Computer method for simulating relative density distributionanalysis with application to gravity separations

K. A. lewis and P. Tucker

SynopsisA computer-based technique has been developed forthe determination of specific gravity distributionprofiles. It has been designed to overcome many ofthe practical and technical problems that are oftenencountered in conventional methods of determiningspecific gravity profiles, such as heavy-liquidanalysis. The new technique relies on a minimum ofeasily obtainable input data (assays and particulatespecific gravity measurements of bulk products) topredict the weights and associated assays of thefractionated products. Details of the method aregiven with practical examples. Results that werecalculated by use of the model are compared withdata collected from heavy-liquid analyses, enablingits scope and limitations to be assessed, and itsapplications are discussed.

The evaluation of gravity separation processes in theminerals industry depends heavily on the acquisition ofspecific gravity distribution data to characterize the materialthat is undergoing separation. These data have widespreadapplications-for example, in establishing the amenabilityof an ore to gravity separation by dense media cyclones,spiral concentrators, etc. I Specific gravity analysis, in theform of partition or washability curves, may also be used tomonitor plant performance.

Specific gravity distribution data are determinedconventionally by heavy-liquid analysis or very similartechniques, but all such methods have practical limitations.A new technique that is based on a computer approach hasbeen designed to overcome many of these obstacles andprovide an alternative to heavy-liquid analysis. It can alsobe used to supplement the data obtained by existingtechniques.

The major limitations that are encountered with heavy­liquid analysis are, first, that the maximum effectiveworking density is often lower than is required tocharacterize metalliferous ores fully-two of the mostcommonly used reagents, 1,1,2,2-tetrabromoethane (TBE)and di-iodomethane, have maximum densities of 2.95 and3.33 x 103 kg/m3, respectively. Clerici solution (an aqueoussolution of thallium malonate and formate) can be used forseparations at densities as high as 4.2 x 103 kg/m3 at roomtemperature and by working at 90°C this level can,theoretically, be increased to 5.0 x 103 kg/m3, but itbecomes very sensitive to any temperature fluctuation.Second, several of the reagents, particularly Clerici solutionand TBE, are highly toxic; many establishments are notequipped with the necessary facilities or safeguards tohandle them. Third, the separation of very fine material«45 /Lm) is both difficult and time-consuming. Fourth,

Manuscript first received by the Institution of Mining andMetallurgyon 18 July, 1985; revised manuscript received on15 November, 1985. Paper published in Trans. Instn Min. Metall.(Sec!. C: Mineral Process. Extr. Metall.), 95, March 1986.© Crown Copyright 1986.

although heavy-liquid analysis is accurate, rapid and,therefore, not expensive under normal circumstances, itscost-effectiveness for the processing of large quantities ofhigh-weight samples into multiple density fractions may bequestioned.

Some of the alternatives to heavy liquids include two­phase solutions, such as Cargille liquids2 and mercury­bromoform emulsions.3 These allow higher maximumeffective densities--of as much as 7.0 and 7.5 x 103 kg/m3,respectively-to be achieved. Both these solutions are,however, extremely toxic and expensive. Cargille liquids arealso restricted to applications with relatively coarsematerial. Another alternative is the use of paramagneticsolutions.4 These liquids can produce working densities asgreat as 20 x 103 kg/m3, though 12 x 103 kg/m3 is theusual limit. This technique is restricted to non-magneticmaterial and, again, the acquisition of the reagents may beprohibitively expensive. Density tracers are nowcommercially available for the determination of specificgravity partitions at operating plant level, but only forrelatively low densities (<3.6 X 103 kg/m3) and coarsematerial (> 1 mm).

None of the mentioned techniques is free of restrictions,either technical or practical. The new technique that hasbeen developed surmounts the majority of these problems.It has no definite upper boundary with respect to maximumworking densities. It is equally applicable to all particle sizeranges and can provide a more rapid analytical method,relying only on measured data that can be obtained easily(assays and particulate specific gravity measurement).

