Prediction of 3D slurry flow within the grinding chamber and discharge from a pilot scale SAG mill

Transcription

1 Prediction of 3D slurry flow within the grinding chamber and discharge from a pilot scale SAG mill Paul W. Cleary 1 and Rob D. Morrison 2 1 CSIRO Mathematics, Informatics and Statistics, Private Bag 33, Clayton South, 3168, Australia. 2 University of Queensland, JKMRC SMI, 40 Isles Rd, Indooroopilly, 4068, Australia. Abstract Slurry flow, including flow through the charge in the grinding chamber, through the discharge grates, along the pulp lifters and its discharge from the mill is an important contributor to the efficiency of the grinding process within a SAG mill. Poor transport of finer ground material can adversely affect grinding leading to excess energy consumption and over-grinding of fine material. This paper uses a 1-way coupled DEM-SPH method in three dimensions to analyse the full slurry flow for a 1.8 m diameter by 0.6 m long AG/SAG pilot mill. This provides detailed information on the internal flow of slurry within a SAG mill, including the prediction of dry regions and of slurry pooling. The importance of the end walls in generating complex three dimensional recirculation patterns in the charge and its influence on axial flow is demonstrated. Such a full 3D analysis ensures that the slurry is presented to the grate with a realistic spatial distribution and allows prediction of the flow through the grates and then within the pulp chamber. The rate of filling of the pulp chambers, the point of initiation of flow down along the rising pulp lifters, back flow into the grinding chamber and flow onto the discharge cone are all investigated. It is shown that the end wall lifters are critical to the flow through the grate generating pumping and shadowing effects which produce strongly non-uniform flow through different grate panels. Keywords DEM; SPH; SAG mill; grate; pulp lifters; slurry; slurry discharge. 1/19

2 1 Introduction Even though the first commercially successful autogenous mills were operated dry, almost all autogenous (AG) and semi-autogenous (SAG) mills in use today are operated with water and ore fed to the mill in a more or less fixed ratio. The water forms a slurry with the fine ore particles. This slurry is much more efficient than air at transporting ore progeny and finished product (sufficiently comminuted particles) out of the mill. Slurry flow, including flow through the charge, through the discharge grates, along the pulp lifters and its discharge from the mill is an important contributor to the efficiency of the grinding process within a SAG mill. Poor transport of finer ground material can adversely affect grinding leading to excess energy consumption and over-grinding of fine material. As overly fine particles ( slimes ) are usually more difficult to separate, fine progeny should be removed from the mill as soon as possible after generation. The removal process has been the subject of considerable controversy in recent years as inadequate slurry removal capacity can constrain mill throughput (Royston, 2000 and Warder and Davies, 1994). However, if the charge is not kept well laden with slurry, the mill capacity to produce fine particles will be reduced as fewer collisions between large particles will interact with the layer of slurry between them. Inadequate discharge of slurry by the pulp lifters is often a problem for single stage AG or SAG mills because of high circulating loads when operated in closed circuit with a fine classifier. Carryover of coarser particles (past the discharge point onto the discharge cone) can cause extremely rapid wear if these particles then fall back down the leading edge of the pulp lifter. Hence, there is substantial incentive to better understand (and to model) both the hold up and flow through mechanisms within the charge as well as the pulp removal capability of the classifying grates and pulp lifters. The earliest use of Discrete Element Method (DEM) modelling in comminution was the prediction of charge motion in ball mills in two dimensions by Mishra and Rajamani (1992, 1994) and then followed by Cleary (1998a, 2001a, b). Similarly the earliest SAG mill DEM modelling, again in two dimensions, was by Rajamani and Mishra (1996). This was used to assess lifter designs when used in large SAG mills and determine which would cause media overthrow and consequent liner damage and ball breakage. The 2D DEM simulations take some account of multi-particle interactions (where combinations of several rocks and/or balls can still cause ball overthrow and consequent liner damage) and 2/19

