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J. S. A t,. In st M in . M eta l/., vol. 85, no. 4.A pr. 1985. pp. 131-135

Statistical aspects of materials balancing in theminerals industryby C.A. HUNT. and A.L. H INDEt

SYNOPSIST he p rob le m of m ate ria ls b ala ncing is revie we d w ith p articu la r re fe ren ce to th e m in era ls in dustry. A b ro ad sta tistica lfram ew ork is established that, besides extending the classical estim ation problem , allow s questions concerningthe value of sam ple inform ation to be addressed. The follow ing are exam ples of such questions: How m uch moneyshould be spent on sam pling? W hich streams in a plant should be sampled? W hat should be sam pled? The approachadopted makes use of Bayesian analysis. The discussion includes an illustrative exam ple involving a sim ple flota-t ion c ir cu it .SA MEV A T TIN GD ie probleem van m ateriaalbalansering w ord in oenskou geneem m et spesiale verw ysing na die m ineraalbedryf.D aa r w ord 'n b re e statistiese ra am we rk da arg este l w at, b en ew en s d ie uitb reid in g van d ie kla ssie ke ram in gsp ro ble em ,dit oak m oontlik m aak om aandag aan vrae oor die w aarde van m onsterin ligting te skenk. D ie volgende is voorbeeldevan sodanige vrae. Hoeveel geld m oet daar aan m onsternem ing bestee word? Van watter strome in 'n a an leg m oetdaar m onsters geneem word'? W aarvan m ,)et daar monsters geneem word? D ie benadering wat gevolg word, maakge bru ik va n B ayes-o ntle din g. D ie b esp re kin gs slu it te r illu stra sie 'n vo orb ee ld va n 'n e en vou dig e flo ttasie krin g in.

IntroductionIn the m inerals industry, there is a constant need toin cre ase m in era l re co ve rie s su bje ct to g ra de a nd to nn ag ec on stra in ts. T he e ffe ctiv en ess o f a ny stra te gy to mee t th isobjective is clearly dependent on the degree of quan-titative inform ation that is available concerning thes te ady-s ta te b al an ce o f ma te ria l flow ra te s i n a g iv en p lant.O bviously, if no inform ation is available, it is im poss-ible to implement a meaningful strategy. On the otherhand, the installation of sophisticated on-stream in-strum entation and the execution of extensive sam plingcam paigns on a routine basis can be a very costly exer-cise. C learly, a trade-off m ust exist betw een the costs ofobtaining extra inform ation about the m ass balance ofa circuit and the net profits that are likely to accrue asa result of this extra inform ation. Unfortunately, thep rob lem of fin ding an op tim al trade-off is aggravated bythe fact that m ineral processes are inherently stochasticin their b ehavio ur. A ccord ing ly, inform ation obtainedab out the m ass balance of the p lant sh ould be interpretedin statistical rather than d eterm inistic term s. T his paperprovides a basis to m eet this requirem ent, its m ain aimbeing to indicate how a statistical approach can be usedin the estimation of the monetary value of sampleinformation.

P re liminary Cons iderat ionsAt the outset, some of the terms that will be used inthe discussion are defined. Firstly, the plant will beregarded as a network of nodes c on ne cte d b y n umbe re d* A nglovaal L im ited, P .O . B ox 62379, M arshalltow n, 2107 T ransvaal.t Gold E xp lo ita tio n L ab ora to ry , C hamb er o f M in es o f S ou th A fric aR esearch O rganization, P.O . Box 91230, A uckland Park, 2006Transvaal.@ The South A frican Institute of M ining and M etaIlurgy, 1985.SA ISSN 0038 - 223X /$3.oo + 0.00. Paper received 28th A ugust,1984.

