Assessment of deisgn and operation of grinding mill using simulation.pdf

7/29/2019 Assessment of deisgn and operation of grinding mill using simulation.pdf

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ETY OFCAL LER NO. Dl ITTLETON, COL

NEE

ASSESSMENT OF DESIGN AND OPERATION OF GRINDINGMILLS USING SIMULATION

V. Voller K.A. SmithResearch Associate Assistant Professor

K.J. ReidProfessor & Director

Mineral Resources Research CenterUniversity of Minnesota

M. Cross*Visiting Professor

*Currently at CHAM Ltd.London, United Kingdom

For presentation at the SME-AIME Fall Meeting and ExhibitDenver, Colorado - November 18-20, 1981

Permission is hereby given to publish with appropriate acknowledgments,excerpts or summaries not to exceed one-fourth of the entire text of the paper.Permission to print in more extended form subsequent to publication by the Institutemust be obtained from the Executive Secretary of the Society of Mining Engineersof AIME.If and when this paper is published by the Society of Mining Engineers of AIME, itmay embody certain changes made by agreement between the Technical PublicationsCommittee and the author, so that the form in which it appears here is not necessarilythat in which i t may be published later.These preptints are available for sale. Mail orders to PREPRINTS, Society of Mining

Engineers, Caller No. Dl ittleton, Colorado 80123.

PREPRINT AVAILABILITY LIST IS PUBLISHED PERIODICALLY IN' MlNlMG ENGINEERING


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A b s t r a c t . A r e l a t i v e l y s i m p l e mod el o f g r i nd -i n g wh ic h i n c o r p o r a t e s b o t h t h e Bond g r i n d a b i l i t yequa t ions and the popu la t ion ba lance equa t ionshas been extended to form a comprehensive simula-t i on package t ha t can des c r ibe m os t k inds o f con-ven t iona l g r ind ing c i r cu i t s . The package has twoprograms, w it h th e f i r s t c o n s i s t i n g o f a n in p u tp rogram which he lps th e us e r t o s pe c i fy the p re -s c r i b e d c i r c u i t , w h i l e t h e s ec on d p ro gr am p e r-f or ms t h e a c t u a l s i m u l a t i o n.The package serves two main purposes - i) a s acheck on m i l l d e s ig n , w i t h t h e o u t p u t p r o v i d i n gn o t o n l y p re d i c t i o n s f o r t h e p r od u ct p a s s i n g s i z eb u t a l s o of t h e f u l l pr od uc t s i z e d i s t r i b u t i o n ,an d i i ) a s a means t o a s s e s s t h e l i k e l y e f f e c t son t h e p r od uc t s i z e d i s t r i b u t i o n o f o p e r a t i n gu n d e r d i f f e r e n t sets of cond i t ions .

I n t r o d u c t i o nThe Bond equ at ion ,(B ond , 1960, 1961; Rowlandand Kjos, 1978), i s us ed as the ba s i s f o r much o ft h e b a l l a nd r o d m i l l des ign done today. Althoughth er e have been a number of " popul ati on type"

models of gri nd in g proposed, (Reid, 1965; Aus ti n,1971; Luckie and Austi n, 1972; Lynch, 1977) ,the r e has been some d i f f i c u l ty i n app ly ing themt o t h e p r a c t i c a l t a s k of m i l l d e si g n. The es-sence of th e proposed s im ulat ion package i s t oe x p l o i t the method of combining Bond's t h i r d lawof comminution, (Bond, 1960, 1961). w it h th e pop-u la r ion ba lance equa t ion (Aus t in , 1971) . Th i sapproach, f i r s t proposed by Cross and Gwst, (1978)h a s be en u se d i n a s t u d y o f t h e c o n t r o l o f c i r -c u i t s gr in ding mixtures of ores (Holtham, 19781,and has r e ce n t l y been m odi f i ed and va l ida te d byCross (1981). The advanrages of t h i s method i st h a c a r e l a t i v e l y s i m p l e g r i n d i n g mo de l 5s usedwhich not o nly prov ides a qu ick and inexpens ivezk e ck on a Bond m i l l d e s ig n , b u t a l s o p r o v i d es ap r e d i c t i o n f o r t h e p r od u ct s i z e d i s t r i b u t i o n f ro mt h e m i l l , info rmati on tha t t he Bond approach tom i l l das ign does no t p rov ide (Herbs t andFuers tenau, 1980) .

BackgroundH i l l S e l e c t i o n

The f i r s t s t e p i n n i l 1 de si gn i s to de te rm inethe power needed to produce t he de s ir ed gr i nd i nthe chos en o re . The m os t u s ed equa t ion , f o r th i spurpose, i s the empir ical Bond equat ion (Bond,1960, 1961; Rowland and Kjos, 1978).E = 10Wi (,$ A)Wh/short ton (1)%I n t h i s e q u a t i o n , E i s t h e s p e c i f i c e n e rg y r e -qu i r ed f o r the g r ind , and Fa0 and Pa0 a r e thes iz es i n mic romete r s th a t 80% of the we igh t pas s esof the m i l l f eed and p roduc t r es pec t ive ly . Theparameter W i , known as the work index of th e ore ,

i s o b t ai n e d fr om b a t c h be nc h t e s t s f i r s t d e v i s e dby Bond (1961). The power cal cu la te d on us in gequa t ion 1, (Bond, 1961; Rowland and Kjos, 19781, .r e l a t e s t o :1 ) Rod milling - a r o d m i l l w i t h a diameter of2 .44 m eter s , in s i de new l i ne r s , g r ind i ng we t ino pe n c i r c u i t .2) B a l l m i l l i n g - a b a l l m i l l w i t h a d i a m e t e rof 2 . 4 4 m eter s , i n s i de new l i ne r s , g r ind ing we t