3 can provide useful qualitative information (Cleary, 1998a, 2001a, c). Three dimensional (3D) slice models of tumbling mills were used by Herbst and Nordell (2001) and Cleary (2001b) to model a thin central slice of the mill. This captures much more of the dynamics of real particle motion. This type of model has been now used widely over the last ten years. In terms of full three dimensional modelling, a detailed comparison of different DEM models was made in a 600 mm diameter scaled AG mill by Cleary et al. (2003) for a range of mill speeds and fill levels. This demonstrated that 3D simulation with spherical particles provided realistic flow predictions over a wide range of conditions. Full three dimensional modelling of pilot scale SAG mills has also been performed over the last decade (see Cleary, 2004, 2009a; Morrison and Cleary, 2008). It has also been used for modelling of axial flow in ball mills (Cleary, 2006, 2009b). Prediction of slurry flow in a mill is a particularly difficult computational challenge. The rotating mill geometry and the requirement to resolve the free surfaces of the fluid make solution using grid based Eulerian flow solvers problematic. In particular, the fragmenting flow of the many thin streams of fluid passing through the potentially hundreds of grate holes is prohibitive. The fluid method most suited to this application is Smoothed Particle Hydrodynamics (SPH) since it naturally allows prediction of complex splashing and fragmenting free surface flow for the fluid in complex rotating mill geometries. Free fluid flow within a SAG mill grinding chamber (ignoring the solids charge), flow through the grate, pooling in the pulp chamber and discharge flow along the pulp lifters was demonstrated using SPH by Cleary et al. (2007). Recently, CFD models of flow from the pulp chamber have been presented by Rajamani et al. (2011) and Metso et al. (2011). In both cases, the approach taken was to assume an amount of fluid had entered each rising pulp chamber and to then predict the flow of this fluid as the pulp lifter angle become steeper leading to the fluid collecting at the throat of each pulp chamber. Rajamani et al. used a grid based method with a surface tracking method which enables reasonable tracking of the free surface when its behaviour is fairly simple. The Metso approach was to also use the SPH method, also coupled to coarse particulates which were also created in the pulp chamber, in order to track this discharge from the pulp chamber. In both cases, the volume of fluid created in each pulp chamber was sufficient to lead to complete blockage of the throat of each chamber. This type of approach is suitable for examining issues relating to flow restriction at the throat for known required discharge rates. For the many cases where the flow is not limited by the throat, it is necessary to predict the amount of fluid actually entering the pulp chambers through the grate from the grinding chamber. 3/19

4 For processes in which slurry flow does not dominate the coarse particle behaviour, a 1- way coupled model can be used. With this approach a dynamic porous media can be characterised from the DEM analysis by averaging the particles onto a fixed data collection grid. This dynamic porous media is the included in the fluid model to represent the average effect of the solid component of the charge. The fluid method most suited to this application is again the SPH method since it naturally allows prediction of complex splashing and fragmenting free surface flow for the fluid in complex rotating mill geometries, and supports coupling of the porous media model to the fluid in order to predict fluid flow within the dynamically moving particulate matrix. The DEM simulation provides averaged porosities and velocities which interact with the SPH fluid particles via an appropriate inter-phase drag. This is controlled by the permeability of the porous media which in turn results from the porosity distribution predicted by the DEM model. This approach was originally developed by Cleary et al. (2006) for predicting two dimensional slurry flow within the grinding chamber of a mill. It was used to study the distribution of slurry for SAG mills and enabled prediction of dry regions under the shoulder and in the toe region for low to moderate slurry loads and slurry pooling at high slurry loads. It was subsequently extended to three dimensions and applied to the prediction of slurry flow in a tower mill by Sinnott et al. (2011). It has also been used to predict slurry flow on a double deck vibrating banana screen (Fernandez et al. 2011). Here we apply this SPH-DEM modelling method to the full three dimensional analysis of slurry flow within an industry standard 1.8 m diameter by 0.6 m long Hardinge AG/SAG pilot mill. Such full 3D analysis allows the pulp flow patterns within the grinding chamber, through the grate into the pulp lifter chambers, flow along the pulp lifters, congestion at the throat of the pulp chambers and the flow onto the discharge cone and out of the mill to be investigated. It also allows quantification of any back flow from the pulp chamber back into the grinding chamber. The effect of end wall lifter geometry can also be understood. 2 Simulation methodology The problem of slurry transport in mills requires both the ability to model the particulate solids and the slurry. The most suitable method for modelling the coarse particulates in the charge is the Discrete Element Method (DEM), see Cleary (1998a, 2001a, 2001b, 2001c, 4/19