The technique can be applied wherever heavy-liquidanalysis is used. Applications include the estimation ofspecific gravity distributions both at bench scale and inoperating plants, eliminating any need for heavy-liquidanalysis. It can also be used to supplement the measuredresults from a low-resolution separation to achieve a higherresolution, or it can extend the analysis to higher specificgravities. Another important application is that it canproduce data in a suitable format for computer simulationmodels. Many of the gravity process models require feedcharacterization in terms of size and specific gravitydistributions as model input data. Few processing plantshave the resources to collect the necessary data. The use ofthe new technique overcomes this problem and helps tomake computer simulation a more attractive proposition.

An additional important feature of the technique is that itprovides estimates of mineral or metal assays in eachspecific gravity fraction automatically without the need forany specific gravity fractionation of the product.

Method

The approach that has been adopted is flexible in that it canbe applied either to samples of whole products-feed,concentrate or tailings--or, more usually, to samples ofindividual size fractions of those products. The latterapproach has been found to provide a more accurateprediction, especially with respect to the assays. Themethod may be applied simultaneously to all the products

C35

Cw = y (8)

NV = VA + VW (7)

(6)

(5)

(4)

WJK(J = feed) = I WJK(J = other products)J

NV = NA .NF + NF(NP - a)

NE = NP(2 + NA)

Equation 5 can be rewritten as

where VA is total number of unknown assays (the AKMvalues) and VW is total number of unknown partitionedweights (the WJK values). The suite of equations can bewritten in conventional matrix form

where NF is number of specific gravity fractions to begenerated. a equals zero for single products that are treatedindividually (NP = 1) or where the flow streams aroundthe separation device are not all specified. If each flowstream is specified, a complete balance can be obtained. Inthis case the number of unknowns is reduced (a = 1)because of the constraints on product weights

where NP is number of products and NA is number ofmineral species or metals for which assays are performed.The suite of equations will contain a number of unknowns,NV (that is, all the WJK and AKM values), which isgiven by

(1)

or

Conservation of mass

where WJK is weight of material in the Kth specific gravityfraction of the Jth product and TJ is total weight of the Jthproduct.

NF

L WJK= TJK=I

into and from a separating device (e.g. a spiralconcentrator), resulting in a simple balance, or it may beapplied to anyone product individually.

For the purposes of the present explanation 'product' canrefer either to a whole product (i.e. flow stream) or to aspecific size fraction of that product. For each product(J = 1 to NP) the model re'quires measured input data forthe weight (total weight, TJ' of material in that product),the specific gravity (average particulate specific gravity, DJ'of the product) and the mineral or metal assay. At least oneassay, BJ' is required for the product. More can be includedsubject to constraints that are outlined later.

The solution technique is based on fundamentalconservation laws. For fractionation into NF specific gravityfractions these laws can be expressed in the followingforms.

Conservation of mineral (metal) species

NF

L WJK·AKM = TJ.BJMK=I

or

(2)

where AKM is Mth mineral/metal assay related to the Kthspecific gravity fraction and BJM is Mth mineral/metal assayrelated to the Jth product. This equation is generated foreach mineral species or metal for which an assay isperformed (M = 1 to NA).

Conservation of volume

where w is a vector of unknown density partitioned weights,y is a unit vector and C is a matrix whose elements are thecoefficients of W as expressed in equations 1,2 and 3.Constraint equations (equation 6) are included ifappropriate. C is a NE x VW rectangular matrix.

A simple solution of equation 8 is not possible becausethe unknown assays, VA, are elements of the matrix, C,and, in general, C'is not square. A solution can 'be obtained,however, if estimates, VA*, of the unknown assays, VA,are provided (i.e. all elements of C become specified). Thegeneral solution is then given by5

(9)

where CT is transpose of C. Provided that NE is greaterthan VW (i.e. the system is overdetermined), w' is the least­squares solution to equation 8. That is

NF

\' WJK = TJL PJK DJK-I

or

(3)

where DJ is average particulate specific gravity of the Jthproduct and PJK is mean specific gravity of the Kth densityfraction of the Jth product.

