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J S. Af Inst Min Metal/ vol. 86, no. 4.Apr. 1 98 6. p p. 1 13 -1 24 .

br sive nd imp ctive we r of grindingb lls in rot ry m illsby LA. VERMEULEN* and D.D. HOWATt

S Y NOP S ISN ew d istrib ution fu nctio ns are d eriv ed to d escribe th e size distrib utio n o f grin ding e le me nts in b all m ills . T hef or m u la ti on s a re b a se d o n t he a s su m p ti on t ha t a b ra si ve a s w e ll a s i m pa c ti ve i nt er ac ti on s o c cu r d u ri ng b a ll m i ll in g -a na ss um ptio n th at is s up po rte d b y a la rg e b od y o f e xp erim e nta l e vid en ce . It is s ho wn th at th e fu nc tio ns c an b e u se din th e e stim atio n of th e m ag nitud es of a brasive a nd im pa ctive c om po nen ts in the tota l ra te o f b all w ea r.T he r el at iv e m a gn it ud es o f th e w e ar c om p on en ts p ro vi de a b as is f or o pt im iz at io n o f th e c he m ic al a nd m e ta ll ur gic alp ro pe rtie s o f th e b alls in a g iv en m illin g s itu atio n. H ow ev er it is c on te nd ed th at th es e q ua ntitie s a re a ls o u se fu lin dic ato rs o f th e re la tiv e in te ns itie s o f th e a bra siv e a nd im p ac tiv e in te ra ctio ns th at a re o pe ra tiv e in th e s iz e re du c-tio n o f m in era l p artic le s w ith in b all m ills . T he se q ua ntitie s c an b e d ete rm in ed fo r a ny in du stria l b all m ill a nd th eirm a gn itu de s p ro vid e p ra ct ic al g uid elin es f or m il l o pe ra ti on .T he th eo ry is u se d in an an alys is o f sa mp le s o f ba ll c ha rg es fro m tw o in du stria l b all m ills an d it is also a pplie dto a ll th e d ata o n b all-s iz e d is tr ib utio ns fo un d in th e lite ra tu re . T he q ua lita tiv e c orre la tio n b etw ee n th e c alc ula te dv alu es o f th e w ea r c om po ne nts a nd th e re po rte d o pe ra tin g c on ditio ns is g oo d fo r a v arie ty o f in du stria l b all m illsin c on fo rm ity w ith th e h yp othe sis th at th e relative m ag nitud es of th e w ear co mp one nts a re re la ted to th e m illingconditions. .S A ME V A TTINGD a ar w ord n uw e verd elingsfun ksie s a fg ele i om die groo tte ve rd eling va n m a alele m en te in b alm e ule te b eskryf.D ie form u lerin gs w ord g eg ro nd o p die aa nna m e d at d aar sow e l sku ur- a s sla gw isse lw erkin g tyde ns balm a lin gplaa svind - n aa nn am e w a t de ur n g root ho evee lh eid e kspe rim entele g etuie nis ge staa f w o rd. D a ar w o rd ge toonda t d ie fu nksies g eb ru ik kan w o rd o m d ie gro otte van d ie skuu r- e n slagkom po nen te in die to ta le tem po van ba lslyta siete raa m.D ie relatiew e g ro otte va n die slytko m po ne nte verska f n g ro nd slag vir die o ptim ering van d ie chem ie se e nm etallurg ie se eie nskap pe van d ie b alle in n g eg ew e m aa lsitu asie. D a ar w ord egte r a an ge voe r da t h ie rdie groo the deoo k nu ttig e a anw ysers is van d ie rela tie w e in tensite it van d ie sku ur- e n sla gw isse lw erkin g w at pla asvind tyde nsdie verkle in in g van m ine ra alpa rtike ls in ba lm eule . H ie rd ie g ro othe de kan vir en ige ind ustriele ba lm eu l be pa al w orden verskaf praktie se rig lyne vir m eu lb ed ryf.D ie te orie w ord ge bruik o m m o nste rs van b allad ings w a t uit tw ee in du strie le ba lm eu le verkry is te on tle ed e nw o rd to eg ep as o p a l die d ata o or b algroo tte ve rd eling s w at in die lite ra tu ur g evind is. D ie kw a litatiew e ko rrela sietussen die be re ke nd e w aa rde s va n die slytkom p on en te en die g erap po rte erde b ed ryfstoe stan de is g oe d vir nve rske ide nhe id ind ustriele b alm eu le in oo ree nste mm ing m et die hip otese da t d ie rela tie we groo th ed e van die slytko m-po nen te m et die m aa lto estand e ve rb an d ho u.

IntroductionB all m illing has been em ployed for m ore than a hun-dred years in the fine grinding of ores, coal, cem ent, andother m aterials. The w orld s consum ption of grindingballs used in this w ay is about 500 kt a year, and the con-sum ption of balls per ton of m aterial m illed varies w ide-ly . Ball c on sumptio n con stitu te s a s ig nific an t p ro po rtio nof the costs of fine grinding, rising markedly for hardand abrasive ores. In attem pts to reduce ball consum p-tion and the high cost of fine grinding, m any studies havebeen directed to the analysis of factors causing ball w ear.These studies have alm ost invariably attributed all thew ear to one of the tw o w ear m echanism s that operate inball m illing: abrasive w ear and im p a ctive w ear.That abrasion plays a large part in all fine-grindingoperations has long been know n and acknow ledged. O verforty years ago, Prentice reported tests aim ed at pro-v id ing ev id en ce th at a ll th e w ear o f g rind in g b alls is c au sedby abrasion. M uch additional evidence has since been. SpecialistScientist ,Council for MineralTechnology(Mintek) ,Priva te Bag X3015, Randburg , 2125 Transvaal .t Emer it us P ro fe ss or , Un iv er si ty o f t he W i twa te rs ra nd ; n ow a t M i nt ek

