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Minerals Engineering Vol. 1 No.2, pp. 97-111,1988Printed in Great Britain 0892-6875/88 3.00 0.00Pergamon Press pIc

DEVELOPMENT OF A MULTICOMPONENT-MULTISIZE LIBERATION MODEL

J.A.HERBST, K.RAJAMANI, C.L.LIN & J.D.MILLERDept. of Metal lurgy, Univers i ty of Utah, Sa l t Lake City , Utah 84112Received 29 Apr i l 1987; r ev i s ion 5 November 1987

ABSTRACTThis paper is concerned with the development of a kinetic model formineral l iberat ion by grinding. Existing population balance sizereduction models are extended to include two minerals with an arbitrarynumber of locked particle fractions. The validity of th is approach toliberation modelling is demonstrated for the ball mill grinding of acopper ore in batch laboratory and continuous pilot plant mills.KeywordsPopulation balance; size reduction; liberation; grinding model

INTRODUCTIONLibera t ion of valuable minera l s by comminution pr ior to concent rat ion may bethe most impor tan t s e r i e s o f process s teps occurr ing in a minera l processplan t . Although s i g n i f i c a n t advances have been made in terms of modelling andcharac te r iza t ion of both s i z e reduct ion and concen t ra t ion processes dur ing thepas t few decades , t h e s tudy o f l i b e r a t i o n , by c o n t r a s t , has r e c e i v e dre l a t ive ly littl at ten t ion . The pr inc ip le reasons for th i s i na t t en t ion appearto be the ted ious exper imenta t ion requi red and the lack of a methodology fort he q u a n t i t a t i v e i n t e r p r e t a t i o n of the r e s u l t i n g d a t a . Since p a r t i c l eminera log ica l compos i t ion as we l l a s s i z e i s impor tan t in c o n c e n t r a t i o nprocesses, a l ibera t ion model i s es s en t i a l i f the s imulat ion of in tegra ted oredressing plan ts i s to ever be developed.The first documented at tempt to develop a l i b e r a t i o n model was repor ted byGaudin [1] . He proposed a model to explain l i be ra t ion by size reduct ion in abinary mineral system cons i s t ing o f a r egula r a r r ay of e lements . Later Wiegel[2,3] randomised the arrangement of the mineral gra ins in the Gaudin model ,thus y ie ld in g a more r e a l i s t i c descr ip t ion of the l i b e r a t i o n process . Thisl ibe ra t ion model has a l so been combined with a simple ba tch g r i n d i n g modelthrough t he use o f d i r e c t i o n a l co e f f i c i en t s , which give the quant i ty o fl iberated mineral gra ins produced by the reduct ion of large locked p ar t i c l e s(4) . I n s t e a d o f an ac t u a l p h y s i ca l model f o r minera l l i b e r a t i o n , o th e rinves t igators have out l ined methods which use computerized image process ingsystems fo r charac te r iz ing the ex ten t o f l ibera t ion in mul t iphase minerals .King 5) developed a model based on the l i nea r in te rcep t length through theminera l g r a in s o f an unbroken o r e sample . These d a t a a r e then a d j u s t e dmathematically and t ransfo rmed i n to a l i b e r a t i o n model. This model has beenrecent ly discussed and evaluated by Finch and Petruck [6 ] .In another approach spec ia l a t t e n t i o n has a l so been given to the determinat ionand use o f s p e c i f i c s u r f ace a r e a of mineral in tergrowth for assess ing thedegree of l ibera t ion [ 7 ] . R e c e n t l y , us ing t he Monte Car lo s i m u l a t i o n andassuming random f rac ture , Klimpel and Austin [8] developed a l ibera t ion modelfor a coal -ash system and for a s y n t h e t i c o r e o f p o l y s t y r e n e , p y r i t e andquartz system.Andrews and Mika 9) combined l ibera t ion with s i ze reduct ion kine t i c s in acomprehensive model fo r l i b e r a t i o n in a ba tch mi l l . This work has beendiscussed r e c e n t l y by s ev e r a l i n v es t i g a to r s [10 ,11] . In more recent work,using a s i mi l a r approach , Ruebush , Herbs t and Rajamani [12) developed aM 1/2 A 97