arcs. T he nodes represent the physical units in the plants uch a s f lo ta tio n cells , g rin ding mills , a nd hyd ro cyclone s.The arcs represent the pulp flows between units. One isu su ally in te re ste d in c erta in species in the pulp. For ex-ample, for W itw atersran d o res, species of interest m igh tbe gangue, gold, sulphur, and possibly particle size. Ingeneral, product and recycle stream s can be sam pled toobtain m easurem ents of the co ncentratio n of the variou sspecies. E stim ates of the m ass flow rates of each speciesin these stream s, relative to the m ass flow rates of the cir-cuit feeds, can then be calculated from thesem easurem ents. In practice, it is virtually im possible toextract perfectly rep resentative samples from a plant, letalone m ain tain the p lant und er co nditions o f ideal steadystate. It is therefore not surprising that redundant andre plic ate mea su rements a re u su ally in co nsiste nt a nd le adto uncertainty with regard to the estimate of a uniquemate ria ls b ala nc e. T he e sta blishment o f a n a cc ep ta ble a ndlogical procedure for the adjustment of the rawm easurem ents to give a self-consistent m ass balance isreferred to as the ma te ria ls b ala nc e p robl em .The ta sk o f a dju stin g c on ce ntra tio ns o r a ssa ys h as b ee ndescribed in m any papers, a good review being that byReid et al. I. The general approach in these papers is thedesign of a com puter-based algorithm to perform a non-linear m inim ization of som e error criteria, such as thesum of squares of adjustments made to the observeda ssa ys. T hu s, th e a lg orithm y ield s a n e stimate o f th e 'tru e'balance at the tim e sam ples w ere taken. Som e w orkers2also provide an idea of the error in the estim ate.T he em phasis of the present paper is on the statisticalinterpretation, rather than algorithm ic aspects, of themate ria ls b alan ce p ro blem .

Principles of the V alue of Sam ple Inform ationT his section gives an overview of basic concepts andthe general structure of the approach. Som e specific ap-JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY APR IL 1 98 5 131

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plications of the approach are outlined.Varia nc e Costs Mone yThere are m any sources of error when a plant is sam p-l ed to e stablis h its s te ady- sta te compos itio n. The se s ou rc esinclud e ran dom proce ss disturban ces, im perfect sampl-ing, and assaying or instrum entation errors. G y3 gives afull treatm ent of the system atic and random errors thatare likely to occur in practice.C ontrol action is usually taken as a result of inform a-tion obtained about the m aterials balance. If no actionis ever taken, there would of course be little point insampling at all. Control action may take a variety ofform s, such as a change in the set-point of an autom aticcontroller or a change in reagent dosage to a bank offlo tatio n ce lls. If the ba lan ce is ex actly true (w ith an errorvariance of zero), correct action can be taken and no lossis incu rred . H ow ev er, the le ss p recise (w ith a hig her erro rv aria nc e) th e mate ria ls b ala nc e, th e g re ate r th e e co nom iclo ss b ec au se , o n a ve ra ge , th e re su ltin g c on tro l a ctio ns w illdepart from the ideal. It follow s that the error distribu-tion of the estim ated m aterials balance determ ines the

e xp ected eco nomic loss.In the quantification of the expected loss, it is im por-tant that em phasis should be placed on the variables thatare critical for the control of the circuit. In this paper,it is assum ed that the solids recoveries at each separatornode, given by the vector 9, are th e critical va riablesabo ut w hich in form atio n is so ug ht. T hus, th e estim ationof the error v arianc e o f 9 w ill d ete rm in e the exp ecte d lossof a g iv en samp ling situatio n. T his is a co nv enient rath erth an a n ecessary assumption . A simple a ssumption a lon gthese lines would be as follows:Loss = k(wl uT + Wz u~ + . . . + Wnu~ , (1 )

where n is the num ber of separator nodes, uf is the errorvariance of the estim ated solids recovery at node i, andW j is the relative weight attached to uf. The factor ktranslates lo ss into mon eta ry term s an d de pen ds on man yitem s, such as the econom ic value of the plant products,the response of the plant to various design and opera-tional changes, and the nature of the control strategy.From Equation (1) it follow s that perfect inform ation(w ith samp lin g-e rro r v aria nc es o f z ero ) wou ld imply z eroloss; but the greater the error variances, uf, the greaterthe econom ic loss.Even if no idea about the loss function is available,the theory of this paper can still be used to provide usefulinform ation concerning confidence regions for 9 underdifferent sam pling schem es, w hich can be com pared ona q ua lita tiv e b asis. T his a pp ro ac h is u se d fo r c on ve nie nc ein the illustrativ e example giv en later, w here it is assum -ed that loss is proportional to the area of the 80 per centconfidence region for 9.

