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Estimating energy in grinding using DEM modelling
Minerals Engineering 85 (2016) 23–33
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Estimating energy in grinding using DEM modelling N.S. Weerasekara ⇑, L.X. Liu, M.S. Powell University of Queensland, JKMRC, 40 Isles Road, Indooroopilly, QLD 4068, Australia
a r t i c l e
i n f o
Article history: Received 28 May 2015 Revised 19 October 2015 Accepted 19 October 2015 Available online 27 October 2015 Keywords: DEM Energy Comminution Tumbling mills Breakage
a b s t r a c t The latest state of the art on Discrete Element Method (DEM) and the increased computational power are capable of incorporating and resolving complex physics in comminution devices such as tumbling mills. A full 3D simulation providing a comprehensive prediction of bulk particle dynamics in a grinding mill is now possible using the latest DEM software tools. This paper explores the breakage environment in mills using DEM techniques, and how these techniques may be expanded to provide more useful data for mill and comminution device modelling. A campaign of DEM simulations were performed by varying the mill size and charge particle size distribution to explore and understand the breakage environment in mills using DEM techniques. Analysis of each mill was conducted through consideration of the total energy dissipation and the nature of the collision environment that leads to comminution. The DEM simulations show that the mill charge particle size distribution has a strong influence on the mill input power and on the way the energy is distributed across the charge. The smaller particles experience higher energies while the larger experience less, but this variation is strongly dependent on the mill size. The results also showed that the average particle collision energy increases with increasing mill size, whereas its distribution over particle size is strongly influenced by the mill content particle size distribution. The simulations also captured the energy distribution within different regions of the tumbling charge, with the toe impact region having higher impact energies and the bulk shear region having higher tangential energies. Regardless of the mill size most of the energy is consumed by the particles in the mid-size range, which has the highest percentage mass of the total charge distribution. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The motion of grinding media and the energy distribution have a profound influence on the comminution of particles in tumbling mills. It allows the calculation of the trajectories of individual entities in the entire grinding charge as they move in the mill and collide with one another. The discrete element method (DEM) also allows numerical simulation of the dynamic interaction of the mill charge media with the mill liners and lifters (Powell et al., 2011). The calculations are based on the fundamental laws of motion and can take into account of the geometry, dimensions and material property of each individual steel ball, chunk of rock, mill liner, and lifters (Mishra, 2003a). DEM codes take advantage of parallel processing capabilities to scale up the number of particles or length of the simulation. The use of DEM in modelling grinding mills has been successful over the last couple decades following the introduction of this technique by Cundall and Strack in 1979. The earliest DEM model ⇑ Corresponding author. E-mail address: [email protected] (N.S. Weerasekara). http://dx.doi.org/10.1016/j.mineng.2015.10.013 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.
of a mill was in 2D by Mishra and Rajamani (1992, 1994). This was then extended to multiple scales ranging from pilot to industrial by a number of authors like Cleary (1998, 2001a,b), Herbst and Nordell (2001), and in more recent years by Djordjevic et al. (2006), Cleary (2004), Cleary et al. (2006), Djordjevic et al. (2006), Kalala et al. (2005a,b), McBride and Powell (2006), Morrison et al. (2006), Weerasekara and Powell (2008), Weerasekara et al. (2010), Powell et al. (2011), to name a few. A review of the current status of DEM usage in comminution is described by Weerasekara et al. (2013). However, discrete element methods are relatively computationally intensive, which limits either the length of a simulation or the number of particles (Weerasekara et al., 2013). It is also a computationally intensive task to keep track of the breakage of millions of particles and their progeny. More convenient approaches such as mass population balance are required with DEM to model and simulate the particle mechanical environments, log impact energies and model breakage (Austin and Luckie, 1972; Barrios et al., 2011; Datta and Rajamani, 2002; Lichter et al., 2009; Tavares and de Carvalho, 2009; Weerasekara and Powell, 2008; Weerasekara and Powell, 2010; Whiten, 1974).
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Nomenclature E F G m R
v
d
e l
effective Young’s modulus force effective shear modulus effective mass effective radius velocity overlap coefficient of restitution coefficient of static friction
Although incorporating breakage into DEM model increases the computational effort, and could be considered impractical with the available computer hardware, the ability to predict the motion of grinding media and the energy distribution is very important in understanding their influence on the comminution of particles in tumbling mills. The potential of the dynamic interaction of the mill charge media with the mill liners and lifters allows calculation of the trajectories of individual entities in the entire grinding charge as they move in the mill and collide with one another and the mill. These impact energy and force distributions can be used to calculate the grinding rate in mills, as presented by some authors in this area, for example Mishra (2003b), Cleary (2001b), Datta and Rajamani (2002), Herbst (2004), Powell (2006) and Powell and Weerasekara (2010). This work presents the ongoing exploration and understanding of the breakage environment in mills through DEM simulation. In this paper, we develop series of DEM models that allowed simulating charge motion in SAG mills ranging from pilot scale to industrial scale with varied mill load. This work also extends to a series of 3D DEM simulations that were performed on a generic SAG mill for lifter geometries in various stages of wear. The characteristic charge structures for this study are estimates for the head, shoulder, bulk toe and impact toe, together with the analysis of DEM modelled breakage energy environment for multiple scale of mills. This type of analysis and comparison will guide in understanding the breakage energy environment in mill scale-up process. This information can also be used as part of a strategy in mill control and design targeted at a specific size reduction. 2. DEM model
Subscripts n normal components t tangential components Superscripts rel relative
Fig. 1. Particle contact model.
Table 1 Spring stiffness and Damping coefficients used in the contact model.
Spring stiffness constant (K) Damping coefficient (C) Sn ¼ 2E
Normal direction (n) pffiffiffiffiffiffiffiffiffiffi K n ¼ 43 E R dn qffiffi pffiffiffiffiffiffiffiffiffiffiffi C n ¼ 2 56b Sn m
Tangential direction (t) pffiffiffiffiffiffiffiffiffiffi K t ¼ 8G R dn qffiffi pffiffiffiffiffiffiffiffiffiffiffiffi C t ¼ 2 56b K t m
pffiffiffiffiffiffiffiffiffiffi ln e R dn , b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ln
eþp2
2006) in modelling the contact between particles. Hertz’s theory of elastic contacts provides compact relations for the normal direction, deriving from integration of the normal pressure distribution over the contact area (Di Maio and Di Renzo, 2005). The resulting total force (Fn) is then a sum of elastic force and a dissipative (damping) force given by (Cleary, 1998; EDEM, 2006);
F n ¼ K n dn þ C n v rel n
ð1Þ
DEM solves Newton’s equations of motion to resolve particle motion and using a contact law to resolve inter-particle contact forces. Forces are typically integrated explicitly in time to predict the time history response of the material using an appropriate quadrature method. The DEM includes a family of techniques that use radically different treatments for the element geometry and the form of contact forces. A non-linear Hertz–Mindlin no-slip model (Cundall and Strack, 1979; EDEM, 2006; Mindlin, 1949; PFC3D, 1999) was employed to solve for the contact between colliding particles. While the motion of each individual particle is governed by the laws of conservation of linear momentum, angular momentum was resolved by solving Newton’s second law of motion.