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98 J. A HERBST t al

s i m p l i f i ed t h r ee component s i z e r educ t ion -mine ra l l i be r a t i on model . The l a t e rmodel combines both l i be r a t i on and s i z e r e duc t i on i n t o one s e t of k i n e t i cequat ions which i s s imple enough to admi t a closed form a n a l y t i c a l s o l u t i on . As imula tor , c a l l e d SIMPLANT which uses t h i s s i z e r educ t ion -mine ra l l i be ra t ionmodel , was a l so developed t o p red ic t the performance o f an i n t eg r a t ed gr indingand f l o t a t i on c i r c u i t . Most r ecen t ly , a parameter es t imat ion program c a l l e dESTILIB was developed by Peterson [131 to reduce the number o f experimentst ha t a r e needed to f ind the l a r ge number o f model parameters from l i be ra t ionda ta . Fur t he r , a new exper imenta l t echnique which involves an image analys issystem t ha t can be used to measure t he pa r t i c l e l i be r a t i on quan t i t a t ive ly wasdeveloped a t Unive r s i t y o f Utah [14-161.In t h i s p a p e r , e x t e n s i o n s o f t h e t h r e e componen t s i z e r educ t ion -minera ll i be r a t i on model a r e cons idered . Volumetr ic grade d i s t r i bu t i ons es t imated froml i ne a r grade measurements are used to es t ima te t he model parameters . Fur ther ,a copper ore i s used to eva lua te t h i s t ype of model from batch gr inding andp i l o t p lan t open c i r c u i t t e s t s .

MODEL DEVELOPMENTThis s ec t i on de s c r i be s t he work done to extend s t anda rd p o p u l a t i o n ba lancemode l s f o r s i z e r e duc t i on to accoun t f o r l i b e r a t i o n k i n e t i c s i nvo l v i ngmixtures of mine ra l s . For s ing le component s i z e r educ t ion a s i ze d i s c r e t i z ede q u a t i o n i nvo lv ing s e l e c t i o n func t ions and breakage func t ions f o r each s izec l a s s i s used. The l inea r s i z e - d i s c r e t i z e d model for t he breakage k ine t i cs ofa s ing le component in a batch m i l l i s obta ined by div id ing the assembly beingground i n to N narrow s i z e i n t e r va l s , Xi, Xi+1), i 1 , 2, - -N. A mass balancefor the m a t e r i a l i n t he i - t h s i ze i n t e r v a l a t t ime t y ie ld s for i = 1, 2, --N:d Hmi t )d t 1

Here, mil t ) i s t he mass f r ac t ion of m a t e r i a l in the i - t h s i z e i n t e r va l and Hi s t he t o t a l mass of mater ia l hold-up) be ing ground. The s i z e d i sc r e t i z e ds e l e c t i o n func t ion , Si denotes the f r a c t i o n a l r a t e a t which m at e r i a l i s brokenou t o f t he i - t h s i ze i n t e r va l . The s i z e d i s c r e t i z e d breakage func t ion , bi j ,r ep r e s en t s t he f r ac t ion of t he primary breakage produc t genera ted from the Jth s i z e i n t e r va l t ha t appea r s in the i - t h s i z e i n t e r va l .This concept can be extended to the gr inding of a multicomponent mixture . Fora binary minera l sys tem, M c l a s s e s o f pa r t i c l e s can be r ep re sen t ed based onmine ra log i ca l compos i t ion: a f r ee A type pa r t i c l e a va luab le minera l ) , a f reeB t ype p a r t i c l e gangue ) , and M-2 s e t s o f locked AB p a r t i c l e s . When f reepa r t i c l e s break, only smal l f r e e pa r t i c l e s o f t he same m i ne r a l s pe c i e s arep r oduced . On t he o t h e r hand, when locked pa r t i c l e s break, they can producef r ee A, f r ee B, and a l l M-2 types of locked p a r t i c l e s . Figure 1 schemat ica l lyillustr tes t h e c o n c e p t o f g r i n d i n g w i t h r e s p e c t to b o t h s i z e andmine ra log i ca l compos i t ion .Us ing E q . 1 ) and t he above d e s c r i p t i o n , a s e t o f s i m i l a r e q u a t i o n s forl i b e r a t i o n / s i z e r e duc t i on can be w r i t t e n f o r each o f t h e M c l a s s e s o fp a r t i c l e s , i = 1 , 2 , - - ,N:For f r ee A-type pa r t i c l e s :