Each equation is repeated for each product in turn (i.e.for J = 1 to NP).

The laws that are detailed above provide a suite ofsimultaneous equations. The number of equations, NE, ineach suite is given by the expression

(10)

is minimized. If this solution is then embedded in an

optimization routine, to optimize the assay estimates, a setof assay values, VA', that corresponds to the minimum ofthe least-squares solutions can be generated. The valuesVA', -w' (for VA = VA') make up the model-calculatedresults. The optimization technique that was used in thiswork was the simplex method of Neider and Mead. 6 Matrixinversion was performed by application of the Gaussianelimination method.

If NE is greater than or equal to NU, the techniqueguarantees the best-fit least-squares solution. A satisfactorysolution is, however, still possible even if NE is less thanNV, provided that the inequality NE > VW remainssatisfied. The solution that is identified in this case will bethe real solution so long as a realistic initial assay estimate,

C36

naterials of a tin ore were sampled from a spiral:oncentrator. The samples were divided into six sizeractions, each of which was subdivided into three specific:ravity fractions. All 18 categories were assayed for tin. Itan be seen that the product assays compare well with eachther and with their associated feed assays for low andItermediate grades of tin. For higher grades the scatter inIe points increases. This can, however, be attributed inart to the limited weights of the samples-that is, sample:presentation was probably poor.

1.-8 5-0I.-I.1.-03-632

Specific gravity

28

Fig_ 2 Estimation of distribution curve with model usedindependently of heavy-liquid analysis

100

90

80

70

~ 60u8. 50

_~ 1.0

o 30:;E 20"

<">10

o2-0

Partition analysisThe results presented so far are all examples of an extremeuse of the model-the generation of specific gravitydistribution data without heavy-liquid analysis at all. Fig. 3,a partition curve to spiral concentrate for the + 180-,...mfraction of another tin ore, illustrates the application of themodel in conjunction with limited heavy-liquid analysis. Asmooth curve was obtained in this example, whichdemonstrates that precision was maintained. It should beremembered that assay values (not shown) have also beengenerated for each calculated point.

ValidationValidation of the model was carried out with a variety ofores. A summary of the results compounded from the

Results

C37

Specific gravity distributionsThe model was tested with a variety of materials. Table 1shows the results for a silver slag. Feed and productmaterial were collected and partitioned into three sizefractions. The mean particulate specific gravity of eachproduct-size category was then measured with an airpycnometer. Each product-size category was assayed forsilver. The model was applied to these data, simulating aspecific gravity partition at sp. gr. 3.3. After prediction ofthe results a heavy-liquid analysis was carried out with puredi-iodomethane (sp. gr. = 3.3) and the predicted andmeasured results were compared. Very close agreement wasobtained between the measured and calculated partitioned ­weights and little discrepancy is seen when the assay resultsare compared.

Tables 2 and 3 show the results of a similar exerciseapplied to two size fractions of a tin ore (-125 +90 and- 180 + 125 ILm, respectively). Three specific gravityfractions were generated, simulating partitions at specificgravities of 2.8 and 3.3 (Table 2) and 3.3 and 4.2 (Table 3).It can be seen that, although the predicted weights areaccurate, some loss of precision occurs with the predictedassays when three specific gravity fractions are generated.The predicted assays, however, still serve as an excellentguideline to the underlying assay trend.

The sequential application of the model is exemplified inFig. 2, which shows the specific gravity distribution in thespiral concentrate of a tin ore: the model was usedindependently of any heavy-liquid analysis. Although asmooth curve was not obtained, a reasonable estimate ofthe behaviour of the ore on being subjected to gravityseparation has been made. The horizontal bars show thedistribution as determined by heavy-liquid analysis.

