mathematical modelling for optimization of mineral processing

Transaction

Paper

Introduction

In this paper we focus on the use of a variable,which is generally underexploited in themineral processing world: temperature. Verylittle mention of it is made in texts dealingwith milling or flotation, for example.However, the unique characteristics oftemperature, that

� it is an indicator of the energy content ofthe material concerned, and

� it can be very easily, cheaply andaccurately measured

impart to it roles which, if recognized, can leadto important insights into process behaviour,and possibly substantial improvements in ourability to control and optimize these processes.

It is of interest to inquire what is theunique contribution of chemical engineering inthe processing of minerals and materials. Aunifying aspect of all processes is the presenseand unique behaviour of particles. The typicalgraduate chemical engineer is presumablymore than familiar with basic transferprocesses (momentum, mass and heat),thermodynamics, reactor design, control, etc.Generally an ability to apply these skills toparticulate systems will be emphasized. Theimportance of the particle disciplines isemphasized when one glances at the contentspages of volume 2 of the Coulson andRichardson1 series. In the latter half of thispaper we therefore review some recent workon the application of discrete element

modelling (DEM) to the modelling of milling,fluidized beds and chutes.

Analysis and control of flotation columns

Fortuitous results2 arose from a researchprogramme aimed at developing ameasurement of interface level for flotationcolumns. The method involved themeasurement of temperature at a number oflocations in the froth phase spanning therange of froth heights to be used in thecolumn.. The method relied on sharp changesin temperature at the interface, caused by thefact that the wash water used in the plant wasat 4ºC while the feed slurry was near 20ºC. Themethod worked well, providing measurementsof level, which were much more accurate thanthose provided by a pneumatic method.However, the temperature probes were toodelicate for the application and a conductivity-based method superceded the temperaturemethod.

It is worth pointing out that a simpleanalysis of the behaviour of the froth phaseshows that the froth washing inefficiencydefined as the ratio: (water in the concentrateoriginating from the feed slurry)/ (total waterin the concentrate–from the feed and the washwater) is given by

[1]

where Tf, Tw and Tc are the temperatures ofthe feed, wash water and concentrate, respec-tively. These can easily be measured and theratio is a measure of the effectiveness withwhich the feed water (and its associatedentrained gangue particles) has been replacedby the wash water (a low value implieseffective washing).

W W T T T Tfc c c w f w/ /= −( ) −( )

Mathematical modelling foroptimization of mineral processingoperationsby M. Moys*, M. Bwalya*, and W. van Drunick†

Synopsis

The use of temperature measurements for unusual applications inthe analysis and control of column flotation and milling isdiscussed. This is followed by a review of some of the authorscurrent research in the application of the discrete element method tothe modelling of particulate processes, specifically milling, fluidizedbeds and chute flow.

* Centre of Material and Process Synthesis, School ofProcess and Materials Engineering, University ofthe Witwatersrand, Johannesburg, Wits, SouthAfrica.

† Anglo American Research Laboratories.© The South African Institute of Mining and

Metallurgy, 2005. SA ISSN 0038–223X/3.00 +0.00. Paper received Jun. 2004; revised paperreceived Apr. 2005.

s663The Journal of The South African Institute of Mining and Metallurgy VOLUME 105 REFEREED PAPER NOVEMBER 2005

Mathematical modelling for optimization of mineral processing operations

Firstly, interesting results were obtained from an analysisof the temperature profiles in the froth phase. They revealedthe distributed nature of the mixing process in the frothphase, as shown in Figure 1. Temperature profiles are shownat two different locations in the froth phase.

The profile sensed by probe 2 shows a much colder froththan that sensed by probe 1, indicating that the frothwashing process near probe 2 was much more effective.Presumably the wash water addition mechanism was due foran overhaul. The effect of a step increase in the wash waterrate on the froth temperature profiles is shown in Figure 2and reveals how the wash water effectiveness increases afterthe change.

Secondly, a neat correlation between the reducedtemperature of the concentrate and the grade of theconcentrate was obtained, as shown in Figure 3. A cheap andrapidly responding measure of concentrate grade is madeavailable by simple temperature measurements.