i n open c i r c u i t .When the g r ind ing cond i t ions d i f f e r f rom thes es p e c i f i e d c o n d i t i o n s , e f f i c i e n c y f a c t o r s ( Rowlandand Kjos, 1978) have t o be used i n c o n j u n c t i o nwi th equa t ion I. I n g e n e r a l , t h e r e f o r e , t h e re -qu i r ed m i l l power i s ca lc u la t ed us ing th e fo l low-i n g e q u a t i o n

where n* i s t h e number of efficient? f ac to r s , EF i ,used and f i s t h e f e e d r a t e o f new o r e t o t h em i l l . ~ h e O ~ o w e ra lcu la ted f rom equa t ion 2 can belooked up i n pub l i shed t ab l es (Rowland and Kjos,1978) and t h e c o r r e c t m i l l s i ze and type can bes e l e c t e d .Note th a t any e f f i c ien cy f ac to r s can be droppedfrom equat ion 2 o n s u b s t i t u t i n g t h e wo rk i n de x ,W i , i n e q ua t io n 1 by an e f f ec t i ve work index, w i ewhere

Populat ion HodelsThe population balance model of comminutiond i v i d e s t h e p a r t i c l e s i z e r a ng e o f t h e ro ug h o r ef e e d i n t o a number o f s i z e i n t e r v a l s ( n) a ndc a l c u l a t e s t h e r a t e o f c h a ng e o f t h e w e i gh t f r a c -t i o n i n e ac h s i z e i n t e r v a l . ( No te t h a t t h e l a r g -e s i p a r t i c l e s a r e p l a ce d i n i n t e r v a l 1 and small-e s t i n i n t e r v a l n ). On m a k i n g a d i s c r e t i z a t i o ni n t i me , t h e p o p u l a t i o n e q ua t i o n c a n b e w r i t t e n i nma tr ix form (Aust in, 1971; Lynch, 1977; Cros s,

1981) , as :

where

E q ua t io n 4 r e l a t e s t h e w e i gh t f r a c t i o n v e c t o r o ft h e g r i n d i n g m i l l c o n t e n t s , W, a t t im e ( p +l ) b t t ot h e we ig ht f r a c t i o n v e c t o r a t t ime p j t . Thisrep re s en t a t io n o f t he popu la t ion ba lance approacht o com minu tion may be r e fe r r ed to as the "un i tg r i n d o p e r a t i o n " f o r m u l a t i o n , e a c h u n i t g r i n dc o r re s p on d i ng t o on e ti me i n t e r v a l 6 t . The matrix[ I ] , i n e q u at i o n 5, i s t h e i d e n t i t y m a t r i x , [K] isa d iagona l m at r i x w i th e lem en ts kj, t h e f r a c t i o n sb ro k en o u t of e a c h s i z e i n t e r v a l J i n t i m e 6 t , a n d[B] i s a lower t r i ang u la r m at r i x whose e lem entsa r e b ji , t h e pr o p o r t i o n of p a r t i c l e s b ro ke n o u t ofs i z e ~ n t e r v a l t h at f a l l i n t o s i z e i n te r va l j .The Cross-Owst Model

Befor e t h e Cross-Owst model (Cros s and Gwst,1978; Holtham, 1978; Cross , 1981) can be used,the con t inuous opera t ion o f a g r in d ing m i l l has tobe approx im ated by a s e r i es o f s h or t ba t ch g r inds .With the t i m e chosen for each approximating batchg r i n d c o r r es p o nd i n g t o a s p e c i f i c e n e rg y i n p u t , e ,t o t h e mi l l .I f a t t ime t = 0 the 80% pas s ing s i ze o f the


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f eed , Fao , and t h e e f f ec t i v e wo rk in d ex , w i e , oft h e o r e i n t h e m i l l a r e known, th e 80% pass ings i z e o f t h e p r od u c t a f t e r o ne a p pr o x i ma t i n g b a t c hgr in d , Pa0 can be found on rewr i t ing t he Bonde q u a t i o n a s

Fu rth er mor e . t h e en e r g y in p u t , e , w i l l co r r e sp on dt o a n umber ( q ) o f u n i t g r i n d s i n th e p o p u la t io ne q u a ti o n . T h er e fo r e, i f t h e w e ig h t d i s t r i b u t i o ni n t h e m i l l a t t im e t * 0 ( i . e . , W(0)) is known,c he w ei g ht d i s t r i b u t i o n a f t e r o ne a p pr o xi m at i ngb a t c h g r i n d of t i m e 6 t i s

w he re t h e 8 0% p a s s i n g s i z e o f t h e p r o d u c t inW(6 t ) i s g iv en b y eq u a t io n 6. Hence, i f t he en--e r g y i n p ut , e , a nd t h e i n i t i a l w ei gh t d i s t r i b u -t i o n i n t h e m i l l ar e known, th e val ues of q , whichmay no t be in te ge rs , (Cross , 1981) and the weightd i s t r i b u t i o n s , W, a f t e r e a ch a p p r ox i ma t in g b a t c hgr i nd may be ca lc u la ted on repeate d use of equa-t i o n s 6 an d 7 . I n th i s way, ( wi th some f u r t h e rman ip u la t io n d esc r i .b ed l a t e r ) , i f t h e we ig ht d i s -t r i b u t i o n in t h e m i l l f e ed i s known, t h e w e i g h td i s t r i b u t i o n i n a c o n t in u o us l y o p e r a t i n g g r i n d i n gm i l l a t d i s c r e t e p o i n t s i n t i m e c s n b e c a l c u l a t e d ,where each t ime s t ep co r r e sp o n d s t o o n e b a tchg r a d .T h er e a r e t h r e e f u r t h e r p o i n t s t o b e m ade c on-cer nin g th e Cross-Owst model:1 ) I n m os t fo r m u l a t i o n s o f t h e p o p u l a t i o nbalance model, th e e lements of t he [K] and/ or [B]m a t r i c e s a r e c a r e f u l l y e s t i m a t e d fr om a s e r i e s o fl a b o r a t a r y e x pe r im e nt s ( H e r b s t , e t a l . , 1 97 1;Austin and Luckie, i971/72; Gardner andSukanjnaj te e , 1972) . In t he Cross-Owst approach,(Cro ss and Owst, 1978; Holtham, 1978; Cro ss , 1981)h ow ev er , s i m p l e e x p r e s s i o n s a r e u t i l i z e d . Th ecomponents of [ B ] a r e ev a lu a te d f ro m a p o p u larf or m o f th e cu mu la t iv e b r eak ag e f u n c t i o n , ( Au s t in ,1971) i .e . ,