( see address above) .@ T he S ou th A fric an In stitu te o f M in in g an d Meta llu rg y, 1 98 6.S A IS SN 003 8-22 3X / 3 .00 + 0 .00. P aper received 8th July, 1 985.

quoted2 to show the significance of three-body abrasiveinteractions in fine grinding, which result in the sizereduction of m inerals and in the w ear of grinding m edia.The wear of grinding balls by impace in a mill is notnearly as evident nor as w idely accepted. N evertheless,there is am ple proof that balls are projected into flightand collide w ith the en m asse cha rg e and o cc as io na llyw ith th e m ill lin in g. Numero us p ho to grap hic stu die s, th erecent techniques of instrum ented bolts4 in m ills, andthe results of tests on balls fitted w ith accelerom eters5have all produced irrefutable evidence of im pact.During impactive processes, the rate of ball wear,w hich is the rate of m ass loss by the balls, is proportionalto the ball mass, Le. the cube of the ball diameter,w hereas the rate of ball w ear during abrasive processesis proportional to the surface area of the balls, or thesquare of the ball diam eter.The general idea of combined wear in whichmechanisms of both abrasive and imp active wear areoperative, can be attributed to Bond6. H is w ork in thisarea can be summarized,,7 by the follow ing em piricalre latio ns hip b etw ee n th e rate o f b all w ear, - dm/dT an dth e bal l d iamete r,

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dm - k' Xq, (1 )dT

where m is the m ass of a ball, T the am ount of m aterialm illed , and ,k I and q are constants. The value of the ex-ponent q, which by hypothesis should lie in the range2 ~ q ~ 3, provides an indication of the relative in ten-sities of the two components in the rate of ball wear.In discussions of the formula given in equation (1),B ond6,s, H ukki9, and Taggart' em phasized the role ofm ill speed and pointed out that, at low speeds, say in then eig hb ou rh oo d o f 6 0 p er c en t o f critic al sp ee d, ca sca din gwill be a prominent feature of the charge m otion and qwill be very nearly equal to 2; at higher speeds, say 85per cent of critical speed and higher, cataracting w ill bea prominent feature and q will be nearly equal to 3.T he idea that the ab rasive and im pactive com po nentsin ball wear are functions not only of the ball materialbut also of the m ill-operating variables is therefore w ellestablished in the literature. A s ball w ear im plies sizereduction during the residence of balls in a mill, it canbe expected that the size distribution of balls in a millw ill be related to the relative intensities of the tw o w earmech an isms. The se v al ue s c an b e o f p ra ctic al s ig nif ic an cein the selection of the best chemical and mechanicalc ha ra cteris tic s o f b alls fo r a g iv en m illin g o peratio n; th eycan also be used to reveal the conditions in the mill inw hich the balls are operating as the grinding m edium . F orthis reason, the p resent investigation is based upon datafrom m easurem ents of ball-size distributions, and in-c lu de s th e d eriv atio n o f ex pre ss io ns to d esc rib e a s te ad y-state ball-size distribution u nder conditions of abrasiveand im pactive w ear, and the determ ination of the relativem agnitudes of the tw o com ponents in the rate of ball w earunder defined con ditions of m illing.Rela tio ns hip s b etw ee n Wear Me chan isms and Ball siz eDistributionOnly b all wear is considered here. The production ofsm aller 'balls' by the fracture of larger ones is beyondthe scope of the present work.Qua litativ e a nd q ua ntitativ e in fo rm atio n re la tin g to th erelative intensities of the tw o w ear com ponents can beobtained from studies of ball-size distributions by theestablishm ent of a relatio nship betw een the num ber den-sity , ,,(X ), of balls of diam eter X and the rate of sizereduction ofthese balls, - dX/dT(millimetre per ton ofm aterial m illed). The num ber density function, w hich iso fte n ca lled th e' fre qu en cy d is trib utio n', is d efin ed b y th erelation

dN = ,,(X)dX, (2 )

where dN is the number of balls in the mill whosediam eters are in the size interval X to X + dX , an d Xis w ithin the range Xo ~ X ~ XmaxFrom equation (2) it can be shown that the mass ofthis aggregate of balls isdM = .~ P7rX ,,(X)dX, (3 )6

where p is the density of the ball m aterial.B all-size d istrib utio n in a g iv en m ill is d ete rm in ed c on -ventionally as follow s: the entire ball charge is rem ovedfrom the mill and screened into a number of size inter-vals; the num ber and m ass in each size interval are thendeterm ined. In this w ay, experim ental values are obtain-ed for the distribution fun ctions n(X) an d m(X), whicha re , re sp ectiv ely , th e cumu la tiv e n umber fra ctio n a nd th ecum ulative m ass fraction of balls passing size X. Thesedistribution functions are related theoretically to thenum ber density function, ,,(X ), acco rding to the follow -ing formulae :

x~ ,,(X)dX

n(X) - Xo , (4)xmax~ (X)dXXo

an d xj XJ (X)dXXom(X) -

xmax~ XJ,,(X)dX

Xo

(5 )