This provides a spring repulsive force and a dashpot to dissipate a portion of the relative kinetic energy. In the tangential direction the possible force–displacement configurations depend on both the normal and tangential loading history. The relative tangential velocity (vt) from the relative tangential motions over the collision, behaves as an incremental spring that stores energy and represents the elastic tangential deformation of the contacting surfaces. The dashpot dissipates energy from the tangential motion and models the tangential plastic deformation of the contact. The total tangential force (Ft) is limited by the Coulomb’s law of friction (Cleary, 1998; Di Renzo and Di Maio, 2004; EDEM, 2006).
2.1. Contact model
F t ¼ minflF n ; K t dt þ C t v rel t g
A non-linear model (Fig. 1), combining Hertz’s theory in the normal direction and the improvement to Mindlin’s no- slip model (Mindlin, 1949) in the tangential direction was employed (EDEM,
The coefficients in Eqs. (1) and (2) are given in Table 1. The total energy dissipation associated with particle interactions can be calculated using the above contact law, with components being normal and tangential. This will also provide the frequency
ð2Þ
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distributions of these energy losses, both in the normal and tangential (shear) directions, from all the individual collision events. Spectra can be calculated for each type of collision event and for each ball and rock size class. These collision energy spectra provide the opportunity to better understand the various contributions to the overall energy dissipation within the mill. 2.2. Data analysis technique A DEM simulation for a typical mill section with several thousands of particles generates massive amounts of output data, running to several Gigabytes. Even with good sampling techniques (Cleary, 2001b) this generates files of Gigabytes. Although some DEM software provides a graphical user interface (GUI) based data analysis environment, their techniques either require a considerable amount of processing power or are not capable of delivering the required inputs for the unified comminution model (UCM) kind of modelling (Powell et al., 2008). Therefore a data logging system developed by Weerasekara and Powell (2010) was employed to fulfil the following objective: (1) provide flexibility in handling huge data sets generated by a typical DEM code; (2) extract particle collision information from the DEM output database; (3) summarise data to provide more useful information for further comminution modelling process. A C/C++ based framework was employed in extracting and analysing the massive data sets generated though the DEM simulation (Weerasekara and Powell, 2008). The framework employs a two stage approach, where stage I of this system interacts directly with the DEM output and generates data sets which can be easily managed and handled by the stage II. The method adapted in stage I varies depending on the DEM output data structure of the DEM code used. In stage II, the data extracted and grouped data in the stage I is further summarised according to the requirements for further modelling work. 3. DEM simulations Different mill operation modes were selected for this DEM modelling study to investigate the energy distribution under different loading and operating conditions. They are described below. 3.1. Impact collision dominant milling mode In this study several aspects of a SAG mill were investigated using DEM. A pilot scale SAG mill operating at 75% critical speed and drawing 9.94 kW of power is used as the ‘‘BaseCase” for the
DEM simulation campaign. This BaseCase mill has an internal diameter of 1696 mm, length of 575 mm and has 11 lifters with 50 50 mm cross-section. A base particle size distribution (PSD) shown in Fig. 2 was used for this BaseCase DEM simulation. A combination of simulations was performed by varying the internal diameter from 1.6 m to 6 m. The internal diameter variation from 1.6 m to 6 m was a linear increase with the same linear increase (proportional to diameter increase) for lifter dimensions. The sizes 3 m and 6 m are considered as a direct scale up of 1.6 m pilot mill with no other extra scale up criterion. Other conditions in the mills are the same. Three distinct PSD’s as shown in Fig. 2 was used. This gave a matrix of simulations as presented in Table 2. This table presents the code for each test that is referred to throughout this paper. G1–3 are the three mill sizes and D1, BD, D3 refer to the charge size distributions. In all these simulations the mills were run at 75% critical speed with rock charge filling of 21.5% by volume, and a steel ball filling of 6% to give a total charge 27.5%. The steel balls had a size distribution of 40, 60, 80, and 100 mm. The material parameters in Table 3 were used throughout the simulation. A coefficient of restitution of 0.5 and a friction coefficient of 0.4 have been used for rock–rock and a coefficient of restitution of 0.7 and a friction coefficient of 0.2 for steel–steel collisions, and a coefficient of restitution of 0.2 and a friction coefficient of 0.3 have been used for rock–steel. A coefficient of rolling friction of 0.1 was used for all rolling interactions. The mill shells are assumed to be made of steel. The coefficient of restitution is the highest for steel–steel collisions which are relatively elastic and the lowest for rock–rock collisions which have significant local fracture at contacts leading to low restitution. The friction coefficient is the smallest for ball– ball collisions and the highest for rock-rock ones since the rocks are typically quite rough. 3.2. Sliding/abrasion milling mode Pilot scale mills without balls with operating conditions shown in Table 4 were used for the DEM simulation. These simulations were run to correlate the experimental results reported in the work of Yahyaei et al. (2013). Both the simulations and the experiments were run at a much lower mill speed, which is 40% of the critical speed with experimentally known mill filling. The operating conditions were selected such that particle sliding is the dominant grinding mode during the operation of the mill. Due to the grinding action caused by particles rubbing each other, the ore particle size distribution in the mill changes, which was quantified experimentally by counting the number of ore particles in each size class inside the mill at the start, at a mid-point and at the end of each experiment (Yahyaei et al. (2013). The particle numbers were used in each snap-shot DEM simulation of the mill charge motion, for all the experimental test conditions. A DEM clumped particle was Table 2 Simulation matrix.
Geometry 1 (D1.6 m) Geometry 2 (D3 m) Geometry 3 (D6 m)
Distribution 1
Base distribution
Distribution 3
G1D1 – –
G1BD G2BD G3BD
– G2D3 G3D3
Table 3 Material parameters.
Density (kg/m3) Shear modulus (GPa) Poisson’s ratio Fig. 2. Three different particle size distributions used in the DEM simulations.
Rock media
Steel
2680 1.0 0.3
7800 79.3 0.29
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Table 4 The sliding mill parameters and operating conditions (Yahyaei et al., 2013). Mill diameter (m)
Speed (rpm)
Charge mass at 30% filling (kg)
1.8 1.2 0.8
12.9 15.8 19.4
207 91 34
used to account for the dominant particle shapes and DEM spheres were used for rounded particles (Fig. 3). The same material parameters were used in these simulations.
Table 5 SAG mill dimensions and load. Mill parameter, ft Internal diameter, mm Internal length, mm DEM simulated slice length, mm Number of liners Ball density, kg/m3 Average speed, rpm Average speed, % critical Average mill filling, %
32 9458 4489 330 30 7800 9.72 70 18
3.3. Industrial scale SAG mill An industrial scale SAG mill was used in this work with operational parameters in Table 5 and 3D mill liner geometries (Fig. 4) using MillMapperTM software, from the work of Weerasekara et al. (2010) and Toor (2013). The mill contents are given in Tables 6 and 7. Four experimentally measured liner profiles over the life of the mill liner were utilised, where each model represents a stage in the life of the liner. These models typically represent the following stages in liner life (Table 8): New (Post-reline), Half-life (50% of life), heavily worn (80% of life), fully worn (Pre-reline). For more details on the milling environment used in is used in this modelling work refer to Toor (2013). 4. Results and discussion 4.1. Collision energy spectrum From DEM simulations, the energy dissipation associated with different particle interactions was calculated. Frequency distributions of energy losses, both in the normal and shear directions were determined from all the individual collision events. Spectra were calculated for each type of collision event and for each ball and rock size class. The collision energy is expressed as a specific energy in kWh/t. This energy density is noted by researchers and practitioners in comminution to be the key driver in rock fracture. Thus although a large rock may experience a large absolute collision energy the dissipation of that energy throughout the rock can result in little damage due to its high mass. For the same absolute impact energy a small rock one tenth of the diameter will experience 1000 fold energy density of the large rock and may shatter from the resultant high intensity stress waves (NapierMunn et al., 1996; Shi and Kojovic, 2007). In this paper all references to energy and energy spectra are the specific energy in the unit of kWh/ton.