E rror D istrib ution o f S olids R ecoveriesIn the calculation of the error distribution of solidsrecoveries at each node, it is im portant that there shouldbe an error m odel for the sam pling and assaying errorsof the original data. It will be assumed that the errorsin the measured assays follow a multivariate norm ald is trib ution w ith z ero mean s and covaria nc e ma trix 2: .13 2 APR IL 1 98 5 JOURNA L O F THE SOU TH A FR IC AN IN ST ITU TE O F M IN ING AND MET ALLURGY

The specification of the m odel in this case reduces to aspe cificatio n of the elem en ts o f the co varian ce m atrix 2:,o r e qu iv ale ntly th e e rro r v aria nc e o f e ac h in div id ua l a ssa yand the covariance betw een each pair of assays. T his in-formation can be obtained by the taking of replicatesamples from each stream . Often the model can besim plified if the relative standard error is allow ed to beconstant for a given species. However, it should be ap-preciated that the covariance term s are not necessarilyzero; if tw o m etal species have the sam e m ineralogicalo rigin , o ne w ould e xp ect th eir sampling e rrors at th e sam esam pling point to be positively correlated. Sim ilarly, ifsiz e grad ing s are u sed, the errors in differe nt siz e cla sse sat the sam e sam ple point w ill be negatively correlated.Once the sam pling-error m odel has been derived, thee rro r d istrib utio n fo r th e e stimate d so lid s re co ve rie s4 , 9,can be calculated. Factors that are taken into account in-clude the structure of the network and actual values ofthe assays. T his latter point is illustrated by the fact thata sp ecie s that has th e sam e assay va lue in all stream s givesno inform ation at all about recoveries. In general, them ore m arkedly a species concentrates around a node, them ore inform ation it gives about recoveries in that node.

Implementa tio n a nd A pp lic atio nsThe user must first contribute a loss function and anerror m odel as described above. In som e instances it m ayb e possible for kn ow n inform ation ab ou t 9 to be specifiedbefore any sam ples are taken. O ne obvious assum ptionm ight be that the recoveries m ust lie between zero anduni ty . I n Baye si an5.6 ana ly si s, th is in fo rma tion i s exp re ss -ed in the form of a prior distribution for 9. The qualityof the inputs, of course, w ill determ ine the quality of theoutputs. The net output, after the sam ple data, x, havebeen taken into account, is a posterior distribution for9, writte nf(9 1 x ). T his d istrib utio n summariz es a va ila bleinformation about the unknown 'true' value of 9.W hatever loss function is specified, the Bayesianestimator for 9 is that value El which m inim izes thep oste rior exp ected lo ss; th e m in imum thus o btain ed is theBay esia n risk for the situation, and will depend on thesam pling schem e and the observed sam ple x. The V alueof Sam ple Inform ation (V SI) is defined as the reductionin Bayesian risk achieved by the incorporation of thesam ple inform ation. In other w ords, it is the differenceb etw een prio r B ay esia n risk an d po sterio r B ayesian risk.Thus, VSI com pares tw o states of inform ation: beforethe sam ple is taken, and after. V SI depends on the actualvalues of the data. In an otherwise identical situation,one sam ple m ay yield m ore inform ation than another.The Expected VSI (EVSI) is the expected value of VSIover a prior distribution for the sam ple results, x. EVSIis thus m ore appropriate in a com parison of the averageperform ance of different sam pling schem es.The fo llowin g a re some u se fu l a pp lic atio ns o f th e a bo veprocedure.. The planning of a sam pling schem e for a plant at thedesign state. Som e of the inputs w ould have to com efrom knowledg e gained from p rev io us ex perie nce, orfrom sh ort-te rm re se arc h p ro je cts o n p la nts o f sim ila rdesign.

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Stream Cu, % Zn , % Fe, %1 0,409 9,31 13,472 0,462 11,73 13,563 0,406 8,72 13,454 0,964 38,85 14,485 1,009 37,01 14,476 0,649 51,70 14,56

J o i n R ou gh er c e lls

. The m odification of sam pling schem es on existingplants. EV SI w ill help one to decide, for exam ple,whether to introduce new sam pling points or evendrop all assays of a certain species on economicgrounds. T he sam e statistical fram ew ork could alsobe adapted for an evaluation of the advantages ofdirect estimation of mass flowrates by the use ofm agnetic flowmeters and nuclear density gauges.O th er a pp lic atio ns th at merit c on sid era tio n in clu de th ed etectio n of ou tlyin g assay s and the design ing of tests forbias at a particular sam ple point.

