THE APPLICATION OF MULTIPHYSICS MODELS FOR THE DESIGN OF MILL DISCHARGE SYSTEMS *J.K.Lichter1, M.Suazo2, R.Noriega2, V.Murariu1 1
Metso Mining & Construction Technology Technology Development 621 South Sierra Madre Colorado Springs Colorado, 80903 USA (*Corresponding author:
[email protected]) 2
Metso Mining & Construction Technology Metso Chile Camino Internacional #5725 Concon Viña del Mar Chile
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THE APPLICATION OF MULTIPHYSICS MODELS FOR THE DESIGN OF MILL DISCHARGE SYSTEMS ABSTRACT Current AG and SAG mill circuits have continued the trend to larger, fewer mills. These large mills place a significant additional load on the discharge systems and it has become imperative that the designs of these mill discharge systems are evaluated carefully to ensure maximum efficiency and have the capacity to be able to handle the increased mill discharge volumes. The current generation of multiphysics modeling tools allow a detailed analysis of AG and SAG mill discharge performance. The combination of Discrete Element Method modeling and Computational Fluid Dynamics allows the designer to model in detail all aspects of a mill discharge performance. This includes a detailed description of the build-up of solids and slurry in a mill discharge, the recycle of material back into the mill through the grates, the relative wear rates and the occurrence of overloading at the pinch points. These tools allow intelligent decisions to be made regarding the type of discharge arrangement, pan lifter geometry, and discharge cone design, so as to ensure that the discharge is capable of handling the duties required for a specific installation. This paper details the current state of the art in these modeling tools, and provides an overview of some recent applications. KEYWORDS Semi Autogenous Grinding Mill, Mill Discharge, Smooth Particle Hydrodynamics, Discrete Element Method, SAG, SPH, DEM INTRODUCTION The practice of using fewer, larger mills, and to maximize tonnages per milling line, is likely to continue up until the maximum size of a single mill is determined by manufacturing limits, rather than any other circuit limitations. Currently, SAG mills of 40’ in diameter are in operation, and 42’ diameter mills are under construction. Tonnages in excess of 7000 tph have been processed in single milling lines, and total tonnages in excess of 9000 tph are being allowed for in current mill discharge designs. The drive for additional tonnes, and the almost standard availability of variable speed drives in these large mills, has resulted in SAG mill discharge systems typically being designed to operate at up to 82% of Cs. The combinations of large mill diameters, and high operating speeds, have contributed to the significantly increasing burden on mill discharge systems. Larger mill diameters result in less discharge area per unit volume to be transported, and increased operating speeds significantly reduce the efficiency of mill discharge systems. The complexities of the processes within a mill discharge have resulted in numerous methods having been developed by equipment suppliers and consultants to assist in the design of these systems. In most cases, these models rely heavily on empirical data and are not able to account for some of the design variables for mill discharge systems. Multiphysics based mill discharge models overcome most of the limitations inherent in empirical models as they have an intrinsic knowledge of all aspects of the mill discharge geometry, and the effect of mill operation on the performance of a mill discharge. It is useful to consider the mill discharge as consisting of two distinct processes, each with the potential of limiting the discharge rate, and therefore potentially the mill throughput. The first process is the presentation of pulp and pebbles to the grate, and the subsequent flow through the grate into the pan lifters. The second process is the ability of the pan lifters to discharge the slurry and solids that have been deposited into the pans. Multiphysics models allow these two processes to be modeled individually, either
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to ease the computational burden, or to study the effect of pan lifter and discharge cone design in isolation of the performance of the discharge grate. This paper concentrates on the modeling of rocks and slurry in the pan lifters and the mill discharge cone. The data available from the use of multiphysics models is extensive, and includes detailed quantitative data on the discharge rates of pulp and rock, the recycle rates inside the pan lifters, the recycle back into the mill, the location of the highest wear areas, and the relative rate of wear. This data is available as a function of the pan lifter and discharge cone design, and is specific to the tonnages being treated, and the operating speed of the mill. In addition to a review of the multiphysics models in current use, this paper also reports