Fig. 4. CAD geometry generated using MillmapperTM data (Weerasekara et al., 2010).
Table 6 Ball size distribution. Size, mm
Mean size, mm
% Retained
No of ball particles in DEM slice
125 88 63 44 31
105.1 74.3 52.6 37.1
54 27 13 6
2067 2922 3980 5195
Total in DEM slice Total in full mill
Fig. 3. (a) Clumped and (b) standard spherical DEM particles.
14,164 192,673
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4.2. Pilot scale mills in SAG mode
Table 7 Ore size distribution. Size, mm
Mean size, mm
% Retained
No of ore particles in DEM slice
150.0 106.0 76.0 53.0 37.5 26.5 19.0
126.1 89.8 63.5 44.6 31.5 22.4
0.18 0.33 3.86 11.48 7.06 3.24
2 7 212 1813 3153 4013
Total in DEM slice
9200
Total in full mill
125,147
The collision energy spectra provide the opportunity to better understand the various contributions to the overall energy dissipation within the mill. Knowledge of the changes in these spectra with operational and geometric mill attributes may create the possibility of improving the energy efficiency of size reduction by distributing energy from less useful types of collisions to more useful ones through the change of mill operating conditions.
4.2.1. Collision distribution in different segments of the mill The 6 m mill simulated using the base particle distribution (G3BD) was used to investigate the collision energy spectrums in different regions of the mill (Fig. 5f). The mill was divided into five regions based on the dominant mode and severity of the collision environment. They were identified as bulk shear, cascade impact, cascade in-flight, toe active, and toe impact. Collision spectrums were generated for each region. It was evident from the spectrums that the most active regions of the mill were bulk shear (Fig. 5a), cascade impact (Fig. 5b) and toe active/impact (Fig. 5d and e) in terms of the normal energy of impacts. Similarly, the same regions have registered higher energies in terms of the tangential component. But out of all the five regions the bulk shear region (Figs. 5a and 6a) has the highest energies with a wider energy spectrum especially in terms of tangential energies, which seems to correctly predict the nature of the bulk shear region where shearing is the dominant mode. Another important observation is the higher normal energies of impact in the toe impact region
Fig. 5. Distribution of normal collision frequencies vs collision energy components in 6 m mill sections simulated using the base particle distribution (G3BD), in different regions in the mill.
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(Fig. 5e). It is also observed that the energy spectrums for all the particle classes have a similar relative trend for each zone.
energies reduce considerably (Fig. 7e and f). It is also observed that the particles larger than 43 mm experience higher normal and tangential energies as the mill size is increased. The particles in the mid – size range between 30 and 15 mm seem to have an increasing trend with a peak at the 3 m size and start to decrease gradually at the 6 m mill size. It is evident that for mills in the small to medium size ranges, the smaller particles experience higher normal and tangential energies while the larger particles experience lower impact energies with a wider energy spectrum. On the other hand, when the mill size is increased higher energy impacts were experienced by the larger particles with narrowing energy spectrum across different particle sizes.
4.2.2. Effect of mill size on collision distribution Energy spectrums for the three different mill sizes are compared in Fig. 7 for the pilot scale SAG mill mode. For this comparison simulations were run with the Base Distribution. With the increase in mill size from 1.6 m to 3 m it was observed that smaller particles less than 11 mm experience higher normal and tangential energies, but when the mill size increases further to 6 m the
4.2.3. Particle size distribution The split in energy between size classes is expressed as a percentage of the total input energy and is shown in Fig. 8. It is seen that the distribution of the normal (Fig. 8a) and tangential (Fig. 8b) energies are strongly dependent on the size distribution in the mill (refer to Fig. 2). The mill size has less effect on the normal energy split, but for the tangential component there is a
Table 8 Liner wear life data. Tag name of the scan profile
Description
Date scanned
Time between scans (days)
New (t1) t2 t3 Fully-worn (t3)
New liner Intermediate Almost fully worn Fully worn
08 28 24 06
0 82 57 12
July 2010 September 2010 November 2010 December 2010
Fig. 6. Distribution of tangential collision frequencies vs collision energy components in 6 m mill sections simulated using the base particle distribution (G3BD), in different regions in the mill.
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Fig. 7. Distribution of collision frequencies vs collision energy components in the normal (column 1) and tangential (column 2) directions for the three mill sizes (1.6–6 m form row 1–3).
notable effect. When a fine particle size distribution (D1) is used we get a narrow region with a peak at around the mean particle size. But with a coarser distribution the energy is spread out with changing peak location. This shows that the percentage of the available energy to each size class is strongly affected by the particle size distribution. This behaviour may be anticipated and it is quantified in this analysis. 4.2.4. Particle volume and surface The total energies recorded for each particle class were then normalised by the total particle volume and total particle surface area in each particle class to obtain particle energy density per volume (W/m3) and energy per surface area (W/m2). These results were plotted against the particle diameter for all the simulated conditions (Fig. 8c–f). Fig. 8c and d shows that the smaller particles tend to have higher energy densities, with a peak around the 30 mm size for all the simulated conditions. It is also evident that the tangential component of energy density is about an order of magnitude smaller than the normal component. When the mill size is increased the
energy densities increase. This increase also occurs when a finer particle size distribution is used. When a coarser particle size distribution is used the curves tends to spread out while having a lower peak. Similar trends are observed with the energy per surface area plots (Fig. 8e and f). However, in the surface area base plots the peak shifts to the left to larger particles in the range of 50–100 mm. This is in line with the percentage mass particle size distribution (Fig. 2). This results may indicate that most of the energy is consumed by the particles in the mid- size range having a higher percentage mass out of the total charge distribution. The plots also display that more energy is consumed when the mill is running with a finer particle distribution. This is evident when we compare the mill power (Table 9) predicted using DEM. For the same mill geometry when a relatively fine charge particle distribution is used a higher mill power draw is obtained. 4.3. Sliding/abrasion milling mode The average simulated normal and tangential collision specific energy (Ecs) are presented in Fig. 9. The Ecs here is on a per particle
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Fig. 8. Cumulative normal and tangential energies expressed as (a), (b) percentage of total, (c), (d) energy density and (e), (f) per surface area for each particle size, for some of the simulated conditions in Table 2.
Table 9 DEM power (kW).