Analysis and control of milling

The focus of this project3,4 involved the development of atechnique aimed at improving the current methods ofcontinuous optimization and control of a grinding circuit. Anonline dynamic energy balance has been designed around thedischarge sump, and is used to calculate the mill dischargedensity. This density is approximately representative of theload viscosity, and can be successfully used to optimize milloperation.

The viscosity of the slurry in the mill affects most of theparameters within the mill, and also has a large effect on thedynamic behaviour of the load. The density of the load is

directly affected due to the relationship between viscosity,percent solids and hold-up. For a fixed load volume, anincrease in the slurry viscosity will increase the mass andhence affect the power. It is thus possible to observe andcontrol changes in the load condition by observing the milldischarge viscosity.

Aside from its effect on mill load behaviour and density,the viscosity of the slurry has a strong influence on the

s

664 NOVEMBER 2005 VOLUME 105 REFEREED PAPER The Journal of The South African Institute of Mining and Metallurgy

Figure 1—Effect of position of temperature probes on temperatureprofile in the froth phase of the flotation column

Figure 2—Effect of an increase in wash water rate on temperatureprofiles in the froth phase

Figure 3—Correlation of the %Pb in the concentrate from the columnwith reduced temperature

grinding performance of the mill media. An excessively highviscosity implies a thick slurry inside the mill. The slurryeffectively separates the grinding media such that thegrinding performance decreases. This results in a higheramount of solids reporting to the cyclone underflow andhence a larger recirculating load. On the other hand, whenthe viscosity is too low, the grinding media are allowed tocome into direct contact and result in excessive wear andwastage of the ‘attrition capacity’ of the mill.

Continuous monitoring of the mill circuit and itsoperating conditions requires the best-suited measurementsthat will describe the system and its dynamic behaviour. Avariety of measurements can be used either directly for stateassessment or to determine other useful parameters from theprocess models. Excellent work on viscosity measurementhas been done by Shi5 who formulated a model to accuratelypredict slurry viscosities using a rotating bobbin viscometer.However, reliable control of mill slurry viscosity has not yetbeen successfully implemented. Online viscometers are stillnot yet robust enough to cope with the harsh conditionstypical of a milling environment. Control of the mill dischargeper cent solids is therefore the closest technique available forcontrolling the mill load rheology. Robust measurement ofslurry viscosity remains a tantalizing and worthwhile goal.

The grinding circuit

The experimental data was obtained from AngloGold’sMponeng gold plant, situated in the West Wits region nearCarletonville, South Africa. The entire grinding circuitconsists of three identical SAG mills running in parallel. Eachmill circuit is closed by a hydrocyclone. The optimizingcontrol system (OCS) installed on the plant was calculatingonly a discharge density for Mill No.1 using a mass balanceprinciple when the energy balance study was initiated. Theenergy balance was therefore applied to Mill 1 for thepurpose of comparing the discharge densities from the twomodels.

Figure 4 illustrates the circuit for Mill 1. The energybalance is concerned with measurements around the

discharge sump only. The sump and its properties, however,cannot be isolated as they are intricately related to changes inthis highly dynamic circuit. Four main control strategiesdominate the circuit. The mill discharge density is controlledby the manipulation of the mill feed water flow. The mill feedbelt controls the mill mass. Sump dilution water flow controlsthe sump volume.

Two pumps are used to pump the slurry from the sump tothe cyclone. The variable speed second pump is used tomaintain the specified inlet pressure to the cyclone.

