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DEM simulation of the flow of grinding media in IsaMill
Minerals Engineering 19 (2006) 984–994 This article is also available online at: www.elsevier.com/locate/mineng
DEM simulation of the flow of grinding media in IsaMill R.Y. Yang a, C.T. Jayasundara a, A.B. Yu a
a,*
, D. Curry
b
Centre for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia b Xstrata Technology, Brisbane, Qld 4000, Australia Received 22 December 2005; accepted 8 May 2006 Available online 7 July 2006
Abstract IsaMill is a high speed stirred mill for high efficiency grinding of mineral ores and concentrates. A numerical model based on the discrete element method (DEM) was developed to study flow of grinding media in IsaMill. The DEM model was first verified by comparing the simulated results of the flow patter, mixing pattern and power draw from those measured from a 1:1 scale lab mill. Then the flow properties were analysed in terms of flow pattern, flow velocity, force field and power draw. The effects of parameters relating to particle material (i.e., sliding friction coefficient and damping coefficient) and operational conditions (i.e., rotation speed and solid loading) were investigated. While the damping coefficient showed a negligible effect for the range considered, other three parameters had strong effects on the flow properties. Increasing the sliding friction caused the flow velocity and compressive force to have minimum values due to the competitive mechanisms for energy transfer and dissipation, but increased the power draw. The increase in the rotation speed and solid loading also increased the flow velocity, compressive force and power draw of mill. The particle scale information obtained would be useful to understand the fundamentals governing the flow of grinding media in IsaMill. 2006 Elsevier Ltd. All rights reserved. Keywords: Grinding; Discrete element modelling; Process simulation; Mineral processing
1. Introduction Grinding is the single largest energy consumption process in the mineral industry. Traditional grinding in tumbling mills (e.g., ball mills) is a low efficiency (1–2%, typically) process and can account for up to 40% of the direct operating cost of a mineral processing plant (Joe, 1979; Wills, 1992). Furthermore, many of the base and precious metal deposits discovered are now fine-grained and complex, and there is a need for grinding them to very fine sizes to permit sufficient mineral liberation, for example, 80% of particles (P80) should be below 20 or even 10 lm. Grinding to such fine sizes economically is beyond the capability of conventional grinding technologies. IsaMill is a high speed stirred mill developed by Mount Isa Mines (Xstrata) in Australia for economically grinding *
Corresponding author. Tel.: +61 2 93854429; fax: +61 2 93855956. E-mail address: [email protected] (A.B. Yu).
0892-6875/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2006.05.002
minerals to fine and ultra-fine size at an industrial-scale (Gao and Forssberg, 1995). It consists of a horizontally mounted shell and rotating grinding discs mounted on a shaft which is coupled to a motor and gearbox and rotates with disc tip speeds of 10–23 m s1. The grinding discs agitate the media and ore particles in a slurry that is continuously fed into a feed port. The product separator (dynamic, centripetal classifier) keeps the grinding media inside the mill allowing only the product to exit. Simple control strategies based on power draw enable the IsaMill to produce a constant target product size. Comparing with the conventional grinding mills such as ball mill and tower mill, IsaMills can significantly reduce total comminution circuit energy cost (Curry and Clermont, 2005) and reduce the size of mineral particles to as fine as P80 passing 7 lm (Gao and Forssberg, 1995). However the IsaMill is still a new technology and most relations established between the quality of the final products and operating parameters of the mill are empirical, and are inadequate to comprehensively
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understand the grinding process. As a result, its optimum control and scale-up need to rely on empirical methods, experience and trial and error testing, rather than detailed scientific principles. There is a need for research into the grinding process to be predicted and modelled based on knowledge of the characteristics of the mill and the properties of the grinding materials. The bulk behaviour of media particle flow depends on the collected outcome of the interactions between particles and between particle and mill. As a result, a better understanding of the flow at an individual particle scale would greatly facilitate the design and scale-up of IsaMill. However, it is a very difficult, if not impossible, task using experimental techniques to obtain microdynamic information such as voidage, force and velocity distributions within the mill. On the other hand, simulation based on Discrete Element Method (DEM) (Cundall and Strack, 1979) has been extensively used in the study of particle packing and flow for various systems and has been demonstrated as an effective way to link microscopic information with macroscopic behaviour of particle flow (for example, see Yu, 2004). However, despite the usefulness, DEM simulation studies have not been applied widely to the high speed stirred mill system like IsaMill. This paper presents a numerical investigation of the flow of grinding media in a simplified IsaMill, aiming to examine the feasibility of using DEM simulation to predict particle flow in a high speed stirred mill. The proposed DEM model is first verified with a 1:1 scale experimental setup which allows qualitative or quantitative comparison of flow and mixing patterns and power consumption. Then, the flow of particles will be analysed in terms of velocity, force and power draw. The effect of some variables related to key materials properties and operational conditions are also investigated. 2. Simulation method and conditions In a DEM simulation, the translational and rotational motions of a particle of radius Ri (=R for mono-sized spheres) and mass mi can be described by dvi X n ¼ mi ðFij þ Fsij þ mi gÞ ð1Þ dt j and Ii
dxi X ^ iÞ ¼ ðRi Fsij lr Ri jFnij jx dt j
ð2Þ
where vi, xi and Ii are, respectively, the transitional and angular velocities, and moment of inertia of the particle. Ri is a vector running from the centre of the particle to the contact point with its magnitude equal to particle radius Ri. The first part of the right side in Eq. (2) is the torque due to the tangential force Fsij , the second part is the rolling friction torque arising from the elastic hysteresis loss or viscous dissipation, and lr is the coefficient of roll-
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ing friction (Tabor, 1955; Brilliantov et al., 1996). This frictional resistance has been demonstrated to play a critical role in achieving physically or numerically stable sandpile, viz. the unconfined packing of particles although it is still unclear which model can best describe rolling friction (Zhou et al., 1999). Fnij and Fsij represent, respectively, the normal contact force and the tangential contact force imposed on particle i by particle j, given by (Mindlin and Deresiewicz, 1953; Johnson, 1985; Langston et al., 1994; Brilliantov et al., 1996) pffiffiffi pffiffiffipffiffiffiffiffi 3 2 Fnij ¼ E Rn2n cn E R nn ðvij ^nij Þ ^nij ð3Þ 3 Fsij ¼ sgnðns Þls jFnij j½1 ð1 minðns ; ns;max Þ=ns;max Þ
ð4Þ
~2 Þ, R ¼ Ri Rj =ðRi þ Rj Þ, ns;max ¼ ls ½ð2 where E ¼ Y =ð1 r ~Þ=2ð1 r ~Þnn , Y = Young’s modulus, r ~ ¼ Poisson ratio, r ^nij ¼ ðRi Rj Þ=jRi Rj j, cn = normal damping coefficient, ls = sliding friction coefficient, ns = total tangential displacement. Details about the force models used in this work can be found from elsewhere (Yang et al., 2000, 2003a,b). Simulations were performed in a stirred mill (Fig. 1) consisting of a fixed chamber of inner diameter of 110 mm and several discs mounted on a rotating shaft of diameter of 25 mm. The disc is 90 mm in diameter and 9 mm in thickness, and is about 30 mm apart from each other. Each disc has five holes of 18 mm diameter. The mill was filled with 4 mm mono-sized glass beads. In practice, grinding media mixed with fine slurry are continuously fed into the mill from one end and the fines exit from the other end. However, as a first step to develop a comprehensive DEM model for IsaMill, no fine slurry was included in the present simulation, and the grinding media were sealed in the mill and therefore no flow in and out were considered. Two types of simulations with different boundary conditions were used in this work. In the first type of simulation (Fig. 1a), the mill has a chamber with two side walls and three discs inside. This would be mainly for model validation by comparing with a 1:1 scale experiment. In the second type of simulations (Fig. 1b), only one disc was considered in the mill but the periodic boundary conditions were applied along the axial direction. This set up corresponds to the region B in Fig. 1a, and allows large scale simulations with a small number of particles. Table 1 lists the values of the parameters used in the simulation. Parameters with a varying range were changed to study their effects on the particle flow. Unless otherwise specified, the effect of each variable was examined when others were fixed at their respective base values. Note that the Young’s modulus used in the simulations is four orders of magnitude smaller than the real glass bead (100 GPa) to reduce the simulation time. The current value ensures that the maximum overlap is less than 1% of particle diameter and has been demonstrated not affect the final results much (Zhou et al., 1999; Yang et al., 2000, 2003a,b). The
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(a)
(b)
Fig. 1. Schematic illustration of simplified IsaMill.