on three examples of the use of multiphysics models to model the behaviour of mill discharge systems. The first two are case studies on operating mills in Chile, the third is a design study for large (40’ diameter and above) SAG mills. MULTIPHYSICS MODELING Multiphysics based mill discharge studies are typically designed to evaluate the various design components of a mill discharge, including the vane geometry, volume of the pan lifters, and the design of the mill discharge cone. In particular, the design is evaluated specifically for excessive recycle rates in the mill discharge (i.e. pulp and solids not leaving the pan lifters), excessive recycle back into the mill through the grates, and volumetric restrictions in the design such that they would physically limit the full volume of the required discharge to exit the mill. The multiphysics models in use have evolved considerably in the last decade. Early models represented the mill discharge only in two dimensions (2D), and only modeled dry particulates. They allowed an evaluation of the trajectories of rocks in the pan lifters of a mill. These models are however of limited use. 2D DEM, by definition, knows nothing of the changing cross sectional areas in a mill discharge, and therefore offers little help in evaluating volumetric constraints. 2D DEM also misrepresents the path length to the mill discharge for most mills, and is not able to account for the majority of the frictional forces that the rocks are subjected to as they leave the mill. 2D DEM therefore allows a rudimentary evaluation of the effect of vane length and mill speed on the relative efficiencies of the mill discharge, but is incapable of representing the volumetric constraints that are of interest, or estimating the correct fraction of material that will exit a mill discharge per revolution. Figure 1 shows a typical visualisation of a 2D DEM mill discharge. The effect of the mill discharge cone is not modeled in 2D DEM.
Figure 1 – 2D DEM Mill Discharge – No slurry – CCW rotation As model sophistication improved, and computer hardware allowed larger problems to be modeled efficiently, 3D DEM models were introduced; initially modeling just the rocks, without the
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presence of slurry, and later with the introduction of slurry. Slurry is significantly more mobile than the solids, and exhibits very different flow characteristics within a mill discharge. Dry DEM models are not able to correctly predict total recycle rates in a mill discharge, or to predict the recycle rates of the rocks independent of slurry. The influence of the slurry therefore has to be included. Figure 2 shows typical 3D DEM visualisations, with and without the slurry being modeled. The slurry is depicted in blue in the right hand image.
Figure 2 – 3D DEM Mill Discharge – With and without slurry – CW rotation Single revolution models such as depicted above have to estimate the volume of material in the mill discharge. A significant fraction of the rock is recycled in the slurry and this has to be accounted for in the initial estimates. By definition, single revolution simulations are not able to accurately estimate the recycle rates and therefore are not able to fully answer the question regarding volumetric constraints in a mill discharge. They are however useful in evaluating the relative efficiencies of different mill discharge designs if computational capacity is limited. Considering all of the limitations imposed by the simplifications described above, it is necessary to model in 3D, multiphase with rocks and slurry accounted for, over multiple revolutions, if a complete picture of mill discharge performance is required. The examples described in the balance of the paper use this complete model. THE NUMERICAL MODELS Two distinct numerical models are employed to model mill discharge systems. These include Discrete Element Method (DEM) and Smoothed Particle Hydrodynamics (SPH). Attempts to model mill discharge systems using only one of these models fail substantially in being able to reproduce the transport of material in a mill discharge system. A coupled model for both the solids and the slurry is therefore an essential requirement. The Discrete Element Method (DEM) (Campbell, 1997) is a numerical technique in which the equations of motion of every particle in a system are integrated numerically at every time step. Thus the motion of every particle in a system, and all of its interactions with the mill discharge and other particles is calculated in great detail. DEM is therefore applied to all rocks in the mill discharge system to model their passage through the pan lifters and out of the mill. Slurry flow has a strong influence on the transport of material through the mill discharge, and therefore has to be included in the simulations if mill discharge performance is to be correctly modeled. The numerical method used to model the slurry is called Smoothed Particle Hydrodynamics (SPH).