The temperatures of all of the inlet and outlet streams arerequired for the energy balance. PT100 temperature probeswere installed at the mill discharge stream, the sump dilutionwater and the sump underflow. The cyclone overflowtemperature was also measured to ascertain whether or not asignificant amount of energy is gained or lost between thesump outlet and the cyclone overflow. The slurry gainsenergy via pumping, pipe friction, etc. The slurry also losesenergy via conduction/convection to the environment. A netgain or loss would therefore be very difficult to quantifymathematically, and so the temperature of the cycloneoverflow was measured to allow an assessment to be made.A comparison of the cyclone overflow and the sumpdischarge temperatures provides an idea as to whether thenet energy change from the sump to the cyclone is positive ornegative. The reliability and variance of the two temperaturesprovide the energy balance model with a choice of input, asone of the measurements may ensure better modelperformance. Another factor influencing the measurement ofthe cyclone overflow temperature is that the environment forprobe insertion is favourable compared to the sumpdischarge. The life of the cyclone overflow probe wouldtherefore extend beyond that of a probe that is subjected tothe highly turbulent conditions in the sump.

The energy balance

The energy balance is based on mass and energy flows in thesystem, thus making it possible to calculate the per centsolids of the mill discharge.

Mathematical modelling for optimization of mineral processing operationsTransaction

Paper

Figure 4—Mponeng milling circuit



The dynamic energy balance around the sump, as shownin Figure 5, is as follows:

[energy into the sump] = [energy leaving the sump] +[energy accumulated in the sump] MJ/hr.

[2]

The symbols are defined as:ψ[Water] – specific heat capacity of pure waterψ[Solids] – specific heat capacity of quartziteFm[Solids] – mill discharge solids flowrateFm[Water] – mill discharge water flowrateTm – mill discharge temperatureFw – sump dilution water flowrateTw – sump dilution water temperatureFs[Solids] – sump underflow solids flowrateFs[Water] – sump underflow water flowrateTs – sump underflow temperatureEsump – energy of the sump contents

The inputs of the energy balance consist mainly of thespecific heat constants and the measured process variables.The mass flowrate of the mill discharge, however, needs tobe determined. This is done using a dynamic mass balancearound the sump. The only ‘unknown’ value in the process isthe flowrate of gland service water. The error introduced byignoring this was found to be negligible, as the value of thegland service water is generally less than 1.3% of the totalflowrate through the sump.

The energy balance should provide improved accuracy(over the mass balance) because flow and density meters arenotorious for incurring significant measurement errors.Temperature probes possess higher levels of inherentaccuracy and thereby decrease the total compounded error inthe measurements and models. Temperature probes are farless prone to drift than density and flow meters, thus makingthem more reliable in the long-term.

In addition to the enhanced accuracy of the energybalance, the capital cost of the temperature probes isinsignificant relative to the cost of the extra instrumentationrequired to perform the mass balance (in fact the energybalance allows one to dispense with one flowratemeasurement).

A detailed discussion of all these issues and the variousequations used in the model are beyond the scope of thispaper; interested readers are encouraged to contact the firstauthor by e-mail for a document which illustrates the use ofthe energy balance concept in more detail.

Results—application of the energy balance

In order to test the model’s performance, the required inputdata was obtained from the industrial circuit. Manualsamples were taken from the mill discharge to allow directassessment of the accuracy of the model. The mill dischargewas sampled over two hours while step changes were madeto the mill feed water flowrate between the values of 10 and90 t/h, which covers all of the typical values during normaloperation. The residence time of the mill is approximately 10minutes and the effects of a change in the mill feed watershould therefore be allowed to manifest in the form of adynamic change in the density of the slurry entering thesump. The sump has a mean residence time of approximatelyone minute. The step changes in mill feed water flowratewere therefore maintained for a minimum of twenty minutesbefore the next change was made. Enough time was thusallowed for the effects of the step change to be registered bythe energy balance around the discharge sump.

Figure 6 clearly shows that the energy balance predictsthe behaviour of the manually sampled densities very well ifthe offset is ignored. The standard deviation of the variancebetween the energy balance output and the manual samplesis less than the deviation of the OCS RD calculation. The OCSuses a Kalman filtered mass balance to model the entiregrinding circuit, thus producing a mill discharge density