Geometry 1 (D1.6 m) Geometry 2 (D3 m) Geometry 3 (D6 m)
Distribution 1
Base distribution
Distribution 3
11.4 – –
10.6 52.4 358.2
– 52.7 291.4
basis for all the different particle size and shape and mill size combinations. It is seen that, for all the simulated conditions, the normal and tangential Ecs decreases as the particle size increases. Similarly as the mill size decreases the Ecs decreases. This is due to the fact that with decreasing mill size the available space and energy for particles decrease, creating a low energy milling environment. Fig. 9a and b for C-Ore shows that as a particle becomes rounded in shape its collision Ecs is reduced. This could be due to the particle shape effect, as particles with angular shape could experience higher local collision force than a rounded/spherical particle. The magnitudes of tangential Ecs are small enough that they could be negligible, making normal Ecs the significant energy
component for all the rounded (R) test cases. In the case of angular (A) particles, the normal to tangential energy ratio averaged across all the tests is approximately four, resulting in normal collision force to be the more dominant force. This means that as the particles are ground from angular to round; during each collision event the tangential component on each collision becomes less significant. The very low tangential energy for the sliding mill mode is unexpected as the mills are run at very low critical speed. But it is observed in Fig. 9b that when the mill size is increased from 1.2 m to 1.8 m, the tangential Ecs tends to increase with mill size. This means that as the mill size increases the tangential Ecs might become significant. On the mill sizes used, it is clear that the normal force is dominant. A maximum of 0.006 kWh/t was observed for C-Ore 34.4 mm particle in 1.8 m mill (Fig. 9a) and an overall average of 0.0036 kWh/t was obtained from all the tests. 4.4. Industrial scale SAG mill Frequency distributions of energy losses, both in the normal and shear directions were determined from all the individual collision events, for new and fully worn lifter arrangements. Spectra
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Fig. 9. DEM calculated average specific energy (Ecs) per particle for C-Ore (a) normal (b) tangential. A here stands for the ‘‘clumped” shape and R stands for rounded shape in Fig. 3.
Fig. 10. Distribution of collision frequencies vs collision energy components in 32 ft industrial SAG mill (a) and (b) new lifter, (c) and (d) fully worn lifter.
were calculated for each type of collision event. A wider collision energy spectra (Fig. 10a and b) was observed when the mill was run with new lifter. But for fully worn case narrow and lower frequencies were observed (Fig. 10c and d). This indicates that as the mill lifters wears higher energy collisions and frequency starts reducing. This will result in decreasing grinding efficiency. 4.5. Comparison of collisions across different scales Frequency distributions of energy losses across the different scales are compared in Fig. 11. It is observed that the 1.8 m pilot scale mill in sliding/abrasion mode has very low energy distribution, which is as expected. When the pilot mills (1.6, 1.8, 3 and 6 m) were run in SAG mode, higher and wider energy distributions
are observed. It is noticed that these energy distributions are in the same range as the operating regime of industrial scale 32 ft mill. This indicates that pilot scale mills run in SAG mode could produce similar energy environment as the industrial SAG mills. The collision energies were then averaged and plotted against particle size distribution for each mill size (Fig. 12). As expected the average energy distribution in the 1.8 m sliding/abrasion mill across the particle sizes are quite low. Whereas the industrial mill with new lifters has the highest energy distribution across the particle sizes, with a peak around 50 mm. Fig. 12 also shows that the average energy distribution in the pilot SAG mill are considerably lower than that of industrial mill. It also shows that the energy available for particles larger 120 mm is considerably less. These results clearly show that the average particle collision energy
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Fig. 11. Distribution of collision frequencies vs collision energy in 32 ft industrial SAG, 1.6 m, 3 m, and 6 m mills.
Fig. 12. Average particle collision energy distribution in 32 ft industrial SAG, 1.6 m, 1.8 m, 3 m, and 6 m mills.
increases with increasing mill size, while its distribution over the particle size is strongly influenced by the mill content size distribution.
5. Conclusions DEM simulations were performed by varying the mill size and charge particle size distribution, to explore and understand the breakage environment in mills. Analysis of each mill was conducted through the total energy dissipation and the nature of the collision environment that produces comminution. The DEM results show that the mill charge particle size distribution has a strong influence on the mill input power and on the way the energy is distributed across the charge. The smaller particles experience higher energies while the large experience less, but this variation is strongly dependent on the mill size. The simulations also captured the energy distribution within different regions of
the tumbling charge. The prediction shows that the toe impact region has higher impact energies, while in the bulk shear region the tangential energies are higher. Furthermore, the DEM predicted that regardless of the mill size most of the energy is consumed by the particles in the mid- size range, which has the highest percentage mass of the total charge distribution. These results clearly show that with the mill size increased the average collision energy on particles increases. This work also shows that the average particle collision energy could form a continuum surface that changes with mill contents particle size distribution and mill size.
Acknowledgements This project is carried out under the auspice and with the financial support of the Vice Chancellors strategic funding for fundamental research. Additional funding from within the JKMRC is also acknowledged.
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References Austin, L.G., Luckie, P.T., 1972. Methods for determination of breakage distribution parameters. Powder Technol. 5 (4), 215–222. Barrios, G.K.P., de Carvalho, R.M., Tavares, L.M., 2011. Modeling breakage of monodispersed particles in unconfined beds. Miner. Eng. 24 (3–4), 308–318. Cleary, P.W., 1998. Predicting charge motion, power draw, segregation and wear in ball mills using discrete element methods. Miner. Eng. 11 (11), 1061–1080. Cleary, P.W., 2001a. Charge behaviour and power consumption in ball mills: sensitivity to mill operating conditions, liner geometry and charge composition. International. J. Miner. Process. 63, 79–114. Cleary, P.W., 2001b. Recent advances in dem modelling of tumbling mills. Miner. Eng. 14 (10), 1295–1319. Cleary, P.W., 2004. Large scale industrial DEM modelling. Eng. Comput. 21, 169– 204. Cleary, P.W., Sinnott, M., Morrison, R., 2006. Prediction of slurry transport in SAG mills using SPH fluid flow in a dynamic DEM based porous media. Miner. Eng. 19 (15), 1517–1527. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29 (1), 47–65. Datta, A., Rajamani, R.K., 2002. A direct approach of modeling batch grinding in ball mills using population balance principles and impact energy. Int. J. Miner. Process. 64 (4), 181–200. Di Maio, F.P., Di Renzo, A., 2005. Modelling particle contacts in distinct element simulations: linear and non-linear approach. Chem. Eng. Res. Des. 83 (11), 1287–1297. Di Renzo, A., Di Maio, F.P., 2004. Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chem. Eng. Sci. 59 (3), 525–541. Djordjevic, N., Morrison, R., Loveday, B., Cleary, P., 2006. Modelling comminution patterns within a pilot scale AG/SAG mill. Miner. Eng. 19 (15), 1505–1516. EDEM, 2006. EDEM User Manual. Herbst, J.A., 2004. A microscale look at tumbling mill scale-up using high fidelity simulation. Int. J. Miner. Process. 74 (Suppl. (0)), S299–S306. Herbst, J.A., Nordell, L., 2001. Optimization of the design of sag mill internals using high fidelity simulation. In: M. e. et al. (Ed.). SAG Conference, pp. 150–164. Kalala, J.T., Bwalya, M., Moys, M.H., 2005a. Discrete element method (DEM) modelling of evolving mill liner profiles due to wear. Part II. Industrial case study. Miner. Eng. 18 (15), 1392–1397. Kalala, J.T., Bwalya, M.M., Moys, M.H., 2005b. Discrete element method (DEM) modelling of evolving mill liner profiles due to wear. Part I: DEM validation. Miner. Eng. 18 (15), 1386–1391. Lichter, J., Lim, K., Potapov, A., Kaja, D., 2009. New developments in cone crusher performance optimization. Miner. Eng. 22 (7–8), 613–617. McBride, A., Powell, M.S., 2006. A structured approach to modelling SAG mill liner wear – numerical modelling of liner evolution. In: Mular et al. (Ed.). International Autogenous and Semiautogenous Grinding Technology 2006, SAG06, CIM, pp. 120–132. Mindlin, R.D., 1949. Compliance of elastic bodies in contact. J. Appl. Mech. 16, 259– 268. Mishra, B.K., 2003a. A review of computer simulation of tumbling mills by the discrete element method: Part I—contact mechanics. Int. J. Miner. Process. 71 (1–4), 73–93.