5 2004, 2009a&b), Cleary et al. (2004) and Morrison and Cleary (2008) for details and comminution examples using the DEM code used in this paper. Smoothed Particle Hydrodynamics (SPH) is a powerful method for modelling complex, splashing, free surface fluid flows, (see Monaghan 1994, Cleary 1998b, Cleary et al for details and application examples). SPH will be used for modelling the slurry within the SAG mill including flow from the grinding chamber into the pulp chamber. The methodology used here to predict the flow of slurry in mills uses a sequential or 1- way coupled DEM-SPH model. It is described in detail in Cleary et al. (2006) and Cleary and Morrison (2011). The process has three key steps: Perform a DEM simulation to predict the particle flow and average the particle data onto a cylindrical grid to obtain steady state volume fraction and velocity distributions that then well characterise the charge distribution and motion. Apply the continuum porosity and velocity information from the DEM simulation in the SPH simulation. The mill geometry used by the two methods is the same but the charge representation changes from discrete to continuous. Perform an SPH simulation to predict the motion and distribution of slurry in this charge using the slurry viscosity and a coupling based on the Darcy law for porous media. This sequential model is possible because the particulate flow is in steady state and the coarse particulates represented in the DEM are only weakly affected by the slurry motion. The slurry motion through the charge is dominated by the flow of the particulates in the charge. Therefore this one way coupling is able to capture the majority of the key physics required to predict slurry behaviour without the expense of the full 2-way coupling. It allows prediction of slurry distribution within the mill (both in the radial and axial directions) as well as axial transport, including discharge through the grates and flow in the pulp lifters and on the discharge cone. This porous media approach allows us to consider slurry pooling and slurry flow into pulp lifters without the computational expense of a fully coupled model. The pulp lifter model requires only an SPH slurry model unless pebble ports are also required. 3 SAG Mill geometry and operating conditions To explore three dimensional slurry flow in a SAG mill, including discharge through the grates and flow in the pulp chamber, we use a pilot scale Hardinge 6 ft (D) x 2 ft (L) (1.8 5/19

6 m x 0.6 m) mill. The model contains the complete mill geometry including the feed end, the main/belly/circumferential lifters, the end wall lifters, the grate, the pulp lifters, the discharge cone and the external mill shell. This configuration has 16 belly lifters with 10 o face angles, 16 lifters on the feed end cone and 8 lifters on the discharge end. This particle flows for this mill, predicted by DEM, have previously been reported in Cleary (2004), Morrison and Cleary (2008) and Cleary (2009a). Here we use exactly the same arrangement for the coarse particulates and the mill as used in these references. A representative SAG mill charge rock and ball size distribution was used, with a rock top size of 122 mm and a ball size range from 15 to 105 mm. The fill level is 39% (by volume), with a 5% ball load and a 6 mm bottom size cut-off for the particle size distribution. The rock mass used is then kg and the ball load is kg. This means that all of the ball size distribution and more than 90% of the measured rock size distribution (PSD) are included correctly in the DEM model with the remaining mass of fine material added to the smallest resolved class. The total number of particles used was 425,000. The mill rotation rate used was 24 rpm corresponding to 76 % critical. The friction coefficient used for contacts/collisions of all material combinations was 0.5. The coefficient of restitution is material dependent with values 0.3, 0.5 and 0.8 for rock-rock, rock-steel and steel-steel collisions respectively. The normal spring stiffness used in the collision model is 4x10 5 N/m. This produces acceptable accuracy with average overlaps in the range % of each particle diameter. Slurry viscosity has been observed to range from Pa s for primary ball mill discharge and SAG mills up to 0.1 Pa s for AG mills which have much higher concentrations of very fine (-37 micron) particles (Shi and Napier-Munn, 1996). In general, the slurry rheology is non-newtonian with a strong variation of viscosity on the shear rate, dependence on increasing solids fraction and on the fineness of the product particles being transported in the slurry. In this preliminary 3D SAG mill work, we will neglect the flow and fine particulate dependence of the viscosity and use constant representative viscosities in order to understand the principle effects of the slurry on the charge motion. The SPH model can easily be extended to include these extra slurry effects. 6/19

7 4 DEM prediction of solids flows in a full 3D pilot SAG mill 4.1 The Coarse Particulate Flow Figure 1 shows the steady state particle distribution of charge motion. In Figure 1a, a cutaway view of the mill is given with the sectioned cylindrical opening on the left being the feed. The discharge grate can be seen on the far side. The particles are coloured by diameter. Figure 1b and c show the mill, sectioned in half by an axial plane and looking along the axis towards the feed end. The particles are shaded by diameter and speed respectively. A typical mill flow pattern is visible, with the particles carried upwards by the motion of the mill liner until they reach the shoulder position and then flow down the cascading free surface of the charge. For these lifters and mill speed, there are significant coherent cataracting streams that are composed almost entirely of fines due to strong radial segregation. The space between lifters acts like a bucket being tipped producing a coherent stream. This leads to significant numbers of impacts on the liner above the toe position. The blue band along the center of the charge in Figure 1c is slowly moving and is sandwiched between the upward moving layers of charge below and the downward moving layers above. This generates significant shear and much of the energy dissipation occurs as a result of this sliding of the particles over reach other. This leads to the abrasion, rounding and chipping of particles (see Morrison and Cleary, 2008 for details of these mechanisms). The upper sections of the cascading stream and the cataracting stream produce stronger impacts that can break particles either with one impact or with the accumulation of damage over many successful weak impacts (again see Morrison and Cleary, 2008 for details of this mechanism). 4.2 Continuum Representation of the Charge from DEM Data A fixed cylindrical grid is placed over the DEM flow in the mill and particle information is averaged over both time and spatially within grid cells to produce smooth continuum representations of the charge distribution and motion. In this case the grid used has 30 radial sub-divisions, 40 angular ones and 20 axial ones. Figure 2 shows the time averaged particle speed and solids volume fraction from the DEM prediction of the flow of coarse particulates in an axial slice through the center of the mill. The high solid fraction regions (red) in Figure 2b clearly correspond to the location of the bulk of the charge in the DEM simulation (Figure 1). The dark blue regions represent areas of low or zero occupancy by the solid particles. The cataracting stream, although 7/19