Table 1 Physical parameters in the simulation Parameter
Base value 3
Particle density, q (kg m ) Number of particles Young’s modulus, Y (N m2) ~ Poisson ratio, r Sliding friction coefficient, ls Rolling friction coefficient, lr Normal damping coefficient, cn (s1) Solid loading, J Rotating speed, X (rpm)
3
2.5 · 10 3000 2.0 · 107 0.29 0.3 0.005 5 · 106 80% 1000
Varying range – 1500–3000 1 · 107–2 · 108 – 0.01–1.0 0.001–0.01 5 · 106–2 · 105 40–80% 500–2000
mill was assumed to have the same materials properties as particles. 3. Results and discussion Fig. 2 shows the snapshots of particle flow at different times in the single disc mill. Each simulation begins with a packed bed (Fig. 2a), then the disc rotates at a given speed and the particles are agitated and driven to move outwards (Fig. 2b), finally the flow reaches a macroscopically stable state after a certain time (Fig. 2c and d). Analyses of the flow velocity and power draw show that the time for a flow to reach such a stable state depends on the particle properties and the rotational speed, but generally is
less than 0.5 s. Unless otherwise stated, our following analysis is based on the cumulative data obtained at macroscopically stable state. Starting from t = 1 s when the flow reached the stable state, the samples were collected every 0.05 s till the simulation finished (10 sets of samples totally). Then these samples were averaged to calculate the flow velocity and compressive force. In the following sections, the DEM model will first be verified by comparing the results obtained from simulations and experiments. Then, based on the simulated results, the particle flow will be analysed in terms of the flow pattern, velocity and force fields, and power draw. The effects of particle properties such as sliding friction coefficient and damping coefficient and operational conditions such as mill filling level and rotating speed on the flow properties will be discussed. Note that the effects of other particle properties such as Young’s modulus and rolling friction coefficients were also examined and found to be negligible for the ranges considered in this work. Therefore their results are not shown for brevity. 3.1. Macroscopic observation and model validation To verify the proposed DEM model, simulations corresponding to physical experiments as shown in Fig. 1a were performed. The simulations and experiments were carried out with the mill about 60% loaded and rotating at various
Fig. 2. Snapshots at different times in the simulations with the variables at their base values: (a) t = 0.0 s, (b) t = 0.01 s, (c) t = 0.5 s and (d) t = 1.0 s.
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speeds. Qualitative and quantitative comparisons in terms of flow and mix patterns and power draw are shown in Fig. 3. The flow patterns at X = 300 rpm (Fig. 3a) indicate that most particles stay at the bottom of the mill and behave like bulk solid with little movement and only a relatively small number of particles are in the upper half of the mill. Vigorous flow of particles can be obtained when the rotation speed is high. Low rotation speeds were used here to produce visualisation of high quality. A more detailed comparison was performed in the radial mixing of two coloured particles. The particles were initially separated into two layers in vertical direction. As discs rotate at 100 rpm, two coloured particles gradually mix together. Fig. 3b shows the mix pattern at 10 s, indicat-
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ing that the particles near the discs move much faster due to the lift of discs. Therefore the mixing process is much faster in the central part of the mill and more dark particles are found mixed with light particles in this region. Power consumption, or power draw, is one of the most important indexes to characterise the mill performance. The total electrical energy input Ein is by definition the product of voltage and current across the DC motor which were measured in the experiment. The input energy cannot fully be transferred to particles, because of the energy losses in running the motor Em1 and the mill Em2. That is, Ein Em1 = Em2 + Ep, where Ep is the energy transferred to the particles in the mill. Ep can be calculated from the DEM simulation. This energy, as the power draw by
Fig. 3. Comparisons of the physical and numerical experiments: (a) flow pattern with X = 300 rpm and J = 60%; (b) mixing pattern at t = 10 s with X = 100 rpm and J = 60%; and (c) power draw from experiment (point) and simulation (line) as a function of rotational speed with J = 60%.
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particles, is the product of rotation speed and total torque acting on the discs and shaft. We have estimated the energy loss Em1 by calibrating the mill at zero loading at a given rotation speed. Therefore, the energy loss Em2 is the only unknown energy. Fig. 3c shows the power draw as a function of rotation speed from the simulation (Ep) and experiment (Ein). Clearly, the power draw increases with the speed of rotation. The overall comparison is quite good except for the slightly larger discrepancy at high speeds (>1000 rpm). The difference can be attributed to the energy loss in the form of heat, sound and others which have not been considered. While the present experimental technique is not perfect, it does provide a reasonable result for model verification. The good agreement between the simulated and measured results in Fig. 3, either qualitative or quantitative, confirms the validity of the present DEM model, at least at a macroscopic scale. 3.2. Effects of material properties Sliding friction coefficient and damping coefficient are two important variables relating to material properties. Sliding friction coefficient relates to the particle roughness, and damping coefficient is due to the inelasticity of particles and relates to the particle restitution coefficient. In practice, different materials, such as river sands and ceramic beads, have been chosen as grinding media. These materials have different sliding frictions and damping coefficients which may affect the grinding performance. Therefore, the study of their effects is relevant to the industry. Fig. 4 shows the flow patterns and the force structure for different sliding friction coefficients. The force structure is characterized in terms of the normal compressive force
P n acting on individual particles, defined as fi ¼ jFij j. As the tangential force (related to abrasive forces) is proportional to the normal force at a contact point, their variations are in the same trend and not shown in this paper. Fig. 4a shows that for all cases, the peak velocities of the flows were near the holes, indicating the boundaries of the holes act as lifters to drive particles. Increasing the sliding friction coefficient causes the dilation of the particle flow. It is interesting to note that the flow has the lowest mean velocity at ls = 0.1. For ls smaller or larger than 0.1, both mean velocity and velocity gradient along the radial direction increase. This is because the sliding friction plays two roles in the flow, energy transfer and energy dissipation. The two mechanisms are competitive against each other. Before the critical value, the dissipation mechanism is dominant so the velocity (kinetic energy) of flow decreases with increasing ls. After the critical value, more energy is transferred with increasing ls, and the flow becomes faster. The same trend can also be observed from Fig. 4b, which shows the spatial distribution of the compressive force. The compressive forces first decrease as ls increases from 0.01 to 0.1 and then increase with further increase in ls. The observations from Fig. 4 are further confirmed from the quantitative analyses of the particle velocities and the compressive forces, as shown in Fig. 5. For ls = 0.01, the particle velocity has a mean value of 2.8 m/s and a distribution with a peak value at 3 m/s. When ls increases to 0.1, the mean velocity decreases to 1.65 m/s and the distribution becomes narrower and shifts to the smaller value with a peak value of 1.2 m/s. Further increase in the sliding friction coefficient causes the distribution to become broader and shift to the right side with the peaks of 1.9 m/s for ls = 0.3 and 2.7 m/s for ls = 1.0. This indicates that, at
Fig. 4. Snapshots of (a) particle flow and velocity distribution; and (b) velocity vector and force field for different sliding friction coefficient: (from left to right) 0.01, 0.1, 0.3 and 1.0. The colour represents the magnitude of particle velocities (top) or the compressive forces acting on particles (bottom). (For interpretation of colour in this figure legend, the reader is referred to the web version of this article.)