Il lu st ra tiv e ExampleThis example (Fig. 1), which is adapted from Sm ithand Ichiyen2, serves to illustrate som e of the conceptsdescribed in the previous sections. Fig. 1 show s the net-w ork un der c on sideration . T his ne tw ork comprises a zincrougher-flotation cell in closed loop w ith a cleaner cell.Assays are taken on all the arcs (num bered from 1 to 6),and in each arc the percentages of zinc, copper, and ironare determ ined. T his circuit is param eterized in term s of8 = (It,(h), where 01 is the solids recovery to the con-centrate for the rougher and 02 is th e c orre sp on din grecovery for the cleaner. T he data used are summarizedbelow:

(leaner cell s

@10,6491[51,70 )114,56f

Legend:0 Stream or arc designation

I 1 Per cent copper assay[ ) P er cen t zin c assay1 f Per cent iron assay8 , R ou g h e r so li ds r ecove ry ,s tr eam C9/s tr eam @8 , Cleaner solids recovery, st ream@/st reamC9

F ig .1 -D eta ils o f th e c irc uit fo r th e illu stra tiv e e xample

These data have already been sm oothed and corres-pond to the solids recoveries 8 = (0 ,1 0 ,1 25 ). Smooth -ed data are used because the actual smoothing process

is not central to the theme of this paper; rather, the in-formation value of the data is being em phasized. Oneconsequence of the use of sm oothed data is that the peakposterior probabilities in Figs. 2, 3, and 7 all occur atthe value 8 = (0,1 0,125), w hich m akes com parison be-tw een all the diagram s som ew hat easier.It is assum ed that the econom ic loss is proportional tothe area of the 80 per cent joint confidence region for01 and 02 . The greater the area of an 80 per cent jointconfidence region, the vaguer, and hence the lessvaluable, is the information about 8. The figure of 80per cent is merely a convenient choice. In Figs. 2 to 7,sev eral con to urs o f th e join t distribu tion o f 01 an d 02 aregiven to help the reader visualize the surface. The con-tours are draw n at fractions of the peak height; thus the0 ,6 con to ur co nn ects all p oin ts h aving a p ro bab ility den -sity eq ua l 6 0 per cen t o f the pe ak prob ability d ensity . T heexact probability content of any particular contour canb e d ete rm in ed by nume rical inte gration, bu t, b y an alo gyw ith the b iv ariate no rm al d istrib ution , th e ap prox im atecontent of the k-contour is 1 - k. Thus, t he app roximatecontent of the 0,2 contour is 80 per cent.

F or the example th e following assumptio ns w ere m ade .. N o p rior inform ation ab out 8 is assum ed.. V arian ces of assaysw e re taken as k now n w ith arela tiv e stand ard erro r (c oefficie nt of varia tio n) o f10 p er cent for copper, and 5 per cent for both ironand zinc.. Except where otherw ise stated, no correlations be-tw een assay errors w ere assum ed.F rom th e a bo ve in fo rmatio n, th e p oste rio r d istrib utio nf(8 Ix) can be calculated, where x represents the dataused. T he details of this calculation require a B ayesianapp ro ach to multivariate statistics4 .7. A summary of th e

procedure for the calculation of f(8 Ix) is outlined below .. For each 8 and each species, w rite the m ass balanceequations in the formB(8)x = 0,where B(8) is the m atrix

[HL 6j 01 - O281(1 - OJ g,]1 - 01 010 -10 0-10-1and x is the vector:x' = (Xl' X2' X3' X4 , XS ' xJrep re se nting the tru e assay Xi in each stream i.

. Find an orthonorm al basis W (a 6 x 3 matrix) forth e feasible sp ace of x, such th atBW = 0

and W' W = I, the 3 x 3 identity m atrix.A lth oug h B and W both depend on 8, this fact issuppressed to m ake the notation clearer.