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The form of the DEM models used in minerals applications has been extensively covered in literature, the details are therefore omitted. The use of SPH is less common, the general form of the model is therefore reviewed below. Smoothed Particle Hydrodynamics (SPH) The basic idea describing SPH is well reviewed by Monaghan (1992). SPH is a numerical method for obtaining approximate solutions of the Navier-Stokes equation of fluid dynamics by replacing the fluid with a set of particles. These small pieces of fluid interact with their neighbours through pressure and viscous forces. Each fluid particle situated at position ra interacts to varying degrees with its neighbours at | and the value of the kernel function position r based on the distance between fluid particles | , . The variable h represents the smoothing length, which is of the order of the initial separation distance of the SPH particles. An example of a typical kernel function is: | |
,
(1)
Where d has a value of 2 or 3, depending on the dimension of the calculation. SPH interpolation of a quantity A, which is a function of the spatial coordinates can be calculated based on the integral interpolant: ,
(2)
The interpolant reproduces A exactly if the kernel is a delta function. In practice, the kernels are functions which tend to the delta function as the length scale h tends to zero. When the kernel is integrated over the entire domain, the result is unity. In numerical simulations, the integral interpolant is represented by a sum over the b SPH neighbouring particles: ∑
,
(3)
Where mb, ρb and rb are the mass, density and position of the neighbour b, respectively. Ab is the value of property A at position rb. The summation is over all the particles but, in practice, it is only over near neighbours because the kernel function falls off rapidly with the distance. The gradients of the property A are computed in terms of the gradients of the kernel function by differentiation of Eq.(3): ∑
(4)
is the gradient of , evaluated with respect to position ra. The mass of each where particle is calculated from the total mass within the computational domain and the number of SPH particles. The continuity equation · can be written using the identity · · · / (Monaghan (1994) and Morris et all (1997)). Thus, the density of a particle at position ra can be written as: ∑
·
(5)
The Eq.(5) shows that the density associated with that particle can change and therefore the SPH simulation is naturally compressible. Since this work is focused on the incompressible flow, an artificial equation of state can be used that approximates incompressibility by ensuring that the Mach number ( / ) of a fluid flow is low. The equation of state used is that of Morris et all (1997):
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(6) where P is the pressure and ρ0 is the fluid reference density. In the current work, the Mach number is specified as an initial parameter such that its value is below 0.1. The reference density is chosen as the initial density of the fluid or the fluid–solid mixture. The equations of motion from Newtonian fluid SPH particles are derived from Navier-Stokes equation (Morris et all ,1997): ∑
∑
(7)
where Pa and Pb are the pressure at position ra and rb, respectively. The viscous term using the results of the work by Morris et all (1997) and Cleary and Monaghan (1999): ·
where
and
is calculated
(8)
are the dynamic viscosity of the fluid corresponding to positions ra and rb, respectively.
The SPH model is using the XSPH correction to the velocity of the particle so that the particle at position ra is moved using an average velocity given by: 0.5 ∑ where
0.5
(9) .
It should be noticed that SPH, as described here, is prone to a “tensile” instability that can result in the clustering of the SPH particles (Segle, 1995). This instability limits the situations that can be modeled using SPH numerical method. There is an ongoing concern to eliminate it (e.g. Monaghan, 2000). APPLICATIONS OF THE MILL DISCHARGE MODEL In this section, three case studies are discussed. Two are case studies for operating mills in Chile. The mill discharge models were employed to evaluate current mill discharge designs, to quantify their performance, and to modify the designs to minimize recycle rates and improve overall mill performance and mill discharge wear life. The third case study is a design study to determine the affect of mill speed on mill discharge performance for large diameter SAG mills (40’ and above). Case study One – Existing SAG Mill - Chile The 28’x14’ that was evaluated on this property has been in operation for approximately 15 years. The mill discharge design includes an atypical feature in the design that allows the return of steel media back into the mill via ball return ports. This mill discharge was subject to excessive highly localized wear in the mill discharge, which increased maintenance requirements and reduced mill availability. The current mill discharge design has two distinct features. The first is the incorporation of two steel media return ports on the discharge cone, see figure 3a. The second is a reversed conical trunnion liner designed to allow media to roll back towards these ports, see figure 3b. This design, while reasonably effective in returning steel media directly into the SAG mill, does have a significant negative impact on the recycle rates.
