a pilot plant investigation of a kinetic model for flotation

7/30/2019 A Pilot Plant Investigation of a Kinetic Model for Flotation

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A pilot-plant investigation of a kinetic modelfor flo tationBy R . P. KING*, B.Se. (E ng.), M .S e. (Eng.) (R and), Ph.D . (M ane.), M .I. C hem . E ., M .5.A.I.

C hem. E . (F ello w)SYNOPSIS

The flotation behaviour of phoscorite from the Phosphate D evelopm ent C orporation w as m easured in a pilotplant, the flotation cells in the plant being varied in configuration so as to operate w ith or without circulatingloads. The ore w as floated in a batch cell so that the param eters for a kinetic m odel could be estim ated, and theseparam eters w ere then used in a flotation-plant sim ulator, w hich is based on the kinetic m odel, in the prediction ofthe perform ance of a pilot flotation plant operating on phoscorite. T he sim ulator predictions com pare w ell w ith them easured perform ance of the ore in the pilot plant, but the com parison show s that the kinetic model is not com -pletely adequate for the prediction of perform ance in continuous flotation plants. T he m odel is w eakest in its abilityt o mode l t h e b ehav iour o f t h e f ro th phase .The f lo ta tio n b ehav iour o f t h e o re cou ld b e adequate ly d es cr ib ed when itw as assum ed that the apatite and gangue each consist of floatable and unfloatable com ponents.The effect of particle size on the specific flotation-rate constant of apatite w as found to have a m axim um at 85fLmand to fall to zero at 400 fLm; that for the gangue had a maximum at 50 fLm and was zero at 350 fLm.SAMEVATTING

D ie flottasiegedrag van foskoriet afkom stig van die Fosfaat O ntginningskorporasie is gem eet in 'n proefaanlegterw yl die konfigurasie van die flottasieselle in die aanleg verander is sodat dit m et of sonder sirkulerende laste kanw erk. Die erts is in 'n lotsel geflotteer sodat die param eters vir 'n kinetiese m odel beraam kon w ord en hierdiep aram eters is to e in 'n flo ttasiea an leg na bo otser w at o p d ie kine tiese mode l g eb ase er is, ge bru ik v ir d ie v oorspe llin gv an d ie p re stasie v an 'n proe fflottasieaa nle g w at m et fo sko riet w erk . D ie v oorspe llin gs m et b eh ulp v an d ie n ab ootse rvergelyk goed met d ie g eme te p re st as ie v an d ie e rts in d ie p ro ef aanl eg , maar d ie v er ge lyking toon dat d ie k in et ie semodel nie heeltem al voldoende vir die voorspelling van die prestasie in deurlopende flottasieaanlegginge is nie.D ie m odel is die sw akste w at betref sy verm oe om die gedrag van die skuim fase te m odelleer. D ie flottasiegedrag vandie erts kan behoorlik beskryfw ord w anneer daar aangeneem w ord dat die apatiet en die aarsteen elkeen uit floteer-b are k ompon en te b estaa n.D aar is g ev ind d at d ie u itw erk ing v an d ie p artike lgro otte o p d ie sp esifieke flottasietem po ko nstan te v an ap atiet 'nm ak sim um b y 85fLm het en tot nul daal by 4O0fLm; vir die aarsteen w as dit 'n m aksim um by 50 fLm en nul by 350 fLm.Introduction

Kinetic models for the flotation process have beenunder investigation for many years, but much of theexperim ental evidence has been obtained alm ost entirelyfrom investigations of single-batch or continuousflotation cells. A kinetic m odel to describe the behaviourof single cells should be capable of providing a quantita-tive description of an entire flotation plant; otherwise, itwould not be of much use as a tool for plant design andoperation. The investigation described here was plannedspecifically to show how far a kinetic model can be usedin the description of the steady-state operation of acom plex flotation plant.

A brief reference to the utility and lim itations ofkinetic models in plant simulation is appropriate here.That they are not necessary for successful plant designis obvious in that many plants that currently operatesuccessfully were designed, not by use of kinetic m odels,but by the conventional route of bench-scale batch testsfollowed by extensive pilot-plant proving work, which isa route that is not likely to be supplanted by any basedon mathematical models alone. However, the kineticmodel has two major contributions to make to plantdesign and operation: it helps in the interpretation andextrapolation of pilot-plant results (an importantconsideration when the labour and time required forextensive pilot-plant tests is considered), and it perm itsthe developm ent of a quantitative relation between plantperformance and the important operating variables, e.g.*Formerly National Institute for Metallurgy, Randburg,Transvaal; now Department of Metallurgy, University of theW itw atersrand, Johannesburg.

recirculating loads, aeration rates, froth flow-rates, andlaunder wash-water rates. These variables determ ine thedesign and size of a plant, and are used by plant opera-tors for control purposes. These matters, and others,have been discussed in some detail in papers that de-scribe the use of a kinetic model for plant-simulations tu die s1 , 2 .

It should be borne in mind that the present investiga-tion was not designed to confirm the superiority of theparticular kinetic model used, but rather to show howfar the model can be used in the application of resultsfrom bench-scale batch tests to the prediction of plantperformance. It is possible that other kinetic modelswould be as effective as that described here. The modelchosen is one that recognizes the basic kinetic nature offlotation and is capable of predicting all the importantplant variables, which include mass flow-rates andgrades in all the streams of the plant, internal as well asexternal. The distributed-param eter model developed bythe NIM Chemical Engineering Research Group1, 3,which classifies particles into discrete classes by size,m ineral content, and flotation-rate constant, was chosenbecause it recognizes that the particles of an ore differand therefore must be treated differently. This modelenables comparisons to be made between the predictedand the m easured particle-size distributions and gradesin the different size classes in concentrate streams. Thishas im portant im plications for the prediction of perform -ance in a full-scale plant, where poor grades are oftenfound only in certain size groups.