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Mishra, B.K., 2003b. A review of computer simulation of tumbling mills by the discrete element method: Part II—Practical applications. Int. J. Miner. Process. 71 (1–4), 95–112. Mishra, B.K., Rajamani, R.K., 1992. The discrete element method for the simulation of ball mills. Appl. Math. Modell. 16 (11), 598–604. Mishra, B.K., Rajamani, R.K., 1994. Simulation of charge motion in ball mills. Part 1: Experimental verifications. Int. J. Miner. Process. 40 (3–4), 171–186. Morrison, R.D., Loveday, B., Powell, M., Djordjevic, N., Cleary, P.W., 2006. Applying discrete element modelling to different modes of breakage in AG and SAG mills. In: Mular et al. (Ed.). International Autogenous and Semiautogenous Grinding Technology. SAG 2006, CIM. Napier-Munn, T.J., Morrell, S., Morrison, R.D., Kojovic, T., 1996. Mineral Comminution Circuits: Their Operation and Optimisation. JKMRC, Queensland, Australia. PFC3D, 1999. Particle flow code in 3 dimension, Ver 2.0 User’s Manual, In Minneapolis. Powell, M.S., 2006. The Unified Comminution Model – A Conceptually New Model. In: XXIII. IMPC, Istanbul, Turkey, pp. 1783–1788. Powell, M.S., Govender, I., McBride, A.T., 2008. Applying DEM outputs to the unified comminution model. Miner. Eng. 21 (11), 744–750. Powell, M.S., Weerasekara, N.S., 2010. Building the Unified Comminution Model. In XXV International Mineral Processing Congress, Brisbane Australia, pp. 1133– 1142. Powell, M.S., Weerasekara, N.S., Cole, S., LaRoche, R.D., Favier, J., 2011. DEM modelling of liner evolution and its influence on grinding rate in ball mills. Miner. Eng. 24 (3–4), 341–351. Shi, F., Kojovic, T., 2007. Validation of a model for impact breakage incorporating particle size effect. Int. J. Miner. Process. 82 (3), 156–163. Tavares, L.M., de Carvalho, R.M., 2009. Modeling breakage rates of coarse particles in ball mills. Miner. Eng. 22 (7–8), 650–659. Toor, B., 2013. Quantifying the influence of liner wear on SAG mill performance In JKMRC. University of Queensland, pp. 122. Weerasekara, N.S., Powell, M.S., 2008. The new energy logging from the discrete element method. In: CSRP’08 2nd Annual Conference, Brisbane, Queensland, Australia. Weerasekara, N.S., Powell, M.S., 2010. Exploring the Breakage Environment in Mills with DEM. In XXV International Mineral Processing Congress, Brisbane Australia, pp. 1357–1366. Weerasekara, N.S., Powell, M.S., Cleary, P.W., Tavares, L.M., Evertsson, M., Morrison, R.D., Quist, J., Carvalho, R.M., 2013. The contribution of DEM to the science of comminution. Powder Technol. 248, 3–24. Weerasekara, N.S., Powell, M.S., Cole, S., Favier, J., 2010. Modelling breakage environment in tumbling mills using DEM and analyzing the outputs. In: The Fifth International Conference on Discrete Element Methods, Queen Mary, University of London, London, UK. Whiten, W.J., 1974. A matrix theory of comminution machines. Chem. Eng. Sci. 29 (2), 589–599. Yahyaei, M., Weerasekara, N.S., Powell, M.S., 2013. Characterisation of abrasion breakage using multi size pilot mills. In: 13th European Symposium on Comminution and Classification, Brunswick, pp. 229–232.
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Estimating energy in grinding using DEM modelling N.S. Weerasekara ⇑, L.X. Liu, M.S. Powell University of Queensland, JKMRC, 40 Isles Road, Indooroopilly, QLD 4068, Australia
a r t i c l e
i n f o
Article history: Received 28 May 2015 Revised 19 October 2015 Accepted 19 October 2015 Available online 27 October 2015 Keywords: DEM Energy Comminution Tumbling mills Breakage
a b s t r a c t The latest state of the art on Discrete Element Method (DEM) and the increased computational power are capable of incorporating and resolving complex physics in comminution devices such as tumbling mills. A full 3D simulation providing a comprehensive prediction of bulk particle dynamics in a grinding mill is now possible using the latest DEM software tools. This paper explores the breakage environment in mills using DEM techniques, and how these techniques may be expanded to provide more useful data for mill and comminution device modelling. A campaign of DEM simulations were performed by varying the mill size and charge particle size distribution to explore and understand the breakage environment in mills using DEM techniques. Analysis of each mill was conducted through consideration of the total energy dissipation and the nature of the collision environment that leads to comminution. The DEM simulations show that the mill charge particle size distribution has a strong influence on the mill input power and on the way the energy is distributed across the charge. The smaller particles experience higher energies while the larger experience less, but this variation is strongly dependent on the mill size. The results also showed that the average particle collision energy increases with increasing mill size, whereas its distribution over particle size is strongly influenced by the mill content particle size distribution. The simulations also captured the energy distribution within different regions of the tumbling charge, with the toe impact region having higher impact energies and the bulk shear region having higher tangential energies. Regardless of the mill size most of the energy is consumed by the particles in the mid-size range, which has the highest percentage mass of the total charge distribution. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The motion of grinding media and the energy distribution have a profound influence on the comminution of particles in tumbling mills. It allows the calculation of the trajectories of individual entities in the entire grinding charge as they move in the mill and collide with one another. The discrete element method (DEM) also allows numerical simulation of the dynamic interaction of the mill charge media with the mill liners and lifters (Powell et al., 2011). The calculations are based on the fundamental laws of motion and can take into account of the geometry, dimensions and material property of each individual steel ball, chunk of rock, mill liner, and lifters (Mishra, 2003a). DEM codes take advantage of parallel processing capabilities to scale up the number of particles or length of the simulation. The use of DEM in modelling grinding mills has been successful over the last couple decades following the introduction of this technique by Cundall and Strack in 1979. The earliest DEM model ⇑ Corresponding author. E-mail address: [email protected] (N.S. Weerasekara). http://dx.doi.org/10.1016/j.mineng.2015.10.013 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.
of a mill was in 2D by Mishra and Rajamani (1992, 1994). This was then extended to multiple scales ranging from pilot to industrial by a number of authors like Cleary (1998, 2001a,b), Herbst and Nordell (2001), and in more recent years by Djordjevic et al. (2006), Cleary (2004), Cleary et al. (2006), Djordjevic et al. (2006), Kalala et al. (2005a,b), McBride and Powell (2006), Morrison et al. (2006), Weerasekara and Powell (2008), Weerasekara et al. (2010), Powell et al. (2011), to name a few. A review of the current status of DEM usage in comminution is described by Weerasekara et al. (2013). However, discrete element methods are relatively computationally intensive, which limits either the length of a simulation or the number of particles (Weerasekara et al., 2013). It is also a computationally intensive task to keep track of the breakage of millions of particles and their progeny. More convenient approaches such as mass population balance are required with DEM to model and simulate the particle mechanical environments, log impact energies and model breakage (Austin and Luckie, 1972; Barrios et al., 2011; Datta and Rajamani, 2002; Lichter et al., 2009; Tavares and de Carvalho, 2009; Weerasekara and Powell, 2008; Weerasekara and Powell, 2010; Whiten, 1974).