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a
b
Figure 3 – a) Current C discharrge cone 3D model m with two ports b) Curreent conical trunnnion liner The value of multiphyssics modeling to quantify, and visually demonstrate, mill dischargge perforrmance is clearrly demonstrateed in this exam mple. Figure 4 shows the devvelopment of thhe recycle in thhe mill discharge d over the first 6 revvolutions of thhe mill. The mill m head has been b removed for clarity. Miill directiion is clockwisse. Slurry is deepicted in blue.
Fiigure 4 – The Current C mill disscharge at 1.5, 3.0, 4.5, and 6.0 6 revolutions from start-up showing the development of the recycledd material
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The high reecycle rates ressulting from this design contrributed significcantly to the hiigh wear rates in i the mill m discharge. These wear arreas predicted by the mill discharge modeel were verifieed by inspectinng worn discharge com mponents from m the mill. Figgure 5 shows the predicted areas of highh wear from thhe modelling exercise, and a figures 6 annd 7 show phootographs of thee respective miill discharge coomponents.
Figgure 5 – Wear map predictionn of the dischaarge cone and pulp p lifters withh the field validdation points
Point 2
b point 1 inn wear map preediction - b) Middle M vane, pooint 2 in wear map m prediction Figuure 6 – a) Pan base,
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Poiint 3
Figuree 7 – a) Inner longer vane, pooint 3 in wear map m predictionn - b) Inner longger vane discharge cone, poinnt 4 in wear map predicttion Once recyccle rates and wear w patterns are a establishedd for the curreent mill dischaarge design, thhe resultss are reviewed to determine the t cause of pooor mill dischaarge performannce. In this partticular examplle, the caause was easy to identify. Thhe ball ports inn the discharge cone, togethher with the use of the reversse conicaal trunnion lineer, hinder the discharge d of maaterial from thee mill and conttribute significaantly to the higgh recyclle rates. Two mechanisms m coontribute to thiss high recycle. The first is thhe material thatt passes througgh the baall return ports back into the mill. m The seconnd contributor is the effect off the reversed conical c trunnioon liner, that t results in a sufficient vollume of materiial building up in the trunnionn area to permiit it to flow bacck into thhe mill discharg ge (both of theese streams cann be seen in figgure 8 a). A revised mill m discharge arrangement was w proposed and modeled that t removed both b the conical trunniion liner and th he ball return ports (shown inn 8b). This desiign exhibited neither n of these recycle stream ms. The modified m design n also included some changes to the lengtths of the vanees in the pan lifters l to further minim mize the recyclee rates.
a
b
gure 8 – a) Currrent design, shoowing recycle streams b) Moodified design Fig When mod deling a mill discharge, d it is modeled at thhe required opperating tonnagge, either at thhe currennt operating con ndition, or alteernatively at soome value that is required for future tonnagee increases. Miill dischaarge size distrib butions, circulaating loads, sluurry and % sollids are all takeen into consideeration. The miill dischaarge is “loaded d” with the appropriate a vollume of rock and slurry onn a continuouss basis, and thhe simulaation is then ru un until steady state is reacheed and the mill discharges thee required tonnnage of rock annd slurry. If the dischaarge does not reach r a steadyy state conditioon where it diischarges the same s volume of o
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material that is loaded into it, then it is understood that the discharge design (at the operating conditions modeled) will not be able to transport the required tonnage. Such tonnage limitations can occur for two primary reasons, either excessive recycle due to vane geometry and mill operating speed, or volumetric restrictions where the pinch points in a mill discharge have insufficient cross sectional area to pass the volume of material required. The use of a multiphysics based mill discharge model allows one to not only detect such limits, but also helps to pinpoint the reasons for high recycle or the pinch points that restrict flow. Design changes can be tested in a virtual environment before any field modifications are made. Figure 9 shows the development of the recycle for the first six revolutions of the modified discharge arrangement.