The charge of a ball m ill has been shown1,J,9to tendto a s ta ,b iliz ed c on ditio n a fte r th e m ill h as b een o pe ra tin gu nd er s te ad y-s ta te c on ditio ns fo r a s uffic ie ntly lo ng tim e.S te ad y s ta te implies a constant feed rate of ore to a milland a steady rate of addition of top-size balls to com -pensate for the depletion of the ball charge by w ear. Thetim e req uired for the stabilized conditio n to be achievedis a bo ut-3M ,(dM/d1) (dT/dt)

where M is the total m ass of balls w ithin the m ill, dM / dTis the rate of ball consum ption (kilogram s per ton of orefe d to th e m ill), dT/ dt is th e m illin g rate , a nd th e n eg ativ esign appears because the m ass of balls decreases duringthe m illing operation. It is assum ed that, in the stabiliz-ed condition, the ball-size distribution in a m ill does notvary .J; nam ely, that during operation the num ber andmas s o f b alls in a ny s pe cific siz e in te rv al rem ain c on stan t,a lth ou gh th ese q ua ntities m ay v ary s ub sta ntia lly from o nes iz e in te rv al to a no th er. T he n umbe r d en sity , ,,(X ), is th enrelated to the rate of ball w ear by a consideration of thenumber of balls in a size interval, as follows.If N j is the number of balls in the mill whosediam eters are in the size range Xi to ~, where Xo ~ Xi< ~ ~ Xmax then from equation (2)

XiN, = j ,,(X)dX. (6 )xi

If ~ is constant, its derivative with respect to theam ount of m aterial m illed, T, is zero, Le. 4 APRIL 98 6 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

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dN d Xj - - \ v (X) dXdT dT XidXV(X) dT

dX ,V(X) - = O. (7)dTSince equation (7) holds good for any X < X withinI Jthe range Xo to Xmax it follows that

- veX) dX is constant. (8)dTFrom a consideration of the largest size group, it canbe shown that the constant in equation (8) is n , th enum ber of top-size balls of diam eter X that are add-ed to the mill in the time required for th~ mill to grinda unit mass of new feed. Equation (8) can therefore bewritten in the formV (X ) = - ~. (9 )

dX/dTEquation (9) shows that, in a stabilized mill, thenumber de nsity of b alls of size X i s inversely proport ionalto the rate at which their diam eters are being reduced.This relation, together with equations (2) to (5), cantherefore be used to supply inform ation on the relativeintensities of the w ear m echanism s that are operative inany g iven stabilized m ill.T he exponent q in the em pirical form ulation of the rateof ball wear given in equation (1) provides im portant,altho ugh on ly qu alitative, inform ation on the relativ e in-tensities of the rates of abrasive and im pactive wear in

ball m illing. This is because a value of q between 2 and3 represents com bined w ear, in w hich the m echanism s ofboth abrasive and imp a ctive wear are operative. Thedeterm ination of q from m easurem ents of the ball-sized istribu tion requ ires th e d erivation of th e nu mb er de nsi-ty fu nction an d the distrib ution fun ctions app ro priate tothis formulation, followed by fitting of the deriveddistribution functions to the measured values of thenum ber and m ass fractions of balls passing size X.From Bond s formulation of the rate of ball wearw hich is expressed by ,

- dm = k Xq, (1 )dTthe follow ing is obtain ed:

- dX = kXq-2, (10)dTwhere k = Zk / p7f. Substitution of equation (10) intoequ atio n (9 ) then yield s the follow ing e xpression fo r then umbe r d en sity fu nc tio n:

nvc(X,q) = -, (11)kXQ-2

w here the subscript C denotes com bined wear.T able I g ives ex pression s for Nco the total num ber ofballs in the m ill, and Mco their total mass, and for thefunctions nc(X,q) and mc(X,q), which are number andm ass distribu tio n fu nction s resp ectively . T hese expres-sions w ere obtained sim ply by the substitution of equa-tio n (1 1) in to e qu atio ns (2 ) to (5 ), fo llowed b y th e re qu ire dintegrations. Table I also show s the form s to w hich theexp re ss ions r educe under hypoth et ic al condit io ns o f pur e-ly abrasive wear and purely im pactive wearJ, underwhich the exponent q has the lim iting values of 2 and 3respectively.As an example of the applicability of the theory, then umb er d istrib utio n fu nc tio n, nc(X,q) (equation 14 inT able I), w as fitted to m easured values7 of the ball-sizedistribution in the charge of a primary ball m ill atM arievale (T able A -I in the A ddendum ). T he fitting pro-cedu re yielded the formula

nc(X,q) - XO,J6 - XoOJ6 , (16)X~;~6 - XoO,J6

where Xo = 1,0 inch and Xmax = 4,0 inches. For pur-p ose s o f c ompa riso n, th e re sp ec tiv e n umbe r d istrib utio nfunctions nA(X) an d n (X) for purely abrasive and pure-ly im pactive wear (Table I), w ith the sam e values for Xan d Xmax are also depicted in Fig. 1. The functio~nc(X,q) for com bined w ear gives the best description ofthe m easured b all-size distributio n. E quatio ns (14) an d(1 6) sh ow th at q , a s d ete rm in ed from th e n umbe rd istrib u-tion function, is

q(nc) = 3 - 0,36 = 2,64.A sim ilar analysis, in w hich the m ass distribution func-tion for com bined w ear mc(X,q) and the m easured m assdistribution w ere u sed, yielded