For the phoscorite ore used in this investigation, itwas found that the rate constant has to be distributed

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY JULY 1978 325


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over only two classes, i.e. a non.floatable component anda floating component for each mineral species. Only onemineral, apatite, was present w ith the gangue, and,because liberation was complete even in the largest sizes,particles were classified as pure m ineral or pure gangue.Its sim ple particulate structure m ade phoscorite particu-larly suitable for this initial attempt to demonstrate thatkinetic models are useful for bridging the gap betweenbench-scale batch tests and the design of complexco ntin uo us p lan ts.Early experimental work4 on the pilot plant of theResearch Group had demonstrated that the distributed-parameter model, once due recognition had been given tothe individual character of each particle, w as sufficientlyversatile to describe the operation of a complex plantprovided that the kinetic parameters were estimatedfrom data obtained on the plant itself. The next prob-lem was to ascertain whether the kinetic parameters ofthe model had absolute values that were independent ofthe physical conditions of the pulp such as aerationrate, and, more important, that were independent of thephysical nature of the flotation cell. The object was adetermination of whether the kinetic parameters couldbe estimated from experiments in a batch cell and beused successfully in the prediction of behaviour in acontinuous plant. The Group's other experimental workon pilot and full-scale industrial plants has already beenreported6,6.

The estimation of the parameters in the kineticmodel from batch or continuous data can be accom-plished by the use of nonlinear regression techniques 7.Computer programmes8 make the estimation a simple

Tailings

. S am pling p aintF = FlowM= MassL = LevelD = DensityI = IndicatorC = Controller

F la pp er g ate

routine matter, although care is required. to ensure thatexperimental data are satisfactory for the estimationo f th e p arame te rs .

Pilot-Plant LayoutThe construction of the individual cells and the layout

of the flotation sections in the pilot plant ofthe ResearchGroup have been reported4. The chief operationaldifficulty experienced initially involved the m echanismfor supplying dry-ground ore at a fixed rate for thesteady-state operation of the plant. An elaboratecomputer-control system9 was implem ented to produce aslurry of fixed density at a fixed volumetric flow-rate,and, before the present investigation was undertaken,an extensive solids-handling and storage system wasinstalled in the pilot-plant laboratory of the Departm entof Chemical Engineering of the University of Natal,where the investigation was conducted. Because thissystem is ideally suited to the task of providing a fixedflow-rate of ground solids, it was adopted as the solids-feed system to the pilot plant. The plant, which isshown in Fig. 1, has the standard rougher-cleaner-recleaner configuration of flotation plants. The alterna-tive routes for the cleaner and recleaner tailings streamsare shown as broken lines, and were used so that theplant could be operated without recycling the tailingsstreams.

The ore was delivered by road, and the storage bunkerwas protected against the weather. This prevented theweathering of the dry-ground solids during storage, aphenomenon that had caused operating problems anddifficulties in analysis during earlier investigations of this

Water NaOH1IIIIII.JI+RecleanwtailingsRecleanerconcentrate

O re d eliv ery~Bucketelevator Storagebunker

326 JULY 1978Fig. I-Flotation pilot plant

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Particle-sizerange,J-Lffi 39< 45 > 45< 53 > 53< 63 > 63< 75 > 75< 90 > 90< 106 > 106< 125 > 125< 180 > 180< 250 >250Me an s iz e, J -L ffi 30 42 49 58 69 83 98 116 15 2 21 5 30 0Fractionaldistribution 0,2449 0,0278 0,0229 0,0263 0,0427 0,0506 0,0464 0,0549 0,1306 0,1255 0,2270Apatite, % 40,1 40,6 40,1 39,5 38,7 36,6 38,0 37,0 34,2 26,1

TABLE IAVERAGE PARTIOLB-SIZJI DISTRIBUTION AND APATITE ASSAY (BASED ON SEVERAL SAMPLES OF THE FEED MATERIAL)

ore4. The ore was transferred from storage bunkers intotwo of the three blending bins, and thorough blendingwas achieved by withdrawal of the ore from two binssimultaneously and discharge of the mixed-ore streaminto one bin. The normal procedure started with twobins half-filled and the other completely filled. Themixed stream was discharged into one half-filled binuntil it was full, the discharge was then switched to thebin containing the least solids, and the ore was with-drawn from the other-two bins. The blending processwas continued for about 8 hours, giving about 20 tonsof well-blended ore for the pilot-plant studies. Thisensured that the experiments conducted in the pilotplant were free, as far as possible, from the effects ofvariable feed properties. Samples of the feed slurry thatwere taken during the experimental programme showedonly very small variations in grade and particle-sizedistribution. The feed material assayed 32,7 per centapatite on average. The particle-size distribution andapatite assay are shown in Table I.The feed system was designed to operate under auto-matic control so that a very steady stream of slurry

would be supplied to the flotation section. W ith theexception of the level controller in the lower slurry tankand the flow controller on the flotation feed line, all thecontrol loops were closed through a CDC 1700 process-control computer. This proved to be a versatile andvery effective automatic control instrument, and it wasused for data logging during the operation of the plantand for the automatic collection of samples. The closedcontrol loops shown in Fig. 1 were all simple feedbackloops with proportional integral action in the controller.The mass-flow indicator on the dry-solids hopper wasbased on a load cell that indicated the load in the hopper.