Figure 9 – Modified mill discharge at 1.5, 3.0, 4.5, and 6.0 revolutions The significant reductions in the recycle rates are apparent from these visualizations. One advantage of multiphysics based models is that they provide quantitative results, not just qualitative. It is therefore possible to provide values for the recycle rates for each of these mill discharge arrangements. Table 1 shows the data for two of the arrangements modeled; the Current discharge design, as shown in figure 4, and the Modified design as shown in figure 9. The Recycle – Pan material is that mass (or volume) of material that recycles inside the pan lifters expressed relative to the mass of material discharged. The Recycle – Mill material is that mass (or volume) of material that returns back into the mill via the grate slots (apertures), expressed relative to the mass of material being discharged.
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The currentt arrangement had h not reacheed steady state after 9 revoluttions, whereas we typically seee steadyy state reached d in 5 to 6 revoolutions. This is another inddicator of high recycle rates with w the currennt dischaarge. The tablee shows the daata after 6 revoolutions. The Modified M dischharge had reachhed steady staate and was w discharging g all of the feedd introduced intto the mill disccharge system. Table T 1 – Resuults from mill discharge d simuulation after 6 revolutions r Discharrged Run
Recycle - Pan
Recycle - Mill
Current
Modified
Current
Modified
Current
Modified
Slurry, m3/hr
401.6
615.9
794.7
317.1
298.1
28.4
S Solids, tph
599.8
866.5
1925.1
991.6
301.6
122.9
The resultss obtained withh the proposedd designs werre considerablyy better than with w the currennt dischaarge system. After A 6 revolutioons, there was an increase off 50% in the diischarge capaciity of slurry annd solids. The change in i the mill discharge arrangeement also redduced recycle rates r (expresseed as a % of thhe materiial discharged)) from ~ 320% % for the currennt discharge arrrangement to ~ 120% for thhe proposed miill dischaarge design. Allso of interest is the mass of material that recycles r througgh the grate baack into the milll. The modified m mill discharge desiign reduced thhose rates by more than twoo thirds. For the t current miill dischaarge design, 30 01 tph of coarsee solids are passsing back intoo the mill for eaach ton discharrged, i.e. 0.5 toon of matterial passes baack through the grates into thhe mill for eachh ton that is disscharged. Thiss number can be b compaared to just 0.14 ton per ton discharged d for the t modified design. d Figure 10 shows the buuildup of thee slurry and solids s recycle for the two mill dischargge arranggements as a fu unction of the number n of revoolutions since mill m startup. Thhe potential im mprovements arre clearlyy evident.
Figure 10 – Slurry and soliids recycle build-up, expresseed as a functionn of the mill diischarge w be installeed The proposed “modifiedd” design is cuurrently in the manufacturingg process and will later this t year. In parallel p with thhis effort, the mine m is reviewing the currentt flowsheet to determine whhat the im mpact will be of o increased mill m discharge performance p o the balance of the circuit.. Following thhis on installlation, there wiill be a follow up to verify performance im mprovements annd to further validate v the toools and methods m used.