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Nomenclature E F G m R
v
d
e l
effective Young’s modulus force effective shear modulus effective mass effective radius velocity overlap coefficient of restitution coefficient of static friction
Although incorporating breakage into DEM model increases the computational effort, and could be considered impractical with the available computer hardware, the ability to predict the motion of grinding media and the energy distribution is very important in understanding their influence on the comminution of particles in tumbling mills. The potential of the dynamic interaction of the mill charge media with the mill liners and lifters allows calculation of the trajectories of individual entities in the entire grinding charge as they move in the mill and collide with one another and the mill. These impact energy and force distributions can be used to calculate the grinding rate in mills, as presented by some authors in this area, for example Mishra (2003b), Cleary (2001b), Datta and Rajamani (2002), Herbst (2004), Powell (2006) and Powell and Weerasekara (2010). This work presents the ongoing exploration and understanding of the breakage environment in mills through DEM simulation. In this paper, we develop series of DEM models that allowed simulating charge motion in SAG mills ranging from pilot scale to industrial scale with varied mill load. This work also extends to a series of 3D DEM simulations that were performed on a generic SAG mill for lifter geometries in various stages of wear. The characteristic charge structures for this study are estimates for the head, shoulder, bulk toe and impact toe, together with the analysis of DEM modelled breakage energy environment for multiple scale of mills. This type of analysis and comparison will guide in understanding the breakage energy environment in mill scale-up process. This information can also be used as part of a strategy in mill control and design targeted at a specific size reduction. 2. DEM model
Subscripts n normal components t tangential components Superscripts rel relative
Fig. 1. Particle contact model.
Table 1 Spring stiffness and Damping coefficients used in the contact model.
Spring stiffness constant (K) Damping coefficient (C) Sn ¼ 2E
Normal direction (n) pffiffiffiffiffiffiffiffiffiffi K n ¼ 43 E R dn qffiffi pffiffiffiffiffiffiffiffiffiffiffi C n ¼ 2 56b Sn m
Tangential direction (t) pffiffiffiffiffiffiffiffiffiffi K t ¼ 8G R dn qffiffi pffiffiffiffiffiffiffiffiffiffiffiffi C t ¼ 2 56b K t m
pffiffiffiffiffiffiffiffiffiffi ln e R dn , b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ln
eþp2
2006) in modelling the contact between particles. Hertz’s theory of elastic contacts provides compact relations for the normal direction, deriving from integration of the normal pressure distribution over the contact area (Di Maio and Di Renzo, 2005). The resulting total force (Fn) is then a sum of elastic force and a dissipative (damping) force given by (Cleary, 1998; EDEM, 2006);
F n ¼ K n dn þ C n v rel n
ð1Þ
DEM solves Newton’s equations of motion to resolve particle motion and using a contact law to resolve inter-particle contact forces. Forces are typically integrated explicitly in time to predict the time history response of the material using an appropriate quadrature method. The DEM includes a family of techniques that use radically different treatments for the element geometry and the form of contact forces. A non-linear Hertz–Mindlin no-slip model (Cundall and Strack, 1979; EDEM, 2006; Mindlin, 1949; PFC3D, 1999) was employed to solve for the contact between colliding particles. While the motion of each individual particle is governed by the laws of conservation of linear momentum, angular momentum was resolved by solving Newton’s second law of motion.
This provides a spring repulsive force and a dashpot to dissipate a portion of the relative kinetic energy. In the tangential direction the possible force–displacement configurations depend on both the normal and tangential loading history. The relative tangential velocity (vt) from the relative tangential motions over the collision, behaves as an incremental spring that stores energy and represents the elastic tangential deformation of the contacting surfaces. The dashpot dissipates energy from the tangential motion and models the tangential plastic deformation of the contact. The total tangential force (Ft) is limited by the Coulomb’s law of friction (Cleary, 1998; Di Renzo and Di Maio, 2004; EDEM, 2006).
2.1. Contact model
F t ¼ minflF n ; K t dt þ C t v rel t g
A non-linear model (Fig. 1), combining Hertz’s theory in the normal direction and the improvement to Mindlin’s no- slip model (Mindlin, 1949) in the tangential direction was employed (EDEM,
The coefficients in Eqs. (1) and (2) are given in Table 1. The total energy dissipation associated with particle interactions can be calculated using the above contact law, with components being normal and tangential. This will also provide the frequency
ð2Þ
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distributions of these energy losses, both in the normal and tangential (shear) directions, from all the individual collision events. Spectra can be calculated for each type of collision event and for each ball and rock size class. These collision energy spectra provide the opportunity to better understand the various contributions to the overall energy dissipation within the mill. 2.2. Data analysis technique A DEM simulation for a typical mill section with several thousands of particles generates massive amounts of output data, running to several Gigabytes. Even with good sampling techniques (Cleary, 2001b) this generates files of Gigabytes. Although some DEM software provides a graphical user interface (GUI) based data analysis environment, their techniques either require a considerable amount of processing power or are not capable of delivering the required inputs for the unified comminution model (UCM) kind of modelling (Powell et al., 2008). Therefore a data logging system developed by Weerasekara and Powell (2010) was employed to fulfil the following objective: (1) provide flexibility in handling huge data sets generated by a typical DEM code; (2) extract particle collision information from the DEM output database; (3) summarise data to provide more useful information for further comminution modelling process. A C/C++ based framework was employed in extracting and analysing the massive data sets generated though the DEM simulation (Weerasekara and Powell, 2008). The framework employs a two stage approach, where stage I of this system interacts directly with the DEM output and generates data sets which can be easily managed and handled by the stage II. The method adapted in stage I varies depending on the DEM output data structure of the DEM code used. In stage II, the data extracted and grouped data in the stage I is further summarised according to the requirements for further modelling work. 3. DEM simulations Different mill operation modes were selected for this DEM modelling study to investigate the energy distribution under different loading and operating conditions. They are described below. 3.1. Impact collision dominant milling mode In this study several aspects of a SAG mill were investigated using DEM. A pilot scale SAG mill operating at 75% critical speed and drawing 9.94 kW of power is used as the ‘‘BaseCase” for the
DEM simulation campaign. This BaseCase mill has an internal diameter of 1696 mm, length of 575 mm and has 11 lifters with 50 50 mm cross-section. A base particle size distribution (PSD) shown in Fig. 2 was used for this BaseCase DEM simulation. A combination of simulations was performed by varying the internal diameter from 1.6 m to 6 m. The internal diameter variation from 1.6 m to 6 m was a linear increase with the same linear increase (proportional to diameter increase) for lifter dimensions. The sizes 3 m and 6 m are considered as a direct scale up of 1.6 m pilot mill with no other extra scale up criterion. Other conditions in the mills are the same. Three distinct PSD’s as shown in Fig. 2 was used. This gave a matrix of simulations as presented in Table 2. This table presents the code for each test that is referred to throughout this paper. G1–3 are the three mill sizes and D1, BD, D3 refer to the charge size distributions. In all these simulations the mills were run at 75% critical speed with rock charge filling of 21.5% by volume, and a steel ball filling of 6% to give a total charge 27.5%. The steel balls had a size distribution of 40, 60, 80, and 100 mm. The material parameters in Table 3 were used throughout the simulation. A coefficient of restitution of 0.5 and a friction coefficient of 0.4 have been used for rock–rock and a coefficient of restitution of 0.7 and a friction coefficient of 0.2 for steel–steel collisions, and a coefficient of restitution of 0.2 and a friction coefficient of 0.3 have been used for rock–steel. A coefficient of rolling friction of 0.1 was used for all rolling interactions. The mill shells are assumed to be made of steel. The coefficient of restitution is the highest for steel–steel collisions which are relatively elastic and the lowest for rock–rock collisions which have significant local fracture at contacts leading to low restitution. The friction coefficient is the smallest for ball– ball collisions and the highest for rock-rock ones since the rocks are typically quite rough. 3.2. Sliding/abrasion milling mode Pilot scale mills without balls with operating conditions shown in Table 4 were used for the DEM simulation. These simulations were run to correlate the experimental results reported in the work of Yahyaei et al. (2013). Both the simulations and the experiments were run at a much lower mill speed, which is 40% of the critical speed with experimentally known mill filling. The operating conditions were selected such that particle sliding is the dominant grinding mode during the operation of the mill. Due to the grinding action caused by particles rubbing each other, the ore particle size distribution in the mill changes, which was quantified experimentally by counting the number of ore particles in each size class inside the mill at the start, at a mid-point and at the end of each experiment (Yahyaei et al. (2013). The particle numbers were used in each snap-shot DEM simulation of the mill charge motion, for all the experimental test conditions. A DEM clumped particle was Table 2 Simulation matrix.