The rate of change of hopper load was determ ined by thedifference in the load at IQ-second intervals. The hopperwas filled interm ittently by discharge of solids from thetop of the solids elevator as shown in Fig. 1, and theflapper valve was actuated by the computer, whichallowed the hopper to be filled whenever the leveldropped below a lower preset lim it. Filling was stoppedat an upper preset limit. Solids were circulated con-tinuously through one of the storage bins during theexperiment.Commercial fatty acid (Unitol DSR) was used as

collector, while polyglycol ether (Berol EM U) togetherwith sodium silicate was used to control the properties(quantity, mobility, and stability) of the froth. All theconditioning chemicals were added at the conditioningtank, and no interstage addition was used.Particular care was taken to ensure good sampling ofthe process streams. Automatic mechanical sampling

devices were installed at all the sampling points shownin Fig. 1. The eight samplers on the concentrate streams

automatically deflected the entire stream into the sam plecontainer for a short time as it cascaded over the frothlip. Cuts of 6 seconds were made every minute so thatlO per cent of the streams were sampled. The samplerslOwere actuated by the computer, and the total time forw hich sam ples w ere collected w as determ ined accurately.The samplerslO on the three feed streams and on thecleaner and recleaner tailings streams were narrowsample-cutters that were actuated mechanically and cutthrough the entire feed stream approximately once persecond, and the average flow down the sample line wasapproximately 3 per cent of the total flow. The sampleron the plant tailings stream was a narrow sam ple-cutterllof somewhat larger dimensions than the others, and wasactu at ed automatic all y.A piping layout was provided on the flotation plant

that was sufficiently flexible to allow m any different cellconfigurations to be used. The tw o configurations studiedduring this investigation are shown in Fig. 1. The basicc onfigura tio n w as a stan da rd ro ug he r-c le an er-re cle ane rconfiguration of the type that is normally used in in-dustry for the flotation of apatite. The recycling oftailings from the cleaner to the rougher and from therecleaner to the cleaner is an important aspect of in-dustrial plant configurations, and is usually essential togood overall plant recovery and a high grade of finalconcentrate. It is these recycle streams that are likelyto strain a kinetic model most severely, and a plantconfiguration w ithout recycle w as investigated to providea com parison w ith the standard configuration.

B atc h E xpe rim en tsThe parameters of the ore that were to be used in the

model were estimated from batch tests. A comprehen-sive set of such tests was conducted on slurry that wasmade up of dry-ground solids and had been conditionedin the batch cell. Because small but significant differ-ences in flotation performance had been noted withvariations in conditioning procedure, batch tests werealso conducted on conditioned slurry that had been fedto the pilot plant during continuous plant tests. Thisallowed the parameters to be estimated under the slurryconditions that actually existed in the plant duringthe tests. In two further tests, an assessment was madeof whether the kinetic parameters maintained the samevalues in a cleaner cell as in a rougher cell. This is ofparamount importance because the simulation method isbased on the assumption that the kinetic parameters donot change from stage to stage. In the first of thesetests, the total concentrate was collected from four batchtests and was used as the charge for simulated cleanerbatch flotation. In the second, a sample of the cleanerfeed was cut from the pilot plant and was used as thecharge in the batch test.



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tn the estimation of the kinetic parameters from thebatch tests, cumulative samples of the concentrate werecollected, and the total mass, mass of dry-ground solids,and apatite content were determined for each sample.In the determination of the effect of particle size on therate of flotation, the concentrate samples were graded bysize and each size class w as assayed for apatite.A batch flotation cell that was identical to the cells inthe pilot plant was used for the batch flotation tests(Fig. 2), the volume of air-free slurry in the cell being0,021 m3 at the start of each batch test. The experimentalprocedure was straight-forward and involved chargingof the cell w ith slurry to within 0,25 cm of the froth lipafter aeration had commenced. The slurry consisted of4 kg of well-blended dry-ground solids from the pilot-plant feed system and water. In the experiments forthe estimation of the kinetic parameters in the made-upfeed slurry from the plant, sufficient conditioned slurrywas cut from the plant feed line to fill the cell. No furtherconditioning of this slurry was required. The slurry madefrom dry-ground ore was conditioned without aerationfor 2 minutes with sodium hydroxide (1,27 kgft), followedby 2 minutes with sodium silicate (1,7 kgft) a nd fin allyby 2 minutes with Unitol (0,5 kgft) and Berol EMU(0,13 kgft). During this conditioning period, a layer ofheavily loaded froth built up on the surface of the slurryand was discharged very rapidly over the froth lip assoon as aeration started, which necessitated a slightzero-time correction to each of the batch experiments.The concentrate was collected in cumulative samplesfor about 6 minutes. The first sample was normallycollected in 10 seconds, and the sample time waslengthened as the flow of concentrate abated. The zero-time correction was calculated as follows: the flow-rateof water in the concentrate was plotted against the timeat which the sample was taken, and the curve wasextrapolated through the second and subsequent pointsto the time for the first sample. This defined the rate atwhich water would have been collected had no frothaccumulated during the conditioning. The rate of wateractually collected compared with this extrapolated rategave an estimate of the effective flotation time for thefirst sample, and the origin was adjusted accordingly. Atypical set of data and a sample calculation are shown inFig. 3.

The cumulative recovery of gangue and apatite wascomputed and plotted against time for each of the batchexperiments.