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2.5 2.0
μs =0.01 μ s =0.3
1.5
3.0
.
μs =0.1
3.0
Mean velocity, (m/s)
Probability distribution, P(v)
3.5
μ s =1.0
1.0 0.5
2.6 2.2 1.8 1.4
0.0
1.0
0
1
(a)
2 3 Velocity, v (m/s)
4
5
0
Mean compress force, (N)
10 Probability distribution, P(f)
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0.001 0.1 0.3 1
1
0.1
0.01
0.001 0
(b)
1 2 3 4 Compress force, f (N)
0.2 0.4 0.6 0.8 Sliding friction coefficient, μ s
1
0.8
0.6
0.4
0.2
0 0
5
0.2
0.4
0.6
0.8
1
Sliding friction coefficient, μs
Fig. 5. The effect of sliding friction coefficient on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocities for different ls; and (b) the probability density distribution (left) and mean value (right) of the compressive forces on particles for different ls.
ls = 0.1, the particles move with the slowest but quite uniform velocities. Fig. 5b shows that the distribution of the compressive forces acting on particles decays exponentially, although forces for smaller ls fall off more quickly. Note that this phenomenon was observed in other static systems such as packing of particles (Liu et al., 1995; Mueth et al., 1998). Our results indicate the exponential tail at the large force seems very robust and can be found in the dynamic systems. The mean compressive force also shows a minimal value at ls = 0.1. Fig. 6 shows the effect of the sliding friction coefficient on the power draw of the mill. The flow takes less time
to reach the stable state for particles of larger sliding friction coefficient. When ls = 1.0, the power draw decreases rapidly as the mill begins to rotate and reaches a stable value of 45 W after less than 0.1 s. For a smaller sliding friction coefficient of 0.1, the power draw reaches a stable value only after 0.3 s. However, the power draw for ls = 0.1 at t = 1.0 s is about 5 W. Decreasing the sliding friction coefficients to 0.01 does not change the power draw significantly. Fig. 6b indicates that the mean power draw of the mill increases with ls. Three damping coefficients were used in the present work, i.e., cn = 5 · 106 s1, 2 · 105 s1 or 5 · 105 s1.
100
60 0.001 0.1 0.3 1
50 Power, (W)
Power, (W)
80 60 40 20
30 20 10 0
0 0 (a)
40
0.3
0.6
0.9
Time, t (s)
1.2
0
1.5 (b)
0.2 0.4 0.6 0.8
1
1.2
Sliding friction coefficient, μs
Fig. 6. The effect of sliding friction coefficient on power draw: (a) variation of power draw with time for different ls; and (b) mean power draw as a function of ls.
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The corresponding restitution coefficients of particles at the impact velocity of 1 m/s are 0.87, 0.60 and 0.28. Our results show that there are no evident differences in terms of flow pattern (not shown here). Quantitative analysis of the particle velocity and the compressive force for different damping coefficients are plotted in Fig. 7, showing the distributions and the mean values of the velocities are almost the same although particles with a small damping coefficient (large restitution coefficient) do have a slightly broader velocity distribution. The distribution of compressive forces acting on particles also shows an exponential decrease, and the mean force drops slightly from 0.313 when cn = 5 · 106 s1 to 0.30 when cn = 5 · 105 s1. The increase in damping coefficient also slightly reduces
Probability distribution, P(v)
1.2
γ n=5e-5
the power draw of mill, as seen from Fig. 8. However, compared to the effects of other variable considered in this work, the effect of cn is negligible. 3.3. Effects of operational conditions Rotation speed and solid loading of the mill are two operational variables considered in this work. Fig. 9a shows the particles with different rotation speeds, indicating that as the rotation speed increases, particles are driven outwards and form a faster flow with an increased velocity gradient along the radial direction. This suggests that a high rotation speed can produce a dense particle packing. The compressive force on particles (Fig. 9b) also increases
2
γ n=2e-5
1.0
Velocity, v (m/s)
γ n=5e-6
0.8 0.6 0.4 0.2
1.8
1.7
1.6
0.0 0
1
2
3
4
0
5
Mean compress force, (N)
10 5e-6 2e-5
1
5e-5 0.1
0.01
0.001 0
0.5
1
1.5
2
2
3
4
5
6
2.5
0.32
0.31
0.30
0.29
0.28 0
3
1
2
3
4
5
Damping coefficient, γ n (x10-5 s -1)
Compress force, f (N)
(b)
1
Damping coefficient, γ n (x10-5 s -1)
Velocity, v (m/s)
(a)
Probability distribution, P(v)
1.9
Fig. 7. The effect of damping coefficient on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocities for different cn; (b) the probability density distribution (left) and mean value (right) of the compressive forces on particles for different cn.
20
5e-6 2e-5 5e-5
16
Power, (W)
Power, (W)
100
10
12 8 4 0
1 0
(a)
0.3
0.6
0.9
Time, t (s)
1.2
0
1.5
(b)
2 4 6 Damping coefficient, γ n (x10-5 s -1)
Fig. 8. The effect of damping coefficient on power draw of mill: (a) power draw as a function of time; and (b) mean power draw as a function of cn.
R.Y. Yang et al. / Minerals Engineering 19 (2006) 984–994
with increasing speed. In fact, as shown in Fig. 10a, the distribution of particle velocities is broader and shows a bi-modal curve for rotation speed of 2000 rpm. Also note while the mean velocity of particles increases with the rota-
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tion speed, the ratio of the particle velocity to disk tip velocity shows a peak at 1000 rpm. The results indicate that the increase of the flow velocity with the rotation speed is not linear. As the energy dissipation within the mill
Fig. 9. Snapshots of (a) particle flow and velocity distribution; and (b) velocity vector and force field for different rotation speeds: 500 rpm, 1000 rpm and 2000 rpm (from left to right). The colour represents the magnitude of particle velocities (top) or compressive forces acting on particles (bottom). (For interpretation of colour in this figure legend, the reader is referred to the web version of this article.)