d tA A i 1 A-A-So Hm t) + E b i jj =1M-2+ Ek=1

SA AJ Hmj t)

2.1)


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Multicomponent-multisize liberation model 99

CONCEPTUAL REPRESENTATION OF LIBERATION PROCESS

WNn

j 1

j 1

RADE I MINERALOGICAL COMPOSITION JFree A AB 1 ocked AB AB-2 Free B

~ - 1

Fig . Pa r t i c l e i den t i ty change due to s ize reduct ionFor f ree B-type p ar t i c l e s :

d H m ~ t )d t

For locked

d HmlAB)k

dt

- 5 ~

M 2+ Ek=1

AB)k

( t )

H m ~ t ) +

type, k

AB)k

M 2 i 1E E1= j =

i 1 B-BE bi jj =

1 , 2, M 2

AB)kHmi ( t )

AB)k- AB), AB), AB),b i j Sj Hmj ( t )

2.2)

2.kl

where H i s constan t in a batch b a l l mil l . The super sc r ip t in Eq. 2) ind ica testhe type of p ar t i c l e to which the parameter appl i e s , and in the case of thebi j , the type of breakage t h a t i s occurr ing . For ins tance:

appl ies to locked AB p ar t i c l e s in the k c lass composition) of s i ze j breakingto l i be ra ted A o f s i ze i The mass f rac t ions are def ined so t h a t a t any t imet :


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100

N:i=1

A B[m i t) + mi t)

J. A. HERBST t 01

M-2 AB)k+ : mi t )k=1 3)

cons t ra in t s are imposed upon the breakage funct ions in order to sa t i s f y themass conse rva t ion equa t ions for the mass of mater ia l and t he mass of eachcomponent. These cons t ra in t s take the form:N A-A: b i j for each j , 1 :i j :i N-1i=j 1 4.1 )N B-B: b i j for each j , 1 :i j :i N-1i=j 1 4 . 2 )

and for locked AB)k type k = 1 , 2, ..... , M-2N AB)k-A AB)k-B M-2 AB)k- AB)1: b . . + b i j + : b i ji=j+1 1=1

For each j 1 :i j :i N-1 4.k)When the se lec t ion and breakage funct ion are independent of both s i ze cons i s tin b a l l m i l l and t i m e , t h e model i s sa id t o be l i n e a r wi th cons tan tcoe f f i c i en t s . This wi l l be t he case cons idered here. Under such condit ions , as e t of MxN s imultaneous l inear d i f f e r en t i a l equat ions der ived from Eqs. 2.1) ,2.2) and 2 .k) , can be represented as a s ing le matr ix equat ion of the form:d ,(t)d t

= - - ~ ~ ~ ( t ) A t) 5Figure 2 shows a l l the matr ices o f dimensions MN x MN. Fi n a l l y , Figure 3i l l us t r a t e s how t hese matr ices are combined to form the overa l l matrix

JiASjAB Q. . . s=_ BA rAB M 2 B M ' 2 - A B M ' 2 0- = =aAB1 A SA A-= :::=:: == oMNxMN MNxMN

Fig .2 Def in i t ion and s t ruc ture of the matr ices used in Equation 4


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Multicomponent multisize liberation model 101