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Case study two – Existing SAG Mill - Chile Two separate design studies were undertaken for the 36’ diameter SAG mill at this site. The initial concern expressed by the site was that the discharge cone open area was restricting mill capacity and generating mill discharge problems. An initial review of the mill discharge geometry however raised some concerns about the design of the pan lifters, and the discharge cone, as being potential contributors to poor discharge performance. There was a long history of different discharge designs at this site, using different materials of construction, as a result of the varying opinions of the vendors of mill discharge components. Some of the variations in discharge cone design and vane lengths were the subject of the initial study. The results of the study were used for field validation of the Modified discharge design installed at the time of this first study. The Original discharge design for this mill was a symmetric short-long design of cast steel construction. Over the years, the design had developed into an asymmetric design consisting of 4 different vane lengths in the long-short-middle-short configuration. Construction was a mixture of cast steel and rubber components. This evolution from the as supplied Original mill discharge to this Modified mill discharge design, employed shorter vanes in the hope of increased mill discharge rates. This did not materialize. The first mill discharge design study included multiple alternatives starting with the Modified asymmetric design (Alt1), the Original OEM design (Alt 2) through multiple design options with variations in the mill discharge cone (different materials of construction) and changes in the pan lifter geometries. The final Recommended design (Alt 9), included improvements to both the discharge cone and increases in the vane lengths. Figure 11 shows snapshots from the visualizations of these three discharge designs.
Figure 11 – Cross sectional view of a) Modified design (Alt 1) b) Original design (Alt2) and c) Recommended design (Alt9)
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Table 2 sho ows the resultss for the recyccle rates for three of the geometries modelled, Alt 1, Alt 2 and Alt A 9. Mill discharge simulation after 5.5 revolutioons for coarse distribution (SAG2) Table 2 –M Dischargged Run
Recyccle-Mill
Recyclee-Pan
Alt A 1
Alt 2
Alt 9
Alt 1
Alt 2
Alt 9
Alt 1
Allt 2
Alt 9
Slurrry, m3/hr
1192
1160
1160
1165
10002
986
30
1 19
17
Sollids, tph
1502
1539
1581
3050
28002
2644
319
269
29
Following the completionn of this first study, a reviseed design wass installed intoo the SAG millls whichh incorporated some (but not all) of the dessign changes suuggested for Alt A 9. This desiigned is referreed to as the t Current dessign, and is a syymmetric 4 intto 1 design. As part of the t model valiidation processs, field photogrraphs were takken after the Cuurrent dischargge designn was installed d and had comppleted one full wear cycle. The correlation of the observeed wear with thhe prediccted wear areass was strong annd can be seen in figures 12 and a 13.
Figuure 12 – Wear map predictionn of the Currennt discharge cone and pulp liffters with the field fi validation points
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Figurre 13 – Longerr vane with foccused wear, preedicted by poinnt 1, 2 and 3 in wear map. Poiint 3 shows thee higheest focused weaar One interessting result of the t installationn of the Currennt mill dischargge was the effe fect of extendinng the lonng discharge vanes v to the ennd of the dischaarge cone. Thee Modified dessign in use at the t beginning of o this exxercise suffereed from a relatiively poor design decision too not extend thhe long discharrge vanes to thhe end off the dischargee cone. This alllowed significaant slurry recyccle back into thhe mill discharrge as a result of o bypass of the dischaarge cone. Wheen this design flaw was correected in the Modified M designn, mill dischargge trajecttories increased d significantly. See figure 144. The mine haad to extend the mill dischargge chute to cater for thee increased trajjectory of the mill m discharge
a
b
b Side view Fiigure 14 – a) Side view snapsshot of Alt1 (leeft) showing thhe pulp and soliids trajectory. b) snapshot of the Current designn showing the pulp and solids trajectory
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Subsequent to the installation of this Modified design, the mill discharge rates improved significantly, showing an increase in the discharge capacity, as well as an improvement in the life of the mill discharge cone from 6 months to 10 months. This resulted in a second study being commissioned to review potential for further additional improvements. For the second study three radial discharge arrangements were modeled, these being the Current design, and two additional layouts labelled Proposed and a design with a Flared discharge cone. The Proposed design is a symmetric 4-into-1 discharge, with a longer center pulp lifting vane and angled end pockets. The Flared design is similar to the Proposed design but with a flared discharge cone and a longer center vane. The Proposed design without the flared discharge cone was selected for the next generation of mill discharge in these mills, and is currently being manufactured. It is hoped to be able to report on the results from this latest generation at some future date after the discharge has been installed and performance monitored. Case study three – Design, Large SAG mill discharge Multiphysics models are now routinely used to improve and validate the designs of key components of new mills under design or construction. This exercise focussed on an evaluation of mill discharge performance as a function of the mill speed, and was targeted at large SAG mills, (the next generation of large mills, 40’ in diameter and above). The mill discharge tonnage used for this modeling exercise was ~4000 tph of coarse solids, and ~4180 m3/hr of slurry. Three speeds were modeled, 74% of Cs, 79% of Cs and 84% of Cs. The design of the mill discharge modeled is shown in figure 15.