Velocity, (m/s) .
1000rpm
0.9
0.36
4
500rpm 1.2
0.6
2000rpm
0.3
3 0.35 2 0.34 1
Mean velocity
Velocity ratio,/vtip
Probability distribution, P(v)
1.5
Velocity ratio
0
0.0 0
1
2 3 4 5 Velocity, v (m/s)
(a)
6
0
7
Mean compress force, (N)
Probability distribution, P(f)
500 rpm 1000 rpm 2000 rpm
0.1
0.01
0.001 0
(b)
1
2
3
0.33 1000 1500 2000 2500
Rotation speed, Ω (rpm)
10
1
500
4
Compress force, f (N)
5
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
500
1000 1500 2000 2500
Rotation speed, Ω (rpm)
Fig. 10. The effect of rotation speed on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocity for different X; and (b) the probability density distribution (left) and mean value (right) of the compressive force on particles for different X.
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The variation of power draw of the mill with time for different rotation speeds can be seen from Fig. 11a. It shows that the flow takes a slightly longer time (t = 0.2 s) to produce a stable power draw for X = 500 rpm, comparing to t = 0.12 s for X = 2000 rpm. It is noted that the fluctuation of power draw also increases with the rotational speed. Fig. 11b shows that the mean power draw increases as a power function with rotational speed. Fig. 12 shows that increasing the solid loading from 40% to 80% increases the flow velocity and compressive forces on particles. Qualitatively this result makes sense, because reducing the loading reduces the interaction of the media with stirring discs. The quantitative results can be seen from Fig. 13, which suggests that the mill needs to be operated with the maximum allowable load of media to be
increases with the flow velocity, the flow velocity may not increase anymore when the energy dissipation overtakes the input energy. Consequently, there exists an optimum rotation speed for the maximum velocity ratio. This optimum rotation speed may vary with flow and operational conditions, which should be studied further in the future. Fig. 10b shows that the distribution of compressive force on particles has an exponential decay while a large rotation speed corresponds to a slow decay. The mean compressive force is proportional to the square of rotational speed. Since the compressive force is associated with the stress intensity of grinding media, this observation is actually comparable to Kwade’s stress intensity analysis of the stirred mills (Kwade, 1996, 1999).
1000
1000
Power, (W)
Power, (W)
500rpm 1000rpm 2000rpm 100
10
100
1 0
(a)
0.3
0.6
0.9
Time, t (s)
1.2
10
1 100
1.5
(b)
1000 10000 Rotation speed, Ω (rpm)
Fig. 11. The effect of rotation speed on power draw of mill: (a) power draw as a function of time and (b) mean power draw as a function of X.
Fig. 12. Snapshots of (a) particle flow and velocity distribution; and (b) velocity vector and force field for different solid loading: J = 40%, 60%, and 80% (from left to right). The colour represents the magnitude of particle velocities (top) or compressive forces acting on particles (bottom). (For interpretation of colour in this figure legend, the reader is referred to the web version of this article.)
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Probability distribution, P(v)
3.5
2.5
40%
3.0 Velocity, v (m/s) .
2.0
2.5
60%
2.0 80%
1.5 1.0 0.5
0
(a)
1 2 3 Velocity, v (m/s)
1.5 1.0 0.5 0.0 0.2
0.0 4
Mean compress force, (N)
10 Probability distribution, P(f)
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40% 60%
1
80% 0.1
0.01
0.001 0
1
2
3
0.4 0.6 0.8 Solid loading, J
1
0.1
0.01
0.001 0.2
0.4
Compress force, f (N)
(b)
1
0.6
0.8
1
Solid loading, J
Fig. 13. Effect of solid loading on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocities for different J; and (b) the probability density distribution (left) and mean value (right) of the compressive forces on particles for different J.
16
100
12 Power, (W)
Power, (W)
40% 60% 80%
10
(a)
0.3
0.6
0.9
1.2
1.5
Time, t (s)
4 0 0.2
1 0
8
(b)
0.4
0.6
0.8
1
Solid loading, J
Fig. 14. The effect of solid loading on power draw of mill: (a) power draw as a function of time; and (b) mean power draw as a function J.
energy efficient. Operating mill with less loading not only decreases the capacity but also reduces the efficiency of grinding. Fig. 14 shows that doubling the solid loading has a six-fold increase in the power draw which makes the power draw per unit weight three times larger with 80% loading than 40%. 4. Conclusions A DEM model has been developed and verified to simulate the flow of grinding media in a simplified IsaMill. The flow properties are characterised in terms of the flow pattern, velocity and force fields, and power draw. The effects
of materials properties and operational condition of the mill on the flow properties are investigated. The results show while the damping coefficient of particles has a negligible effect for the range considered, other parameters, such as the sliding friction coefficient, rotation speed and solid loading of the mill, have significant effects on the flow properties. Increasing the sliding friction can decrease the flow velocity or compressive force to a minimum before increasing either of them due to the competitive mechanisms for energy transfer and dissipation, but increases the power draw. The increase in the rotation speed and solid loading also increases the flow velocity, compressive force and power draw of mill. The results would be useful
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to understanding the flow of grinding media and the performance of IsaMill operation. Acknowledgements The authors would like to thank Australia Research Council and Xstrata Technology for financial support of this work. The permission granted by Xstrata Technology to publish this paper is gratefully acknowledged. References Brilliantov, N.V., Spahn, F., Hertzsch, J.M., Po¨schel, T., 1996. Model for collisions in granular gases. Phys. Rev. E 53, 5382–5392. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47–65. Curry, D.C., Clermont, B., 2005. Improving the efficiency of fine grinding – development in ceramic media technology. In: Randol Innovative Metallurgy Conference, Perth, WA. Gao, M.W., Forssberg, E., 1995. Prediction of product size distributions for a stirred ball mill. Powder Technol. 84, 101–106. Joe, E.G., 1979. Energy consumption in Canadian mills. CIM Bullet. 72, 147. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge. Kwade, A., 1996. Motion and stress intensity of grinding beads in a stirred media mill. Part 2: Stress intensity and its effects on comminution. Powder Technol. 86, 69–76.