F T F T

F T F T F

T dE dt

m Solids Solids m m Water Water m

w Water w s Solids Solids s s Water

Water s sump

[ ]• [ ]• [ ]• [ ]•

• [ ]• [ ]• [ ]• [ ]•

[ ]•

+ +

= +

+

ψ ψ

ψ ψ

ψ /

s


Figure 5—Sump schematic

Mill Discharge Launder

Cyclone Feed

Mill DischargeSumo

GlandService Water

Fixed Speed Pump

SumpDilution Water

Ts

Tm, Fm

Fp,RDp

TwFw

estimation. The energy balance outperforms the massbalance in its ability to account for the dynamic lag that themill imposes on the change in the mill feed water flowrate.An offset does, however, exist between the energy balanceand the manually sampled data. A large amount ofconfidence is placed in the manually sampled RD data, andcalibration errors in flow or density must be the cause of thisdiscrepancy.

As previously mentioned, the cyclone overflowtemperature was measured to assess its use as a replacementfor the sump temperature measurement. The cycloneoverflow is in most cases easier to access and presents less ofan abrasive environment to the temperature probe.

The sump temperature signal has a noisy-lookingvariance over very short time intervals. This is believed to bedue to poor mixing in the sump, thereby resulting in poorrepresentation of average sump temperature. The averagetemperature difference between the cyclone overflow and thesump was 0.1°C and indicates that there is a net positive gainin enthalpy of the stream (due to energy input by the pump)by the time it leaves the cyclone. Figure 7 shows the effect ofa disturbance in the temperature of the sump contents. Theaverage time lag between the two signals is 39 seconds. It isnow possible to include the temperature difference and timelag into the dynamic model so that the cyclone overflowtemperature can replace the sump underflow temperature.

Modelling the dynamics of the entire mill circuitThe above work was focused on modelling the behaviour ofthe mill discharge sump. A second project is now focused onthe development of a Kalman filter based on an energybalance approach for the entire milling circuit.

Shortcomings of the mass-balances have been identified.Firstly, the mass-balance displays a large positive offset fromthe manually sampled discharge densities—as shown inFigure 6.

Secondly, the mass-balance assumes that the mill isperfectly mixed and therefore does not allow the controlscheme to deal with the dynamic changes in the loadappropriately. The new energy balance models are based onthe ‘two-tank’ model for the mill load. The model assumesthat a portion of the slurry in Tank 2 flows back into Tank 1,thereby allowing the model to cope with a variable amount ofmixing (a large backflow implies the mill is essentiallyperfectly mixed, while no backflow implies that two mixers inseries describes the mixing behaviour).

Online measurements will allow the determination of theappropriate amount of backflow. The temperature of Tank 2will be assumed equal to the measured mill dischargetemperature.

The temperature of all streams (slurry, rock and water) inthe plant and ambient temperature will be measured. Theenergy of all of the streams entering and leaving the mill will


Paper

Figure 6—The energy balance density estimate compared with the mass balance and manual samples

Figure 7—The Cyclone O/F and Sump U/F Temperature Responses



therefore be accounted for. The energy loss via conductionand convection through the mill shell will be accounted for byquantifying the heat transfer coefficient of the shell allowingthe models to account for convective loss.

The OCS© mass balance already computes many usefulparameters, such as the fractional loading (ore, balls, water),the load RD, the discharge coefficient, etc. The energybalance is utilized to refine these estimations based on amass-balance/energy balance combination. This system willthen be capable of producing more meaningful parameters forstate estimation. The Kalman filter (currently applied to themass balance) observes the mill mass and power and othermeasurements such as the water and feed flowrates and thecyclone feed density, all of which are subject to significanterrors (of the order of 5%). Kalman filtering will be applied tothe energy balance in the same way to take advantage of themuch more reliable information in the temperaturemeasurements.

The energy balance has shown remarkable results andpromises to be a useful application in the control of SAG millcircuits. A more detailed discussion of all these issues andthe various equations used in the model is beyond the scopeof this paper; interested readers are encouraged to contact thefirst author for a document which illustrates the use of theenergy balance concept in more detail.

Discrete element method (DEM) modelling ofparticulate processes

We turn now to the modelling of the behaviour of particles(large—grinding balls—and small) using the DEM. The DEMwas first introduced by Cundall and Strack6 and hasgradually developed to become a highly promising initiativein the simulation of particulate processes. The past decadehas seen major strides in DEM development, for which theincrease in the power and affordability of desktop computershas been critical.