JOURNA L O F TH E SOU TH A FR IC AN IN ST IT UT E O F M IN ING AND MET ALLURGY APR IL 1 98 5 13 3

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. If E is the covariance matrix for the species, writez = E-v,x

and U = E-Y'W,where E - Y,is a symm etric square root of E -I. Then,the posterior density can be calculated as

f(9 Ix)oc IU' U I -v , exp[ - Yzz' (1- U(U' U)-l U')z].T his calculation m ust be done for each species, andth e p ro du ct o f th e re su ltin g p oste rio r d en sitie s fo rm -ed, on the assum ption that the errors for each speciesare independent of those for other species. For eachof Figs. 2 to 7, f(9 Ix) was calculated at a grid ofpoints, and approxim ate contours were drawn in.Figs. 2, 3, and 7 show the posterior distribution for9 based only on assays for copper, zinc, and iron respec-tively. It can be seen that the zinc assays contain far m ore

inform ation about 9 than the copper assays do, w hile theiron assays are virtually useless. T he uselessness of theiron assays arises from the fact that these assays arealm ost uniform throughout the netw ork.The fact that zinc assays give m ore inform ation thancopper assays is partly due to the assum ption in the errormodel th at the cop per a ssa ys ha ve a g rea ter re lative error,and partly due to the fact that zinc separates m ore selec-tively in the netw ork.0,6

0, 5

0, 2;:- 04'" .>0u~'"0

D. 4

~ 0,3 0,6~'":"'"aJN 0,2

0,1

0,00,0 0,3 0,4,1 0,2

8, : R o u g h e r s olid s re co ve ryF ig . 2 -C on to urs o f p os te rio r d istrib utio n fo r s olid s re co ve rie swhen only copper assays are used

;:- 0, 3'"0u'"'"0~ 0, 2'""'"Q) 0, 1

0, 4

0, 00,05 0,10 0,15

8, . R ou gh er so lid s reco veryF ig . 3 -C on to urs o f p os te rio r d is trib utio n fo r s olid s re co ve rie sw hen only zinc assays are used

It is important to notice that, while the error modelallows for sm all relative erro rs in the assay s, the relativestandard errors for the solids recoveries (especially (}2)are relatively la rg e. T hu s, th ere is more unc ertain ty in thee stimate d re co ve rie s th an in th e o rig in al a ssa ys. A no th erway of looking at this is that small adjustments to theassay s d o n ot im ply that th e co rresp on din g rec ove ries areclose to the 'true' values.Fig. 4 arises from the com bination of copper and zincassays. W hen Figs. 3 and 4 are compared, it can be seenthat not much extra information (VSI) about 9 is pro-vided by the inclusion of copper assays w hen zinc assaysa re a lre ad y a va ila ble .

~ 0,3'"0~

'"0~ 0,2'""'"cD 0,1

0,4

0, 00,05 0,15,10

8, : R o u g h e r s o l i d s r e c o v e r y

F ig . 4 -C on to urs o f p os te rio r d is trib utio n fo r s olid s re co ve rie sw hen copper and zinc assays are used and no error correlationI s assumed13 4 APR IL 1 985 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

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Fig. 5 gives an analysis for zinc assays only, with nosam ple taken from arc 3 (rougher tailings). W hen Figs.3 and 5 are compared, it can be seen that, by the omis-sion of sam ple 3, the inform ation is decreased substan-tially (and hence the expected loss is increased). In-terestingly, b ecause the contou rs are exten ded mostly inthe direction ()I = ()2' the inform ation about ()I + ()2isaffected m ore drastically than is the inform ation about() ! - () r

0,4

;:- 0 ,3QI>0uQIV>-c51 0,2~QIc:

'"QIUID 1

0,00,05 0,10 0,1581 : R o u g h e r s olid s r ec ov er y

F ig . 5 -C on to urs o f p oste rio r d is trib utio n fo r so lid s re co ve rie swhen only zinc assays are used and no sample is taken fromstream 3

0,4

;:- 0,3QI>0uQI

0,15

V>~~ 0,2~QIc:

'"QIU

cD ' 0,1

0,00,05 0,10

81: R ougher solids recovery

F ig . 6 -C on to urs o f p os te rio r d is trib utio n fo r s olid s re co ve rie sW hen zinc and copper assays are used and an error correlationof 0,60 is assum ed

Figure 6 show s the effect of a change in the error m odelto allow 0,6 correlation between the error in the copperand zinc assays at the same sample point. Figs. 4 and 6show that in this case only a sm all difference in the VSIoccurred. Thus, although correlations should be inclu-ded in the m odel if they are known to exist, their exclu-sio n may n ot n ec essa rily c ha ng e a ny c on clu sio ns from th eanalysis.1,0

0,4

>-QI>0~V>-c0V> 0,5-QI