Estim ation of Param etersThe distributed-constant kinetic model for flotation

required the estimation of several parameters. It is veryeasy to produce highly correlated estimates of para-meters that would be useless in testing the predictivecapabilities of the model. The distributed-constant m odelrequired the particles in the feed to be classified accordingto size and mineral content, as well as according toinherent flotation-rate constant. Fortunately, the classi-fication by size and mineral content can be done by.It is shown later that even the simple ore used in this investiga-tion required different form s of cP (D,g) for apatite and gangue.328 JULY 1978

straight-forward physical examination: sieving for sizeclassification, and microscopic examination and assayfor mineral content. M icroscopic examination revealedthat the ore was almost completely liberated, even in thelargest sizes. W hen liberation w as incom plete, it resultedfrom finely disseminated inclusions of magnetite in theapatite. These inclusions rarely exceeded 5 per cent, andthey were ignored in the classification of the ore. Eachsize group could therefore be classified quantitativelyinto an apatite and a gangue fraction by assay alone.Apart from apatite and magnetite, the major mineralsin the ore were phlogopite, calcite, and serpentine. Theparticle-size distribution and assays w ere averaged overseveral different samples as shown in Table I, and theaverage values w ere used in all the calculations.Parameters Associated with Particle Size

O ne further aspect of the particle-size distribution thathad to be considered was the effect of size on the rate offlotation. This effect was modelled on the assumptionthat all the particles having the same mineral contentare affected in the same relative way by particle size,independently of the actual flotation-rate constant ordistribution of rate constants that must be assigned tothat m ineral class12. This assum ption w as accommodatedby factoring out of the specific flotation-rate constant, afunction cP(D,g) that is a function of size and mineralcontent of the particles. Thus, the specific flotation rateof particles of size D and m ineral content g is given bykcP(D,g)SA S-l, w here k is the flotation-rate constant,cP(D,g) is a function showing the dependence of the rateconstant of particles of mineral content g on p artic lesize*, A is the bubble surface area per unit volume ofpulp, and S is the fraction of the bubble surface areathat is not covered by adhering particles.

T he fu nc tio n c P ( D , g ) assumes a very important role inthe model because it defines the flotation properties ofthe particles in the different size groups. The questionof how best to determine the nature of the functionc P ( D , g ) has been given considerable attention by theResearch Group over the years. W oodburn e t a l.1 2 wereable to propose a plausible functional form after ananalysis of the micro-processes of impaction, adhesion,and detachment. This function has two parameters anda unimodal shape, reflecting the fact that neither verysmall nor very large particles float. It has the formcP(D ,g )=2 ,33 (EfD2 )t exp (- EfD2) {1 - (Df6)l,5}.

The last factor in this function accounts for thedetachment of particles once they have adhered to abubble, the parameter 6 representing the largestparticle that can remain attached under the conditionsprevailing in the pulp. The remainder of the expressionfo r cP models the impaction and adhesion processes, andthe constant 2,33 is chosen so that this factor has amaximum value of unity. The function depends on gbecause both the param eters E and 6 take differentvalues for particles having different mineral contents.That this should be so for 6 is obvious because largerparticles of gangue, being held less tenaciously at theair-water interface, are detached more easily than arestrongly adhering m inerals. H ow ever, E should dependvery largely on the turbulent forces in the pulp, and it

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,'." ,",~,'

A SHAFT. I AIR INLET FEED BOXTANDPE FROTH LIP

---

CROSS -SECTION ON A-A FRONT VIEW HALF CROSS-SECTIONScale: Icm = 5 cm

Fig. 2-Details of the Fagergren-type flotation cells

G)-~~G)-0~

Expt 823/1Effectiveorigin

~~of-~-cG)uc0u.5

W ater collected in first sample = 1,59 kgW ater rate in first sample by

extrapolation = 0,082 1,59Effective time for first sample =0,082 =19,4

+100 200 300

Time, sFig. 3-W ater rate as a function of sample time in batch test, showing calculation of effective origin(1 9,4h shou ld read 1 9,45 )JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY JULY 1978 329


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is not so obvious that it should depend on the mineralcontent of the particle. Such a. d ependence can be justi-fied by noting that the first factor in the function rPshould model the adhesion as well as the impactionprocess, and that adhesion is certainly influenced bysurface properties. Careful experim entation in this studyestablished that both parameters are functions of themineral content of the ore that was used.The estimation of and 6. should be done very care-fully. Direct measurement of 6. is possible because it is

the largest particle of a particular mineral type that canbe floated and can be determined by particle-sizeanaly sis of the concen trates, altho ugh the differentiationbetween mineral types is difficult. W oodburn et alPsuggested a rapid method of estimation in which thevalue of is inferred from the flotation behaviour ofp articles o f d iffe ren t siz e. N on lin ear re gre ssio n te ch niq ue shave been used exclusively up to now , but these leavemuch to be desired in precision of estimation andconvenience7. A very reliable direct method was de-veloped by the Research Group, and this is now recom-mended as the standard estimation procedure for thefunction cp. The method has the great advantage thatthe entire form of the function is estimated, whichliberates the model from the constraints of a particularfunctional form in which only the parameters arevariable. The method requires data from a single batchtest or a single test in a continuous cell, and a sampleof concentrate must be graded and each size classassayed so that the recovery of each g-class can becalculated in each of these size classes. In the batchtest, the analysis is best conducted on the first sampleof concentrate that is collected in the test. If the con-tinuous cell forms part of a bank of cells or is part of aplant, the data should be collected from the first roughercell.The theoretical development of the method has been

described elsew herel3. It is show n that, in a continuouslyoperatin g cell,R(g,D)rP(D,g)=C(g) I-R(g,D)' """"" (1 )