DEM has been used in a wide range of applications:

� Soils—stability, subsidence, creep, avalanching � Powders—mixing, flow� Fragmented and loose solids—chutes, bins, mills, block

caving systems� Brittle solids—tunnels, dams, foundations, stone

structures; rock fracture; fragmentation and heave inblasting

� Other applications—fluidized beds (incorporating bothcontinuum methods and DEM), ice fields.

The method has been shown to naturally reproducemacroscopic behaviour, which hitherto has requiredcomplicated explanations, such as the peculiarities ofstress/strain behaviour and fracture in brittle solids, or thephenomenon of ‘memory’ in loaded soils.

Nature exhibits what has been called emergentbehaviour: that simple relationships on the microscale lead tocomplex macroscopic behaviour. The fundamental laws ofphysics are simple and few in number, but the behaviour ofmatter in the large is extremely complex. Standard mesh-based codes solve equations derived for continuous ratherthan discrete systems and have had remarkable successwhen the equations, or constitutive relationships, have beenknown, and have been known to apply to the material beinginvestigated. One has only to consider the success of

computational fluid dynamics (CFD) codes, which solve theNavier-Stokes equations for fluids, or of large deformationhydrocodes, with their dozens of material models, whichenable them to model shock effects in a wide range ofmaterials, to appreciate the power of traditional, meshed-based methods. However, there is a vast range of materialbehaviour for which constitutive laws are almost impossibleto obtain, or severely limited in applicability when they areobtained. Examples are the deformation and failure ofgranular assemblies and brittle solids subjected tocomplicated patterns of loading and unloading, or the flow ofgranular materials in general. It is in just these areas that onewould turn to DEM.

In addition to standard DEM investigations aimed atsolving particular industrial problems, discrete elementsimulations can be used to create a numerical laboratory, andthus be incorporated into powerful new research programmesutilizing experiment and numerical simulation. A discretecode can be primed with relatively simple contact laws, whichqualitatively reproduce the interparticle behaviour of a givensystem. Through study of the evolution of the system,patterns of behaviour, including patterns leading to newconstitutive relationships, can be isolated, observed andinterpreted. This potential, to explore areas which areeffectively inaccessible to experiment alone and beyond thebounds of current engineering knowledge, is one of the mostexciting prospects for the future of DEM.

Despite what has been said, issues remain to be resolvedand obstacles overcome before DEM matures into a fullengineering tool. Firstly, DEM analysis is computationallyexpensive. A 3D simulation of a bench blast, which wouldsimulate a large enough section of ground with enough detailto give meaningful results, would require, at a bareminimum, around 200 000 particles. Simulation time wouldbe measured in weeks. A second issue is that of materialparameterization, which is laborious and time consuming.The parameters used in DEM codes have no clear relation tothe macroscopic variables traditionally used to characterizematerial. Determination of parameter values is an inverseproblem.

Simulations of simple experiments, such as uniaxial ortriaxial tests, are set up, and parameters varied until experi-mental results are reproduced. Ways will have to be found toautomate the inverse process, or thorough investigations ofthe DEM parameter space will have to be conducted and theresults documented.

A third issue concerns scale. A DEM ‘particle’ isinvariably much larger than the particles in the granularassembly or the grains in the brittle solid being simulated.DEM simulations have been shown to qualitatively reproducecomplex material behaviour and in some applications, suchas the flow of coal or ore in chutes, have been used verysuccessfully as a guide to chute design and in correcting poordesign. In general, however, questions of the ability of DEMto give accurate numbers when the DEM particle size isappreciably larger than characteristic scale lengths of thematerial being simulated, remain to be answered.

Milling

Rajamani7 was one of the first researchers to apply DEM tothe modelling of the milling process. His Millsoft package has

s


been widely used by the milling fraternity. It models a two-dimensional approximation featuring one layer of balls withtheir centres on a plane perpendicular to the mill axis.Eskom, our main funder in the DEM area, sponsored a visitby Rajamani to Wits in 1996. He assisted with the firstexperiments performed in our experimental ‘2D-Mill’.