100-1 1-0

Grade Sn02 in feed

0-010-01

Fig_ 1 Comparison of product assays for individual size-specific~ravity categories (dOlS, concentrates; plus signs, middlings; crosses,

:ailings)

U 1-0"'0ea.~0'c­

oo

..'0o~ 0-1

10

UA*, is chosen. Convergence to the solution has been rapidin all applications to date; convergence to a rogue solutioncan easily be identified.

The foregoing solution technique has proved mostaccurate when the generation of specific gravity fractions(NF) is limited to two or three. Additional fractions, ifrequired, can be produced most reliably by a sequential useof the model, though this application is limited to singleproducts (NP = 1). The solution is achieved by using thepredicted results from one step as the input data to thenext. The model will generate all the input data that arenecessary for the subsequent step.

The only assumption that was made during thedevelopment of the technique relates to equation 2. Toobtain a solution to the problem when NP exceeds unity itis assumed that when a category of given size and specificgravity, e.g. the fraction 0.1-Q.2 mm of sp. gr. > 4.0, ispartitioned by a separating device it retains the samemineralogical composition in all products of the separationand only the proportion of this fraction that reports to theproducts is affected by the separation. This assumption willbe rigorously true only if all the minerals that report to thatcategory are liberated or if the size-specific gravitycategories are very narrow. It follows from this that themost accurate results will be obtained if the material is sizedbefore application of the model.

The validity of this assumption was investigated for amore general case (Fig. 1). The feed and other product

Table IComparison of calculated and measured heavy-liquid analysis results for silverslag material

Product

Specific gravityProportion in eachAgassay, %band

band,wt%Measured

CalculatedMeasuredCalculated

-90J-LID

Feed-3.3 84.4184.13

+3.315.5915.87

Concentrates-3.3 79.01

79.96 ~0.160.18

+3.320.9920.04 1.401.48

Tails-3.3 93.9091.90

+3.36.108.10

-180+9O/LID

Feed-3.3 89.6088.71

+3.310.4011.29

Concentrates-3.3 46.32

46.90 ~0.140.16

+3.353.6853.10 1.451.55

Tails-3.3 97.1098.05

+3.32.901.95

+ 180 /LID

Feed-3.3 75.31

75.02 ~+3.324.6924.98

Concentrates-3.3 20.57

21.01 ~0.150.14

+3.379.4378.99 2.222.39

Tails-3.3 89.32

90.10 ~+3.310.689.90

Table 2

Comparison of calculated and measured heavy-liquid analysis results for tin ore

Product

Specific gravityProportion in eachTina5~y, %(-12S +90 1LD1)

bandband, wt%Measured

CalculatedMeasuredCalculated

Feed

-2.817.0715.050.022.8-3.3

70.1372.330.37+3.3

12.8012.624.39Concentrates

-2.89.068.92

om ~.

0.022.8-3.3

38.9938.350.220.37+3.3

51.9452.736.454.33

Middlings-2.818.6018.17

om!2.8-3.3

70.4068.750.25+3.3

11.0013.083.92Tails

-2.815.2515.80

0.02 !2.8-3.3

70.4068.750.19+3.3

5.902.544.06

Table 3

Comparison of calculated and measured heavy-liquid analysis results for tin ore

Product

Specific gravityProportion in eachTin assay, %(-180 + 12S ILm) band

band, wt%Measured

CalculatedMeasuredCalculated

Feed

-3.380.0981.110.213.3-4.2

3.132.894.78+4.2

16.7816.005.55Concentrates

-3.328.5129.28

0.28 t

0.223.3-4.2

17.2318.144.894.14+4.2

54.2652.584.894.41

Middlings-3.375.0091.23

0.22 !3.3-4.2

0.850.652.33+4.2

54.2652.584.38Tails

-3.397.9196.39

0.22 !3.3-4.2

0.260.504.80+4.2

1.843.114.72

.'C38

Fig. 4 Comparison of model predictions with partitioned weightsas measured by heavy-liquid analysis

validation exercises is presented in Figs. 4 and 5. For thepredicted weights (Fig. 4) accuracy is, generally, very high:the discrepancy between the calculated and measuredweights is normally within ± 0.1 %. The isolated points thatdo not give this precision can be attributed to small sample