where R(g,D) is the recovery of particles of grade g an dsize D in the cell, and C(g) is a constant (independentof D) for each g-class. Thus, rP(D,g) is determined up toth e co nsta nt C(g) by the plotting of R(g,D)/(I-R(g,D))against D. This can be done direct from the experimentaldata.A similar analysis can be applied to the batch tesp3:rP (D,g)= -B(g ,t ) In (I-R(g,D) . . . . . . . . (2)Equation (2) defines the functional form of rp up to them ultip ly in g co nstan t B(g,t), and the function rp can be

obtained from a plot of In(I- R(g,D) versus D directfrom the experimental data.Data from two experiments were analysed so that the

function rp (g,D) for apatite and gangue could be estab-lished under the conditions prevailing in the flotationequipment. The data are plotted in Figs. 4 and 5,according to equation (1) for the continuous test (experi-ments C8/2-I and C25/5-I) and according to equation (2)for the batch test (experim ent B3I/1O -I). The continuousdata were obtained from the first cell in a five-cell330 JULV 1978

rougher bank. In the batch test, the data for the firstsample of concentrate are shown together w ith those forthe composite sample of concentrate collected over theduration of the test.In spite of the scatter in the experimental data, the

characteristic unim odal shape is im mediately apparent,and the peak in the curve for gangue appears at a smallerparticle size than that for apatite, confirm ing that thefunction rp is different for the two mineral species. Theestimation from these data of the two parameters and6. in the function for rp is straight-forward: 6. is obtainedby extrapolation of the data to cut the horizontal axissince this is the largest size of particle that is recovered,and the value of is obtained m ost conveniently fromthe expression =0,5 D2max, where Dmax is the size atwhich the data have a maximum. The function2,33 (D2)texp (-D2) {I-(D/6.)1,5}

with the appropriate parameter values is plotted in Figs.4 and 5 for comparison with the experimental data. Thisfunction w as considered to be a satisfactory representa-tion of the data for gangue, but the small particles ofapatite showed a distinct tendency to float more rapidlythan predicted by the function of Woodlurn et a l.1 2.This tendency can be observed in other oresl4, and canbe very pronounced if there is a significant degree ofentrainment of fine particles in the froth. This has beenobserved with phoscorite ore under conditions of highfroth stability4 . The functional form of rp was accordinglyaltered in the fine particle-size range, as shown by thebroken line in Fig. 4, and this modified form of functionrP(D) was used in all further estimations of parametersand plant simulations. The modification of the standardcomputer programmes MREGR8 and FLOTE15 toaccommodate this empirical function was a simplematter. The function rp that was used in the model isgiven in Table 11 for both apatite and gangue.

TABLE IITHE EFFECT OF PARTICLE SIZE

Mineraltype rP(g,D)rP=2,33 (D2)texp (_D2) (1-D/6.1,5) D?: 58JLm=4,05xlO-o 6.=400xlO-oApatite(g=l)

Gangue(g=o)

4> .=0 ,60 when D=49 JLmq,=0,47 when D=42 JLmcp=0,28 when D=30 JLmrP=2,33 (D2)t exp (-D 2) (1-(D /6 .)1,5 ) fo r a ll D= 1,5 X 10-0 6.=350x lO-om

A significant conclusion to be drawn from Figs. 4 and5 is that the data from the batch and continuous testsare substantially the same. This is an encouraging step inthe application of batch data to the continuous operationof a plant.The K inetic Param etersAs already stated, the main purpose of this investiga-

tion was an assessment of whether the flotation modelwas capable of correctly predicting the performance ofa continuously operating plant from kinetic data ob-tained from standard batch flotation tests. The basicdata that were used for the estimation of the kineticparameters were the cumulative recovery of each m ineral



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..O m a x

!20 30 50 100 150 200Par t ic le size, um

1,28~ 1,1:!:! 1,0a: a:~'::=:' I I 0,9~ Cl - -~ C;""' '" 0,8a:~~~I a:ClCl07c !..C ;~'.5 ~ a: a: 0, 6in - 0 10 0,5r


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0,50,4go

'c 0,3'0e."~c 0,2,~""

~0,1

0,06

species as a function of time in the batch test. Thebubble surface area per unit volume of pulp was calcu-lated from the known aeration rate, average bubble size,and average bubble residence time. The bubble sizedistribution and average size in the cells was measuredby Moys7, and the bubble residence time was measuredas the time taken for the steady-state bubble distribu-tion to appear at the surface of the pulp after a suddeninitiation of aeration. The aeration rate was measuredby a rotameter in the air line. Typical experimental dataare plotted in Figs. 6 and 7, which show the fraction ofeach mineral that remains in the batch cell as a functionof time. These curves show the characteristic curvatureon the logarithmic-arithmetic coordinates, which is anindication of the departure of the process from linearkinetic behaviour. This departure could be ascribed tothree possible causes: distribution of the rate constant,gradual reduction in the froth production capacityduring the course of the experiment, and true nonlinearkinetic behaviour. The last of these three possibilitieswas excluded because, once due allowance had beenmade for the distribution of particle sizes in the feed andfor the decrease in froth production rate w ith time, what

G a n o u e

95% confidence region

< Apatite

F roth transm issio n coefficien t r

e Experimental data- Model10 0 20 0 30 0

Time, SFig. 6-D ata for batch experim ent B8/2

A !ltite

,--r-

0,4O~~o,2~

le Experime11tal dataI.:-- Mode l p rl 'd ic ti onC, I 10 0

c!