_ l_BAB1AB1IsAB, BABM'2AB1SABu2= = = = =

A=-( 1-8 IS===: t ~ A B M 2 ~ A ~ a A ~ A SABl= _ ~ _ ~ A A I ~ A

_[I_B8-BIS B= = =

Fig .3 Combination of the matr ices of Figure 2

SOLUTION OF MODEL EQUATIONSFor a batch mil l with i n i t i a l s i ze d i s t r i b u t i o n ~ A T C H O ) , a formal ana ly t i ca lsolut ion to Eq. 5) can be obta ined using s tandard matr ix techniques, i . e . ,

~ A T C H t ) = exp t} ~ A T C H O ) 6The matr ix exponent ia l appearing in Eq. 6) i s not in genera l eas i ly evaluatedcomputat ional ly. Two cases , involv ing one locked c lass and then more than onelocked c l a s s , are t rea ted below.Case I : Three Component SystemFor a t h r ee component system, there i s only one locked c l a s s . In t h i s caseprogeny locked p ar t i c l e s wi l l be cons idered to have the same m i n e r a l o g i c a lcomposit ion as the parent locked p a r t i c l e s . Then, the matr ix A of Eq. 5) i slower t r i angula r in form as shown in Figure 4. A more c o n v e n i e n t s o l u t i o nar i ses for th i s case making a s imi l a r i t y t ransformat ion on matrix A of Eq. 5)such t h a t :

7 )Here i s t h e mat r ix of eigenvector s o f A and A i s a diagonal matr ix ofe igenvalues of A In order to use th i s t ransformation, t i s necessary t h a t as e t of 3N l i n ea r ly independent eigenvectors be iden t i f ied for the A matr ix .Based on the study by Ruebush [121, three l inear ly independent e igenvec tors doe x i s t fo r t he ze ro-va lued eigenvalue of mul t ip l ic i ty t h ree . Therefore, theana ly t i ca l so lu t ion of Eq. 6 ) , f o r three component system, can be r ew r i t t enas:

8The matr ix t ) i s a diagonal matr ix which i s equal to exp { ~ t } .

_[ 1_ BAB ABI SAB= = = o=_[J,-t AJA ~

_[I_BB BISB= = =Fig .4 A matr ix of Equation 5 fo r th r ee component case


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102 J. A. HERBST t al

Case I I : Four Component SystemI f the number of locked f rac t ions i s grea te r than one, the matr ix a of Eq. S)s no longer lower t r iangular . The so lu t ion o f equa t ion 6) c a n e i t h e r beo b t a i n e d n u m e r i c a l ly o r w i th s p e c i a l man ip u la t i o n o f A. The specia lmanipula t ion of 4 for a four component system s given as follows:

The matr ix 4 can be wri t ten as:

AB -AB ABlB 2

AB -A1

o o= =

o=

With the use of an elementary t ransformation , column 2 of the matr ix can bemult ipl ied by:

and added to column 1 .This opera t ion can be expressed as follows:

f * i:1 x

al l 0 0 0 I r 12 0 00 0 0 I 0 0

same as A 0 0 0 I 00 0 0 I

where

AB2AB2 AB 2 -AB2 ]-112 = l [Therefore = ~ - 1where = II 0 0r l2

0 I 0 00 0 I 00 0 0 I


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Here -1 and a are upper and lower t r iangular matr ices , respect ively. In t h i scase the so lu t ion of Eq. 6) can be expressed as:~ t )

I t i s p o s s i b l e to s i m p l i f y t h e m a t r i x e x p o n e n t i a l s by a s i m i l a r i t yt ransformation to give :

~ - 1 = ~ R ~ - 1~ t ) = ~ ~ t ) ~ - 1 I .(O) + ~ t ) ~ - 1 I .(O)where ~ , 1:~ , ~ = matr ices of e igenvec tor of a- 1 and a respect ively,= diagonal matrix of eigenvalues of ~ - 1 and respect ivelyand ~ t ) i ~ t ) are equal to x p { ~ } and e x p { ~ } .The recurs ion formula as for the elements of the Eigenvector Matr ix aregiven in Appendix 1.For the case of M grea te r than four components, a simi lar procedure can befollowed.