Kwade, A., 1999. Determination of the most important grinding mechanism in stirred media mills by calculating stress intensity and stress number. Powder Technol. 105, 382–388. Langston, P.A., Tu¨zu¨n, U., Heyes, D.M., 1994. Continuous potential discrete particle simulation of stress and velocity fields in hoppers: transition from fluid to granular flow. Chem. Eng. Sci. 49, 1259. Liu, C.-H., Nagel, S.R., Schecter, D.A., Coppersmith, S.N., Majumdar, S., Narayan, S., Witten, T.A., 1995. Force fluctuations in bead packs. Science 269, 513–515. Mindlin, R.D., Deresiewicz, H., 1953. Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344. Mueth, D.M., Jaeger, H.M., Nagel, S.R., 1998. Force distribution in a granular medium. Phys. Rev. E 57, 3164–3169. Tabor, D., 1955. Mechanism of the rolling friction. Proc. Roy. Soc. A 229, 198. Wills, B.A., 1992. Mineral Processing Technology – An Introduction to the Practical Aspects of Ore Treatment and Mineral Recovery. Pergamon Press, New York. Yang, R.Y., Zou, R.P., Yu, A.B., 2000. Computer simulation of packing of fine particles. Phys. Rev. E 62, 3900–3908. Yang, R.Y., Zou, R.P., Yu, A.B., 2003a. Effect of material properties on the packing of fine particles. J. Appl. Phys. 94, 3025–3034. Yang, R.Y., Zou, R.P., Yu, A.B., 2003b. Microdynamic analysis of particle flow in a horizontal rotating drum. Powder Technol. 130, 138– 146. Yu, A.B., 2004. Discrete element method – an effective way for particle scale research of particulate matter. Eng. Comp. 21, 205–214. Zhou, Y.C., Wright, B.D., Yang, R.Y., Xu, B.H., Yu, A.B., 1999. Rolling friction in the dynamic simulation of sandpile formation. Physica A 269, 536–553.
DEM simulation of the flow of grinding media in IsaMill R.Y. Yang a, C.T. Jayasundara a, A.B. Yu a
a,*
, D. Curry
b
Centre for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia b Xstrata Technology, Brisbane, Qld 4000, Australia Received 22 December 2005; accepted 8 May 2006 Available online 7 July 2006
Abstract IsaMill is a high speed stirred mill for high efficiency grinding of mineral ores and concentrates. A numerical model based on the discrete element method (DEM) was developed to study flow of grinding media in IsaMill. The DEM model was first verified by comparing the simulated results of the flow patter, mixing pattern and power draw from those measured from a 1:1 scale lab mill. Then the flow properties were analysed in terms of flow pattern, flow velocity, force field and power draw. The effects of parameters relating to particle material (i.e., sliding friction coefficient and damping coefficient) and operational conditions (i.e., rotation speed and solid loading) were investigated. While the damping coefficient showed a negligible effect for the range considered, other three parameters had strong effects on the flow properties. Increasing the sliding friction caused the flow velocity and compressive force to have minimum values due to the competitive mechanisms for energy transfer and dissipation, but increased the power draw. The increase in the rotation speed and solid loading also increased the flow velocity, compressive force and power draw of mill. The particle scale information obtained would be useful to understand the fundamentals governing the flow of grinding media in IsaMill. 2006 Elsevier Ltd. All rights reserved. Keywords: Grinding; Discrete element modelling; Process simulation; Mineral processing
1. Introduction Grinding is the single largest energy consumption process in the mineral industry. Traditional grinding in tumbling mills (e.g., ball mills) is a low efficiency (1–2%, typically) process and can account for up to 40% of the direct operating cost of a mineral processing plant (Joe, 1979; Wills, 1992). Furthermore, many of the base and precious metal deposits discovered are now fine-grained and complex, and there is a need for grinding them to very fine sizes to permit sufficient mineral liberation, for example, 80% of particles (P80) should be below 20 or even 10 lm. Grinding to such fine sizes economically is beyond the capability of conventional grinding technologies. IsaMill is a high speed stirred mill developed by Mount Isa Mines (Xstrata) in Australia for economically grinding *
Corresponding author. Tel.: +61 2 93854429; fax: +61 2 93855956. E-mail address: [email protected] (A.B. Yu).
0892-6875/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2006.05.002
minerals to fine and ultra-fine size at an industrial-scale (Gao and Forssberg, 1995). It consists of a horizontally mounted shell and rotating grinding discs mounted on a shaft which is coupled to a motor and gearbox and rotates with disc tip speeds of 10–23 m s1. The grinding discs agitate the media and ore particles in a slurry that is continuously fed into a feed port. The product separator (dynamic, centripetal classifier) keeps the grinding media inside the mill allowing only the product to exit. Simple control strategies based on power draw enable the IsaMill to produce a constant target product size. Comparing with the conventional grinding mills such as ball mill and tower mill, IsaMills can significantly reduce total comminution circuit energy cost (Curry and Clermont, 2005) and reduce the size of mineral particles to as fine as P80 passing 7 lm (Gao and Forssberg, 1995). However the IsaMill is still a new technology and most relations established between the quality of the final products and operating parameters of the mill are empirical, and are inadequate to comprehensively
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understand the grinding process. As a result, its optimum control and scale-up need to rely on empirical methods, experience and trial and error testing, rather than detailed scientific principles. There is a need for research into the grinding process to be predicted and modelled based on knowledge of the characteristics of the mill and the properties of the grinding materials. The bulk behaviour of media particle flow depends on the collected outcome of the interactions between particles and between particle and mill. As a result, a better understanding of the flow at an individual particle scale would greatly facilitate the design and scale-up of IsaMill. However, it is a very difficult, if not impossible, task using experimental techniques to obtain microdynamic information such as voidage, force and velocity distributions within the mill. On the other hand, simulation based on Discrete Element Method (DEM) (Cundall and Strack, 1979) has been extensively used in the study of particle packing and flow for various systems and has been demonstrated as an effective way to link microscopic information with macroscopic behaviour of particle flow (for example, see Yu, 2004). However, despite the usefulness, DEM simulation studies have not been applied widely to the high speed stirred mill system like IsaMill. This paper presents a numerical investigation of the flow of grinding media in a simplified IsaMill, aiming to examine the feasibility of using DEM simulation to predict particle flow in a high speed stirred mill. The proposed DEM model is first verified with a 1:1 scale experimental setup which allows qualitative or quantitative comparison of flow and mixing patterns and power consumption. Then, the flow of particles will be analysed in terms of velocity, force and power draw. The effect of some variables related to key materials properties and operational conditions are also investigated. 2. Simulation method and conditions In a DEM simulation, the translational and rotational motions of a particle of radius Ri (=R for mono-sized spheres) and mass mi can be described by dvi X n ¼ mi ðFij þ Fsij þ mi gÞ ð1Þ dt j and Ii
dxi X ^ iÞ ¼ ðRi Fsij lr Ri jFnij jx dt j
ð2Þ
where vi, xi and Ii are, respectively, the transitional and angular velocities, and moment of inertia of the particle. Ri is a vector running from the centre of the particle to the contact point with its magnitude equal to particle radius Ri. The first part of the right side in Eq. (2) is the torque due to the tangential force Fsij , the second part is the rolling friction torque arising from the elastic hysteresis loss or viscous dissipation, and lr is the coefficient of roll-