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S pecific rateconstant for N onlinear 95 % Fraction of Nonlinear 95 %Experiment floatable component confidence floatable confidenceno . Component m/s X 10' interval component intervalB8/2 Apatite 1,308 1,18 to 1,44 0,844 0,814 to 0,875Gangue 0,205 0,197 to 0,247 0,998 0,834 to 1,00

--. B1O/5-1 Apatite 1,81 1,61 to 2 ,03 0,850 0,820 to 0,878Gangue 0,297 0,281 to 0,365 1,00 0,828 to 1,00

measurement of y. tn this work, y was estimated fromthe ratio of water flow-rate over the lip to the waterflow-rate that was measured in a cell containing fullyconditioned feed pulp to a level just below the froth lip,i.e. the conditions that obtained at the very start of abatch test or in the first cell of a rougher bank in acontinuously operating open-circuit plant. This stand-ard water flow-rate is a measure of the froth productioncapacity of flotation cells when operating on fully condi-tioned pulp. This method for the estimation of y ha salready been used successfully4, 7,1. However, asshown later, it is not considered satisfactory, particu-larly for cleaner and recleaner cells. Other meth::>ds forthe direct measurement of y are currently underinvestigation.During the batch tests, the water production rate

decreased steadily, and consequently the froth trans-m ission coefficient as calculated from the water ratealso decreased. The calculated values of y, which areshown on the graphs, were used in the theoretical modelfor the estimation of the kinetic constants. As shown onthe graphs, y was assumed to be constant during eachs amplin g p erio d.The standard regression computer programme7, R

that was used in the estimation of the kinetic constantsfor the gangue and the apatite incorporates all thenecessary statistical tests to determ ine the number ofsignificant values over which the rate constant must bedistributed. Earlier work on this ore by DaveyI7 andMoys7 indicated that the apatite and the gangue wereeach characterized by two kinetic parameters: thespecific rate constant for the floatable component, andthe fraction of floatable component represented by theu nfloa ta ble c ompon ent.

Simulation of Continuous Plant PerformanceThe crucial test of the usefulness of the model was the

performance of a continuously operating plant based onthe parameter estimates obtained from batch experi-m ents. The pilot plant was operated in several configura-tions, and samples of the various concentrate andtailings streams were assayed for comparison with themodel predictions. In each experiment, a sample of theconditioned feed was floated in the batch cell, and theparameters estimated are shown in Table Ill. The modelprediction of the plant performance is compared with theexperimental data in Figs. 8 to 11. These data give animmediate indication of the ability of the simulator totranslate the data from the batch test to the operation

of a plant m ade up of several cel1sin different conftgura.-tions. Com parisons are shown for the concentrate gradesfrom each cell and for the flow-rates of solids in the cellconcentrates. Of these, the former can be measured withgreater accuracy. The collection of reliable data from acontinuously operating plant is extremely difficultbecause the plant is not at a true steady operating state.The greatest possible precautions were taken to ensureaccurate sampling and steady operation of the plantduring the test, but small variations in froth stabilityand m obility from cell to cell have a disproportionatelylarge effect on the flow-rate of pulp in a plant. This isone of the real difficulties in the developm ent of reliableplant simulators. Not much of the evident lack of fitbetween the predictions and the experimental data canbe ascribed to uncertainty in the estimation of para-meters. The 95 per cent confidence regions around thepredictions that are generated by the 95 per cent confi-dence lim its in the parameter estimates are shown inFigs. 8 to 11. The residual lack of fit must be ascribed toexperimental uncertainty, and to inadequacies and un-certainties in the m odel used for sim ulation.A significant source of uncertainty in the model is the

value of the froth transm ission coefficient, y, in each ofthe cells. In the rougher bank of the open-circuit con-figuration, y was calculated as the ratio of the measured

80 E xp erim ent C 8/2 -1Parameters fram batch test 88/24::".:

-.e

70..,

60

50

.; 40..,0~ 30~'E:t 2010

2 5 84 6 7Cell num ber

Fig. 8--Comparison between simulator predictions andexperimental observations in plant w ith no recycle

..------

TABLE IIIKINETIC PARAMETERS ESTIMATED FROM BATCH TESTS

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY JULY 1978 33 3


10/14

.5;on~ on~ 'E 0,0081; .:! ~ 0,0060 ..~ c~ ~ 0,004;;:: 0u"0"0I-

Experiment C8/2-1Param ete.. fram batch0,012 tut B 8/2

0,010

0,002

0 4 .7 .83 6C en nu mb er

Fig. 9-Comparison between simulator predictions andexperimental observations in plant with no recycle

water rate in the concentrate to the known maximumwater production rate with the ore in the cells used.This was completely consistent with the method usedfor the estimation of the flotation parameters from thebatch tests. The cells in the cleaner and recleaner bankproduced froths of significantly different characterfrom those in the rougher bank. The greater hydrophobi-city of the solid in the cleaning stages seems to preventthe return of solid material from the froth to the pulp.

;!. 60..C l)

-g 50"-l

~ 40-Q.et 30

334 JULY 1978

90 Experiment CI0/5-2Parameters from batch test 810/5-180 Experimental70

20Rougher ...

10 2 3 4 5C ell num ber

although the rate of return of water is high owing tothe rapid rate at which the froth breaks. Under thesecircumstances, the froth transmission coefficient isgreater than the ratio of measured water rate to thestandard rate, i.e. more solid than liquid is transmittedby the froth. Thus, y lies between the water ratio and1,0. It was observed that the transmission of solids bythe cleaner froths was very high, and y appeared to beclose to unity. In the absence of a direct measurement ofy, a value of unity was assumed in each of the cells inthe cleaner and recleaner.