w h i l e t h e b r e ak a ge r a t e p a r am e t e rs a r e e s t i m a t e dfrom

where a and m a r e p a r a m e t e r s and d j i s some rep-r e s e n t a t i v e measure f o r t h e s i z e of p a r t i c l e s i ni n t e r v a l j . The Cross-Owst model i s not depen-dent on th e forms of B(x,y) and k and a us er cans u b s t i t u t e o t h e r fo- f o r t h e s e $ < n ct i o n s.2 ) As an a l t e r n a t iv e t o th e Bond g r in d in g law,r e p r e s e n t e d by e q u a t i o n s 1 an d 6 , B i t t e n g e r s o rKicks laws (Lynch, 1977) may b e u s ed i n t h e C ro ss -

Owst f o r mu la t io n .3 ) Two major assumptions a r e made i n usi ng th eCross-Owst model. Assumption 1: t h e m a t e r i a lbeing ground i s homogeneous, i . e. , th e work inde xi s a c o n s t a n t f o r a l l pr o d uc t s i z e s (Bond, 1 9 61 ) .Asslimption 2 : the Bond equation i s a p p l i c a b l e t o

Eke conten ts of a gr ind ing sill when t h e m i l l i s

o p er a te d f o r a s h o r t t i m e i n t h e b a t c h mode.The MRRC Gr in d in g H i l l Simu la t io n Pack age

Gen er a l Desc r ip t io nThe phi losophy in th e development of t he MRRCg r i n d i n g s i m u l a t i o n p ac ka ge wa s t o b u i l d i n t e r -a c t i v e s o f t w a r e t h a t c o ul d be u se d a s a n i n -

ex p en s iv e means o f p r o v id in g a semi - q u an t i t a t iv ech eck o n a g r in d in g m i l l d e s i g n . I n a d d it i o nt h e s o f t wa r e i s d e s i gn e d t o " s l o t i n " t o a gen-e r a l min e r a l p r o ces s in g p ack age n ow u n de r g oin gd ev el op me nt a t t h e MRRC.The sof tw are , wr i t te n i n BASIC+, and imple-mented on a Dec 11/60 computer , co ns is ts o f th re emain programs:1) in put program: - g r d a t2) we l l mixed f low simulat ion : . - g r mix3 ) p l u g f l o w s i m u l a t i o n : - g r p l ugTh ese p r o g r ams an d th e s imu la t io n a lg o r i th ms a r ed i scu ss ed b e lo w. Note th a t i n some g r in d in gsim ulat ions , (Luckie and Aus t in , 1972; Holtham,19781, a r e s i d e n c e t i m e d i s t r i b u t i o n m ode l i sused. In t he MRRC si mul at i on package, however ,on ly t he two ex tremes of we l l mixed and p lugf low (Luckie and Aust in , 1972) th rough th e m i l la r e co n s id e r ed . Th i s app r o ach r ed uces th e amou nto f in p u t in f o r ma t io n an d co mpu ta t ion r eq u i r e d i nt h e s i m u l a t i o n . P r e d i c t i o n s o b t a i n e d f r om as i mu l at i on a r e f o r t h e m i l l p r o du c t s i z e d i s -t r i b u t i o n , t h a t i s t h e s i z e d i s t r i b u t i o n a f t e rg r i n d i n g b u t b e f o r e s e p a r a t i o n .The Input Program

I n p u t i s r e q u e s t e d f r om t h e u s e r b y t h e p r o -gram g r d a t . T h i s pr og ra m r u n s i n a n i n t e r a c t i v econ ve rsa t i ona l mode; inc lud es on l in e documen-t a t i o n , a nd c he c ks t h a t t h e i n p u t i n f o r m a t i o n i sc o n s i s t e n t an d o f t h e c o r r e c t fo rm . The r e q u i r e di n p u t i n f o r m a t i o n i s summarized i n Ta ble 1. Notet h a t :1 ) Resul ts o f a Bond m i l l d e s i g n a r e n e ed edb e f o r e t h e r e q u i r e d i n p u t i n f o r m a t i o n i s complete .2) The work index of th e or e i s n o t r e q u e s t e dby t h e p ro gr am g r d a t . I n s t e a d t h e e f f e c t i v ework index i s c a l c u l a t e d , u s i n g t h e known m i l lpower , by th e Bond equa t io n . In th i s way thework index used i n subsequent programs of t hes i m u l a t i o n c o r r es p o nd s t o t h e m i l l i n q u e s t i onand no t t he 2.44 meter d iameter m i l l t o w h ic h t h eBond work index corresponds.

3 ) The no min a l s i z e o f s ep a r a t io n co r r e sp o n d st o a c l a s s i f i e r w hi ch o p e r a t e s w i t h 100% e f f i -c i e n c y .Well Mived Flow

Th e d e t a i l s o f t h e a l g o r i t h m s u se d in t h ep ro gr am g r mix can b es t b e u n d e r s to o d b y f i r s tc o n s i d e r i n g a g r i n d i n g m i l l , i n wh ic h f l o w i sp e r f e c t l y m ix ed , r u n n i n g i n a c o n t i n u ou s c l o s e dc i r c u i t . In a d d i t i o n l e t t h e d i s c h a r g e fr om t h em i l l b e c o n t r o l l e d by a n o v e rf l o w we i r . Th en e c e s s a r y m o d i f i c a t i o n s t o t h e a l g o r i t h m s f o r t h ec a s e s o f an o pe n c i r c u i t a nd g r a t e d i s c h a r g e w i l lb e o u t l i n e d a t t h e e nd of t h i s s e c t i o n .T he t i me , 6 t , f o r e a c h o f t h e a p p r o x i m t i n gb a t c h g r i n d s i s c h o s e n s o t h a t