Figure 15 – 40’ SAG mill discharge, mill head removed for clarity The discharge and recycle rates and their respective percentages for the three simulations are presented in Table 3. Percentages shown are relative to the achieved discharge rates.
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Table 3 –Simulation results Discharged
Recycle-Pan
Recycle-Mill
Speed (%Cs)
74%
79%
84%
74%
79%
84%
74%
79%
84%
Revolution (s)
5.5
6
5.5
5.5
6
5.5
5.5
6
5.5
Slurry, (m3/hr)
1150
2790
5095
417
591
868
4180
4126
3770 27.5%
67.6%
135%
10.0%
14.3%
23.0%
Solids, (mtph)
2313
5090
8935
424
588
827
4000
57.8%
131%
268%
10.6%
15.1%
24.8%
3898
3331
It is clear from the table above that as the speed of the mill increases, the recycle of slurry and solids increases dramatically. At 84% of critical speed, the mill is effectively unable to discharge the required slurry and solids rate due to the very high recycle rate. Target mill discharge rates (4000 mtph solids, 4180 m3/h slurry) are achievable at 74% and 79% of critical speed. Figures 16 through 18 show discharge and recycle statistics for slurry and solids for each of the three simulations. Slurry and solids recycle to pan is the material that either remains in the mill discharge per discharge cycle, or that flows back into the pan lifters from the trunnion. Recycle to the mill is the material returning to the mill via the grates (see figure 19 for clarity). Note in the figures below that steady state is achieved faster (i.e. after fewer revolutions) at lower speeds.
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Discharge and Recycle Rates
Solids Mass Rate (mtph) and Slurry Rate (m3/h)
Run 1 ‐ 74% Critical Speed Solids Target
Solids Discharge
Solids Rec to Mill
Solids Rec to Pan
Slurry Target
Slurry Discharge
Slurry Rec to Mill
Slurry Rec to Pan
6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Revolution
Figure 16 – Mill Discharge and Recycle Statistics at 74% Cs
Discharge and Recycle Rates
Solids Mass Rate (mtph) and Slurry Rate (m3/h)
Run 2 ‐ 79% Critical Speed
6000
Solids Target
Solids Discharge
Solids Rec to Mill
Solids Rec to Pan
Slurry Target
Slurry Discharge
Slurry Rec to Mill
Slurry Rec to Pan
5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Revolution
Figure 17 – Mill Discharge and Recycle Statistics at 79% Cs
17
5.5
6
Discharge and Recycle Rates
Solids Mass Rate (mtph) and Slurry Rate (m3/h)
Run 3 ‐ 84% Critical Speed
9000
Solids Target
Solids Discharge
Solids Rec to Mill
Solids Rec to Pan
Slurry Target
Slurry Discharge
Slurry Rec to Mill
Slurry Rec to Pan
8000 7000 6000 5000 4000 3000 2000 1000 0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Revolution
Figure 18 – Mill Discharge and Recycle Statistics at 84% Cs Figures 19 through 21 show snapshots of the mill discharge operating at 74%, 79% and 84% of critical speed. The velocity of the coarse particles is indicated by the color legend. The slurry is depicted in magenta. Note the increase in recycled material as the speed increases.