8 easily visible in the DEM figure, is quite dilute and so when time averaged leads to a low solid fraction above the free surface of the upper cascading layer. The free surface region in the middle of the mill is quite sharply defined (changing from red to dark blue in a very short distance with only very narrow bands of yellow, green and light blue). As one approaches the shoulder the free surface becomes much less distinct with the solid fraction varying from densely packed to empty over progressively broader areas. This reflects the fluidisation of the charge in the shoulder region when the centrifugal and gravity forces come into balance and the particles are able to spread out and become less concentrated and is shown by the broad green and light blue region above the densely packed (red) part of the bed below. This is the dynamic upper parts of the cascading flow from the shoulder. There is a separated light blue band above which represents a moderately dilute but coherent flow in the lower parts of the cataracting stream. These variations in solid fraction are very important for the subsequent flow of slurry through the solid charge as they strongly control the degree of coupling between the phases and therefore the predisposition of the slurry to locally follow the coarse particle flow or to move more independently under gravity. Figure 3 shows the time averaged particle speed on the axial slice through the center of the mill with red being high speed, green intermediate and dark blue being stationary or where there is no particle data. Superimposed are velocity arrows showing the direction of the particle flow at selected grid points. The regions near the mill shell are moving at speeds close to that produced by the rigid body motion of the shell. The speed decreases with distance from the shell as charge layers slide over each other. The center of recirculation is clearly visible with a recirculation cell centered on the dark blue region on the left. At the center of recirculation, there is, on average, no motion of the charge. Above the center of recirculation, the charge flows from the region of the shoulder towards the toe. The speed of the downward flow increases with distance from the center of recirculation. The upper parts of the cascading stream are shown as yellow. The cataracting stream occupies a large volume of space above the bed with high speed gravity driven flow. The small dark blue region in the upper right indicates locations where charge never reaches and so there is no average speed. In order to understand the three dimensional nature of the charge distribution and the flow field we show an isosurface of charge volume fraction 0.4. This is closed surface within the mill, all the points on which have a volume fraction of 0.4. The region inside this volume has a higher volume fraction, that is, it is more densely packed. The regions of the charge outside this surface are more dilute. The 0.4 level was chosen because it differentiates between the dense packed shearing bed part of the flow and the dilute upper 8/19

9 cascading and cataracting part of the flow. In a sense, it is a representation of the steady state charge free surface. The surface is shaded by the average flow speed of the charge at each location. The elevated shoulder of the charge can be clearly seen at the rear of the mill. The dense core of the cascading flow can be seen leading from the shoulder down towards the center of the mill. This joins a gently sloped surface that leads down towards the toe of the charge. At lower heights, this isosurface extends down to the mill shell and axially from the feed end to the discharge end walls, indicating that the majority of the charge is densely packed into the lower half of the mill. Two important structural features can be observed in this isosurface: 1. The shoulder level is not constant along the axis of the mill, but is moderately higher at the discharge end and sharply higher at the feed end. These higher shoulder positions demonstrate the strong lifting action of the end wall lifters. 2. The raised central tongue like structure representing the cascading stream flowing from the shoulder converges quite strongly, so that it is less than half the axial length of the mill at the point where it falls onto the flat free surface leading to the toe. This focusing of the cascading stream, which is not identifiable in the conventional DEM particle visualisations, is a coherent flow structure that demonstrates significant axial motion. The delivery of the majority of the cascading stream onto this small area before the toe will lead to much higher loads on the mill shell below and will lead to much higher wear in the middle of the shell lifters. Figure 5 shows different isosurfaces of constant volume fraction coloured by the axial speed. The VF=0.1 isosurface shows the charge volume that has a moderate particle density and above. The elevated shoulder at the feed end is even higher than in Figure 4 reflecting the strong lifting of the end wall lifters. The converging cascading structure is well defined and in the feed half is coloured yellow-red indicating that there is strong flow away from the feed end cone. It is coloured light blue in the discharge end indicating that there is a reasonably strong flow away from the discharge end. These strong axial velocity components are the cause of the narrowing of the coherent stream as it falls towards the gently sloped charge surface below. The axial flow in the upper part of the flat gently inclined surface leading to the toe shows flow away from the feed end in the feed half of the mill and flow away from the grates in the discharge half of the mill. These reverse near the toe with strong flow back towards both ends of the mill. The other isosurfaces show different levels progressively deeper inside the charge. The axial flow patterns remain strong and consistent with those observed at the surface. Together these show that the end walls generate strong three dimensional flow within the mill with particles driven away 9/19