The predicted weights and the predicted assays bothappear to be more sensitive to the specific gravityparameters-the specific gravities of the products and themean particulate specific gravities (PJK) for the partitions.Although large errors in the latter would produceunpredictable results, they are unlikely fer narrow densitybands and an intelligent estimate is sufficient to yieldreliable results. More restrictive is the error that is caused

by uncertainties in the specific gravities of the products(Fig. 7). This is the most sensitive of the input parametersand large errors could be introduced in this way. A 5°"error in input data may produce an error of IS°" in thepredicted results. With an air pycnometer a specific gravitymeasurement that is accurate to ± 2°" can generally beachieved, which allows sufficient accuracy to be maintained.

A sensitivity analysis was applied to investigate the effectthat the accuracy of the measured input data has on theaccuracy of the results. In the simultaneous solution ofmodel equations for the products around a separatingdevice errors in the weight of a specific product (e.g. thefeed) will cause errors in the weight predictions for thefractions of that product and also in those for the fractionsof the other products (concentrate and tailings) and theirassociated assays. These interactive errors are, however,minimal. Errors in product assay data also have a minimaleffect on the predicted weights, but may introduce errorsinto the predicted assays. The error in the predicted resultsis approximately equal to the error in the measured data(Fig. 6).

50

4540~ 35

'x x/~ 30

~x /~250~ 20u

~~ 15 lL 10

5

xx

a0

203040-40 -30-20-10 10

Sensitivity of method

Fig. 6 Effect of errors in product assays on predicted partitionedweights-plot of results of sensitivity analysis

weights, which would cause poor sample representationand, thus, poor measured data.

The discrepancy between the measured (mean for allproducts) and predicted assays (Fig. 5) is higher than thatseen for the weights. The results for the relatively lowassays are normally excellent, but for the higher assaysmore scatter occurs. The results that were obtained are stilluseful, however, as they provide a reasonable quantitativeestimate and do so very cost-effectively.

The same trend of increasing scatter on the points withincreasing mineral concentration was identified in Fig. 1and explained through an assumption about mineral assays.As this scatter also correlates with the coarser size ranges,poor mineral liberation is indicated. The predicted assaysare also sensitive to errors in the product assay data, andthis may well be responsible for some of the scatter. Amaximum relative error of ±5% for the predicted assay cannormally be achieved with this method.

10

100

9

90

x

8

8070605040

3 4 5 6Measured assay per cent

30

Measured weight per cent

2

20

Fig. 5 Comparison of assays predicted by model with measuredassays

>- •o::: 5o~4":;~ 3ou

10

9

_ 7c•.u<; 6a.

c•.~ 70•.a.

80

90

8

100

20

Specific gravity

Fig. 3 Generation of partition curve from model in conjunctionwith some heavy-liquid analysis (M, measured points; C, model­calculated points)

1·0 /~C09•.

08

~ 07

X

~ 061Mv

g 0·5XC

vX{~Q-4 c: 0-3 /.Q Z 02

fa·,

/,xM x---1 Ma

2-02-42-83-23·64,04-4485·0

L 6001

-iij

~ 50"0•.C 40:iv;3 30

C39

When the use of an air pycnometer is not practical, i.e. withsmall sample weights, a density bottle can provide therequired accuracy. Although this sensitivity could berestricting, it has not yet been found to be a prime source oferror.

50x

x45 \ /40

~ 35•• 30

..g' 25Cx

~ 20 '"x

~ 15 /x,,-

10 x/x5~

0-12

-8-404812

Percentage change

Fig. 7 Effectof errors in product densities on predictedpartitioned weights-plot of results of sensitivityanalysis

Major errors that have been encountered have alwaysbeen linked to poor representativeness of the samples. It isprobable that in such circumstances the results calculatedwith the model are more accurate than those achievable byconventional measurement.