q(mc) = 2,28.D espite a discrepancy of about 15 per cent betw een thetw o estim ates o f q , the resu lts su ggest fairly strong ly th atthe balls in the m ill were subjected to a significant im -pac tive component .It is clear that exponent q is a number, and that thedeterm ination o f its v alu e p ro vid es only a q ualitativ e in-dication of th e relative intensities of abrasiv e and impa c-tive wear in a given mill.Bond s em pirical relation, which was shown in equa-tion (1), can be replaced by a m ore detailed analysis thatleads to the determ ination of the m agnitudes of the tw ocom ponents in the rate of ball w ear. The rates of abrasivean d impactiv e w ear a re su perpo sed in the fo llo wing m an -n er: durin g ab rasio n the rate of ball w ear is p ro po rtion alto the surface area of a ball , Le.

dm- - ex m2/J;dTduring im pact the rate of ball w ear is proportional to themass of a ballJ, Le.

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TABLE IE XP RE SS IO NS T O C HA RA CIE RIZ E A S TA BILIZ ED B AL L-S IZ E D IS TR IB UT IO N U ND ER C OMB IN ED WE AR A ND U ND ER T HE H YP OT HE TICA LCOND IT IONSOF PURELY ABRAS IVE AND PURELY IMPACTIVEW EAR ACCORDING TO THE BOND FORMULAT ION OF THE RATE OF BALL WEAR

Quantity

Form ula for w ear rate

N um be r de nsity fu nc tio n

Number of balls in themill

M ass of balls in themill

Numb er d is tr ib ut io nfun ction o f b allssm aller than size X

Mass d is tr ibu ti onfu nc tion o f b allssm aller than size X

G ene ra l e xp re ssio n fo r c om bine d(a bra sive a nd im pa ctiv e) w ear'

-~;OCxq.............

- n'vc(X,q) - kXq-2 (11)

n'(X,;;;;.q - x;-q)Nc(q) = k(3 - q) . . .. (12)

P urely ab ra siv e w ear7(limitingformasq~2)

P ure ly im pa ctive w ea r3( lim it ing form as q ~ 3)

n' pJt(X;;;.q - X:-q)Mc(q) = 6k(6- q) . , . . (13)

(1 ) - dm oc mdT dm- dT oc m

x3-q - X.q-3nc(X,q) = x~xq - X; q . . . . . . (14)t

vA(X) = ~k n',(X) = kX

x6-q - X:-qmc(X,q) = X.:..q - X: q . . . . . . (15)t

n'NA = k (Xmax X.) n', = T(lnXmax-lnX.)- n'pJt 4 4MA - 24j{(Xmax-X.) - n' pi t ( 3 3 )M, - r s Xmax - X.

X-X.nA(X) = Xmax- X.InX-lnX.n.(X) = InXmax-lnX.

X4 - X.4mA(X) = X':' - X.X3 - X;m,(X) = X~ - X;

' The p re sen t i nve st ig at ionS ub sc rip t C d en ote s c om bin ed w ea r, S ub sc rip t A d en ote s p ure ly a bra siv e w ea r, S ub sc rip t I d en ote s p ure ly im pa ctiv e w ea rt Au stin a nd K limp el2 0 re ce ntly a ls o d er iv ed th es e f un ctio ns b y d if fe re nt m e th od s

x ~ 1 .0 inXm ~4.0in0 ~ Experim ental data'- n,(X) ~ Numbe r f ra ct io n f or p ur el y imp acUve we ar

nA(X) ~ N umber fraction for purely abrasive w earm.. ndX,q) ~ N umber fraction for combined w ear,when q ~ 2 ,6 4

1,0

0,8

~'

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TABLE 11EXPRESS IONSTO CHARACfERIZE A STABIL IZED BALL-S IZEDISTRIBUTIONUNDER CONDITIONSOF COMBINED WEAR

Quantity G eneral expression for co mbined (abrasive and im pactiv e) w ear

Form ula for w ear rate dm 2;- - = \m+~m;>dT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (17)

N um ber d en sity fu nc tio n vc(X,I.) = \ ln~l.) (22)

Number of balls in the mill 3n ( Xmax + A)c(l.) = a- In Xo + A , 9~Mass of balls in the m ill M (A ) = n pJ[ [ ...L x> -X LI. X 2 _X2)+ 1.2 X -X)-I.>ln (Xmax+I. )J .. . . . .. (24)c 2(\ 3 max 0 2 max 0 max 0 Xo + AN um ber d is trib utio n fu nc tio n In X + A) - In Xo + A)nc X,I.) = In Xm ax+I.)-ln Xo+l.) (25)

M as s d is tri bu tio n fu nc tio n + X> - X; ) - t I. X2 - Xo2) + 1.2 X - Xo ) - 1.>ln ( :.:~ )mc X,I.)= ...L X 3 -X 3) LI. X2 _X 2)+ 1.2 X -X)-I.>ln (Xmax+l. )3 max 0 2 max 0 max 0 Xo + A . . . . . . . . . . (26)Based upon the superposition of abrasive and im pactive wear rates

vc X, A) = 3n . (22)a X + A)[his function is expressed as a function of A , because thelatter h as im po rtan t p hy sical sig nifican ce w ith in th e p re-sent context, w hich w ill be discussed further.E xp ressio ns eq uivalen t to tho se listed in T ab le I, d eriv -ed by the subi titution of equation (22) into equations (2)to (5), are shown in Table 11. In addition to theseequations-(23) to (26)-expressions can be derived forthe cumulative number and cumulative mass of ballssm aller or larger than a given size, the num ber and m assof balls in a size interval, the m ean ball diam eter, the sur-face area of the ball charge, and so on.The param eter A in the expressions above is propor-tional to {31a so that, if {3tends to zero, then A a lso tendsto zero. The expressions in Table 11 then reduce to thecorresponding expressions that are applicable underh yp oth etical co nd itio ns o f p urely im p a ctiv e w ear (T ab leI). On the other hand, if {3is finite and a tends to zero,then A t ends to infinity. T he expressions in T able 11 thenreduce to the corresponding expressions that are validunder hypothetical conditions of purely abrasive w ear(T ab le I).