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ing friction (Tabor, 1955; Brilliantov et al., 1996). This frictional resistance has been demonstrated to play a critical role in achieving physically or numerically stable sandpile, viz. the unconfined packing of particles although it is still unclear which model can best describe rolling friction (Zhou et al., 1999). Fnij and Fsij represent, respectively, the normal contact force and the tangential contact force imposed on particle i by particle j, given by (Mindlin and Deresiewicz, 1953; Johnson, 1985; Langston et al., 1994; Brilliantov et al., 1996) pffiffiffi pffiffiffipffiffiffiffiffi 3 2 Fnij ¼ E Rn2n cn E R nn ðvij ^nij Þ ^nij ð3Þ 3 Fsij ¼ sgnðns Þls jFnij j½1 ð1 minðns ; ns;max Þ=ns;max Þ
ð4Þ
~2 Þ, R ¼ Ri Rj =ðRi þ Rj Þ, ns;max ¼ ls ½ð2 where E ¼ Y =ð1 r ~Þ=2ð1 r ~Þnn , Y = Young’s modulus, r ~ ¼ Poisson ratio, r ^nij ¼ ðRi Rj Þ=jRi Rj j, cn = normal damping coefficient, ls = sliding friction coefficient, ns = total tangential displacement. Details about the force models used in this work can be found from elsewhere (Yang et al., 2000, 2003a,b). Simulations were performed in a stirred mill (Fig. 1) consisting of a fixed chamber of inner diameter of 110 mm and several discs mounted on a rotating shaft of diameter of 25 mm. The disc is 90 mm in diameter and 9 mm in thickness, and is about 30 mm apart from each other. Each disc has five holes of 18 mm diameter. The mill was filled with 4 mm mono-sized glass beads. In practice, grinding media mixed with fine slurry are continuously fed into the mill from one end and the fines exit from the other end. However, as a first step to develop a comprehensive DEM model for IsaMill, no fine slurry was included in the present simulation, and the grinding media were sealed in the mill and therefore no flow in and out were considered. Two types of simulations with different boundary conditions were used in this work. In the first type of simulation (Fig. 1a), the mill has a chamber with two side walls and three discs inside. This would be mainly for model validation by comparing with a 1:1 scale experiment. In the second type of simulations (Fig. 1b), only one disc was considered in the mill but the periodic boundary conditions were applied along the axial direction. This set up corresponds to the region B in Fig. 1a, and allows large scale simulations with a small number of particles. Table 1 lists the values of the parameters used in the simulation. Parameters with a varying range were changed to study their effects on the particle flow. Unless otherwise specified, the effect of each variable was examined when others were fixed at their respective base values. Note that the Young’s modulus used in the simulations is four orders of magnitude smaller than the real glass bead (100 GPa) to reduce the simulation time. The current value ensures that the maximum overlap is less than 1% of particle diameter and has been demonstrated not affect the final results much (Zhou et al., 1999; Yang et al., 2000, 2003a,b). The
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(a)
(b)
Fig. 1. Schematic illustration of simplified IsaMill.
Table 1 Physical parameters in the simulation Parameter
Base value 3
Particle density, q (kg m ) Number of particles Young’s modulus, Y (N m2) ~ Poisson ratio, r Sliding friction coefficient, ls Rolling friction coefficient, lr Normal damping coefficient, cn (s1) Solid loading, J Rotating speed, X (rpm)
3
2.5 · 10 3000 2.0 · 107 0.29 0.3 0.005 5 · 106 80% 1000
Varying range – 1500–3000 1 · 107–2 · 108 – 0.01–1.0 0.001–0.01 5 · 106–2 · 105 40–80% 500–2000
mill was assumed to have the same materials properties as particles. 3. Results and discussion Fig. 2 shows the snapshots of particle flow at different times in the single disc mill. Each simulation begins with a packed bed (Fig. 2a), then the disc rotates at a given speed and the particles are agitated and driven to move outwards (Fig. 2b), finally the flow reaches a macroscopically stable state after a certain time (Fig. 2c and d). Analyses of the flow velocity and power draw show that the time for a flow to reach such a stable state depends on the particle properties and the rotational speed, but generally is
less than 0.5 s. Unless otherwise stated, our following analysis is based on the cumulative data obtained at macroscopically stable state. Starting from t = 1 s when the flow reached the stable state, the samples were collected every 0.05 s till the simulation finished (10 sets of samples totally). Then these samples were averaged to calculate the flow velocity and compressive force. In the following sections, the DEM model will first be verified by comparing the results obtained from simulations and experiments. Then, based on the simulated results, the particle flow will be analysed in terms of the flow pattern, velocity and force fields, and power draw. The effects of particle properties such as sliding friction coefficient and damping coefficient and operational conditions such as mill filling level and rotating speed on the flow properties will be discussed. Note that the effects of other particle properties such as Young’s modulus and rolling friction coefficients were also examined and found to be negligible for the ranges considered in this work. Therefore their results are not shown for brevity. 3.1. Macroscopic observation and model validation To verify the proposed DEM model, simulations corresponding to physical experiments as shown in Fig. 1a were performed. The simulations and experiments were carried out with the mill about 60% loaded and rotating at various
Fig. 2. Snapshots at different times in the simulations with the variables at their base values: (a) t = 0.0 s, (b) t = 0.01 s, (c) t = 0.5 s and (d) t = 1.0 s.
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speeds. Qualitative and quantitative comparisons in terms of flow and mix patterns and power draw are shown in Fig. 3. The flow patterns at X = 300 rpm (Fig. 3a) indicate that most particles stay at the bottom of the mill and behave like bulk solid with little movement and only a relatively small number of particles are in the upper half of the mill. Vigorous flow of particles can be obtained when the rotation speed is high. Low rotation speeds were used here to produce visualisation of high quality. A more detailed comparison was performed in the radial mixing of two coloured particles. The particles were initially separated into two layers in vertical direction. As discs rotate at 100 rpm, two coloured particles gradually mix together. Fig. 3b shows the mix pattern at 10 s, indicat-
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ing that the particles near the discs move much faster due to the lift of discs. Therefore the mixing process is much faster in the central part of the mill and more dark particles are found mixed with light particles in this region. Power consumption, or power draw, is one of the most important indexes to characterise the mill performance. The total electrical energy input Ein is by definition the product of voltage and current across the DC motor which were measured in the experiment. The input energy cannot fully be transferred to particles, because of the energy losses in running the motor Em1 and the mill Em2. That is, Ein Em1 = Em2 + Ep, where Ep is the energy transferred to the particles in the mill. Ep can be calculated from the DEM simulation. This energy, as the power draw by
Fig. 3. Comparisons of the physical and numerical experiments: (a) flow pattern with X = 300 rpm and J = 60%; (b) mixing pattern at t = 10 s with X = 100 rpm and J = 60%; and (c) power draw from experiment (point) and simulation (line) as a function of rotational speed with J = 60%.