10 from the end walls both in the cascading stream and in the flow below this and that there is a counter-balancing flow back towards the end walls in the toe region. This shows that the end wall effects in such low aspect ratio SAG mills are strong and extend all the way to the center of the mill. One consequence of this is that the often used DEM slice model is clearly not a good approximation for what is happening within the charge in this mill. This model is only valid in the middle of the mill well away from any end effects. These results demonstrate that the edge effects propagate into and beyond the middle of the mill, so that there is no such middle section. This means that there is strong axial flow throughout the mill. 5 Slurry Flow within the grinding chamber of a SAG Mill The behaviour of slurry flow in the porous media results from the interaction between two dominant forces: The natural tendency of the fluid to settle under gravity The drag arising from the moving porous media through which it passes. In dense regions of charge, the porosity is low leading to low permeability and limited ability of the slurry to move relative to the solid charge. In such regions slurry and solid charge essentially move together. As the charge becomes dilated, the porosity rises, (as is observed in the shoulder region) and the permeability increases. The fluid then becomes substantially more mobile and is increasingly able to move independently of the solid charge, responding to gravity and to therefore drain down into the charge. This leads to formation of a slurry shoulder that is lower than the solids shoulder and causes the slurry to flow progressively deeper into the charge moving down away from the free surface. These behaviours were clearly demonstrated in two dimensions in Cleary et al. (2006). Figure 6 shows the slurry flow within the grinding chamber at four times after steady state is reached for a slurry viscosity of 0.1 Pa s and a using fluid resolution of 5 mm. This model contains 1.84 million fluid particles. The amount of fluid added to the mill charge is chosen to just be enough to fill the pore space within the charge. The slurry here is coloured by its speed and the solid component of the charge is not shown. The general shape of the slurry distribution is similar to that of the particulates, but the pulp shoulder is slightly lower than the solids shoulder. This occurs because the increasing dilation of the charge near the shoulder rapidly reduces its solid fraction and therefore the strength of the coupling of the fluid motion to the particulate motion. So the slurry becomes able to move independently and responds to gravity more quickly than the solids turning over and 10/19

11 settling lower into the cascading layer of the charge. This behaviour is consistent with that observed in the two dimension slurry flow model (Cleary et al. 2006). Substantial amounts of fluid are lifted both by the belly lifters and the solid particles that form the cataracting stream and are thrown on parabolic trajectories above the cascading upper layer of the solid charge. So there is significant free fluid observed in the cataracting stream in this SAG mill. The slurry moves at around 2.5 m/s near the mill shell near the bottom of the charge. This is the same speed as that of the solid charge and occurs because the high packing in these regions leads to low permeability and little ability of the slurry to move independently of the solids. Comparing this to the charge velocity distribution (Figure 1c) we see some important differences in the surface of the fluid compared to the surface of the solid charge. Specifically, the slurry slumps more in the upper parts of the charge near the middle of the mill. This occurs because of the moderate packing of the charge here which gives the slurry the ability to flow faster than the solid charge and therefore settle further towards the toe. The angle of the slurry surface leading to the toe is much steeper than for the solid charge and the slurry toe position is lower. This reflects the ability of the slurry to again drain through the porous and drier parts of the charge near the toe settling to densely fill the pore space below. This leads to the toe being relatively dry. Almost all aspects of the flow are steady at the point. One non-steady feature is the filling of the spaces between the feed end and the discharge end lifters. Figure 6a shows the situation when one of the spaces between feed end wall lifters has pushed down below the surface of the charge. This region is empty of slurry before it rotates down to be adjacent to the slurry under the toe. Slurry then starts to flow into this space at a rate determined by the permeability of the nearby solid charge. At the time shown, the slurry has just started filling the space. Note that the leading fluid moves at high speed (5 m/s) and it races into and swirls around within this space. Figure 6b shows the situation 0.1 s later when this space has become completely filled and the residual swirling flow has almost dissipated. Figure 6c shows the situation near the end of the filling of the space between the next pair of lifters. The fluid free surface is nearly vertical within this space and is travelling rapidly (at more than 5 m/s) towards the front of the trailing lifter. Figure 6d shows the filling of the following lifter space when it is around half filled. The phase of the filling is slightly later than that shown in Figure 6a, but it is clear that the structure of the flow is very consistent with each new lifter that pushes into the charge. This swirling recirculation pattern between the end wall lifters is a strong and consistent flow pattern and could be expected to generate strong wear on the grates, feed end wall and end wall lifters. 11/19

