having a slightly higher velocity. No significant difference is observed, however, between 3H and 5H stirrers in terms of flow pattern and porosity distribution. In the case of a pin stirrer, the normal impact from stirrers to particles is far greater than that from disc holes, so particles have high kinetic energy and centrifugal velocities toward the mill drum and leave an empty space in the 1055



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Figure 10. (a) Mean collision frequency and collision energy, (b) impact energy and power draw for different dh when J = 80% and Ω = 1200 rpm.

Figure 11. Particle flow for different stirrer geometries: (a) 0H; (b) 3H; (c) 5H; and (d) pin.

Figure 12. Spatial distributions of velocity and porosity for different stirrer geometries when J = 80% and Ω = 1200 rpm: (a) 0H; (b) 3H; (c) 5H; and (d) pin.

middle. The velocity profile in the radial plane is quite uniform and its magnitude is considerably larger than the other three types of stirrers. It is interesting to compare the axial views of particle velocity for the mills with 5H and pin stirrers. Figure 13 shows that for the

5H disc mill, the particle velocity points inward near the shaft and outward near the discs, causing a circulating flow between the discs. On the other hand, the flow field for the pin stirrer mill (Figure 13b) is significantly different and does not show the circulating flow field in the axial plane, resulting in a better mixed 1056


Industrial & Engineering Chemistry Research flow. The particles also have higher velocities due to their direct contacts with pin surface. Figure 14a and b show the mean collision frequency and collision energy for different stirrers. For the disc stirrer, increasing number of holes increases both mean collision frequency and collision energy. The pin stirrer, on the other hand, has the highest ones. Figure 14b shows Pin and Ei for different stirrer geometries. Looking at 0H through 5H, there is a slightly disproportionate increase in Ei with Pin, i.e., as the number of holes is increased, Ei increases faster than Pin. Therefore, among the 0H, 3H, and 5H stirrers, 5H is expected to show the highest grinding performance. The pin stirrer exhibits the highest impact energy intensity, but its energy consumption is also the highest. In fact, its power draw is much higher than other stirrers. Therefore, in terms of energy efficiency, the 5H stirrer may be more favorable than the pin stirrer. However, the pin stirrer gives the highest throughput. This issue will be further discussed in the next section.

4. GRINDING PERFORMANCE Grinding is a very energy consuming process. In fact, the energy efficiency of a typical mill is dramatically low.30,31 How to optimize the operating conditions for a given material would benefit the mineral industry enormously. Generally this is achieved based on physical experiments which are laborious and costly, and sometimes extremely difficult to conduct. On the other hand, while DEM simulations can be used to obtain microdynamic information such as impact energy, direct simulation of particle breakage is still computationally prohibitive. Only recently tentative attempts have been made, for relatively small systems, to implement empirical comminution function into DEM for such purpose.32,33 Here we

Figure 13. Spatial distribution of velocities at section XX0 for (a) 5H stirrer, and (b) pin stirrer when J = 80% and Ω = 1200 rpm.

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show that DEM simulations, supported by some simple experiments, can be used to assess grinding performance. 4.1. Size Reduction Kinetics. Different stirrers produce different flow patterns and hence different collision environments. In terms of grinding, what is important is the impact between particles which leads to fragmentation of ground materials. For example, for a given grinding material, one would expect a similar grinding rate for different stirrers as long as their impact energies are the same. In this light, we experimentally investigated the actual particle breakage with different types of stirrers. Physical experiments were performed in a stirred mill made of a transparent chamber and three polyurethane agitator discs. The mill was driven by a DC motor with a speed controller to control the agitation speed. Glass beads were used as grinding media, and sea salt (mean size around 1000 μm) or calcium carbonate (mean size around 50 μm) were used as ground materials. For each experimental run, the amount of calcium carbonate or sea salt used was 180 or 250 g, respectively. At different time intervals, the size distribution of calcium carbonate powders was analyzed by a Malvern particle analyzer, whereas size distribution of sea salt was obtained by a standard sieving technique. This procedure was done at different mill speeds. The power draw was obtained by measuring voltage and current across the motor. The same experiments were carried out for different stirrers, corresponding to the numerical experiments. It was observed that the power draws from experiments were comparable with the simulated ones, further confirming the validity of the model.

Figure 15. Cumulative particle size distribution of calcium carbonate at different times for pin stirrer when J = 60% and Ω = 800 rpm.

Figure 14. Comparison of different stirrers: (a) mean collision frequency and collision energy, and (b) impact energy and power draw when J = 80% and Ω = 1200 rpm. 1057



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Figure 17. Correlation between grinding rate constant and total impact energy for D80 passing size.