XmaxWA(charge) = ) WA X)V X,A)dX

Xo= Y z n p1T A [Y 2 X~ ax

The physical meaning of A becomes apparent fromequations (18), (19), and (21) w hen the late of abrasivewear is equated to that of impactive wear. This showsthat A represents a ball size for w hich, under the givenm illin g cond itio ns, th e ra tio o f ab ra siv e to impac tiv e wearis unity. If AI X max > 1, then all the balls in the mill ex-perience prim arily abrasive w ear, Le. the action of them ill is p rim arily ab rasiv e, b ut th ere is a fin ite im pactiv ecomponent as well. If AIXmax< 1, then all the balls inth e m ill w ith d iam eters larg er th an A e xp erien ce p rim ari-ly im p a ctiv e w ear, an d all th os:: w ith d iam eters less th anA u nd erg o p rim arily ab rasiv e w ear. T he actio n o f th e m illis then clearly such that balls are subject to both wearmechanisms, but the impactive component is nowenhanced relative to the previous case, in w hich A IXmaxwas greater than 1.Equations (18) and (19) give the respective rates ofabrasive and im pactive w ear on balls of size X. Summa-tion over all the balls in the mill gives the values forVA(charge) and W ,(charge), the respective rates ofabrasive and im pactive w ear on the w hole ball charge,as shown in equations (27) and (28).

- X 2) - A(X . - X ) + A2ln X max + A J . (27)0 max 0 Xo + ASimilarly,W,(charge) = Y z n p1T [1/3(X~ax - X~ ) - Y2A X ~ax - X ~) + A2(Xm ax- Xo ) - AJlnXmaX+ A J . (28)Xo + A

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H ence, the ratio 0 f the rates of abrasive to im pactive w ear on the ball charge isV 2A ~ a x - X ~) - A 2 X m ax X J + A Jln Xmax + AXo + A

l/3 X~ax - X~) - V2A X~ax - X~ ) + A 2 X m ax- Xo) - AJln Xmax + AXo + A

R A l charge) -

E quation 29) show s that the ratio R AI charg e) is in-dependent of lX , 3, a nd the rate o f ball consum ption. Itis a function on ly of A , X o, an d Xmax T he quantity Ac an th er ef ore b e u se d to e xp re ss q ua ntita tiv ely th e re la tiv em agnitudes of the rates of abrasive and im pactive w earthat are operative on the w hole ball charge. It is contend-ed that n ot only are these quantities related to ball w ear,but they are also im portant indicators of the m echan ism sof size reduction of m ineral particles in a giv en m illingsitu atio n. H en ce , th eir c alcu lated m ag nitu de s sh ou ld c or-re la te w ith th e m illin g c on ditio ns , e .g . m ill d ia me te r, m illspeed, liner configuratio n, pulp density, ball size.F urtherm ore, A c an also be related to the exponent qin the B ond form ulatio n of ball w ear. T he relationshipis found by the use of equations 12) and 13) in T ableI in conjunction w ith equatio ns 23) and 24) in T able11 , to yield equation 30):

. . 29) ff

m as s u nle ss all th e g rin din g e le me nts a re tru ly s ph eric al.T he d eg re e o f u nc ert ain ty to b e a ss oc ia te d w ith th eo re tic alassessm ents of the abrasive and im p a ctive com po nentsin ball w ear can be estim ated by the use of this differencebetw een the m easured and the calculated ball m asses.

T he N ature of B all W ear in Som e Indu strial M illsT he determ ination of the ball-size distribution in anin du strial m ill is a le ng th y p ro ce ss in vo lv in g d is ru ptio nof the m illing process and possibly som e loss of produc-tio n. A lthough the alternative p rocedure, L e. sam plingof the ball charge, introduces an elem ent of uncertainty,it w as carried out in tw o ind ustrial m ills-a 9 ft by 10 ftball m ill at the L ibanon G old M ine and an 8 ft by 8 ftball m ill at the M arikana M ine of W estern P latinumL im ited-and the data obtain ed w ere analysed. In add i-tion , the literature w as searched for all the relevant data

l/3 X~ax - X~) - V 2A X ~ax - X~) + A X m ax - Xo) - AJln Xmax + AXo + A

1 n X m ax + A ) - 1n Xo + A )T his relationship sh ow s that, if the value of A i s know n,q can be determ ined, and v ic e v er sa .