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particles, is the product of rotation speed and total torque acting on the discs and shaft. We have estimated the energy loss Em1 by calibrating the mill at zero loading at a given rotation speed. Therefore, the energy loss Em2 is the only unknown energy. Fig. 3c shows the power draw as a function of rotation speed from the simulation (Ep) and experiment (Ein). Clearly, the power draw increases with the speed of rotation. The overall comparison is quite good except for the slightly larger discrepancy at high speeds (>1000 rpm). The difference can be attributed to the energy loss in the form of heat, sound and others which have not been considered. While the present experimental technique is not perfect, it does provide a reasonable result for model verification. The good agreement between the simulated and measured results in Fig. 3, either qualitative or quantitative, confirms the validity of the present DEM model, at least at a macroscopic scale. 3.2. Effects of material properties Sliding friction coefficient and damping coefficient are two important variables relating to material properties. Sliding friction coefficient relates to the particle roughness, and damping coefficient is due to the inelasticity of particles and relates to the particle restitution coefficient. In practice, different materials, such as river sands and ceramic beads, have been chosen as grinding media. These materials have different sliding frictions and damping coefficients which may affect the grinding performance. Therefore, the study of their effects is relevant to the industry. Fig. 4 shows the flow patterns and the force structure for different sliding friction coefficients. The force structure is characterized in terms of the normal compressive force
P n acting on individual particles, defined as fi ¼ jFij j. As the tangential force (related to abrasive forces) is proportional to the normal force at a contact point, their variations are in the same trend and not shown in this paper. Fig. 4a shows that for all cases, the peak velocities of the flows were near the holes, indicating the boundaries of the holes act as lifters to drive particles. Increasing the sliding friction coefficient causes the dilation of the particle flow. It is interesting to note that the flow has the lowest mean velocity at ls = 0.1. For ls smaller or larger than 0.1, both mean velocity and velocity gradient along the radial direction increase. This is because the sliding friction plays two roles in the flow, energy transfer and energy dissipation. The two mechanisms are competitive against each other. Before the critical value, the dissipation mechanism is dominant so the velocity (kinetic energy) of flow decreases with increasing ls. After the critical value, more energy is transferred with increasing ls, and the flow becomes faster. The same trend can also be observed from Fig. 4b, which shows the spatial distribution of the compressive force. The compressive forces first decrease as ls increases from 0.01 to 0.1 and then increase with further increase in ls. The observations from Fig. 4 are further confirmed from the quantitative analyses of the particle velocities and the compressive forces, as shown in Fig. 5. For ls = 0.01, the particle velocity has a mean value of 2.8 m/s and a distribution with a peak value at 3 m/s. When ls increases to 0.1, the mean velocity decreases to 1.65 m/s and the distribution becomes narrower and shifts to the smaller value with a peak value of 1.2 m/s. Further increase in the sliding friction coefficient causes the distribution to become broader and shift to the right side with the peaks of 1.9 m/s for ls = 0.3 and 2.7 m/s for ls = 1.0. This indicates that, at
Fig. 4. Snapshots of (a) particle flow and velocity distribution; and (b) velocity vector and force field for different sliding friction coefficient: (from left to right) 0.01, 0.1, 0.3 and 1.0. The colour represents the magnitude of particle velocities (top) or the compressive forces acting on particles (bottom). (For interpretation of colour in this figure legend, the reader is referred to the web version of this article.)
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2.5 2.0
μs =0.01 μ s =0.3
1.5
3.0
.
μs =0.1
3.0
Mean velocity,
Probability distribution, P(v)
3.5
μ s =1.0
1.0 0.5
2.6 2.2 1.8 1.4
0.0
1.0
0
1
(a)
2 3 Velocity, v (m/s)
4
5
0
Mean compress force,
10 Probability distribution, P(f)
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0.001 0.1 0.3 1
1
0.1
0.01
0.001 0
(b)
1 2 3 4 Compress force, f (N)
0.2 0.4 0.6 0.8 Sliding friction coefficient, μ s
1
0.8
0.6
0.4
0.2
0 0
5
0.2
0.4
0.6
0.8
1
Sliding friction coefficient, μs
Fig. 5. The effect of sliding friction coefficient on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocities for different ls; and (b) the probability density distribution (left) and mean value (right) of the compressive forces on particles for different ls.
ls = 0.1, the particles move with the slowest but quite uniform velocities. Fig. 5b shows that the distribution of the compressive forces acting on particles decays exponentially, although forces for smaller ls fall off more quickly. Note that this phenomenon was observed in other static systems such as packing of particles (Liu et al., 1995; Mueth et al., 1998). Our results indicate the exponential tail at the large force seems very robust and can be found in the dynamic systems. The mean compressive force also shows a minimal value at ls = 0.1. Fig. 6 shows the effect of the sliding friction coefficient on the power draw of the mill. The flow takes less time
to reach the stable state for particles of larger sliding friction coefficient. When ls = 1.0, the power draw decreases rapidly as the mill begins to rotate and reaches a stable value of 45 W after less than 0.1 s. For a smaller sliding friction coefficient of 0.1, the power draw reaches a stable value only after 0.3 s. However, the power draw for ls = 0.1 at t = 1.0 s is about 5 W. Decreasing the sliding friction coefficients to 0.01 does not change the power draw significantly. Fig. 6b indicates that the mean power draw of the mill increases with ls. Three damping coefficients were used in the present work, i.e., cn = 5 · 106 s1, 2 · 105 s1 or 5 · 105 s1.
100
60 0.001 0.1 0.3 1
50 Power, (W)
Power, (W)
80 60 40 20
30 20 10 0
0 0 (a)
40
0.3
0.6
0.9
Time, t (s)
1.2
0
1.5 (b)
0.2 0.4 0.6 0.8
1
1.2
Sliding friction coefficient, μs
Fig. 6. The effect of sliding friction coefficient on power draw: (a) variation of power draw with time for different ls; and (b) mean power draw as a function of ls.
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The corresponding restitution coefficients of particles at the impact velocity of 1 m/s are 0.87, 0.60 and 0.28. Our results show that there are no evident differences in terms of flow pattern (not shown here). Quantitative analysis of the particle velocity and the compressive force for different damping coefficients are plotted in Fig. 7, showing the distributions and the mean values of the velocities are almost the same although particles with a small damping coefficient (large restitution coefficient) do have a slightly broader velocity distribution. The distribution of compressive forces acting on particles also shows an exponential decrease, and the mean force drops slightly from 0.313 when cn = 5 · 106 s1 to 0.30 when cn = 5 · 105 s1. The increase in damping coefficient also slightly reduces
Probability distribution, P(v)
1.2
γ n=5e-5
the power draw of mill, as seen from Fig. 8. However, compared to the effects of other variable considered in this work, the effect of cn is negligible. 3.3. Effects of operational conditions Rotation speed and solid loading of the mill are two operational variables considered in this work. Fig. 9a shows the particles with different rotation speeds, indicating that as the rotation speed increases, particles are driven outwards and form a faster flow with an increased velocity gradient along the radial direction. This suggests that a high rotation speed can produce a dense particle packing. The compressive force on particles (Fig. 9b) also increases
2
γ n=2e-5
1.0
Velocity, v (m/s)
γ n=5e-6
0.8 0.6 0.4 0.2
1.8
1.7
1.6
0.0 0
1
2
3
4
0
5
Mean compress force,
10 5e-6 2e-5
1
5e-5 0.1
0.01
0.001 0
0.5
1
1.5
2
2
3
4
5
6
2.5
0.32
0.31
0.30
0.29
0.28 0
3
1
2
3
4
5
Damping coefficient, γ n (x10-5 s -1)
Compress force, f (N)
(b)
1
Damping coefficient, γ n (x10-5 s -1)
Velocity, v (m/s)
(a)
Probability distribution, P(v)
1.9
Fig. 7. The effect of damping coefficient on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocities for different cn; (b) the probability density distribution (left) and mean value (right) of the compressive forces on particles for different cn.