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where n i s a n i n t e g e r > 1 an d r t i s t h e r e s i d e n c et ime o f th e o re in t h e m i l l . Th e r e s i d e n c e t i m ean d l i k e w i s e 6 t d ep en ds o n t h e f u l l m i l l f e e dr a t e ( i .e ., f r e s h f e ed p l u s c i r c u l a t i n g l o ad ) .The t ime s t ep, 6 t , i s a s s o c i a t e d w i t h an energyi n p u t t o t h e m i l l , v i a t h e f r e s h f ee d. I n i t i a l l y ,i . e ., when th e c i r c u l a t i ng loa d i s z e r o , t h i se ne rgy inpu t i s

where

i s t h e e n e r g y s u p p l i e d t o t h e m i l l i n kWh p e r t o no f f r e s h f e e d. A f t e r t i me t = f r t i / n ( i . e . ,

a - 7A- Aa f t e r j a p p ro x i m at i n g b a t c h g r i n d s ) , i f r i s t h ep r op o rt i on o f c i r c u l a t i n g l o ad i n t h e f u l l m i l l .f e e d, t h e e n er g y i n p u t f o r t h e n e x t b a t ch g r i n di s

The a lgor i thm i s i n i t i a l i z e d o n p u t t i ngwnew = a nd us ing the e ne rgy inpu t , e o , 07 thef i r s t a p p ro x im at i ng b a t c h g r i n d . T he s te ps ou t -l i n e d a b ov e a r e t h e n r e p e a t e d u n t i l c on ve rg en ce ,i .e ., u n t i l t h e v e c to r pewemains unchanged fore a c h a pprox imat ing ba tc h g r ind .khe n th e d i s c ha rge f rom th e p i l l i s c o n t r o l l e db y a g r a t e , l / n t h o f t h e m i l l c o n t en t s i s s t i l lr emoved a t t he e nd o f e a c h ba tc h g r in d , bu t nowt h e r e i s t h e r e s t r i c t i o n t h a t t h e s i z e of m a t e r i a lremoved must be be low th e gr a t e s i ze . Thi s ap-p roac h d e v i a te s f rom pe r f e c t l y mixe d f low be ca usean y p a r t i c l e i n t h e m i l l f e e d wi t h s i z e a bo ve t h eg r a t e o p en i ng s i z e m us t r em ai n i n t h e m i l l f o r aminimum t im e, i . e. , u n t i l t h e p a r t i c l e s i z e h asbe en re duce d to th e g r a t e s i z e . Note , however ,t h i s r e s t r i c t i o n d o es n o t n ec e s s a r i l y n eg a t e t h ea ss um pt i on t h a t t h e m i l l c o n t e n t s a r e p e r f e c t l ymixed a t a l l t i me s. W it h a g r a t e d i s ch a r g ec ha nge s ne e d t o be made t o t he e qua t ion s p re se n te da bove . T he we igh t d i s t r i bu t i on in t h e m i l l i s nol o n g e r e q u a l t o t h e we ig ht d i s t r i b u t i o n i n t h em i l l p r od u ct . T h e re f o r e, i n t h e c a l c u l a t i o n o ft h e m i l l r e t u r n , e q u a t i o n 1 4 , wield i s r e p la c e dby P i , whe re

N ot e t h a t , t h i s e n er g y i n p u t r e f l e c t s t h e c ha ng e 0 i c gin t h e r e s i d e n c e t i m e o f t h e m i l l .At th e end of any energy inpu t , i . e . , an y Pi = (17)a p pr o xi m at i ng b a t c h g r i n d , t h e f o l l o w i n g s t e p s o l da r e t a ke n i n g r m i x . Time i s ' f r o z e n ' an d l / n t h Wi / nw i>_gof the m i l l c o n t e n t s i s c ons ide re d to be ' r e -moved ', a s the mi l l produc t . The we ll mixed f lowa s s u m p t i o n i m p l i e s t h a t t h e m i l l c o n t e n t s a r epe r fe c t l y mixed . Hence , e we igh t d i s t r i bu t i oni n t h e m i l l produc t i s i . e . , t h e w e i gh td i s t r i b u t i o n i n t h e m i l l c o nt e nt s a t t h e p o i n t

w is t h e p r op o r t i o n o f t h e m i l l c o n t e n t s s m a l l e rt h a n t h e g r a t e s i z e , an d g i s t he f i r s t s i z ei n t e r v a l w i t h a r e p r e se n t a t i v e s i z e l e s s t ha n t h eg r a t e o pe ni ng . F u r t he r , t h e w e ig h t d i s t r i b u t i o ni n t h e m i l l a t t h e s t a r t of a n ew en e rg y i n p u t i s->.- - . -t i m e i s f r o ze n . The w e ig h t d i s t r i b u t i o n i n t h e given O yc i r c u l a t i n g l o ad a t t i m e t , r , i s found f rom the

m i l l p r od u ct w e i gh t d i s t r i b u t i o n o n w r i t i n go l d i < s

i = l

bwhe re th e uppe r bound on the pa r t i c l e s i n s i z ei n t e r v a l , s , i s t h e ' n o m in a l' s i z e o f s e p a r a t i o n( a t p r e se n t , a c l a s s i f i e r m odel i s be ing de ve l -ope d a nd in the me an time c l a s s i f i c a t i on i s con-s i d e r e d t o b e p e r f e c t ) . The w ei gh t d i s t r i b u t i o ni n t h e f u l l f ee d f o r t h e n e x t e ne rg y i n p u t t h enfol lows f rom

where f i s t h e d i s t r i b u t i o n i n t h e ro ug h m i l lfe ed . -ithe fe ed f r e p l a c e s t h e l / n t h o f t h e m i l lcontents removed from the m i l l . T his modi f ie st h e we ig ht d i s t r i b u t i o n in t h e m i l l , s o t h a t

