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expected yields a t specified heat valuesor ash values.
However, this conventional method hadthe following disadvantages:a) The method was tedious and time
consuming;b) the data could not be analyzed into
satisfactory depth, due to time l imitat ions; and
c) only ideal separations could beinduced, while the extrapolation topractical expected organic andscreening eff iciencies proved to becomplicated and time consuming.
The need therefore existed to ut i l i ze acomputer simulation model in order toeliminate the disadvantages of the handmethod. The model had to be able toa) accept the raw data in the format
available, calculate the informationrequired to draw washability curvesand present resul ts in a summarizedformat.;
b
c)
d)
simulate the heavy-medium and sizeseparation uni t processes by calculat ing par t i t ion factors, using a history of standard process parametersaccumulated on the Grootegeluk cokingcoal beneficiat ion plant;allow a choice of Epm values, organiceff iciencies and other parameters inorder to simulate, non- ideal ly ,thespecific heavy-media separationcess required a t each point in protheflowline. Simulation of the non-idealscreening process had to be fac i l i -tated by the choice of screeningeff iciencies;compute the total sink/f loatscreen analyses of both productseach separation step, in orderpermit a s tep-by-step developmenta logical flowline.
andoftoof
The model therefore had to be able to296
compute ful l part i t ion curves for ei ther adensity or a screening operation. Thecomputation of separation product charact e r i s t ics could then be fac i l i ta ted .
To evaluate borehole data in addit ionto the above) this model had to be able toa) compute blended washability data for
different boreholes; andb) compute expected yields a t specified
ash, heat value or density cutpoints .By using this model t is clear tha t the
supplied data from borehole and bulksamples could be manipulated extensively,maximising the usage of given data, thuspermitting the development of a more accura te and practically orientated plantflowsheet than would have been possibleotherwise.
Overview of existing modelsThe specif ic requirements as discussedabove, together with the given time cons t raints vir tual ly excluded the use ofan exist ing simulator package. l houghIscor Wd.S negotiating the purchase ofl DDSIM from Prof. R.P. King< 1 of theUniversity of the Witwatersrand, thepackage was not available a t tha t stage,as t was s t i l l being prepared for d is t r i -bution.
The use of other simulation packages wasexcluded owing toa) inabi l i ty to cater for the format ofthe raw data;b) equipment information required by
such packages, which was unavailablea t tha t stage of the design; and
c) the time constraints involved, i . e .purchasing and commissioning.
The decision was therefore taken to usean available APPLE I I microcomputer anddevelop a model in-house. The APPLE wasl a te r replaced by an Olivet t i M-24.
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ModellingHeavy medium separationIscor Grootegeluk had been using a coalwasher performance program for weeklymetallurgical audits for some time already . This program i s similar to the onedescribed by Wizzard(2). An extensivedatabase, including Tromp curve cutpoints ,probable errors and Wolf curve cutpointswas therefore available for incorporationinto the model.Performance curve fittingIn order to determine the metallurgicalperformance of a coal washer, samples ofthe feed, product and tai l ings are takenand applied to a sink-f loat analysis.Discrete Tromp part i t ion factors are thencalculated in the conventional way 3 ) (This i s referred to as the observed part i t ion factors) .
The performance evaluation program f i t san arctangent curve t thepart i t ion factors accordingfollowing equation 4,5):
t = lOO(PIwhere
Arctan(P2(d P3)PI - P4
observedto the
[ ]
t = f i t t ed Tramp part i t ion factor (ie.the probability tha t a coal part icleof a given density, d, will report tothe overflow ) (
d = density ( g/cm3 )PI - P4 = parameters describing the shapeof the specif ic separation curve.
The general shape of this curve(referred to as the f i t ted curve) is givenin Figures 1 and 4.
The use 01 he:: c;. ' ; : . ' i : ~ e n t function dif -1 0 0 r ~
Hx; - Shift for asymmetry
o1,450WWg
1,470
Ideal Epm - no asymmetry
1,490DENSITY (g/cm3)
1,510
FIGURE 1. Plot of the overflow Tramp distribution factor against density showing the development ofthe HMS simulation modelPoints A (d ,,,)
B (d ,5o Tramp cutpointC (d'5,25)I (d;. t; Arctan in flection pointW True mass based Wolf cutpointWg Two-dimensional Wolf cutpoint based on graphic integration
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fers from the method followed by Wizzavd,who made use of a Weibull distr ibution.The arc angent approach is preferred,since t culminates in a two- dimensionalsearch, instead of four, thus reducing thenumber of i terat ions required from +/-1000 to +/- 120, whilst s t i l l obtainingcorrelat ion coefficients of 0,995 andhigher.