Figure 16. Normalized particle sizes of calcium carbonate ground samples as a function of grinding time for different stirrers when J = 60% and different mill speeds (, 400 rpm; O, 600 rpm; 1, 700 rpm; 9, 800 rpm; and ), 1000 rpm; numbers represent the grinding rates Kp (min1)).

The experimental results showed that the change of particle size with grinding time follows the first-order kinetics, given by Dt D∞ ¼ expKp t D0 D∞

ð5Þ

where D0, Dt, and D∞ are the original particle size, particle size at grinding time t, and the limiting size after an infinite grinding time, respectively; Kp is the grinding rate constant. Similar correlations have been reported elsewhere for planetary ball mills.3436 No such a study has been reported, however, for stirred mills with different stirrer types. As part of the present study, therefore, physical experiments were performed with different stirrer types at different mill speeds. Figure 15 shows the cumulative size distribution of calcium carbonate ground samples measured at different times for a pin stirrer when J = 60% and Ω = 800 rpm. Similar distributions are also obtained for other stirrers at different mill speeds. The distribution curves shift progressively toward the finer side as the grinding continues. As used in normal practice, 80% passing particle size D80 was selected to represent the particle size in order to quantitatively investigate grinding performance under different conditions.37 The loglinear plots of (D80,t D80,∞)/(D80,0 D80,∞) against grinding time t in Figure 16 follow straight lines for all cases, although different slopes are observed for different operating conditions. This indicates that the grinding process indeed follows the first-order kinetics and particle size decays exponentially with time according to eq 5. 4.2. Role of Impact Energy. Figure 17 shows the correlation between grinding rate constant Kp and impact energy Ei for calcium carbonate or sea salt. Note that owing to extremely low impact energy observed in the 0H disc, it is practically impossible to determine the grinding rate constant and hence the results for 0H disc have not been included. Both curves show that the grinding rate increases with the impact energy. For a given impact

Figure 18. Comparison of impact energy to power ratio at different mill conditions: (a) particledisc sliding friction coefficient; (b) distance between stirrers; and (c) hole size.

energy, sea salt shows a higher grinding rate than calcium carbonate. Salt coarse particles (≈ 1000 μm) are easier to break than calcium carbonate fine particles (≈ 50 μm). Unlike coarse 1058



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Figure 19. Comparison of mill efficiency for different stirrers: (a) sea salt and (b) calcium carbonate.

particles, fine particle breakage requires more energy38 and hence shows a reduced grinding rate. The present results confirm that for a given material, there is a well-defined correlation between impact energy and grinding rate which is given by Kp = k0Ein where k0 = 0.039, 0.078 and n = 0.714, 0.728 for calcium carbonate and sea salt, respectively. The dependences of grinding rate on materials properties and size have also been observed in previous studies.39,40 Further work is required for the quantitative description of such dependence in stirred mills. The efficiency of a mill depends on many parameters related to operating conditions and material properties. Mill efficiency can generally be measured by the energy requirement to produce a unit mass of product.31 Here we demonstrated that the impact energy is proportional to the grinding rate. Consequently, the ratio of the impact energy to input energy can be used as a measure of the mill efficiency. Figure 18 shows Ei/P ratio for the mill conditions considered. Ei and P are obtained from the simulation. In the simulations, power draw is the product of the rotation speed and the total torque acting on the discs and shaft. At a particular time, each contact between a particle with the rotating shaft or discs produces a torque on the mill, which is the product of the contact force and distance between the contact point and centerline of the drum. The individual torques are summed to give the total torque which, multiplied by the angular mill velocity, gives the power draw of the mill at that particular time. Because of the impulsive nature of the interactions between the particles and discs, fluctuation occurs in the power draw. Averaging the power draw over a certain time gives a relatively invariant value. Experimental power was obtained by measuring voltage and current across the direct current (DC) motor and deducting the no load power measurements. In fact, simulated and experimental power consumptions are comparable. When particle disc sliding friction coefficient is high, a relatively large amount of energy is being wasted through the particledisc friction surface. As a result, Ei/P ratio decreases. On the other hand, when the distance between discs increases and the size of disc hole increases, Ei/P ratio increases. Therefore, while changing these mill properties may improve impact energy, the mill efficiency may decrease. In fact, too large a hole may not be useful as the hole can be enlarged due to wear leading to structural failure. Likely other material properties will also play an important role in controlling the mill efficiency. Figure 19 shows the comparison of mill efficiency for different stirrers under different mill speeds. For the grinding of salt, Ei/P ratio increases with speed up to 800 rpm