MODEL VERIFICATION FOR THREE COMPONENT CASEThe genera l approach followed for model ver i f i ca t ion involved conduc t ing aser ies of bench scale wet batch gr inding experiments using a copper porphyryare from which model parameters could be est imated based on volumetric gradedis t r ibu t ion . These parameters were then used to ca lcu la te model predic t ionsof pi lo t scale open c i r cu i t gr inding and compare them to ac tua l open c i r cu i tgr inding of copper porphyry ore .Model for the Est imat ion of Volumetric Grade Dis t r ibu t ionModel p a rame t e r e s t i m a t i o n sh o u ld be b a se d on t h e v o l u m e t r i c g r ad ed i s t r i b u t i o n o f t h e p a r t i c l e p o p u l a t i o n . The measured l i n e a r gradedis t r ibu t ions must be converted i n to three-dimensional information in order top r ed i c t t r u e minera log ica l composi t ion dis t r ibu t ion . A model [17] has beendeveloped and t es ted to convert the one- or two-dimensional resu l t s in to threedimensional in format ion . For a monosize sample , t he r e l a t i o n sh ip betweenl i n e a r grade d i s t r i b u t i o n f 91) and volumet r i c grade d i s t r i b u t i o n peg)r e q u i r e s i n f o r m a t i o n about t h e t r an s f o r ma t io n f u n c t io n , H 91\g,Nn) , acondit ional probabi l i ty funct ion. The t ransformation can be expressed as:

f 91 )1 H 91\g,Nn)p g)dgo 9 )

The t ransformat ion funct ion has been establ ished from computer s imulat ion. Ont h i s ba s i s , the t r ansformat ion equa t ion , Eq.9, has been solved and t es tedagains t experimental depth prof i l e measurements for di f feren t ore par t i c les a tthe Univers i ty of Utah [16, 171.In pract ice , par t i c les of speci f ied s ize were mounted in a r es in matrix andthe l inear grade d i s t r i b u t io n from a pol ished sect ion was determined as f 91)of Eq.9. Then, volumetric grade d i s t r i b u t io n , peg), is est imated by so lv ingthe t r ans format ion equat ion . True minera logica l composi t ion d i s t r i b u t io n ofspec i f i ed s i ze can be ca l cu l a t ed from t h e e s t i m a t e d v o l u m e t r i c g r ad ed is t r ibu t ion .Wet Grinding Batch TestTo de te rmine t h e se lec t ion and breakage funct ion parameters for the copperare , a ser ies of wet gr ind batch t e s t s were conducted. Four ba tch gr ind ingt e s t s were performed: the mate r i a l was ground for 2 minutes, 4 minutes, 8minutes and 16 minutes respect ively. A bal l mil l of 10 inch diameter and 11.5inch length was f i l l ed to 50 of i t s volume with b a l l s having an equi l ibr iums ize d i s t r i b u t i o n . The mi l l i s equipped w i th a Graham v a r i a b l e sp e e dt ransmission coupled with a BLH torque sensor . The sensor i s connected to a