The objective of the simulation modelwas simple: to reverse the process operformance evaluation This meant thatstart ing with calculated results available, one had to work backwards unti l theproduct and ta i l ings streams have beenfound.
This was approached as follows:a) Determine which information is
b)
c
d)
el
f
g)
required to solve the four parametersPI to P4 ;Solve the parameters and obtain asimulated Tromp part i t ion curve;Transform the continuous part i t ioncurve into discrete intervals, bymeans of integration;apply the discrete part i t ion factorsto the feed stream washability data;calculate the simulated product and
t ~ i l i n g s streams;compare the simulated streams tothose observed and refine the model;replace the historical performancedata with those desired on the newplant .
arameter solvingThe influence of ~ c h of the parameters Plto P4 is bet ter understood by rearrangingEquation [1] as follows:
t =a Arctan[b (d - c)] + e [2]wheret = overflow Tromp distr ibution factor,
as before( )298
d density g/cm3)a = overall efficiency parameterb = parameter describing sharpness of
separationc = horizontal shif t of the arctan in
f lection point Under symmetricalconditions, parameter e would beequal to 50, in which case c would beequal to the Tromp cutpoint)
e = parameter describing the degree ofasymmetry of the part i t ion curve
Four points, distr ibuted evenly on thepart i t ion curve, are required to solve thefour parameters a , b, c and e. Finding theroot of the Arctan equation, Le theinflection point, would solve parameter cimmediately, thus making i t logicalchoice to find. The points (d2s ,25) andd7 5 , 75 are distr ibuted evenly enough
around the inflection point in order toconsider finding them as well. This leavesonly one point not yet defined.
Ideally, one would endeavour to use theTromp cutpoint in order to find the in flection point and the Ecart ProbableMoyen Epn) as a degree of sharpness tosolve d2s and d75 , since these values havefound widespread application in the coalprocessing industry.
Tramp cutpointAt th is stage i t is not possible tosubsti tute the Tromp cutpoint for theinflection point, because of the asymmetryinvolved. However, i f the degree ofasymmetry is known, t can be correctedfor , by using the l inear relat ionshipassumed to apply in the centre region ofthe part i t ion curve.
Wolf cutpointWhen integrating the top error area fromthe le f t of the curve and a t the same time
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integrating the bottom area from theright one arrives t the point of equalerror area intersection. Under synmetricalcondit ions , th is intersection will coincide with the Tramp cutpoint t t = 50.However, the larger the extent of asynmet ry the larger will be the differencebetween these two cutpoints. Since integrat ion from two sides is nothing elsethan a two-dimensional version of the Wolfcutpoint calculation t followed that thethe difference between the Tramp and Wolfcutpoints was a key in the search for theinflect ion point. Therefore the relat ionship between the difference in the Trompand Wolf cutpoints and the abscissa ofthe inflection point had to be established.
This relationship was determined bymeans of l inear regression on 25 observat ions as
ot
54
46;:::;::l:9bo 34
tI = 50 25 - 377 2 xwheretI =Abscissa of inflection pointx =Tramp cutpoint - Wolf cutpoint
[3]
The relat ionship is considered s ignif i cant with a coefficient of correlat ion of0 91 as exemplified in Figure 2. Furthermore, i f no asynmetry is present onewould expect that with x =0, tI would beequal to 50. The constant in Equation [3]is acceptably close to th is theoreticalvalue t 50 25.
harpness o separationIn order to obtain the ordinat e of theinflection point parameter e i t is necessary to find the horizontal differencebetween th is point and the Tramp cutpointdefined as follows:
3 0 ; ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~-0 002 0 002 0 006 0 01 0 014 0 018 0 022 0 026Tramp cutpoint - Wolf cutpoint g/cml)
o Included ExcludedFIGURE 2. Plot of the difference between Tramp and Wolf cutpoints against the distribution factors ofthe Arctan inflection point.
Liner regression: t; = 50,25 - 377,2 x correlation coefficient = 0,91)
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XI = dI - dso [4]
This can be done by determining thes lope in the l inea r sec t ion of the par t i -t ion curve and applying it to the ver t i ca ldif fe rence , which i s already known (tr -50). Assuming tha t the l inea r re la t ionshipholds from dz 5 to d75, the slope, m i sgiven as:
m = (75 - 25) (d7 5 - dzs) [5]For coal process ing, m i s always negat ive.