12 On a more positive note, the discharge end wall lifters stir and dilate the solids charge leading to higher porosity and therefore to greater slurry mobility in the approach to the grate. So the choice of the size and shape of the end wall lifters could potentially be used to enhance slurry draining from the charge into the grates, but at the penalty of increasing the wear on the grate and liner. Depending on whether the mill performance is limited by the pumping capacity of the mill to get slurry out of the grinding chamber or not and whether the current wear life of the grates is long or short, the balance of these factors could be changed in order to produce better outcomes. 6 Slurry flow through the grate and within the pulp chamber Figure 7 shows the flow of slurry from the grinding chamber through the grates and into the pulp chamber. There is very little flow through the grate adjacent to the toe region of the charge. The fluid flows from the charge towards the grate at a rate controlled by the local permeability and therefore the local solid fraction. The charge around the toe is relatively dilated so one may have expected to see significant flow through the grate between the 7 and 8 o clock positions. Due to the high rotational speed of the mill, the actual time that the grate spends passing the toe region is perhaps 0.2 s. This is a very short time for the fluid, constrained still by the solids charge, to respond to the hydrostatic pressure gradient and to start to flow in the axial direction. So although the slurry is relatively mobile here, it still does not have time to pass through the grate. The swirling flow that fills the spaces between the discharge end lifters within the grinding chamber (Figure 6) also contribute strongly to the delay in flow initiation through the grate. The flow that does occur here is irregular and consists of small dribbles of fluid. In the lower parts of the charge (from 7 o clock around to 4 o clock) the charge in the grinding chamber is packed densely and so there is limited mobility of the slurry to drain from the charge. However, the space in front of each end wall lifter is full by the time it reaches the 7 o clock position and so can act somewhat like a buffer allowing reasonable and increasing flow through the grate. The fluid enters the pulp chamber as a large number of small streams through the many grate slots that are adjacent to the fluid. It is resolving this very detailed flow of many small streams that is the most computationally demanding aspect of simulating slurry flow in the mill and is the strongest reason for the choice of SPH for simulating this type of flow. 12/19

13 All these small streams of slurry flow down along the downward inclined pulp lifters in the lower half right half of the mill and collect in pools between the pulp lifter and the shell. This occurs between the 3 and 5 o clock positions in the view shown. As the pulp lifters on the right side of the pulp chamber rise above horizontal, these fluid pools start to flow down along the pulp lifters. Significant flow speeds are not established until the pulp lifters reach the 1-2 o clock positions (see Figure 7b). Once the pulp lifter is vertical (see Figure 7c) the flow accelerates and the fluid mass fragments. The earlier parts of the flow from this lifter have now reached the discharge cone. Fluid is still falling from this pulp lifter even once it has passed vertical and is in the 11 o clock position (Figure 7d). Some of the fluid striking the discharge cone is deflected out of the mill but quite a bit is only slightly deflected and falls down to the shell liner at the bottom of the pulp chamber where it collects and is re-circulated around the pulp chamber. Reasonable volumes of fluid can still be seen on the back of the near horizontal pulp lifter just below the 9 o clock position (in Figure 7d). This slurry is trapped and is forced to recirculate. The small pool at the base of the pulp lifter in the 7 o clock position is fluid that has failed to exit the pulp chamber and which has flow backed down along the lifter and collected as a small pool against the liner. This demonstrates the challenges in clearing slurry from straight radial pulp lifters. For this size of mill (pilot scale), there are no difficulties in obtaining reasonably good discharge even with this identified issues. But as the scale of the mill increases and the volume of charge to grate area increases, these difficulties can only increase in importance. The behaviours observed here reflect the full range of behaviour that can be expected for such pulp lifters. The ability of the coupled DEM-SPH model to make such plausible predictions of slurry flow through the SAG mill is very encouraging. It is important to appreciate that with such a model, the flow of slurry from the grinding chamber through the grates and into the pulp chambers is a direct model prediction. This is in contrast to existing CFD models of slurry flow in the pulp chamber where specific amounts of fluid are assumed to flow in and are instantaneously created high in the pulp chamber. In contrast, for this model the amount of fluid predicted to flow into the pulp chamber is directly controlled by the rotating geometry of the grate and the fluid dynamic forces in the nearby part of the charge in the grinding chamber. The flow rate of slurry through the charge and through the grate is influenced by the viscosity of the slurry. Next we compare the filling of the pulp lifters using two DEM- SPH simulations that are identical except for the viscosity. One uses the viscosity 0.1 Pa s (the same as used in Figures 6 and 7) which is a plausible higher end value of viscosity for an operating SAG mill. The other uses a viscosity of 0.01 Pa s (ten times than of water) 13/19