W hen the plant was operated with complete recyclingof the tailings from the cleaner and recleaner bank, thehydrophobicity of the solids appeared to increasethroughout the plant, and it was only in the feed cellthat the froth had the appearance of a typical rougherconcentrate because of the preponderance of fresh feedin that cell. The solids appeared to fall out of the frothback into the pulp at a significant rate, unlike the situa-tion with no recycle. Thus, values of y were chosen be-tween the values calculated from the water rates andunity in each bank: y was taken to be 0,2 in the lastthree cells of the rougher bank, 0,4 in the cleaner cells,and 0,15 in the recleaner cells.

The level of uncertainty introduced by the uncertaintyin y is shown in Figs. 8 to 11, where the simulated outputis shown for the extreme values of y that could be used,i.e. the water rate ratio as the smallest and unity as the

~701Model prediction with 95%confidence regions inducedby uncertainty inparameter estimates

-Recleane...6 7 8

Fig. IO-Grade in concentrates from each cell In plant with recycleJOURNAL OF THE SOUTH AFRICAN INSTITUTE OF M INING AND METALLURGY


11/14

Speci fi c r at econstant for N onlinear 95 % Fraction of Nonlinear 95 %Experiment f lo at ab le c omponen t confidence floatable confidenceno . Component m Js X 1O. interval component intervalB9Jll-1 Apatite 1,02 0,944 to I,ll 0,979 0,970 t o 0,989Gangue 0,324 0,314 to 0,372 0,841 0,824 to 0 ,858BlOJ5-2 Apatite 0,844 0,743 to 0,959 0,943 0,920 to 0 ,965Gangue 0,320 0,274 to 0,367 0,830 0,791 to 0,868

.......E

..~c..uc:0u.1 :..~I.~.....,::lE

Parameters from botchtest B 1 0/5-10.014 E xperim ent C 10/5-2

0,010

largest.By way of comparIson wIth the sImulator predictionsshown in Figs. 8 to 11, another predictionIO of plantperformance is shown in Fig. 12, in which the kineticparam eters were predicted from the measured recoveriesin the four cells of the rougher bank and the frothertransmission coefficients in the cleaner and recleanercells were estimated by matching measured and pre-dicted recoveries of apatite in these cells. Notsurprisingly, the correspondence betw een the sim ulationand the measured data is excellent.

8

Estimation of Parameters in Cleaner FeedThe kinetic parameters estimated from two batch

flotation tests on cleaner feed material are shown inTable IV and Figs. 13 and 14. The slurry used in experi-ment B9/11-1 was prepared from the combined concen-trates of four batch rougher tests. The slurry used forexperiment B lO/5-2 was cut from the feed to the cleanerduring experim ent 010/5-2, and these param eters shouldbe compared with those estimated from experimentBlO/5-1 (Table Ill). These comparisons show that theparameters estimated from the batch tests on the cleanerfeed are significantly different from the parametersestimated from the plant feed at the 95 per cent confi-dence level. However, it is not certain how much of theuncertainty is due to the method of estimation of Y inthe cleaner tests. In both experiments, a constant valueof 1,0 was used for Y,

" '0,008,'"',,'

0,006 ',' cc,:,' .-, 'I,',:,,>// odel predictions with ',~,:," ,"-:95% confidence regions .;',,' '

Cell n umbe rFig. I I-Mass flow-rates of solids in concentrates in plantw ith recy cle

at 40,.:;~>8 30~..~ 20..Q .Cl :

ae.; 40...~co:! 30liQ.Cl : 20

0,004

0,002

2 4 6

Continuous Plant Performance as a Functionof Particle Size

A simulator that is to be useful for the design ofplants, for the modification of existing circuits, and forplant control should provide an accurate and reliabledescription of the behaviour of individual types of par-ticles. T his is m ost conveniently assessed by com parisonof the predicted and measured grades of apatite in theseveral particle-size classes used in the model. Data ofthis kind were obtained in several of the importantstreams of the plant operating with and without recycle.The experimental data are compared with the simulatorpredictions in Figs. 15 and 16, the parameters used inthe simulations being the same as those used to producethe simulations shown in Figs, 8 to 11, The correspond-ence between the predictions and the plant data is goodand, in spite of slight differences in the shape of thecorresponding curves, the trends in the experimentaldata are satisfactorily predicted by the sim ulator.

Fig. 12-Comparison between experiment and simulation

7

TABLE IVKINETIC PARAMETERS ESTIMATED FROM THE FLOTATION OF CLEANER FEED IN A BATCH CELL

3 5

10

~!/0 'f.'/:I.:& 'i. "'"./...;;%

(;;.--

./ijjji.

50.. , Experiment0--- Simulation


.:--.,....--..,...,,~ ::~o--~'

.....-;:;;.00" "

10--

JULY 1978 335


12/14

...ccos~ 0,1c..g 0,07u" 0,05

...og 0,2osE~c02g 0,1~

i,o0, 90,80,70, 60, 50, 40, 3

E& Experime~tal; d~ tajB~~: I~I '-J- Model0,2

reoi


13/14

100

90

. . Model predic tionsEDExpe rimenta l data

~ 80..Cl )-acL.Cl 70Cl )--Q.et 60 ED5010 20 30 40 50 60 80 100 150 200 300 500

Particle size, pmF ig . IS -G ra de a s a fu nction o f p artic le siz e, p la nt w ith out re cyc le

100

90 F in al c oncentra te80

70

60 Ist rougher concentrate~

.. 50Cl )"CCL.Cl

. . Model predic tion" Exp erim en tal dataCl )- 30

10

-~ 20

010 20 30 40 50 60 70 90100 150 200Particle size, pmFig. I6-Grade as a function of particle size, plant with recycle