Applying the def in i t ion of the EcartProbable Moyen to coa l benef ic ia t ion , wefind:
pm = z 5 - d? 5 ) 2 (posit ive) [6 ]and subst,i tu t ion thereof in to Equation[5 ] , we f ind m in terms of Epm:
I = -25 hpn [7]Solving for asymmetryThe hor izonta l dif fe rence between theTromp cutpoin t and the inf lec t ion point i stherefore:
5Ql [8]m
The in f lec t ion point , dr, may now befound in terms of the Tramp cutpoint , byrearranging [4] :
d I = d s o + X I [9]
Solving d75 and d 5The Ecart Probable i s generally not d i s -t r ibuted symmetrically around dso. Ourresearch h s shown tha t the Epm tends tosh i f t according t the extent of asymmetrypresent and t ha t it i s dis t r ibuted sym-
300
metrical ly around an imaginary axis lyingin an opposi te di rec t ion from ds 0 than thein f lec t ion point , dI , but with equal dis-tance, XI. Therefore, d 7 s and dz 5 may befound by compensating for t h i s s h i f t inasymmetry:
d 7 5 =ds 0 - XI - Epm [10]and
dzs = dso - XI Epll [11]
Solving arctan parametersThe parameters a. b, c and e may now be
solved by ass igning the following i n i t i a lvalues:
a = 100 Irec = dIe = tI
[12][13][14]
leaving b to be solved by subs t i tu t ion in[2] with d, t ) = (d7S. 75).efinement of parameters
fhe above parameters must be re f ined,owing to the approximation IJlade inEquation [3] and the assumption IJlade inEquation [12]. Furthermore, an addit ionalpoint on the Tromp curve i s required s incethe points dz s , ds 0 dr and d 7 5 l i e in anarrow b nd on the curve. Since Equation[12] i s based on an assumption, theaddit ional point required must l i e c lose lyto the end points of the d is t r ibu t ioncurve in order to incorporate the e f fec tof overal l effic iency. The ~ x m u mobserved dis t r ibut ion fac tor was chosenhere, e .g . (1,24 ; 99), r e fe r re d to as dMand tH.
An i t e r a t ive procedure i s then followedwhereby the four calculated da ta pairs aresubs t i tu ted in to Equation [2] , according
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to following four equations:
b =Tan[(t7s-e)/a]d7s -c
a =Arctan[b (dr. : - c)]
and
c = dso - Tan[(tso - el /a]b
[15]
[16)
(17]
[18]
applied in the order as shown.I t was found tha t the se t of parameters
converge within 5 i terations in thosecases where the original ly observed par t i t ion factors were f i t t ed adequately by thearctan curve. However , divergence wasfound in the cases where the observedpar t i t ion curve had exhibited a low ef f i ciency t a i l in the lower density region.This t a i l can often be ascribed to analyt ica l errors made in the laboratory. Thes i tua t ion was rect i f ied by l imiting thenumber of i terations to 2, thus achievinga meta-stable convergence.
iscrete partition factorsThe arctan curve describes a continuouspar t i t ion factor. In practice, discretedensi y intervals are used to obtain thepar t i t ion factors . The par t i t ion factorthus calculated i s associated wi h themidpoint of the density interval . I t istherefore necessary to integrate the simula ted romp curve across the densityinterva l in order to obtain the simulateddiscrete par t i t ion factor for tha t par t i cular interval .
I f T represents the integral of thearctan curve, then
T =a (d-c) Arctan[b(d-c)] e(d-c)a(ln[1+b2 (d-c)2)
2b
which resul ts into
TJ = T(dJ-l }-T(dJ )dJ - dJ-I
[19]
[20]
where TJ is the discrete par t i t ion factorfor the J th interval .Size sep r tionApling 1985) describes a method t.o measure the performance of screens, 6 ) whichis essent ia l ly the same as tha t followedin the coal washer performance program.In this method the natural logaritmn ofscreen aperture is plotted on the ordinateaxis instead of the density.3 and 5 for the general
(See Figuresform of the
dis t r ibut ion curve.)Following the exemplary work of Apling
t was decided to simulate non-idealscreening operations too by means of theTramp par t i t ion curve and to follow asimilar route to the one described abovein generating the par t i t ion curve. Aproblem a t hand was tha t very l i t t l e plant:history wasmethod of
available,computing
since Apling sthe screening
performance curve had not been used at.Grootegeluk a t tha t stage. I t has sincebeen implemented.) The only processparameters available were therefore i)undersize screen efficiency and i i ) thenominal cutpoint.