14 which is at the lower end of the plausible range of viscosity in a SAG mill. Figure 8 shows a comparison of the slurry in the pulp chamber for these two viscosities at 0.5 s. We observe a very similar pattern of slurry distribution within the pulp chamber. So an order of magnitude change in viscosity has not changed the nature of the flow of the slurry within the grinding chamber, the amount or manner of slurry that is presented to the grate or the locations in the grate through which the slurry flows. Close inspection reveals that there is modestly greater filling in each of the pulp chambers for the lower viscosity. In such a mill slurry model, the flow rates of slurry through each grate panel can be individually measured. The flow rates into pulp chambers are shown in Figure 9 for four cases. The top row of the figure are calculated using a coarser spatial of 7.5 mm. The bottom row shows the flow rates calculated using a spatial resolution of 5 mm (as used in Figures 6-8). The left column shows the flow rates for a viscosity of 0.1 Pa s, while the right column is for a viscosity of 0.01 Pa s. The structure of the discharge end of the mill exerts a very strong influence on the structure of the flow through the grate. There are two grate panels in front of each pulp lifter and with both panels contributing to the flow into each pulp chamber. Inside the grinding chamber there is a discharge lifter between the two panels that feed into each pulp chamber. These will be termed the leading and the trailing panels due to their location relative to the discharge end lifter. The flow through the leading panel is shown in blue and the flow through the trailing panel in red. The overall flow rate into the pulp chamber is shown in black and is the sum of the red and blue curves. In each case there is no flow until the leading panel first passes down into the wet part of the charge just under the toe (as was shown in Figure 7). The flow rate rises sharply once slurry is adjacent to the panel. The start of flow into the trailing panel is delayed by the angular difference in the location of the two panels. The flow into each of the pulp chambers shown in Figure 9 has a fairly consistent pattern. The flow through the leading panel is significantly stronger than that through the trailing panel. This is caused by two effects both resulting from the presence of the discharge end lifter. Firstly, the lifter in pushing the fluid and charge ahead of it creates a high pressure which helps to pump slurry from the grinding chamber into the pulp chamber through the leading panel. Secondly, the lifter creates a shadowing effect which makes it much harder for slurry to pass through the trailing grate panel. The combination of these effects leads to the first panel contributing the vast majority of the flow into each of the pulp chambers. For the second pass of each pulp chamber through the charge, an interesting effect is observed with a strong negative flow rate preceding the normal positive flow of slurry 14/19

15 from the grinding chamber. This occurs because the pulp lifters have not completely emptied in the previous revolution of the mill (see the lower left side of the flow at 1.7 s in Figure 7) and the design of the inside of these pulp lifters do not prevent reverse flow of slurry from the pulp chamber back into the grinding chamber. This occurs as the pulp lifters are angled down on their approach to the toe of the charge. This reverse flow represents a strong loss of slurry transport efficiency. Comparing the two rows in Figure 9 allows us to determine that the flow rates are not particularly sensitive to the resolution used in the simulations here. Establishing that the flow rates are not sensitive to the specific choice of resolution in indicates that resolutions used here are numerically adequate to resolve the flow through the grates. Comparing the two columns of Figure 9 allows us to see quantitatively that there is little variation in the nature of the flow through the grates with viscosity. All the key aspects of the flow structure, particularly the dominance of the leading grate panel and the back flow from the grinding chamber occur for both viscosities. The peak flow rates into each pulp chamber are consistently between 2.0 and 2.5 kg/s. The averages of these flow rates are modestly higher for the lower viscosity. This is consistent with the qualitative observations made for Figure 8. 7 CONCLUSIONS It is now feasible to predict slurry motion both within the grinding chamber and within the pulp chamber and most importantly the flow through the grates between chambers for a SAG mills using a 1-way coupled DEM-SPH method. The SPH method has been demonstrated to be able to resolve the extremely complex free surface flow, particularly splashing from the shoulder in the grinding chamber and the flow of many thin streams of slurry through the many holes in the grate. It also has able to naturally deal with the rotating motion of the mill geometry. The end effects of the mill are significant for both the solid charge and the slurry particularly at the corners of the mill where the cylinder liners meet the sloping end liners. However, the combination of the DEM and SPH methods allows us to obtain detailed understanding of the structure of the charge, its motion and the transport of slurry through the mill. As well as providing some degree of lifting to the charge, the lifters in the conical ends of the mill cause quite strong dilation of the charge close to the ends. Hence, the charge surface is carried higher at the ends by the end wall lifters (more so at the feed end which 15/19