The mill is 0.55 m in diameter and 0.023 m long—slightly larger than one ball diameter. It thus allows us to doexperiments, which exactly correspond with the situationmodelled in Millsoft. An example of the data obtained in thisexperimental programme is shown in Figure 8, whichcompares DEM predictions with experimentally measuredpower data over a wide range of mill speeds (10–180% ofcritical speed). Excellent prediction of mill power is obtainedfor speeds between 0 to 80 per cent of critical speed. Abovethis speed centrifugal forces begin to balance gravitationalforces, so power predictions diverge somewhat; however, theoverall pattern is well predicted. Figure 8 shows pictures ofthe load behaviour compared with DEM predictions at threedifferent mill speeds.

At 100 per cent of critical speed a full layer of balls hadcentrifuged between the lifters, while at 160% of criticalspeed two layers of centrifuged balls are observed, explainingthe substantial loss of power that was observed at this speed.Figure 8 showing the complex variation of power and loadbehaviour with mill speed, illustrates well the commentsabove on the ‘emergent behaviour’ of Nature.

An area of major application for DEM models of mills isthe analysis of the effect of liner profile on the behaviour ofthe load in the mill, which in turn affects the kinetics ofmilling and the rate at which the pulverized fuel can beremoved from the mill. All the power used for grindingpasses through the liners, so this significance is notsurprising. The effect of wear of the liner on the loadbehaviour is illustrated in Figure 9, which shows positiondensity plots (PDPs, which represent the average behaviour

of the load over several mill revolutions) for unworn andworn lifters. Note that the unworn lifters tend to project acertain proportion of the grinding media into the air (thesemedia are said to be cascading) above the bulk of the media,which cataract down the surface of the load. Very littlecataracting occurs with the worn lifters. This observationsupports an hypothesis being currently investigatedconcerning the relationship between this aspect of loadbehaviour and the capability of the mill to produce fineproduct at an adequate rate. Minimal cataracting impliesminimal projection of balls and interstitial coal above the loadsurface, leading to minimal opportunity for interactionbetween fine coal particles and the classifying air passingthrough the mill.

The DEM simulations can also be used to study thetendency towards mixing in the load. The behaviour of 10balls in a simulation over twenty mill revolutions using wornlifters is shown in Figure 10. It is remarkable for 6 of thesepictures (nos. 2, 3, 4, 5, 7 and 10 numbered row-wise) howrepeatable these ball trajectories are, implying that the ballstend to follow the same path as time passes. This tendency isevident to a certain extent in the other pictures as well.Obviously a mechanism exists, which results in balls movingoff a trajectory every now and then. This lack of mixing hasimplications for take-up of coal into the load and removal ofPF from the load.

This implies that there is not much movement acrossthese trajectories, so that balls or coal particles that are closeto the centre of the charge will tend to stay there. This hassignificant implications for the effectiveness of the millingmechanism in these rotary mills: coarse coal will havedifficulty entering the load, and fine particles will have a lowtendency to be removed. Coal near the centre of rotation ofthe load will thus tend to be overground. It should be notedthat other designs of mill (e.g. the vertical spindle mill)ensure minimal residence time of coal particles in the mill


Paper

Figure 8—DEM prediction of power drawn by a 2D rotary grinding mill with 12 square lifters showing pictures of load behaviour at interesting speeds

Mill Power Drawn Against Mill SpeedLoad Volume J = 35%

12 Square Litress669The Journal of The South African Institute of Mining and Metallurgy VOLUME 105 REFEREED PAPER NOVEMBER 2005


before a classification process is applied; these mills areapproximately 80% less expensive in terms of their energyutilization (kWh/t of ground product).