Wext, time i s 'unf roze n' and a new batc h gri nd iss imula ted. The 80% p a s si n g s i z e of t h i s b a t chg r i n d p r o d uc t i s found us ing e qua t ions 6 , 7 and13. This v a l u e i s ne ede d i n o rd e r t o imple me ntth e Cro ss- O~s t model , and hence de te rmine t hechange in the m i l l w el gh t d i s t r i b u t i o n a f t e r t h eg r i n d .

as opposed t o e qua t io n 16 i n th e c a se o f a n ove r -f low d i sc ha rge .When th e m i l l is r un ni ng i n o pen c i r c u i t t h er e t u r n p o r t i o n , r , and t h e r e t u r n w ei g ht d i s t r i -b u t i o n , a r e a lw ay s ze r o ; t h e m i l l d i s t r i b u t i o n i sf, a t t h e b e gi n ni n g of a l l b at c h g r i n d s ; a nd t h ee n e rg y i n p u t t o e ac h b at c h g r i n d i s e o .Plug Flow

In the p rogra m gr p lug , p lug f low th rough them i l l i s modeled by assuming the m i l l c o n t e n t s t oco ns is t of n mass segments , each of which cont a inl / n t h o f t h e m i l l ' s mass cap ac i ty , c . Then a sgri nd in g proce eds, th e or e moves down the m i l ls u c h t h a t

where f i s t h e f ee d r a t e t o t h e m i l l , and dmi/dti s t h e r a t e a t w hi ch o r e l e a v e s s eg me nt i. Thet i me f o r e a ch b a t c h g r i n d i s c h o s e n s o t h a t

(cf. e q u a t i o n 1 0 ) . Then i n ea ch b a t c h g r i n d , a l l


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t h e o r e o r i g i n a l l y i n s eg me nt i ha s moved t o seg-ment i + 1, t h e o r e o r i g i n a l l y i n s egme nt 1 hasbeen replaced by new feed, a nd t h e o r e o r i g i n a l l yi n s egme nt n h as l e f t t h e m i l l . As i n t h e w el lmixed model, th e batch gri nd can be as soc iat edwith an energy input . I n i t i a l l y t h e e ne rg y in p u tis e o ( s ee e qu at i on l l ) , a f t e r t im e t the energyinput i s

(cf . equat ion 12) , where r i s now the proportiono f t h e o r e i n t h e l a s t s eg me nt t o l e a v e t h e m i l lwhich has a s i ze l a rg e r than th e nominal s i z e ofsepa ra t ion . A t the end of an energy inpu t , thewe igh t d i s t r ibu t ion 2 i s cal cul a te d using equa-t ion 14. Now, however, ~ l ds replaced byl&O1d, t h e w ei gh t d i s t r i b u t i o n i n t h e n th masssegment. The d i s t r i b u t i o n i n t h e f u l l m i l l feed,which w i l l a l s o b e t h e d i s t r i b ut i o n i n t he f i r s tm i l l segment a t t he beginning of a new energy in-put , ( i .e . , ba tch grind) , i s calcula ted from equa-t ion 15 .In order to short en the computer usage, in-st ea d of gri nding each segment f o r each energyi n p u t o nl y t h e o r e i n t h e l a s t s egmen t i s ground.The grinding energy in th i s cas e becomes

where e iwas the energy input ca l cul a te d fromequation 21 when th e o re in segment i w a s f i r s tf e d t o t h e m i l l . This i s purely a mathematicalconcept of the f low of or e i n a gr inding m i l l .The computational advantage of t h i s method i st ha t a s a "plug" of or e moves down th e m i l l fromsegment to segment the s i ze d i s t r i bu t i on remainse q u a l t o i t s i n i t i a l s i z e d i s tr i b ut i o n f. Re-s u l t s u s i ng t h i s " s i n g l e g r i nd i n g" a pp ro ac h a r ei d e n t i c a l t o r e s u l t s o b t ai n ed i n a p p l yi n g t h egrinding energy input , e i , to each segment foreach approximating batch grind.

A t time t = 0 the re a re no previous energyi n pu ts . T h er ef or e i n o r de r t o i n i t i a l i z e t h eso lu t ion the g r ind ing ene rgy i s given by:

Then E i s updated a t the end of every t ime s t epby sub t rac t ing the ene rgy inpu t a s soc i a ted wi ththe segment th at lea ves t he m i l l , and adding onth e energy in put a sso cia ted with th e new segmentc rea ted by th e m i l l feed . I f th e m i l l i s i nc losed c i rc u i t , the above rocedures a r e repea tedu n t i l t h e d is t r i b u ti o n %'Id remains unchanged.In open c i rc u i t f and E w i l l remain unchanged i ntime, hence, only one ope rat ion i s r e q u i r e d f o rconvergence.

A m i l l with a gr a t e d ischarge cannot be s i m -ula ted using a plug flow approximation. In fac:,a m i l l with a g r a te d i scha rge w i l l au tomat ica l lyi n t ro d uc e mi xi ng , i n t h a t l a r g e p a r t i c l e s a r -r i v i n g a t t h e d i sc h ar g e w i l l remain i n t h e m i l l ,thereby incre asing t he ir res idence t ime abovethe required p lug f low res idence t ime of c /f .