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Recycle to Pan Recycle to Mill
Figure 19 – Mill Discharge at 74% Cs
At 84% of critical speed the recycle rate of solids to the pan is more than 260% of the total solids discharged. Recycled slurry to the pan is greater than 130% of the slurry discharge. As Figure 21 shows, most of this recycled material ends up in the first pocket after a long vane. The amount of recycled slurry and solids is sufficient to completely block the discharge grate in the first pocket. For computational simplicity, mill discharge simulations add new slurry and solids in each pocket at a position above the recycled slurry/solid mass. Since the flow through the grate is not modeled, the effect of recycled material blocking the grate is not taken into account in the discharge statistics. At low speeds, the recycle rate is low enough that grate blockage can be neglected. However, at 84% of critical speed, the fraction of grate area blocked by recycled material may have an effect on the total discharge rates achievable.
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Figure 20 – Mill Discharge at 79% Cs
Simulated addition point for new slurry and solids
High level of recycle material in pocket 1
Figure 21 – Mill Discharge at 84% Cs
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The increase in recycled solids and slurry at higher speeds is evident from these figures. At 74% of critical speed there is sufficient time for most of the slurry and solids in the pan to exit the discharge before the mill rotates past the 3 o’clock position where any leftover material falls back into the pan to be recycled. At higher speeds there is less time for this material to exit the discharge, and recycled material builds up in the bottom of the pans. This buildup continues until the height of recycled plus new material in the bottom of the pans reduces the distance the material has to travel enough to balance the increased speed. When this occurs the simulation has reached steady state. At 74% of critical speed this occurred very quickly (within 4 revolutions), whereas at 84% of critical speed it quickly became evident that the recycle rate was unsustainable. CONCLUSIONS Software advancements together with improvements in the speed of computer hardware have made multiphysics modeling of mill discharge systems feasible and routine. The data available from such exercises is detailed and quantitative, and provides an in depth insight into the effect that mill discharge design and the mill operating conditions have on the performance of a SAG mill. These tools will identify problems resulting from poor pan lifter design, mill discharge design, and volumetric constraints. In addition, modeling allows alternative designs to be tested and proven in a virtual environment without the disruption of production due to experimentation on operating mills. Changes in mill discharge design have been shown to contribute to increased mill capacity, and improvements in mill discharge life of over 60%. The development of multiphysics mill discharge models continues with specific emphasis on improving computational efficiency, thereby allowing larger problems to be solved and to increase the level of fidelity. Simplified models have been shown to be incapable of modeling the detailed dynamics of a mill discharge, and every effort must be made to maintain model fidelity and to not sacrifice accuracy for computational ease. In addition flow through the grate, and the effect of recycled material on that flow, is included in the next generation of these models. REFERENCES Campbell, C.S., Computer Simulation of Powder Flows. Powder Technology Handbook, ed. Gotoh et al, 2 edition, Dekker, New York, 1997, pp.777-793. Monaghan, J.J. (1992). Smooth particle hydrodynamics. Ann. Rev. Astron. Astrophys, 30, 543-574 Monaghan, J.J. (1994). Simulating free surface flows with SPH, J. Comput. Phys., 110, 399-406 Morris, J.P., Fox, J.P. and Zhu, Y. (1997), Modeling low Reynolds number incompressible flows using SPH, J. Comput. Phys., 136, 214-226 Cleary, P.W. and Monaghan, J.J. (1999), Conductions modeling using smooth particle hydrodynamics, J. Comput. Phys., 148, 227-264 Segle, J.W., Hicks, D.L., Attaway, S.W. (1995), Smoothed particle hydrodynamics stability analysis, J. Comput. Phys., 116, 123-134 Monaghan, J.J., SPH without a tensile instability, J. Comput. Phys, 159, 290-311
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