Fluidized Beds

Netshidongololwe, Moys and Horio8 have described theresults of Netshidongololwe’s MSc. research project. Itinvolved the use of an experimental 2D fluidized bed,illustrated in Figure 11, to obtain data for validating Horio’sSAFIRE code, which combines the use of DEM and anelementary CFD model to provide remarkable simulations offluidized bed behaviour. The DEM model is essentially asdescribed above; the CFD model involved the continuityequation for gas behaviour (assumed incompressible) andpublished correlations (such as the Ergun correlation) forcalculating the drag force exerted between the particles andthe gas.

Results obtained are illustrated below. Figure 12 showsthe correlation between experiment and the model forprediction of the pressure drop versus gas velocity for the bedfor a wide range of gas velocities. This is important for thedesign of the bed and the sizing of the fan for forcing the airthrough the bed. The experimental data shows the hysteresisin pressure drop, which is typically observed when the gasrate is increased from zero for a well packed bed or decreasedtowards zero from the fluidized state (this arises because inthe former case the bed will be well-settled and will thereforehave a slightly lower voidage and will require extra pressureto mobilize particles before they enter the fluidized state).

The simulation does not give rise to this behaviour (it couldbe persuaded to if a tighter initial configuration of theparticles were ‘forced’). The minimum fluidizing velocity isaccurately simulated (this is recognized as being fortuitous,since the drag force correlation is subject to large errors).

The ability of the model to simulate the behaviour of abubbling bed is illustrated in Figure 13. In this case the bedwas set up with a single-hole orifice (a uniform distributorwas used in most of the experimental work). The experi-mental bed exhibited a very interesting vortex behaviour justabove the nozzle inlet, which was not simulated by themodel; however, the gross behaviour of the bed is simulatedreasonably accurately. The model should thus be useful forpredicting aspects of fluidized bed behaviour such asresidence time distribution of gas in the bed, mixing of solidparticles, transition from uniform fluidization (typical at gasvelocities just above the fluidizing velocity) to bubblingfluidization, etc. These features have a profound effect onreaction rates, conversion, etc. The model will be particularlyuseful for the simulation of processes that are difficult toobserve, e.g. high pressure fluidized beds.

Chutes

DEM modelling of the flow of coarse, non-cohesive ore inchutes and through transfer points has had notablesuccesses. Both Nordell9 and Dewicki10 have been able touse knowledge gained from DEM analysis of granularmaterial flow at problem transfer points, to recommendretrofits which have significantly reduced belt wear and haveeliminated the problems which were being faced.

s


Figure 9—Position density plots (PDPs) showing the effect of liner wear on the behaviour of the load in a dry coal mill

Figure 10—Single ball trajectories (for 10 balls) in a Lethabo mill with worn lifters indicates that balls generally tend to follow the same track in the mill loadfor a number of mill revolutions

Unworn liner Worn liner


Paper


Figure 11—Front and side views of the experimental 2D fluidized bed rig

Figure 12—Experimental pressure-drop versus gas velocity diagram for 2 mm glass beads with a density of 2 600 kg/m3

Figure 13—The bubbling bed for 2 mm glass beads at Uor/Uor, mf = 1.3 with a single-hole inlet distributor. The top and bottom series are the experimentaland simulation respectively

230 mm

Rotameter

15

Light SourceCamera

Water bottle forhumidifying air

FRONT SIDE

265

1000

Increasing flow

Decreasing flow

Safire

0 0.5 1 Umf 1.5 2

Gas Velocity (m/s)

∆P

(P

a)

∆Pmax

A B C D

5000

4000

3000

2000

1000

0


The COMPS group at Wits has begun a project to modelthe flow of ash in chutes and of coarse coal through 3Dtransfer points. The objective of the project is produce a toolfor chute design, which will more accurately predict thegeometry of coal flow through a three-dimensional chute sothat the geometry of the chute that eliminates spillage can befound reliably. Figure 14 illustrates the effect of placing aguiding bonnet too close, and at too steep an angle, to theexit belt. On the left the particle stream has just begun to hitthe bonnet. The right figure shows the flow some time later.A shock has moved back through the flow to the belt and isdisrupting the material on the belt itself.