16 has double the number of lifters) so that the shoulder position is strongly axially dependent. The cascading stream of charge narrows as it falls (that is, as it velocity increases) and plunges into the charge surface as a narrow jet - well short of the toe region. Complex patterns of coherent axial flow are observed within the charge driven by the conical end wall effects. These patterns include strong axial flow away from the end walls in the upper half of the charge and counter-balancing strong axial flows from the middle of the toe back towards both end walls as the spaces behind their lifters fill and drag particles under the charge resulting in a pair of large recirculation bubbles. These complex recirculation patterns will have strong implications for both slurry transport and for wear in the grinding chamber. Because of these large axial variations, the commonly used DEM slice model is not valid for low aspect ratio SAG mills. The end wall lifters also have an important effect on the slurry flow. As these lifters move into the charge from the toe at high speed, the space between successive lifters is initially empty but fills rapidly with slurry with a swirling motion. These end wall lifters have positive effects in that they stir and dilate the change increasing its porosity and therefore the mobility of the slurry as it approaches the grate. The space between discharge end lifters as acts as a reservoir of predominantly slurry that is the principle supply of slurry to the grate (since draining from the adjacent densely packed charge is too slow). They also have negative effects in that they generate charge motion that will lead to enhanced wear on the grates and axial variations in wear on the shell liner. Understanding the tradeoff between the factors and how they are controlled by the geometry of these lifters offers an avenue for potentially improving discharge behaviour. The flow through the grate into and then within the pulp chamber is predicted by the model. Little flow through the grate was observed adjacent to the toe since there is a response time for slurry to start to flow through the grate and in particular, to fill the spaces between the discharge end wall lifters. This means that slurry does not start entering the pulp chamber until it is around the 7 o clock position. The slurry flow is in the form of many small streams through the individual grate slots that then collect into pools against the mill shell and the rising pulp lifter. Flow along the pulp lifter is initiated around the 2 o clock position and is fully developed by the 1 o clock position. Significant discharge from the pulp chamber is then observed but a reasonable level of holdover is observed with slurry trapped and recirculated within the pulp chamber. The combined SPH/DEM simulation approach allows the flow rates through each panel and parts of panels to be predicted in order to better understand their relative contributions. Each pulp chamber had two panels contributing to their filling. Between 16/19

17 each of these is a lifter on the discharge end of the mill inside the grinding chamber. The flow through the leading panel (in front of the discharge end lifter) was found to be significantly larger with slurry forced or pumped through by the passage of discharge end lifter through the charge. The trailing panel was found to contribute only modestly to the discharge because it is shadowed by the discharge end lifter. Strong back flow was predicted as incompletely emptied pulp chambers flows back into the grinding chamber. The model is therefore able to capture flow both ways through the grate. As this pilot scale mill has a much larger surface to volume ratio than a full scale mill, therefore back flow from incompletely emptied pulp lifters will be proportionally worse in larger mills. The pattern of flow through the grate does not appear to vary with wide changes in the viscosity of the slurry. Modest increases in the flow rates are observed for lower viscosities. Comparison of the quantitative flow predictions for different SPH spatial resolutions demonstrates that the resolutions used in this study are adequate to resolve the flow through the grate and produce resolution insensitive results. Overall, the combined SPH/DEM model should provide a much more realistic framework for investigating ways to reduce slurry transport problems in SAG mills than is possible with slice models or even full 3D DEM dry models. 8 ACKNOWLEDGEMENTS The authors would like to thank Matt Sinnott for his assistance with the visualisations and Phil Owen for the processing of the discharge rates. This paper was presented at SAG2011 and is published with their permission. 9 REFERENCES Cleary, P. W., (1998a), Predicting charge motion, power draw, segregation, wear and particle breakage in ball mills using discrete element methods, Minerals Engineering, 11, Cleary, P.W., (1998b), Modelling confined multi-material heat and mass flows using SPH, App. Math. Model., 22, /19

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