Model assumptionsAs less information was available thanrequired to describe the par t i t ion curveaccurately, the following assumptions hadto be made (see Figure 3):
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100 . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ _
' 80J-..2uojc.9 60::loD5en: a0-E 40J-.
f 2 f22b
Furthermore, by defining
[25]
the to ta lundersize area as the rectangular block100f, the Tramp undersize efficiency,EffuT, may be defined asEffuT = 100f Eu
fBy entering a required
[26]
undersizeeff iciency EffuR, b can be solved byi terat ion unt i l EffuR =EffuT. Since b canvary between 0,1 and 2000 a geometric
interval halving routine was used unti lEffuT EffUR is less than a preselectedtolerance.
iscrete partition factorsThe discre te par t i t ion factors may onceagain be calculated as discussed previously. Once obtained, these are applied tothe s ize intervals of the feed stream inorder to generate the overflow product)and underflow tai l ings) streams.
odel accuracyHeavy medium separation modelA typical simulated par t i t ion curve lshown in Figure 4, with the observedpart i t ion factors and the f i t ted curvecalculated by the performance evaluationprogram. I t can be seen that both the
1 0 0 l F = = = = = = e = ~ ~ = = ~ : : : : ; : : : : : ~ = = ~ 1
Simulated
O ; - - - ~ - - ~ - . - - - r - - - - - ~ - ~ - . - - - r - - , - - - , - - , ~ - , ~ ~ - - 41,2 1,24 1,28 1,32 1,36 1,4 1,44 1,48
Density gl cml o Observed
FIGURE 4 Plot of overflow Tramp distribution factor against density showing the accuracy of the HMSmodelObserved Determined from sampling.Fitted Curve generated by the performance evaluation program.Simulated Curve generated by the HMS simulation model.
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simulated and f i t t ed curve deviate fromthe observed points only in the highdensity region, by more or less equalamounts.
The accuracy of the heavy-med.iumseparation model is further exemplified inFigures 6 to 8, representing the errorsbetween simulated and actual values a tvarious confidence levels, the la t te rbeing detenninedevaluation program.
by the perfonnanceThese resul ts and
others, are further summarized in Table1.
There i s no doubt tha t other simulatorsmight achieve bet ter accuracies, but thesewere considered accurate enough for thepart icular application, with the advantagethat only process paramaters are ut i l ized.The accuracy in predicting the Wolfcutpoint may be enhanced by compensatingfor the difference between the graphicalcutpointvalue i s
and mass basedmore or less
cutpoint.fixed for
Thisthe
application a t Grootegeluk a t 0,003 g/cm3.(See Figure 1 - points and Wg .
The prediction of organic efficiency mayalso be erilianced by ut i l iz ing more
TABLE 1. Accuracy of HMS model
ParameterConfidence level*
Absolute errors
Concentrate ash,Misplaced material,Wolf cutpoint, g/cm3Epm Ecart )Organic efficiency,Clean coal yield,
50
0,240,60,0020,0011,60,5
80
0,481,00,0040,002
95
1,002,00,0060,004
3,0 9,00,9 1,4
To read as follows: in 50 of the~ e s the simulated concentrate ash
304
differed less than 0,24from the actual.
absolute
accurate fonns of interpolat ion than thel inear fonn that was used. Recent invest i -gations a t Grootegeluk showed tha t alogarithmic interpolation i s by far moreaccurate, while interpolation by usingLagrange polynomials yields unpredictableresults .
Size sep r tion modelAcceptable accuracies were obtainedonly
in those cases where t iS known fromexperience tha t the assumptions made inthe screen model were valid. Further workshowed tha t in other cases acceptableaccuracies could be obtained when thenominal cutpoint was replaced with thecutpoint of inf lect ion, and the t ruepar t i ion factor of inf lect ion was usedinstead of 90 . In other words, (dI , tI )had to be known. This is of course al imiting factor since usage of the co-ordinates of the inf lect ion point is anovel concept introduced in this paper.More research wil l therefore be necessaryto establish the relationship between morewidely used parameters, such as thenominal size, Epm values etc. and thosementioned above.
Furthermore, there is no reason not tofollow the same route as had been used inthe MS model, apart from a lack of adatabase where the relevant parametershave been established. This had onlybecome available af ter the bnplementationof the perfonnance evaluation program forscreening operations.
A typical plot of the f i t t ed and simu-lated par t i t ion curves are shown in Figure5, whilst Table 2 summarizes the relativeHMS errors (root mean squared) of sometypical parameters. Relative errors aregiven since a logarithmic transformationwas used to normalize the size ranges.
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100 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ ~
oo ~ ~ ~ ~ ~ ~ T r T ~ ~ ~ T ~ r n T r T r n ~ r n T ~ ~ ~ ~ T ~ r n T ~ ~1,0 1,5 2,2 3 3 5,0 7,4 11,0 16,4
Screen size (mm) (logarithmic scale)o Observed
FIGURE 5. Plot of overflow Tromp distribution factor against screen size showing the accuracy of thescreening modelObserved Determined from sampling.itted Curve generated by the performance evaluation program.