T he new distribu tion functio ns ne X,q), m e X,q),ne X,A), an d me X,A) show n in T ables I and 11th ere fo re p ro vid e v ario us m eth od s fo r th e d eterm in atio nof A , and h ence of the rates of abrasive and im pactivew ear during ball m illing.In m athem atical integratio ns over a rang e of ball sizes,th e q ua ntitie s k , lX , and {3in the tw o form ulations of ballw ear w ere regarded as constants, this assum ption beingcom m on to the p resent w ork and all previously reportedi nv es ti ga ti on sl ,J ,z o. T h es e q ua nt it ie s a re e ss en ti al ly s ha pefactors, im plying that the derived expressions are ap-plicable strictly only to size distributio ns in w hich thegrind ing elem ents rem ain spherical. E xam ination of theb all c ha rg e in a ny in du stria l m ill w ill sh ow th at a su bs ta n-tial proportion of the sm aller grinding elem ents are notspherical, and som e degree of error therefore arises w henthe derived distribution functions are applied to ballcharges.T he c om ple te e valu atio n o f th es e e rro rs is d ifficu lt, b utth e co nsta nt-sh ap e a pp ro xim atio n is g oo d fo r b alls la rg erthan one-third of the size of top-size balls because th eseb alls are fa irly w ell ro un de d. T he sm aller, n on -sp he rica lelem en ts u su ally c on stitu te les s th an o ne -th ird o f th e to ta lnum ber and a m uch sm aller proportion of the total m ass.T he non-sphericity factor can be taken into account ifthe total m ass is calculated according to equations 13)or 24) and the results are com pared w ith the m easuredtotal m ass of the charge. T he value of the calculated m assw ill alw ays be sm aller than the value of the m easured

3 - q) X~:xq - x,~-q) . 3 0) 6 - q) X ~:xq xt 1)that could be used in testing the validity of the theoryof com bined w ear.D ata reported by PrenticeI for the w hole ball chargein a 6 ~ - 2ft by 12 ft m ill at B lyvo oruitsig G old M ine in-cluded the num bers and m asses of balls in every V 2-inchsize interval betw een 1 and 3 inches, the total num ber andtotal m ass of the balls, and the rate of ball consum ption,w hich w as eq uivalent to 300 top-size balls per day . T hem ill h ad b ee n o pe ra tin g u nd er ste ad y-s ta te c on ditio ns fo r8 m onths, and it seem s a safe assum ption that the chargew as stabilized, since about 9,5 tim es the total ball lo adhad been consu m ed in that period. T he throughput ofthe m ill w as not reported, but it has been suggestedlJthat it w as about 450 t/ d, givin g a rate of ball consum p-tion of 0,667 balls or 1,17 kg of balls per ton m illed. T heoriginal datal are reported in T able A -2 of the A dden -dum . S uch com prehensive data provide a basis for il-lustrating the app lication of the various form ulae in thecalculation of the rates of ab rasive and im pactive w earu nd er g iv en m illin g c on ditio ns.T he cum ulative form s of P rentice s data are show n bythe points in F igs. 2 a ) to 2 d ). T he distribution functionsne X,A) an d me X,A), sh ow n in equ ations 25) and 26),w ere adapted to the data. T he fitting procedures, w ithXo equal to 0,6 inch, gav e the follow ing values:

ACne) = 8,1 inches = 206 m mA Cme ) = 8,9 inch es = 226 m m .In a) and b) of F ig. 2, the cum ulative form s o f the

B lyvooruitsig valuesl are com pared w ith graphs of the1 18 A PR IL 1 98 6 JO U R N A L O F T H E S O U TH A FR IC A N IN S TIT U T E O F M IN IN G A N D M ET A L L U R G Y

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R' '-::~';:: i : 0,6;;j';;;.... co~ ,~';:: ~ 0,4. So.='S='

(a ) -=Distribution functionnc X, A) w ith A =8,1 inches0,8 @) = P rent ic e's e xp er iment aldatal

0,2

R' '-EO

(b) -= Dis trib utio n f un ctio n mc X, A)w ith A= 8 ,9 in ch es

0,8 @) = P ren tice 's ex pe rim en tal d ata ld,g ~ 0,6;;j i :

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was subject. For exam ple, for balls of average m ass m(0 ,5 58 k g),W A m ) = 3m2/3 ::: :::4,4 x 10-5 kg t-I,

an dWj m) = am ::::::1,1 x 10-5 kg cl.

These values, for balls of average mass, indicate thatab rasive in teractions w ere predom inant in th e given m ill.That this was characteristic of the whole charge in themill can be confirmed by calculation of the rates ofabrasive wear and impactive wear on the whole ballcharge, viz W A(charge) and W 1(charge). Quick esti-m ates, w hich can be confirm ed by detailed calculationsb ased on e quatio ns (2 7) and (28 ), in dic ate that, since Nmis the total mass of the charge (13 309 kg), the chargeexperienced an im pactive w ear rate of

Wj(charge = aNm ::::::0,26 kg t-I.Hence the abrasive wear rate of the whole charge was

WA(charge) = rate of ball consumption minusW1(charge)- 1,17 - 0,26::::::0 ,9 1 kgt -l.T he ratio R AI(charge) of the rates of abrasive to im pac-tive wear on the ball charge was therefore