Resu l t sThe Test Examples

Befo re the MRRC packa ge can be used, a Bond de-s i gn ex er ci se must be completed . A wet b a l l m i l lwi th 40% vol ume tri c l oad ing and 2.44 met ers diam-e t e r i n s i d e l i n e r s was c hos en a s a t e s t m i l l .Th i s c ho i c e means t h a t t h e o n l y e f f i c i e n c y f a c t o rwhich i s n on -u ni ty w i l l b e f o r t h e m i l l i n openc i r c u i t and t h i s f a c t o r i s equ al t o 1.2 (Rowlandand Kjos, 1978). The s i z e i n m i cr om e te r s t h a t80% of th e weight of th e feed and product p asses ,Fa0 and P80, were kept const ant f o r each gr indin gsim ula tio n a t 1200 and 175 micrometers respec-t i v e l y . The feed r a t e was var i ed i n each simu-l a t i o n i n o r d e r t h a t t h e power p r e d i c t e d f ro mequation 1 corresponded t o th e 2.44 meter diame tert e s t b a l l m i l l . A number of s imu lat ion s corre-s po nd in g t o d i f f e r i n g o p e r a t i ng c o n d i t i o n s f o rt he t e s t b a l l m i l l have been ca rr ie d out . Thesea r e summarized i n Table 2. With th e predi c tedr e s u l t s , t h e e f f e c t of t h e f o l lo w i ng op e r a t i ngcond i t ions on the m i l l p ro du ct s i z e d i s t r i b u t i o ncan be te s ted :

1 ) comparison of w e l l mixed flow and plug flow,2) compar i son o f d i f f e r en t s i ze s o f sepa ra t ion,3) e f f e c t s o f d i s c h ar g e g r a t e s i z e ,4 ) e f fe c t s of ope ra t ing th e mi l l under "non-des ign" cond i t ions .The w e ig ht d i s t r i b u t i o n i n t h e f r e s h f e e d t ot h e m i l l , &, i s g i ve n i n Ta bl e 3 , Th i s d i s t r i -

bu t ion has been chosen a rb i t ra r i ly .Discussion

Study of th e resulFs pr edic t ed from the p lugf low s imula t ion of the te s t m i l l with overf lowd i s ch a rg e i l l u s t r a t e t h e e f f e ct i v e n e ss of t h e MRRCgrind ing s imul at ion package. F igure 1 shows si z ed i s t r i b u t i o n d a t a f o r t h e f e e d and f o r t h e prod-u c t s of t h r e e d i f f e r e n t c o n f i g ur a t i o n s f o r t h em i l l , N ote t h a t t h e s e s i z e d i s t r i b u t i o n s a r e oft h e m i l l product before i t i s separ ated , and ofth e hydrocyclone produc ts assuming 100% ef fi ci en cya t the nomina l s i ze of sepa ra t ion . Curve 2 i s t h ep r e d i c te d s i z e d i s t r i b u t i o n when t h e m i l l i s op-e r a t e d i n a n open c i r c u i t . T h is d i s t r i b u t i o ngives a product wit h a 80% passing s i z e of 177micrometers, which provides confidence in the re-s u l t s of t h e MRRC grind ing models. The ef fe ct s ofd i f f e r i n g s i z e s o f s e p a r a t i o n may be s tu die d bycomparing curves 3 and 4 i n F i g ur e 1 and thec i r c u l a t i n g lo a d v a l ue s i n T ab le 4 , F i r s t it maybe noticed th at th e gr ind becomes co ars er whenthe nomina l s i ze o f se pa ra t ion i s changed from250 t o 125 micrometers (80% passi ng s i z e of 389micrometers f or a 125 micrometer se par at i on asopposed to an 80% passi ng s i z e of 250 micrometerswith a 250 micrometer separation). Th i s c o a r s e rg r ind might be expected because, wi th t he s ha l l e rs i z e of s e p a r a t i o n , t h e c i r c u l a t i n g l o a d i sl a r ge r , and hence, the re s idence t ime o f t he o rei n t h e m i l l i s s h o r t e r , I n a d d i t i o n, w i t h t h e125 micrometer separ at ion , the pr edic t ed weightd i s t r i b u t i o n i n t h e s ep a r a t o r f e e d i s l e s s d i s-p er se d, ( c f . t h e g r a d i e nt s i n s t r a i g h t l i n e p a r t sof curves 3 and 4 ) , which i s a d e s i re d a t t r i b u t ei f th e nex t s ta ge of p rocessing i s f l o t a t i o n .

The s i z e d i s t r i b u t i o n p l o t s and c i r c u l a t i n gloads fo r the same m i l l usi ng th e well-mixedsimu lat io n ar e shown i n Figure 2 and Table 4 ,


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r espec t iv e ly . The behavio r of these p l o t s in -di ca te t ha t wi th wel l mixed f low, changes in t h em i l l operat ion have th e same ef fe ct s on th e prod-u c t w e ig ht d i s t r i b u t i o n s , a s w i t h p l u g fl ow .However, th e predic ted m il l prod ucts a re muchcoar se r i n open c i r c u i t , e .g ., t he 80% pass ings i z e i s 550 micrometers compared t o an 804 pass-ing s i z e of 177 micrometers in the product pre-dict ed using the plug f low simulat ion . Ther ea so n f o r t h i s i s in t h e n a t u r e of t h e w e l lmixed f low assumption. I f th e f low through th em i l l i s assumed wel l mixed, pa r t i c l e s i n th e f eedc an e x i t fr om t h e m i l l i n s t a n t a n e o u s l y . T he re -f o r e , t h e r ed u c ti o n in t he s i z e o f some of thefeed par t i c l es may be minimal , thereby leading t oa coar se p roduc t g r ind . On comparing t he r e su l t si n F i gu r es 1 and 2, it a pp ea rs , a t l e a s t f o r t h ec a s e of a b a l l m i l l with an overf low discharge,a plug flow approximation i s c l o s er t o t h e t r u enatu re of f low through th e m i l l . This i s reason-ab le on s tudy ing r es idence t ime distribution d a t aw hi ch o f t e n e x h i b i t f a i r l y s h a rp p ea ks (Luckieand Au sti n, 1972; Holtham, 1978) .