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1 4 J. A. HERBST t al

char t recorder to measure the torque in the dr ive shaf t between the mi l l andthe t ransmiss ion. For a l l t e s t s the mil l was rota ted a t 60 of i t s c r i t i ca lspeed or 54 rpm. The mil l was opera ted a t 70 weight so l ids for a l l of thet e s t s . I t was charged with 3422 g of copper ore . Added to th i s dry materialwas 1467 ml of water .For each batch t e s t , a represen ta t ive sample was obtained by using a r i f f l es p l i t t e r . Representa t ive samples were taken from four s i ze in tervals (48x65,100x150, 150x200 and 200x270 mesh) fo r minera logica l composi t ion measurement.Minera log ica l compos i t ions were de te rmined us ing microprobe analys i s . Asu f f i c i en t ly la rge number of sec t ioned p a r t i c l e s , suspended in an appropriatemolding r es in , can be examined for the q u a n t i t a t i v e i d e n t i f i c a t i o n of thevar ious cons t i tuen t s . During l i n ea r scanning of the sect ion of the specimen,the an a l y s t may record the leng th of the r e su l t i ng chord t h rough eachi n t e r cep t ed s ec t i o n as wel l as t he l eng th o f t h e same chord a c r o s s thevaluable phase appear ing in t h a t spec i f i c p a r t i c l e s ec t i o n . Therefore , al i nea r grade may be d e f i n e d as the r a t i o between the length of these twochords for each par t i c le . The end resu l t i s a probabi l i s t ic d i s t r ibu t ion oft h i s l i n ea r grade as f g l ) o f Eq.9. The volumetric grade dis t r ibu t ions werethen estimated from these l i n ea r grade dis t r ibu t ions [17]. Both l inear gradeand volumetric grades were used to es t imate the parameters for three-componentmodels. Breakage r a t e s were es t imated by n o n - l i n ea r r eg r es s io n using anunpublished program termed ESTILIB , developed by J .A. Herbst and coworkersa t the Universi ty of Utah. The r e s u l t i n g s p e c i f i c s e l e c t i o n and breakagef u n c t io n s e s t im a tes a r e shown in Figure 5. A di f feren t speci f ic select ionfunc t ion of the locked par t i c les est imated from the l inear grade dist r ibut ioni s observed as shown in Figure 5.

1 0 r - - - - - - - - - ~ - - - - - - - - - - - . - - - - - - - - - - ~ - - -

i~

1

0.1

0.1ILw( J4 :wa:III 0.015::J::JU

10

A, Chalcopyrite

___ --AB Locked_ AB} inear. . ~ B

,,/

100

.,,//

..... B, Gangue

1000

o A-AA B-Bo AB-AB< J AB-A> AB-B

10000PARTICLE SIZE, microns

Fig .5 Estimated speci f ic se lec t ion and breakage func t ions fo rA (Chalcopyri te ) , B (Gangue) and AB (Locked) components


9/15


For batch t e s t s a l l gr ind t imes were converted to t h e i r equiva len t spec i f i cenergy inputs . P lo t s of cumulative mass f rac t ions for each component CAB Aand B) a t d i f f e r en t energy inputs versus p a r t i c l e s i ze are shown in Figures 6,7 and 8 . Depending upon the amount of l ibera t ion occurr ing the t o t a l mass ofAB p ar t i c l e s should remain about the same or decrease while the t o t a l mass ofA and B p ar t i c l e s should remain the same or increase with gr inding t ime. Thisi s t rue for a l l cases as shown in Figures 6 to 8. Using ESTILIB, the bes t itparameter where de te rmined . Plo ts of the bes t model values are a lso shown inFigures 6, 7 and 8. The RMS re s idua l s value fo r t h i s f i t t i n g was 1.98 x 10-2 General ly, the curves it the da ta we l l al though there appears to be moree r r o r in t h e p red i c t i o n s o f A and AB va lues due to t he smal l amount ofmater ial in these two composition classes .1 r - - - - - - - - - - - - - r - - - - - - - - - - - ~ r _ - - - - - - - - - - -

2 min Grinding

a: 0 1wZii:zO

':u. 0 016 AS0 A

BPredicted

0 00110 100 1000 10000

PARTICLE SIZE micronsFig.6 Comparison of exper imenta l and model predicted r e su l t s of componentsobta ined from 10- inch b a l l mil l t e s t - 2 minutes gr inding