Figure 15 shows material moving through a relativelywell-behaved 3D transfer system. The velocity vector of thestream has been smoothly rotated from the x-direction to thez-direction, but spillage of some particles is still evident. TheDEM tool could be used to explore minor changes in thedesign of the chute to eliminate this spillage completely; itcould also be used to select design parameters (such as theslope of the transfer chute), which will ensure that theparticles are deposited on the lower conveyor at a velocitythat is similar to the conveyor velocity (in order to minimizewear of the belt).

These and other similar phenomena can be simulatedusing DEM. Once the model is validated by comparison withmeasurements in existing chutes, it becomes available fordesigning new chutes; for example, we are currently investi-gating the method for simulating flow behaviour in spiralchutes, which have unique applications in certain situations.

Conclusions

Analysis of the behaviour of processes via measurements oftemperature has been shown to provide valuable insightsinto process behaviour. Fault diagnosis (e.g. for theeffectiveness of a wash water arrangement on a flotationcolumn) and process control opportunities for both flotationcolumns and milling circuits have been illustrated.

The discrete element method has tremendous potential asan engineering tool for modelling particulate systems. It iscurrently the focus of very active research, along two generallines. One is development of the method itself. The timerequired to characterize material, and the time to completesimulations, must be reduced, and ways must be found tosmoothly integrate DEM with more established numericalmethods. The second area is exploration of its potential

s


Figure 14—Simulation showing flow disruption resulting from back-up due to an improperly positioned bonnet

Figure 15—Simulation of the flow of coal through a 3D transfer point

through expanding the range of applications investigatedthrough DEM, and the depth and detail of those investi-gations. The next decade will undoubtedly see a vastexpansion in academic research into DEM and in the use ofDEM in modelling industrial processes.

Acknowledgements

Anglogold and ESKOM have supported most of the researchdiscussed in this paper. Their interest and generous supportis acknowledged. The contributions of postgraduate studentsand research staff are also highly valued.

References

1. COULSON, J.M. and RICHARDSON, J.F. Chemical Engineering, vol. 2, 4th(edn.), 1991.

2. MOYS, M.H. and FINCH, J.A. The use of temperature measurements in theanalysis and control of flotation columns. Cam Met Q, vol. 30, no. 3,1991, pp. 131–137.

3. VAN DRUNICK, W.I. and MOYS, M.H. The Use of an Energy Balance toMeasure and Control the Rheology of Mill Discharge Slurry. 2001 SAG

Conference, Vancouver B.C. Volume II of IV, ISBN 0-88865-794-3, 2001.pp. 304–316.

4. VAN DRUNICK, W.I. and MOYS, M.H. Online Process Modeling for SAG millcontrol using temperature measurements, energy balancing andinferential parameter estimation. XXII Int. Miner Proc. Congr., Cape Town,South Africa, 29 Sept–3 Oct 2003.

5. SHI, F. Slurry Rheology and Its Effects on Grinding, PhD Thesis, JuliusKruttschnitt Mineral Research Centre, Department of Mining andMetallurgical Engineering, University of Queensland. 1994.

6. CUNDALL, P.A. and STRACK, O.D.L. A discrete numerical model for granularassemblies. Geotechnique, vol. 29, 1979. pp. 47–65.

7. MISHRA, B.K. and RAJAMANI, R.K. Numerical simulation of charge motion ina ball mill. Preprints of the 7th European Symposium on Comminution, K.Schonert, (ed.), University of Yugoslaviia, Ljubljana, 1990. pp. 555–563.

8. NETSHIDONGOLOLWE, T.R., MOYS, M. H., and HORIO, M. Proc. InternationalFluidisation Conference, South Africa (IFSA), Nov, 2002.

9. NORDELL, L.K. Particle flow modeling. Transfer chutes and otherapplications. Beltcon 9 International Materials Handling Conference. Oct1997, South Africa.

10. DEWICKI, G. (Computer simulation of Granular Material Flows. Powder andBulk Dot Com Newsletter. http://www.powderandbulk.com, Jan 2003. u


Paper


s


mathematical modelling for optimization of mineral processing

Documents