Geometry 1 (D1.6 m) Geometry 2 (D3 m) Geometry 3 (D6 m)
Distribution 1
Base distribution
Distribution 3
G1D1 – –
G1BD G2BD G3BD
– G2D3 G3D3
Table 3 Material parameters.
Density (kg/m3) Shear modulus (GPa) Poisson’s ratio Fig. 2. Three different particle size distributions used in the DEM simulations.
Rock media
Steel
2680 1.0 0.3
7800 79.3 0.29
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Table 4 The sliding mill parameters and operating conditions (Yahyaei et al., 2013). Mill diameter (m)
Speed (rpm)
Charge mass at 30% filling (kg)
1.8 1.2 0.8
12.9 15.8 19.4
207 91 34
used to account for the dominant particle shapes and DEM spheres were used for rounded particles (Fig. 3). The same material parameters were used in these simulations.
Table 5 SAG mill dimensions and load. Mill parameter, ft Internal diameter, mm Internal length, mm DEM simulated slice length, mm Number of liners Ball density, kg/m3 Average speed, rpm Average speed, % critical Average mill filling, %
32 9458 4489 330 30 7800 9.72 70 18
3.3. Industrial scale SAG mill An industrial scale SAG mill was used in this work with operational parameters in Table 5 and 3D mill liner geometries (Fig. 4) using MillMapperTM software, from the work of Weerasekara et al. (2010) and Toor (2013). The mill contents are given in Tables 6 and 7. Four experimentally measured liner profiles over the life of the mill liner were utilised, where each model represents a stage in the life of the liner. These models typically represent the following stages in liner life (Table 8): New (Post-reline), Half-life (50% of life), heavily worn (80% of life), fully worn (Pre-reline). For more details on the milling environment used in is used in this modelling work refer to Toor (2013). 4. Results and discussion 4.1. Collision energy spectrum From DEM simulations, the energy dissipation associated with different particle interactions was calculated. Frequency distributions of energy losses, both in the normal and shear directions were determined from all the individual collision events. Spectra were calculated for each type of collision event and for each ball and rock size class. The collision energy is expressed as a specific energy in kWh/t. This energy density is noted by researchers and practitioners in comminution to be the key driver in rock fracture. Thus although a large rock may experience a large absolute collision energy the dissipation of that energy throughout the rock can result in little damage due to its high mass. For the same absolute impact energy a small rock one tenth of the diameter will experience 1000 fold energy density of the large rock and may shatter from the resultant high intensity stress waves (NapierMunn et al., 1996; Shi and Kojovic, 2007). In this paper all references to energy and energy spectra are the specific energy in the unit of kWh/ton.
Fig. 4. CAD geometry generated using MillmapperTM data (Weerasekara et al., 2010).
Table 6 Ball size distribution. Size, mm
Mean size, mm
% Retained
No of ball particles in DEM slice
125 88 63 44 31
105.1 74.3 52.6 37.1
54 27 13 6
2067 2922 3980 5195
Total in DEM slice Total in full mill
Fig. 3. (a) Clumped and (b) standard spherical DEM particles.
14,164 192,673
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4.2. Pilot scale mills in SAG mode
Table 7 Ore size distribution. Size, mm
Mean size, mm
% Retained
No of ore particles in DEM slice
150.0 106.0 76.0 53.0 37.5 26.5 19.0
126.1 89.8 63.5 44.6 31.5 22.4
0.18 0.33 3.86 11.48 7.06 3.24
2 7 212 1813 3153 4013
Total in DEM slice
9200
Total in full mill
125,147
The collision energy spectra provide the opportunity to better understand the various contributions to the overall energy dissipation within the mill. Knowledge of the changes in these spectra with operational and geometric mill attributes may create the possibility of improving the energy efficiency of size reduction by distributing energy from less useful types of collisions to more useful ones through the change of mill operating conditions.
4.2.1. Collision distribution in different segments of the mill The 6 m mill simulated using the base particle distribution (G3BD) was used to investigate the collision energy spectrums in different regions of the mill (Fig. 5f). The mill was divided into five regions based on the dominant mode and severity of the collision environment. They were identified as bulk shear, cascade impact, cascade in-flight, toe active, and toe impact. Collision spectrums were generated for each region. It was evident from the spectrums that the most active regions of the mill were bulk shear (Fig. 5a), cascade impact (Fig. 5b) and toe active/impact (Fig. 5d and e) in terms of the normal energy of impacts. Similarly, the same regions have registered higher energies in terms of the tangential component. But out of all the five regions the bulk shear region (Figs. 5a and 6a) has the highest energies with a wider energy spectrum especially in terms of tangential energies, which seems to correctly predict the nature of the bulk shear region where shearing is the dominant mode. Another important observation is the higher normal energies of impact in the toe impact region
Fig. 5. Distribution of normal collision frequencies vs collision energy components in 6 m mill sections simulated using the base particle distribution (G3BD), in different regions in the mill.
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(Fig. 5e). It is also observed that the energy spectrums for all the particle classes have a similar relative trend for each zone.
energies reduce considerably (Fig. 7e and f). It is also observed that the particles larger than 43 mm experience higher normal and tangential energies as the mill size is increased. The particles in the mid – size range between 30 and 15 mm seem to have an increasing trend with a peak at the 3 m size and start to decrease gradually at the 6 m mill size. It is evident that for mills in the small to medium size ranges, the smaller particles experience higher normal and tangential energies while the larger particles experience lower impact energies with a wider energy spectrum. On the other hand, when the mill size is increased higher energy impacts were experienced by the larger particles with narrowing energy spectrum across different particle sizes.
4.2.2. Effect of mill size on collision distribution Energy spectrums for the three different mill sizes are compared in Fig. 7 for the pilot scale SAG mill mode. For this comparison simulations were run with the Base Distribution. With the increase in mill size from 1.6 m to 3 m it was observed that smaller particles less than 11 mm experience higher normal and tangential energies, but when the mill size increases further to 6 m the
4.2.3. Particle size distribution The split in energy between size classes is expressed as a percentage of the total input energy and is shown in Fig. 8. It is seen that the distribution of the normal (Fig. 8a) and tangential (Fig. 8b) energies are strongly dependent on the size distribution in the mill (refer to Fig. 2). The mill size has less effect on the normal energy split, but for the tangential component there is a
Table 8 Liner wear life data. Tag name of the scan profile
Description
Date scanned
Time between scans (days)
New (t1) t2 t3 Fully-worn (t3)
New liner Intermediate Almost fully worn Fully worn
08 28 24 06
0 82 57 12
July 2010 September 2010 November 2010 December 2010
Fig. 6. Distribution of tangential collision frequencies vs collision energy components in 6 m mill sections simulated using the base particle distribution (G3BD), in different regions in the mill.