When a gr at e cont rol s th e discharge f rom themi l l , a l though the assumpt ion tha t t h e m i l l con-t e n t s a r e p e r f e c t l y mixed a t a l l t im es c a n beapp l i ed , t he p roduc t i s n o t n e c e s s a r i l y i d e n t i c a lt o t h e m i l l c o n t en t s , s i n c e f e ed p a r t i c l e s l a r g e rt h an t h e g r a t e s i z e c a nn ot e x i t i n s ta n t a ne o u s ly .Hence, a g r a t e d i scharge no t on ly g ives con t ro lover th e mi l l p roduc t bu t it may al so in t rod ucea d i s t r ibu t io n o f r es idenc e times in t o the simu-la t i on . The e f f e c t o f changing the g ra t e open-i n g s i z e o n t h e p r od u ct w e ig h t d i s t r i b u t i o n sfrom t h e t e s t b a l l m i l l i n o pen c i r c u i t may beobserved in Figure 3 w i t h t h e e - q e c t ed r e s u l tt h a t t h e s ma l le r t h e g r a t e s i z e t h e f i n e r t h egr ind . In f a c t , t he p roduc t of a m i l l with 250micrometer grate i s f i ne r than the p roduct f romthe same m i l l with an overf low discharge.

A u s e f u l f e a t u r e of t h e MRRC grinding simu-la t ion package i s t h a t t he e f f e c t s of non-designopera t ing cond i t ions can be ea s i ly t e s t ed . Whent he t e s t b a l l m i l l i s r u n i n o pe n c i r c u i t t h e d e-s i g n fe e d r a t e i s 27.42 tonnes/h our. Figu re 4compares the p red ic t ed s i ze d i s t r ibu t io ns f romt h e open c i r c u i t b a l l m i l l w i th f e ed r a t e s of20.0, 27.62 and 35.0 tonnes/hour resp ect ive ly .A s might be expec ted , t he l a r ge r th e feed r a t et h e c o a r s e r t h e g r i n d i n g m i l l discharge product .

ConclusionsSummary

An in te ra ct iv e computer package to s imu lateconvent i onal gr inding has been out l ined . Thepackage may s er ve t hr ee main purpose s:

I ) a s a check on m i l l des ign2) as a means to asses s the e f f e c t s of

d i f f e r i n g o p e r a t i n g c o n d i ti o n s o n t h e m i l l prod-u c t

3) as a teaching and demonstrat ion ai d .The ve rs a t i l i ty of th e package has been dem-o n s t r a t e d by s i m u la t i ng a b a l l m i l l i n n i n ed i f f e r e n t c o n f i g u r a t i on s . P r e li m i na r y r e s u l t sa r e of t h e c o r r e c t n a t u re and i l l u s t r a t e t h eexpec ted t r a i t s when opera t ing cond i t ions a r echanged.

Further WorkThere ar e a number of st ag es of f ur th er work

to be accompl ished befo re the MRRC model w i l l beready fo r more gen eral use. These include:1) Documentation - i n o rder t o make t he op-e ra t ion o f th e package c l e a r and easy2) Val ida t ion - t h a t i s , a d e t a i l e d t e s t i n g

of the mode l wi th t he p red ic t ed r e su l t s checkedagainst exper iments3) C l a s s i f i e r m od el - t h i s i s needed so thatthe g r ind ing package can be inc luded i n t o th ege ne ral mine ral pro ces sing package now underdevelopment a t t h e HRRC.

4) User def ined func t ion s - in o r d e r f o r au s e r t o e a s i l y s u b s t i t u t e d e s i r e d fo rm s of t h efunc t ions B(x ,y ) and k j i n to t he g r ind ing s imu-la t i on programs g r mix and gr plug.

AcknowledgmentsThe au thor s would l i ke to thank the T i t l e I11

Program of th e Minera l In s t i t u t es Act f o r fund ingt h i s p r o j e c t and Xr. Steven Riemer fo r col l ect *a gt h e g r i n d in g s i m ul a t io n d a t a and f o r c r i t i c a l l yreading t he manuscr ipt .

References1. Au st in , L.G., 1971, "A Review Int roduct ion to

th e Mathemat ical Descr ipt ion of Gr indine as aRa te Pro ce ss ," Powder Tech., Vol. 5, p. 1.

2. Au st in , L.G., and Luck ie, P.T., 1971-1972,"?lethods f o r t he Deter mination of BreakageDis t r ibu t ion Paramete r s I 1 Powder Tech. ,Vol. 5. DO. 215-222...Bond, F.C., 1960, "Confirmation of the h i r dTheory," A M Trans., Vol. 217, p. 139.Bond, F C . , 1961, "Crushing and GrindingC a l c u l a t i o n s , P a r t s 1 and 2,"Al l i s -Cha lmer sr e p o r t .Cross , M . , 1981, "A Method fo r Extra cti ngProduc t S ize Dis t r ib u t ion s f rom Empi r i ca lComminution Models," University of Minnesota,Mimeographed re po rt , Powder Tech., cu rr en tl yi n p r e s s.Cross, )I., and O w s t , A.P., 1978 , "ComputerSimu lati on of th e Comminution of Mix turesof H a te ri a l, " IPC 1978, Proc . UKSC Con. onComputer Sim ulat ion, p. 107.Gardner, R.P., and Suka njna jtee , K., 1972,"A Combined Tracer and Back-CalculationMethod fo r Determining Par t i cu la te BreekageFunct ions i n B a l l M l l i n g . P a r t I , Rat iona leand De sc rip tio n of th e Proposed Method,"Powder Tech ., Vol. 6, pp. 65-74.Her bst, J.A., and Fuerstena u, D .W . , 1980,"Scale up Procedures f o r Continuous GrindingM i l l Design Using Population Balance Hodels,"I n t . J. Hiner. Process, Vol. 7, pp 1-31.Herbs t , J . A . , e t a l . , 1971, "An Approach toth e Estima tion of the Parameter s of LumpedParameter Grinding Models from On-LineMeasurements,' ' Zerkleinern. DECFEIlA Monogr,Vol. 69. DD . 475-514...10. Holtham, P.N., 1978, "Sim ulat ion, Modelingand Co ntr ol of a Closed Loop B al l GrindingC i r c u i t 'I M. Sc. Thesis, UMIST.