RAI(charge) : : :: : : 3,5.Substitution of A into equation (29) yields a value of 3,6for RAI(charge), which is in good agreem ent with theabove.T he value of R AI(charge) clearly indicates that abra-sion was probably the predominant comminutionm echanism in that m ill and explains w hy Prenticel wasable to claim that the Blyvooruitsig data supported histheory of ball wear. H ow ever, the results of the presentwork show that about 23 per cent of the total ball con-sumption in the given mill was due to 'ID impactivecomponent.The influence of the proportion vf non-sphericalgrinding elem ents on the above results should be con-sidered. The m easured m ass of the charge w as 13 309 kg,and the mass, as calculated from equation (24), was12 334 kg, Le. the calculated mass was sm aller than them easured m ass, as expected. The discrepancy of only 8per cent is due to the non-sphericity of the sm aller, w orngrinding elem ents in the charge. This suggests that theerror in the results for the abrasive com ponent is about82/3per cent, Le. 4 per cent, and that the im pactive com-ponent is too high by about 8 per cent. If allow ances aremade for uncer ta in ti es i n 'A and Xo, it can be concludedthat the estim ate of the ratio of the rates of abrasive toimpactive wear in the m ill at B lyvooruitsig is, as far asis known at present, reliable to about 20 per cent.In the present w ork, ball sam ples w ere draw n from thecharges of the m ills at the Libanon Gold M ine and theMarikana M ine of Western Platinum Ltd. The data

describing the size distribution in the two ball sam plesare given in Tables A- III and A- IV of the Addendum .Fig. 3 compares the cumulative forms of the data (Le.the numbe r fraction s of b alls sm aller tha n size X ) relatingto these m ills with curves representing the extrem es ofpurely abrasive w ear and purely im pactive w ear, nA X)and nI X) respectively. The data points are clearly in-termediate between these two extremes, and the bestdescriptions of the data are provided in each instance bythe graphs for the theory of com bined w ear, L e. the func-tion nc X , A) as given by equation (25) in Table 11, withth e v alu es

'ALibanon 149 mm'AMarikana 10,2 mm.The se v alu es o f 'Aa nd th e re le va nt d ata w ere su bstitu te dinto equation (29), and the ratio of the rates of abrasiveto impactive wear on the charge RAI(charge) wascalculated for both ball sam ples. The results (T able Ill)show that the im pactive com ponent accounted for 74 and34 per cent of the ball consum ption in the M arikana andL ib ano n m ills resp ectively . T able III also sho ws th at th e

d ifferences in the m illing c ond ition s at the tw o m ines areq uite co nsisten t w ith these results: th e m ill at M arik anaw as equipped with lifter bars, which provided a strongim pactive com ponent, w hereas the m ill at Libanon hada grid liner, w hich w as effectively sm ooth and w hich ex-hibited m arked circum ferential grooving due to slip andtherefo re m ark ed ab rasion o f the grind ing ch arg e. T heseresults are also consistent w ith those of B lyvooruitsig,w here the m ill speed and diam eter, volum e of the chargeand top-size balls, and the im p active com ponent (23 percent) w ere all less than at Libanon. The relationship be-tween the m agnitudes of the wear com ponents and theoperating conditions of ball m ills is an important co-ro lla ry to th e p re se nt w ork . The sa lie nt m illin g c on ditio nsinclude the diameter and speed of the mill, the size ofthe balls fed, the nature of the mill lining, and the pulpdensity. It is contended that the relative m agnitudes ofthe wear components provide an indication of theg rin ding m ec hanism s that are o perative in b all m ills, andth at the relativ e m agn itu des can be chan ged b y v ariatio nof one or m ore of the salient m illing conditions.T his matte r w as in ve stig ate d fu rth er b y th e a pp lic atio nof the theory of combined wear to all the ball-sizedistributions found in the literature, and the values ofq, 'A , and the relative m agnitudes of the com ponents inthe rate of ball w ear w ere determ ined. T he results of thisanalysis are shown in Fig. 4, which was obtained by useof the fact that 'A is a function of q and of the ball-sizer educ tion rat io , X m a/ X o , as shown in equation (30). Thepoints in Fig. 4 show values of A/Xmaxfor the variousm ills plotted as a function of the corresponding valuesof q. T he con tin uou s curves in F ig. 4 sho w the theo reticalre la tio nsh ip -e qu atio n (3 0)-b etw ee n A/Xmaxand q forv ario us sp ec ific v alu es o f Xm J Xo In spite of the lim itedin fo rm ation and som etim es inad equ ate data (e.g . th e siz erange of the balls in four ball~tube m ills at L ake Shore14w as d iv id ed in to o nly th re e in te rv als), th e mea su re d v alu esexhibit a trend that provides strong confirm ation of thevalidity of the theory of com bined w ear proposed here.A t first sight, the m easured values corresponding to ball

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(a) Libanon (A = 149 mm )Im pactive com ponent 34070

1,0 @ = Expe rimen ta l dat a- = nI X); pure ly impac tiv e w ea r= nA X); pure ly a bra si ve w ea r Inc X, A ); c om bin ed w ear :'/:/,/:/:'/:/:'/: I1/:/:'/:/1/:/~/: I:/: I:'/1// nA X)r4.'/:'/:/1/:/:'1/:/~1/I/1It

0,8

R'ci' 0,60'~ ~.::: 1;\t t

oD bI)EL5a ~

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Mill G ri nd in g b al ls C a lcul at ed pa ramet er sSpeed Volume Ratio of Impactive07 0 of To p Reject abrasive wear

Diameter Length of charge size size 0 impact- % ofM ine ft ft critical % mm mm q A AIXmax iv e w ea r t ot al we arM iami CopperJ 8 2 82 20 50,8 0 2,97 0,0047 Small 0,0033 99,7(Ha rd in ge c on ic al

mill)Marikana 8 8 80 45 60 25 2,78 10,17 0,203 0,26 74Wes te rn P la ti numLimitedMarievale' 8 8 83 47 120 25 2,46 67,5 0,66 0,91 52