Open Circu i t Pi lo t Plant TestThe open c i r c u i t p i l o t p l an t t e s t work was conducted in the Univers i ty ofUtah s Ore Dress ing Laboratory. To conduct an open c i r cu i t gr inding t e s t a l lof the mass flow measurement devices were ca l ib ra ted . Feed r a t e to the c i r cu i twas cont ro l led using an HP1000 computer . Once s teady s t a t e co n d i t i o n s werea t t a i n e d th e t e s t was begun. The se t -po in t fo r the feed streams were 300lbs /hr for so l id and 128.6 lbs /h r for water . Copper ore samples from the openc i rcu i t m i l l were analyzed fo r s i ze d i s t r i b u t io n and mineralogical compositionas previous ly mentioned.ESTILIB was used to s imulate open c i r cu i t gr inding of the copper porphyry ore .The values for se lec t ion and breakage funct ions determined us i ng t he ba tchgr inding da ta were used in ESTILIB a long wi th t he s i ze and mineralogicalcomposition of the feed.The spec i f i c se lec t ion funct ion values from the batch t e s t s were scaled up forpredict ion of the p i l o t sca le continuous t e s t s using the re l a t ionsh ip [18] :

for a l l mineralogical components.


10/15

106

a:wZLL

zo-o


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Multicomponent multisize liberation model 107

Figures 9 and 10 show the pi lo t plan t da ta in the s teady s t a t e form of thecumulative mass f r ac t ion for each component and the overa l l cumulat ive massf r ac t ion as a funct ion o f p a r t i c l e s ize , re spec t ive ly . Also shown in thesef igures a re comparisons between the exper imenta l data and model predic t ions.I t i s noted t h a t a very good fit i s observed between the model predic t ions andac tua l mil l discharge product .

B Ba: PredictedwZLLz 0 1 t=00a:LL

0 01 - - _ . .L . . . - - - - L - - - - - - - - L . . . .L . . .L . .L L-_. . . .L . ._L. . - . . .L . . - . .L . . . -L-L. . . .L . J -2 100 1000P RTICLE SIZE microns

Fig.9 Open c i r cu i t p i l o t plant gr ind da ta and pred ic t ion based onor ig ina l model parameters from laboratory scalebatch grinding t e s t for d i f f e r en t components

CONCLUSIONSThe extended models for s i ze reduct ion-minera l l i b e r a t i o n t h a t were developedin t h i s paper a re su i tab le to provide more informat ion fo r grinding processanalys is and pred ic t ion . For t h ree component case , ESTILIB a program fo rgr ind ing- l ibe ra t ion model s imula t ion/parameter es t imation, made it poss ib le todetermine the model parameters from a minimum of exper imenta l da ta .The exper imenta l v e r i f i c a t i o n of t h e gr ind ing- l ibera t ion model copper oregrinding was found to be s u cces s fu l . The discharge produc t from an openci rcu i t mil l was accura te ly pred ic ted from da ta gathered in a laboratory scalebatch mil l t e s t . This r e s u l t sugges ts t h a t f u r t h e r sca l e - u p to commercia lopera t ions may a l s o be p o s s i b l e . Addit ional work i s necessary to make theg r i n d i n g - l i b e ra t i o n model more g en e r a l . This fu t u re work i n c l u d e s t h edevelopment of mathematical so lu t ion and parameter es t imat ion schemes for thecase of more than one class of locked p a r t i c l e s .


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108

a:wZI.Lz0t-O


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5. King R.P. I n t . J. Min. Proc. 6, 207 1979)6. Finch J.A. Petruk W I n t . J. Min. Proc. 12, 305 1984)7. S t e i n e r , H . J . Libe ra t ion k ine t i c s in gr ind ing ope ra t ions , XI IMPCCagl iar i , Rome 1975)8. Klimpel R.R. Austin L.G. Powder Technology 34, 1219. Andrews J.G.R. Mika T.S. Comminution of a heterogeneous mater ia l :development of a model fo r l ibe ra t ion phenomena, XI IMPC Cagliari , Rome,1975 )10. Cutt ing G.W. The Chemical Engineer 351, 845 Dec. 1979)11. Cutting G.W. Barber S.P. Charac ter iza t ion and pre d i c t i on of minera lre lease during gr inding processes : appl ica t ion to a lead/z inc Ore Prepr in tEuropean Symposium on Par t i c l e Technology ed. K Schoenert Dechema 1980)12. Ruebush J . C . , Herbst J .A. Rajamani K. SIMPLANT-A Program f o r t h eSimula t ion of Ind iv idua l Mineral Behav iour in an I n t e g r a t e d Gr ind ing andF lo ta t i o n C i r c u i t Department of Metallurgy, University o f Utah, Sal t LakeCity 1980)13. Pe te r son R.D. Es t ima t i o n of Parame t e r s for and Ve r i f i c a t i o n of anIn t egra ted Model for Grinding and Flo ta t i on Circu i t S imu la t i on M.S. Thesis,Universi ty of Utah 1983)14. Lin C.L . , Mi l le r J . D . , Herbst J .A . , Sepulveda J .E. Pr i s b rey K.A.Prediction of volumetric abundance from two-dimensional minerals image Appl i edMineralogy ed. W.C.Park e t aI , TMS/AlME, 157 1985)15. L i n C . L . , M i l l e r J . D . , H e r b s t J . A . R aj am an i K. articleCharac t e r i za t i on 4 78 1987)16. Mille r J .D. Lin C.L. Treatment of polished sec t ion da ta for detai ledl i be ra t i on a na lys i s . Conf. on Recen t Deve l opmen t s in Comminut ion Hawaii