and the pin stirrer has higher energy ratio than the other stirrers. Note the highest rotation speed for the pin stirrer is 800 rpm as the current DC motor cannot provide enough power for higher rotation speeds. With calcium carbonate grinding, when mill speed increases, 5H stirrer shows the highest Ei/P ratio and the pin stirrer the lowest Ei/P ratio where the rotational speed has an insignificant effect. Different performance observed in different stirrers could be due to different grinding mechanisms for different materials. In general, there are three breakage mechanisms: (i) abrasion, (ii) impact, and (iii) compression, which of course do not occur in isolation, but with one normally dominating.37,38 Coarse particles have a tendency to break under impact damage, whereas fine particles have a tendency to break under abrasion and compression.38 Because the majority of energy transfer takes place through high impacts in the pin stirrer, impact damage could be dominant. As a result, relatively coarse and brittle materials such as the salt particles used in the present experiments could be broken readily under impact damage, rather than abrasion. On the other hand, energy transfer from 3H and 5H stirrers take place mainly through particledisc sliding friction where abrasion is dominant. Therefore, fine particles such as calcium carbonate particles are more likely to break under abrasion. The results suggest that 3H and 5H stirrers are energy-efficient for particles that can be broken by abrasiondominant grinding, whereas the pin stirrer is energy-efficient for particles that can be broken by impact-dominant grinding. Therefore, material type and its feed size should be taken into account in the selection of stirrer geometry under dry grinding. This will be further investigated in the future.

5. CONCLUSIONS DEM has been used to investigate the granular flow in stirred mills, focusing on the effects of mill material properties and disc configurations on the flow variables, such as velocity, porosity, collision energy, collision frequency, and power draw. Grinding performance of different stirrer types has also been examined experimentally, in connection with the numerical simulations. The results from the present study can be summarized as follows: • When particlestirrer sliding friction coefficient increases, different flow patterns can be observed. When sliding friction coefficient increases, collision frequency increases and collision energy decreases. Compared to impact energy, power draw increases significantly with increasing particlestirrer 1059


Industrial & Engineering Chemistry Research friction. The results also indicate that although particle wall sliding friction coefficient affects the energy transfer from discs to particles, too high a sliding friction may lead to a decrease in energy efficiency. • It is shown that the distance between stirrers significantly affects the circulation of grinding media between discs. When the distance increases, collision frequency decreases whereas collision energy increases. Compared to the impact energy, power draw decreases significantly with increasing distance. As a result, within the distance range considered in this work, the highest distance shows the maximum energy efficiency. • Among the different stirrer types considered, energy transfer is more effective when disc holes are present. When the hole size increases, as a result of more energy being transferred to particles, both collision energy and collision frequency are increased. The energy efficiency increases with bigger holes because the rate of increase in impact energy is higher than the rate of increase in power draw. However, too large a hole may not be preferable as the hole can be enlarged due to wear leading to structural failure • It is shown that the dry grinding of stirred mills follows the first-order kinetics. Although different collision environments exist with different stirrer types, impact energy is a useful index which determines the grinding rate. Regardless of the stirrer type, for a given material, a unique correlation exists between impact energy and grinding rate constant. • Energy efficiency varies with material type because of different dominant grinding mechanisms. For calcium carbonate, when mill speed increases, 3H and 5H stirrers show an increase in mill efficiency, whereas pin stirrer shows the lowest efficiency. For sea salt, when mill speed increases, all three stirrers show an increase in mill efficiency where pin stirrer running at 800 rpm being the most efficient. An important finding from this work is that grinding performance, as described by the macroscopic parameter Kp, can be linked to the microscopic variable Ei, obtained from particle scale simulation. Different materials have different mechanical properties and hence give different EiKp relations. However, the EiKp relation appears to be an intrinsic property of a material. That is, this relation is fundamental and, once established, can be applied to different types of mills which may have different impact energies and hence different grinding efficiency. This finding is very useful, as it may open up a promising direction to assess grinding performance through DEM study, at least for materials following the first-order kinetics in particle size reduction.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank the Australian Research Council and Xstrata Technology for providing the financial support for this work, and Mr. D. Curry and Mr. J. Rubenstein of Xstrata Technology for valuable discussion and feedback at the early stage of this work. ’ REFERENCES (1) Gao, M.; Young, M. F.; Cronin, B.; Harbort, G. IsaMill medium competency and its effect on milling performance. Miner. Metall. Process. 2001, 18 (2), 117–121.

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Discrete Particle Simulation of Particle Flow in a Stirred Mill: Effect of

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