14/14

References1. KING, R. P. Simulation of flotation plants. Trans. Soc.M in. Engrs AIM E, vo l. 2 58. 19 75. pp. 286 -293 .2. KING, R. P. The use of simulation in the design andm odification of flotation plants. FLOTATION. A. M.GAUDIN MEMORIAL VOLUME. Fuerstenau, M. C.(ed.). New York, AIME 1976. vol. 2, p. 937.3. KING, R. P. A model for the design and control of flotationplants. APPLICATION OF COMPUTER METHODSIN THE MINERAL INDUSTRY. Salamon, M. D. G.,and Lancaster, F. J. (eds.). Johannesburg, South AfricanInstitute of M ining and M etallurgy, 1973. pp. 341-350.4. KING, R. P., e t a l. A pilot-plan t inv estigation of a flotationm odel. Johannesburg, National Institute for M etallurgy,R ep ort N o. 1 57 3. 1 97 4.5. KING, R. P., et al. Application of a flotation model to anindustrial plant. Johannesburg, National Institute forMetallurgy, R eport no. 1 56 2. 1 97 3.6. KING, R. P., and JOCHENS, P. R. Characterization of theflotation properties of fluorspar from sm all-scale batch andpilot-plant tests. Johannesburg, National Institute forMetallurgy, R ep or t n o. 1 55 3. 1 97 3.7. Moys, M . H., et al. Estimation of parameters in thed istrib ute d-c on sta nt flo ta tio n m od el. J oh an ne sb ur g, N a tio n-0 .1 In stitu te f or Me ta llu rg y, R ep or t n o. 1 56 7. 1 97 3.8. Moys, M . H., e t a l. Com puter programm e for the estima-

tion of parameters in flotation. Johannesburg, NationalI nstitu te fo r M eta llu rg y, R epo rt no. 1 52 8. 1 97 3.9. KING, R. P. On-line digital computer control of slurryc on ditio nin g in m in era l flo ta tio n. Automatica, vo l. 10 . 1 974 .p p. 5 -14.10. BucHALTER, E. M . Ph.D . Thesis, University of Natal,1973.11. CRAMER, L . C. M .Sc. Thesis, University of Natal, 1974.12. W OODBURN, E. T., KING, R. P., and COLBORN,R. P. Theeffect of particle-size distribution on the perform ance of ap hosph ate flotation p rocess. Meta ll. T ra ns ., vol. 2. 1971.p . 316 3.13. KING, R. P., and HAINES, A. K. A pilot-plant investigationof a kinetic model for flotation. National Institute forMetallurgy, R ep or t n o. 1 84 7. 1 96 7.14. TOM LINSON,H. S., and FLEM ING, M . G. P ro ce ed in gs o f th e6 th In te rn atio na l M in er al P ro ce ss in g C on gr ess , C an ne s, 1963.Oxford, Pergamon Press, 1965. p. 563.15. KING, R. P., e t a l. A computer programme for the simula-tion of the perform ance of a flotation plant. Revised report.Joh an nesburg, N atio nal Institute for M etallurg y, R ep or t n o.1 43 6. 1 97 3.16. HUTCHINSON, P., and Luss, D . Lumping of mixtures withm an y parallel first o rder reactio ns. Chem. Engng J., vol. 1.1970.17. DAVY, W . An experimental investigation of the con-ditioning of an apatite ore for flotation. M .Sc. Thesis,U niversity of Natal, 1973.

D iscussion: Mathematical unification of an equation for soluterecover ies in countercurrent decantationThe above article by E. Barnea, which appeared in

the January 1978 issue of the Journal (vol. 78, no. 6,pp. 143-145), calls for comment in view of the author'sclaims for the novelty of his proof.He states in his article that its main purpose is to give

a mathematically acceptable proof by induction of the'recently proposed' equation

Co-CwCn- I+R+R2+. . . Rn +cw,(equation 1 in his paper)as well as to simplify this into the form

R-lCn- RnH-l (Co-Cw)+Cw. (equation 3 in his paper)This analysis of countercurrent m ultistage separation

processes was first published in the early 1930s byKremserl, and by Souders and Brown2 for the specificcase of absorption from a gas. They also proved theequations derived by induction. The application of these*Chemical Engineering Research Group, C SIR, Pretoria.

F. L. D . CLOETE *equations to washing, or CCD as it is known in mineralprocessing, is trivial and is given in undergraduate textson chem ical engineering3, 4 and elsew here 5.I do not disagree with the mathematical logic ofBarnea's paper but, contrary to his claim , there isnothing new about it at all. H is paper is, however, agood illustration of an observation attributed to KingSolomon6: 'What has been done before will be doneagain ' .References1. KREM SER, A. Theoretical analysis of absorption processes.N atl P etroleum N ew s, vol. 22. 1930. p. 43.2. SOUDERS, M . and BROWN, G. G. Fundamental design ofabsorbing and stripping columns for complex vapours. Ind.E ng ng C hem., vol. 24. 1932. p. 519.3. M cCABE, W . L., and SM ITH, J. C. Uni t o pe ra ti on s o f c hemi ca lengineering. New York, M cGraw Hill, 1956.4. CO ULSO N,J. M ., and R ICH ARD SO N,J. F. Chem ica l eng in ee r-ing, vol. 11. New York, M cGraw Hill, 1955.5. CLOETE, F. L . D., et al. Hydrometallurgy, theory andpractice. Johannesburg, South African Institute of M iningand M etallurgy, Vacation School, Aug. 1976, Lecture X.6. Ecclesiastes 1:9 .

338 JULY 1 9 7 8 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF M INING AND M ETALLURGY

a pilot plant investigation of a kinetic model for flotation

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