1985). To be published in In t .J .Mineral Proc. , 198717. Lin C.L., Mil ler J.D. Herbst J .A. Powder Technology 50,1 55 1987)18. Herbst J.A. Fuerstenau D.W. I n t . J. Min. Proc. 1 1980)APPENDIX 1

Recursion formulae for the elements o f the Eigenvector Matrix I for FourComponent System

i ) Par t i t ion 1) o o oV) I I ) o o

VI) VIII) I I I ) 0VII) IX) o IV)

i i ) [ i=N+l , 2N( II)j=N+l 2N.

For I I ) . I I I ) , IV )[i =2N+ 1 , 3N( I I I )J=2N+l, 3N,

o i


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(V) [ ~ = N + 1 ' 2 NJ-1 ,2 , N

For

o i -N(jo i-N=j

Tij

[i.=2N+1 3N(VI) J=1, N

Tij

o i-2N(joi -2N-lEk=j

J. A. HERBST t al

i -N)j andj ) i -N-2

i-2N)j and j ) i -2N-2

i ~ 2 N - l bik T + i -2N-l bi ,k+2N 5k+ 2N T . i -2N)jk=J 5]..-5J. kj 5i -5 J k+2N,J j i -2N-2k=j+1

[i=3N+l, ,4N

(VII)j=1,2 , N

o i-3N(jo i-3N=j

Tiji -3N-l b ik SkE T . i -3N)j and j ) i -3N-2k=j 5i -S j kJ

i ~ 3 N - l bik 5k T . + i ~ 3 N - l bi ,k+3N 5k+3N Tk=j 5 i -5 j kJ k=j+l 5 i -5 j k+3N,j

[i=2N+l, 2N+2, N

VIII)j=N+l, N+2, 2N

Tij

o i -N joi -N- lk=j

i -3N)j andj i -3N-2

i-N>j and j ) i -N-2

i -N)j andj i -N-2


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Multicomponent-multisize liberation model

[i=3N+l,3N+2 ,4NIX)j=N+l,N+2, ,2N

i-2Nj andj (i-2N-2

I_BAB2 -AB2{

I 0i=j is low t r i angula r matr ix

1Let [I_BAB2-AB2]b ~ 2 A B 2 D j

athen bi ji - I AB2-AB2 } AB2-AB2

- : bik bjk bi i i>jk=j

ijBAB2-ABl

And BAB1 - AB2 [ 0 i jb ~ 1 A B 2 D j

[I_BAB2-AB2] -lBAB1-AB2 = [ i ( ji l[ AB2AB l k- l k-l AB2-AB2 a ABI-AB2]1: bik L - : bk1 b l ~ b mk=J m=J l=m J

For section 1 )[ i=I , - - -Nj=l , - - -N

or

ij

ij

i - I- :k=j

k-l k- l AB2-ABI AB2-AB2 ABI-AB2: : b b b a bm=j l=m ik k l 1m mj i>j

111

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