Conclusions

A new technique that is based on fundamental physicalprinciples and that overcomes many of the difficulties ofconventional methods has been established for thesimulation of specific gravity distribution analysis. Thedifficulties include the limitations with respect to themaximum working densities of the reagents, their toxicity(which often causes handling problems) and also theirprohibitive cost.

The new technique can provide coarse estimates ofspecific gravity distributions in the absence of heavy-liquidanalysis. When, however, the model is used in conjunctionwith some measured heavy-liquid data it can extend andenhance the data, maintaining precision. It also offers anadded advantage of providing a quantitative guideline to theassay of each fractionated product without the need forfractionation and at no additional cost. These specificgravity distribution data can then be used in a manneridentical to that for conventionally collected data.

Applications of the technique include (1) the generationof estimated density distribution curves to predict theresponse of an ore to gravity separation-for example, bydense media methods-a long established practice, asdescribed by Ruff; 7 (2) the checking of measured data foraccuracy and the identification of any errors; (3) the

C40

generation of distribution data for missing or very smallsamples; (4) the extension of analysis to higher specificgravities, finer sizes or both; and (5) the generation of datain a suitable form for gravity process simulation models, 8

which overcomes a previous limitation of these models inreducing the resources that collection of the input data hasdemanded. The last feature enables computer simulationmethods to become a practicality in gravity processingplants; because of the current interest in simulation modelsthat are applicable to the mineral processing industry it mayproduce the most important application of the newtechnique .

To summarize: used with easily obtainable weight, bulkspecific gravity and assay measurements, the model canpredict specific gravity distribution data and associatedassays for each partition. The technique reduces theinvestment of both manpower and reagent that is needed toobtain the data.

The solution technique has been written in FORTRAN77 and is available for use on IBM personal computercompatible systems.

ReferencesI. BignellJ. D. Prediction and assessment of gravity separatorperformance from heavy liquid data. In Proceedings of the EleventhCommonwealth Mining and Metallurgical Cungress, Hong Kong,1978Jones M. J. ed. (London: IMM, 1979), 123-31.2. BrowningJ. S. Heavy liquids and procedures for laboratoryseparation of minerals. In!- Cire. U.S. Bur. Mines 8007,1961,14 p.3. Desnoes A. Utilisation des suspensions denses de mercure dansIe bromoformeau laboratoire. Revue Ind. minlir., 47, 1965,33-8.4. ParsonageP. Small-scaleseparation of minerals by use ofparamagnetic liquids. Trans. Imtn Min. Metall. (Sect. B: Appl.earth sei.), 86, 1977,B43-6..5. Penrose R. A generalisedinverse for matrices. Proe. Cambrophil. Soc. math. phys. Sci., 51, 1955,406-13.6. Neider J. A. and Mead R. A simplex method for functionminimization. Computer J., 7,1965,308-13.7. Ruff H. J. Dense media separation as applied to aggregatepreparation. Quarry Mgrs J., 45, 1961,228-34.8. Tucker P. et a/. A mathematicalmodel of spiral concentrationas part of a generalisedgravity-process simulation model, and itsapplication at two Cornish tin operations. In XVth I1!ternationalmineral processing emigress, Cqnnes, 1985 (St-Etienne: EditionGEDIM, 1985),vol. 3, 3-15.

Authors

K. A. Lewis obtained the degree of B.Sc. in geologyand computersciencefrom the Council for National AcademicAwards in 1975.She has been employedat Warren Spring Laboratory, Stevenage,England, since 1972.Her current position is that of scientificofficerin the software development section of the Minerals andMetals Division of the Laboratory.

P. Tucker graduated in physics from Oxford University in 1974and wasawarded a Ph.D. from the University of Newcastle uponTyne for research in geophysics in 1978.Since 1980he has beenemployedat Warren Spring Laboratory, Stevenage,England. Hiscurrent position is that of senior scientificofficerin the softwaredevelopmentsection of the Minerals and Metals Division of theLaboratory.