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29
Fig. 7. Distribution of collision frequencies vs collision energy components in the normal (column 1) and tangential (column 2) directions for the three mill sizes (1.6–6 m form row 1–3).
notable effect. When a fine particle size distribution (D1) is used we get a narrow region with a peak at around the mean particle size. But with a coarser distribution the energy is spread out with changing peak location. This shows that the percentage of the available energy to each size class is strongly affected by the particle size distribution. This behaviour may be anticipated and it is quantified in this analysis. 4.2.4. Particle volume and surface The total energies recorded for each particle class were then normalised by the total particle volume and total particle surface area in each particle class to obtain particle energy density per volume (W/m3) and energy per surface area (W/m2). These results were plotted against the particle diameter for all the simulated conditions (Fig. 8c–f). Fig. 8c and d shows that the smaller particles tend to have higher energy densities, with a peak around the 30 mm size for all the simulated conditions. It is also evident that the tangential component of energy density is about an order of magnitude smaller than the normal component. When the mill size is increased the
energy densities increase. This increase also occurs when a finer particle size distribution is used. When a coarser particle size distribution is used the curves tends to spread out while having a lower peak. Similar trends are observed with the energy per surface area plots (Fig. 8e and f). However, in the surface area base plots the peak shifts to the left to larger particles in the range of 50–100 mm. This is in line with the percentage mass particle size distribution (Fig. 2). This results may indicate that most of the energy is consumed by the particles in the mid- size range having a higher percentage mass out of the total charge distribution. The plots also display that more energy is consumed when the mill is running with a finer particle distribution. This is evident when we compare the mill power (Table 9) predicted using DEM. For the same mill geometry when a relatively fine charge particle distribution is used a higher mill power draw is obtained. 4.3. Sliding/abrasion milling mode The average simulated normal and tangential collision specific energy (Ecs) are presented in Fig. 9. The Ecs here is on a per particle
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Fig. 8. Cumulative normal and tangential energies expressed as (a), (b) percentage of total, (c), (d) energy density and (e), (f) per surface area for each particle size, for some of the simulated conditions in Table 2.
Table 9 DEM power (kW).
Geometry 1 (D1.6 m) Geometry 2 (D3 m) Geometry 3 (D6 m)
Distribution 1
Base distribution
Distribution 3
11.4 – –
10.6 52.4 358.2
– 52.7 291.4
basis for all the different particle size and shape and mill size combinations. It is seen that, for all the simulated conditions, the normal and tangential Ecs decreases as the particle size increases. Similarly as the mill size decreases the Ecs decreases. This is due to the fact that with decreasing mill size the available space and energy for particles decrease, creating a low energy milling environment. Fig. 9a and b for C-Ore shows that as a particle becomes rounded in shape its collision Ecs is reduced. This could be due to the particle shape effect, as particles with angular shape could experience higher local collision force than a rounded/spherical particle. The magnitudes of tangential Ecs are small enough that they could be negligible, making normal Ecs the significant energy
component for all the rounded (R) test cases. In the case of angular (A) particles, the normal to tangential energy ratio averaged across all the tests is approximately four, resulting in normal collision force to be the more dominant force. This means that as the particles are ground from angular to round; during each collision event the tangential component on each collision becomes less significant. The very low tangential energy for the sliding mill mode is unexpected as the mills are run at very low critical speed. But it is observed in Fig. 9b that when the mill size is increased from 1.2 m to 1.8 m, the tangential Ecs tends to increase with mill size. This means that as the mill size increases the tangential Ecs might become significant. On the mill sizes used, it is clear that the normal force is dominant. A maximum of 0.006 kWh/t was observed for C-Ore 34.4 mm particle in 1.8 m mill (Fig. 9a) and an overall average of 0.0036 kWh/t was obtained from all the tests. 4.4. Industrial scale SAG mill Frequency distributions of energy losses, both in the normal and shear directions were determined from all the individual collision events, for new and fully worn lifter arrangements. Spectra
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Fig. 9. DEM calculated average specific energy (Ecs) per particle for C-Ore (a) normal (b) tangential. A here stands for the ‘‘clumped” shape and R stands for rounded shape in Fig. 3.
Fig. 10. Distribution of collision frequencies vs collision energy components in 32 ft industrial SAG mill (a) and (b) new lifter, (c) and (d) fully worn lifter.
were calculated for each type of collision event. A wider collision energy spectra (Fig. 10a and b) was observed when the mill was run with new lifter. But for fully worn case narrow and lower frequencies were observed (Fig. 10c and d). This indicates that as the mill lifters wears higher energy collisions and frequency starts reducing. This will result in decreasing grinding efficiency. 4.5. Comparison of collisions across different scales Frequency distributions of energy losses across the different scales are compared in Fig. 11. It is observed that the 1.8 m pilot scale mill in sliding/abrasion mode has very low energy distribution, which is as expected. When the pilot mills (1.6, 1.8, 3 and 6 m) were run in SAG mode, higher and wider energy distributions
are observed. It is noticed that these energy distributions are in the same range as the operating regime of industrial scale 32 ft mill. This indicates that pilot scale mills run in SAG mode could produce similar energy environment as the industrial SAG mills. The collision energies were then averaged and plotted against particle size distribution for each mill size (Fig. 12). As expected the average energy distribution in the 1.8 m sliding/abrasion mill across the particle sizes are quite low. Whereas the industrial mill with new lifters has the highest energy distribution across the particle sizes, with a peak around 50 mm. Fig. 12 also shows that the average energy distribution in the pilot SAG mill are considerably lower than that of industrial mill. It also shows that the energy available for particles larger 120 mm is considerably less. These results clearly show that the average particle collision energy
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Fig. 11. Distribution of collision frequencies vs collision energy in 32 ft industrial SAG, 1.6 m, 3 m, and 6 m mills.
Fig. 12. Average particle collision energy distribution in 32 ft industrial SAG, 1.6 m, 1.8 m, 3 m, and 6 m mills.
increases with increasing mill size, while its distribution over the particle size is strongly influenced by the mill content size distribution.
5. Conclusions DEM simulations were performed by varying the mill size and charge particle size distribution, to explore and understand the breakage environment in mills. Analysis of each mill was conducted through the total energy dissipation and the nature of the collision environment that produces comminution. The DEM results show that the mill charge particle size distribution has a strong influence on the mill input power and on the way the energy is distributed across the charge. The smaller particles experience higher energies while the large experience less, but this variation is strongly dependent on the mill size. The simulations also captured the energy distribution within different regions of
the tumbling charge. The prediction shows that the toe impact region has higher impact energies, while in the bulk shear region the tangential energies are higher. Furthermore, the DEM predicted that regardless of the mill size most of the energy is consumed by the particles in the mid- size range, which has the highest percentage mass of the total charge distribution. These results clearly show that with the mill size increased the average collision energy on particles increases. This work also shows that the average particle collision energy could form a continuum surface that changes with mill contents particle size distribution and mill size.
Acknowledgements This project is carried out under the auspice and with the financial support of the Vice Chancellors strategic funding for fundamental research. Additional funding from within the JKMRC is also acknowledged.
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