Simulated Curve generated by the screening simulation model.
TABLE 2 Accuracy o f Screening Model
cutpoint mmcutpoint mm
(Ecar t)U/f ef f i c iency
y ie ldmater ia l
MS er rors
2,13,5
11,91,5 Abs0,8 Abs1,3 Abs
Application of the modeland mining
Waterberg coal f i e ld can be dividedthe Upper and Middle Ecca se r iesthe Lower Ecca i s not developed. The
se r ies cons i s t s o f 11 coalwi th in terbedded shale layers with
zone subdivided in to samples The
Upper Ecca contains br igh t and du l l coalsui table for the production o f cokingcoal . For mining operat ions , 4 bencheshave been developed, with bench 1 as over-burden. The t r ans i t i on zone, bench 5,exhibi ts too high a phosphorus content tobe rendered sui table fo r the product ion o fcoking coal . The Middle Ecca contains nobr ight coal and i s only sui table for theprodL'Ction o f power s t a t ion coal . Thissect ion i s divided in to mining benches 6to 14, o f which bench 14 wil l not be minedas the over ying sandstone l ayer bench13, i s too th ick.
Ten boreholes , spaced over the plannedmining operat ions fo r the next 40 years,were dr i l l ed and analysed, in order todetermine the qua l i ty and expected y ie ldo f the raw coal to be t r e a t ed in theenvisaged benef ic ia t ion p lan t .
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Borehole evaluationThe borehole cores were crushed to -25 DIDand the -0,5 DID fraction removed.Evaluation of the borehole analyses wasdone by using the model to reconst i tutethe various zones from the s mple analyt ica l data, and subsequently the benches.A modified version of the computer modelwas then used to calculate, for eachbench, thea) in-situ characterist ics;b) mass yields a t a density of 2,0
g/crn3 , ideal ly and non-ideally separated;
c)
d)
yields a t other densit ies rangingbetween 1,8 and 2,0 ;yields a t such density where aproduct with a 20 MJjkg heat value isobtained; and
e) sensi t ivi ty analysis on theyields a t other heat values rangingfrom 12 to 22 MJjkg).
The -0, 5 DID fractions were assumed to bebeneficiated by spirals to yield a productof 20 MJ /kg and a discard of 4 MJ /kg; theresults were incorporated into the abovecalculations.
From the in-s i tu characterist ics i t waspossible to determine which benches couldbe mined without any beneficiation apartfrom s ize reduction. From the sensi t ivi tyand constant heat value analyses items dand e above) i t was determined whichbenches could be beneficiated together andwhich had to be beneficiated in separateplant modules.
roduction constraintsThe constraints introduced by the simulta neous adherence to product quality controland pi t development were also el ic i ted, byplanning production, blending options andavailable clean coal stocks for the next40 years. This showed that production306
peaks from the upper benches must occur,during ini i a l production stages, i f thepi is to be developed properly. Thisimplied that cer ta in plant modules had tobe designed for dual purposes in order toeliminate unusable excess capacity oncethese production peaks had been passed.Bulk sample evaluationBulk samples were collected bench by benchand crushed by the primary Bradford breakers in the exist ing plant . Analyticalresults were fed into the computer andextensive simulations performed a t thefollowing cutpoints:a) primary screens: 35, 25 and 15 IIIID,
a t 88 U/f efficiency;b) degradation screens: 30, 20 and 10
IIIID, a t 92 U/f efficiency;c) feed preparation screens: 5, 3 and
1 DID, a t 75 U/f efficiency;d) s ta t ic bath HMS: 1 7 ; 1,8 1,9 and
2,0 g/cm3 a t 0,025 Epm and 90
e)organic efficiency;cyclone HMS: 1,7; 1,8; 1,9 and 2,0g/cm3 a t 0,017 Epm and 90 organicefficiency.
From the above simulations the optimizedcutpoints were chosen and correlated withthe minimum and maximum expected yieldsobtained from the borehole evaluations.Thus the average, minimum and maximum massflowrates could be established for eachstream, the flowline designed in detai land the reticulation balance completed.
Simulation resultsFigure 9 shows a typical simulation studyperformed on bulk sample results frombench 2, which contains 13 coking coal insitu. t shows that with pulp densit iesof 1,8 g/crn3 in the cylcone plant and 1,9g/cm3 in the s ta t ic bath plant , themaximum densit ies achievable with con-
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