11. Luckie, P.T., and Au sti n, L.G., 1972, "A Re-v iew In t roduc t ion to the So lu t ion of theGrinding Equations by D ig it al Computation,"Xi ner al Sc i. Enmg., Vol. 4, No. 2, p. 24.


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12. Lynch, A.J., 1977, Mi ne ra l Cru shi ng and-& i n d in g C i r c u i t s , V ol . 1, E l s e v i e r .13 . Reid, K.J., 1965, "A Sol ut i on t o th e BatchGri ndi ng Equ ati on, " Chem. Eng. Sci ., Vol. 24p . 953.14. Rowland, C . A . , and Kjos , D.M., 1978, "Roda nd B a l l M i l l s, " M i n e r al P r o c e s s i n g P l a n tDes ign, (eds . A.L. Muler and R.B. Bhappu),SME/ADE.


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TABLE 1. INPU T IXFORMATION REQUESTED BY GR DAT- --

FEED INFORMATIONU n i t s o f mss m e as ur em e nt ( k g , t o n n e , l o n g t o n o r s h o r t t o n )8 02 p a s s i n g s i z e o f t h e f e e d8 0% p a s s i n g s i z e o f t h e p r o d u c t ( a s p r e d i c t e d b y B ond e q u a t i o n )F ee d r a t e o f f r e s h o r eNumber o f s i z e i n t e r v a l sS i e v e s i z e s t o d e f i n e s i z e i n t e r v a l sW e i g ht d i s t r i b u t i o n i n f e e dS t a t e o f f e e d ( w e t o r d r y )Z s o l i d s by w e i g h t i n p u l p ( i f a p p l i c a b l e )D e n s i t y of f e e d o r e

CIRCUIT AhXl OPERATIONX i 1 1 t y pe ( r o d o r b a l l )C i r c u i t ( op en o r c l o s e d )O u t l e t c o n t r o l ( dam o r g r a t e )Flow in m i l l ( p l u g o r dam)N om in al s i z e o f s e p a r a t i o n ( w h er e a p p l i c a b l e )S i z e o f g r a t e o p e n i n g ( w he re a p p l i c a b l e )

NILL SIZEL e n g t hD i a m e t e r% v o lu m e o c c u p i e d b y g r i n d i n g m ed i a

POWERU n i t s o f p o we r ( h p o r kW).Xi11 p o w e r

T h i s i n f o m a t i o n i s c om b in ed v i t h p r e v i o u si n p u t a n d t h e ills o r e w e ig h t c a p a c i t yc a l c u l a t e d and o u t p u t f o r t h e i n f o r i n a t i ono f t h e u s e r .


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TABLE 2. CONFIGURATIONS OF TEST BALL MILLCircuit Separation Discharge Simulation Type P w e r Feed RatekW & tonneslhour

open - overflow mixed (11-25) 198 266 27.42plug(n=lO) 198 266 27.42plug(n=lO) 198 266 20.00p ug(n=lO) 198 266 35.00Closed 250 um overflow125 umOpen - grate 707 um mixed(na25)500 "

NOTE THAT: i) The value of l/n in all well mixed simulacbns refers to the portion of the millcontents removed, the value of n in the plug flow simulationscorresponds to thenumber of approximating mass segments.ii) The power drawn by a mill with grate discharge is larger than that of a mill witha dam overflow (Rowland and Kjos, 1978).iii) The elements of :he matrix[K], in all simulations are derived Erom kj = 1/2[(x + X,-~)/~X~]+, here xi is the largest sized particle in the size intervali(c3. equation 9 )

TABLE 3. FEED WEIGHT DISTRIBUTION- - - . -).PISIZE INTERVAL X WEIGHT RETAINED

+ 1410 micrometers+ 1000 micrometers+ 707 micrometers+ 500 micrometers+ 354 micrometers+ 250 micrometers+ 125 micrometers+ 88 micrometers+ 63 micrometers+ 44 micrometers- 44 micrometers


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TABLE 4 . CLOSED CIRCUIT GRI3DIXG MATERIAL KOKS- -WELL-L-MXED FLOW S IW IA T IO N

SI ZE OF FRESH CIRCULATINGLOAD(TPH) (X)

TOTALFEED(TPH)

PLUG FLOW SLXUUTION32 . 9 5 6 . 2 1 7 1


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c u r a s 3: Closed Circuit bll X i l l D i a c b t q e(Em). h s l f i a r O n r f l w (COT). m dCL us if ia r Uodarflav (CW)C u m a 4: Closed Circuic BW, COP, and CT2(123 uu r epara t ion)

I 44 q3 88 125 177 250 351 500 707 1000 la1 0f I I I I , I I I IParc ic le Size (microns)

~i - 1 S l r c d i s t r i bu t i on o f t ne t e e d and pmduczs Era. a b a l l n i l 1 with o v s r f l wdischarge (plug f lw s i m l a r i a n )

Cm r . 1: Opm Clrcuic 9 3Cune. 2: Clmcd Clrcuic StQ, COF a d C'JP(250 e n s.peraclon)Cutves 3: Closed CLe:ufc BXD, COP and CJT(125 un separation)Sasd r a m as in Figure 1.

i i y r c 2 Size d i s t r i b u r i a n cf pra iucr s fro8 a b a l l n i l 1 w i t h ove r f l w d i s c ha r ge[be l l- a ixed f l c r s iaula t io n)


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Curve 1: G r a ce s i r e 2 50 n i c r o n sC u r v e 2: G r a ce s i z e 5 0 0 n i c r o n sC u r v e j: G r a t e s i t e TO7 m i c ro n s

P r r r i c l e S i z e ( n i c r o n s)

F i g u r e 5 S i z e d i s r r i b u t i o n o f p r o d u c t s f ro m an op en c i r c u ~ c a l l m i l l w i t h 3g r a c e d i s c h a r g e

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Assessment of deisgn and operation of grinding mill using simulation.pdf