20
5e-6 2e-5 5e-5
16
Power, (W)
Power, (W)
100
10
12 8 4 0
1 0
(a)
0.3
0.6
0.9
Time, t (s)
1.2
0
1.5
(b)
2 4 6 Damping coefficient, γ n (x10-5 s -1)
Fig. 8. The effect of damping coefficient on power draw of mill: (a) power draw as a function of time; and (b) mean power draw as a function of cn.
R.Y. Yang et al. / Minerals Engineering 19 (2006) 984–994
with increasing speed. In fact, as shown in Fig. 10a, the distribution of particle velocities is broader and shows a bi-modal curve for rotation speed of 2000 rpm. Also note while the mean velocity of particles increases with the rota-
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tion speed, the ratio of the particle velocity to disk tip velocity shows a peak at 1000 rpm. The results indicate that the increase of the flow velocity with the rotation speed is not linear. As the energy dissipation within the mill
Fig. 9. Snapshots of (a) particle flow and velocity distribution; and (b) velocity vector and force field for different rotation speeds: 500 rpm, 1000 rpm and 2000 rpm (from left to right). The colour represents the magnitude of particle velocities (top) or compressive forces acting on particles (bottom). (For interpretation of colour in this figure legend, the reader is referred to the web version of this article.)
Velocity,
1000rpm
0.9
0.36
4
500rpm 1.2
0.6
2000rpm
0.3
3 0.35 2 0.34 1
Mean velocity
Velocity ratio,
Probability distribution, P(v)
1.5
Velocity ratio
0
0.0 0
1
2 3 4 5 Velocity, v (m/s)
(a)
6
0
7
Mean compress force,
Probability distribution, P(f)
500 rpm 1000 rpm 2000 rpm
0.1
0.01
0.001 0
(b)
1
2
3
0.33 1000 1500 2000 2500
Rotation speed, Ω (rpm)
10
1
500
4
Compress force, f (N)
5
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
500
1000 1500 2000 2500
Rotation speed, Ω (rpm)
Fig. 10. The effect of rotation speed on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocity for different X; and (b) the probability density distribution (left) and mean value (right) of the compressive force on particles for different X.
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The variation of power draw of the mill with time for different rotation speeds can be seen from Fig. 11a. It shows that the flow takes a slightly longer time (t = 0.2 s) to produce a stable power draw for X = 500 rpm, comparing to t = 0.12 s for X = 2000 rpm. It is noted that the fluctuation of power draw also increases with the rotational speed. Fig. 11b shows that the mean power draw increases as a power function with rotational speed. Fig. 12 shows that increasing the solid loading from 40% to 80% increases the flow velocity and compressive forces on particles. Qualitatively this result makes sense, because reducing the loading reduces the interaction of the media with stirring discs. The quantitative results can be seen from Fig. 13, which suggests that the mill needs to be operated with the maximum allowable load of media to be
increases with the flow velocity, the flow velocity may not increase anymore when the energy dissipation overtakes the input energy. Consequently, there exists an optimum rotation speed for the maximum velocity ratio. This optimum rotation speed may vary with flow and operational conditions, which should be studied further in the future. Fig. 10b shows that the distribution of compressive force on particles has an exponential decay while a large rotation speed corresponds to a slow decay. The mean compressive force is proportional to the square of rotational speed. Since the compressive force is associated with the stress intensity of grinding media, this observation is actually comparable to Kwade’s stress intensity analysis of the stirred mills (Kwade, 1996, 1999).
1000
1000
Power, (W)
Power, (W)
500rpm 1000rpm 2000rpm 100
10
100
1 0
(a)
0.3
0.6
0.9
Time, t (s)
1.2
10
1 100
1.5
(b)
1000 10000 Rotation speed, Ω (rpm)
Fig. 11. The effect of rotation speed on power draw of mill: (a) power draw as a function of time and (b) mean power draw as a function of X.
Fig. 12. Snapshots of (a) particle flow and velocity distribution; and (b) velocity vector and force field for different solid loading: J = 40%, 60%, and 80% (from left to right). The colour represents the magnitude of particle velocities (top) or compressive forces acting on particles (bottom). (For interpretation of colour in this figure legend, the reader is referred to the web version of this article.)
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Probability distribution, P(v)
3.5
2.5
40%
3.0 Velocity, v (m/s) .
2.0
2.5
60%
2.0 80%
1.5 1.0 0.5
0
(a)
1 2 3 Velocity, v (m/s)
1.5 1.0 0.5 0.0 0.2
0.0 4
Mean compress force,
10 Probability distribution, P(f)
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40% 60%
1
80% 0.1
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0.001 0
1
2
3
0.4 0.6 0.8 Solid loading, J
1
0.1
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0.001 0.2
0.4
Compress force, f (N)
(b)
1
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0.8
1
Solid loading, J
Fig. 13. Effect of solid loading on the velocity and force of particles: (a) the probability density distribution (left) and mean value (right) of particle velocities for different J; and (b) the probability density distribution (left) and mean value (right) of the compressive forces on particles for different J.
16
100
12 Power, (W)
Power, (W)
40% 60% 80%
10
(a)
0.3
0.6
0.9
1.2
1.5
Time, t (s)
4 0 0.2
1 0
8
(b)
0.4
0.6
0.8
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Solid loading, J
Fig. 14. The effect of solid loading on power draw of mill: (a) power draw as a function of time; and (b) mean power draw as a function J.
energy efficient. Operating mill with less loading not only decreases the capacity but also reduces the efficiency of grinding. Fig. 14 shows that doubling the solid loading has a six-fold increase in the power draw which makes the power draw per unit weight three times larger with 80% loading than 40%. 4. Conclusions A DEM model has been developed and verified to simulate the flow of grinding media in a simplified IsaMill. The flow properties are characterised in terms of the flow pattern, velocity and force fields, and power draw. The effects
of materials properties and operational condition of the mill on the flow properties are investigated. The results show while the damping coefficient of particles has a negligible effect for the range considered, other parameters, such as the sliding friction coefficient, rotation speed and solid loading of the mill, have significant effects on the flow properties. Increasing the sliding friction can decrease the flow velocity or compressive force to a minimum before increasing either of them due to the competitive mechanisms for energy transfer and dissipation, but increases the power draw. The increase in the rotation speed and solid loading also increases the flow velocity, compressive force and power draw of mill. The results would be useful
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to understanding the flow of grinding media and the performance of IsaMill operation. Acknowledgements The authors would like to thank Australia Research Council and Xstrata Technology for financial support of this work. The permission granted by Xstrata Technology to publish this paper is gratefully acknowledged. References Brilliantov, N.V., Spahn, F., Hertzsch, J.M., Po¨schel, T., 1996. Model for collisions in granular gases. Phys. Rev. E 53, 5382–5392. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47–65. Curry, D.C., Clermont, B., 2005. Improving the efficiency of fine grinding – development in ceramic media technology. In: Randol Innovative Metallurgy Conference, Perth, WA. Gao, M.W., Forssberg, E., 1995. Prediction of product size distributions for a stirred ball mill. Powder Technol. 84, 101–106. Joe, E.G., 1979. Energy consumption in Canadian mills. CIM Bullet. 72, 147. Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, Cambridge. Kwade, A., 1996. Motion and stress intensity of grinding beads in a stirred media mill. Part 2: Stress intensity and its effects on comminution. Powder Technol. 86, 69–76.
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