Lakeshorel4 7 6 85 45 108 25 2,31 13 8 1,28 1,72 36no. 7 ball m illLibanon 9 10 79 48 108 30 2,30 14 9 1,42 1,88 34,7

Hollinger 15 6Y z 12 80 50 76 0 2,17 12 7 1,67 2,18 30,5Blyvoortuitsig' 6Y , 12 70 25 76 15 2,17 20 7 2,73 3,65 21,5

S ub N ige l' 6Y z 9 68 33 100 0 2,13 48 3 4,75 6,4 13,5

TA BLE III MILLING CONDITIONS AND THE MAGNITUDE OF THE IMPACTIVE COMPONENT IN SOME INDUSTRIAL BALL MILLS

7. 5 ,I:Sub N igelIIIIIIIIIIIIIIIIIIII .I :: \I :: \: \\\ \\1\:11\\ \\\ \\\ \\3 \ \ \I toBlyvo~r. ~\

\ \\\ \. ~\ \\ \ ~\ '. E\ ~..Lakesh ore2 :\ \\ \ ~ ~Hollinger1i> ~ $ ~...~ 1 ~ \ ,\\o~13~e~cE0'2a.

02,0 2.2 2,4 2.6 2,8qF ig . 4 -V alu es of AIXmlX as a function of q for the bail-sizedistributions in a number of mills

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TA BLE A -IBALL-SIZE DISTRIBUTION AT MARIEVALE (AFTER WHITE ')

Ball s ize Number of Prentice Davisinch Mass balls theoryl theory'

3Y2 45,3 2282 2185 I 6853 18,7 1423 2185 19472Y2 17,6 2240 2 185 23042 7,7 1792 2 185 28201Y2 8,6 4268 2 185 36361 2,1 2935 2 185 5124Total 100,0 14940 13110 17 516

T AB LE A -IIBALL -S IZE D ISTRIBUTION AT THE BLYVOORUITS IG GOLD MINE ,18TH OCTOBER, 94 2(AFTER PRENTICE1)G rad ing of Averagecharge Mass T ota l m ass mass per ball Number ofinch 07 0 Ib Ib balls

2Y2 53,34 1 5 6 50 3 114 50262 2225 6528 1,774 36801Y2 17,05 5002 0,890 56231 5,90 1732 0,333 5198

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the r extensio ns. G ratitu de is also ex presse d to ProfessorF.R .N . N abarro, F.R .S., for discussions, criticism , andcomments, to Dr LA. Barker and Dr D. Hulbert fordiscussions, to M rs E. van der Berg, who assisted withsom e com putations, and to General M ining Union Cor-poration Lim ited, who m ade m aterial from the G encorarchives available. D iscussions w ere also held w ith M rW . Flook of G encor and M r N.F. Peverett of G old Fieldsof South Africa Limited, to whom thanks are due forthe friendly interest they show ed in this w ork. G ratefulacknow ledgem ent is m ade of the assistance given by M rR.J. Adey and M r C.L.M . Gough, the m ill superinten-dents at Libanon Gold M ine and the Marikana M ine ofW estern Platinum L im ited respectively. T he vigorouscoo peration o f their m illing p erso nne l is also gratefullyacknowledged.

ReferencesI. P RE NT IC E,T .K . B all w ear in c ylin dric al m ills. J. C hem . M eta ll.M in. Soc. S. Afr., Jan. 1943. pp. 99-116.2. V ER ME UL EN ,oA ., and H OW AT , D .D . A brasive and im pactivewe ar o f h ig h- ch rom ium cast -ir on g rin din g b alls . R andbur g, Counc ilfo r M in er al T ec hnolo gy , Report M 116. 1984. 25 pp.3. D AV IS ,E.W . F ine crushing in ball m ills. T ra ns . A IM E , v ol. 6 1.1 91 9. p p. 2 50 -2 97 .4 . V ER M EU LE N,oA., O HL SO ND E F IN E, M .J., a nd SCH A KO W SK I,.Physical inform ation from the inside of a rotary m ill. J. S . A fr.In st. M in . M eta ll., v ol. 8 4. 1 984 . p p. 24 7-2 53.5. D uN N, D .J., an d M AR TIN ,R .G . M ea surem en t of im pactiv e forcesin b all m ills. Trans. Soc. M in. Eng. AIM E, vol. 264. 1978. pp.

Ion Ex 87Papers are invited for Ion-Ex '87, an InternationalC onference a nd In dustrial E xhibition o n th e in dustrial,analytical, and preparative applications of ionc hro matog rap hy an d ion -e xchan ge p ro cesses, w hich is tobe held on 13th to 16th April, 1987, in W rexham , W ales.

The Conference is supported by the Royal Society ofC hem istry A naly tical D ivision (N orth West R eg ion ) andthe So ciety o f C hem ica l Indu stry, S olv ent E xtraction andIon E xch ang e G ro up, to geth er w ith m ajo r o rg aniza tio nsinvolved in the field.The proposed scope of the meeting is to include thefollow ing general areas, and each will be review ed by arecogn ized au tho ri ty .. I no rgan ic ion ana ly sis. Ion -exchangeres in s, ion-e xc ha ng e p ro ce sse s a nd in strumen ta tio n (R ev iew: D rH am ish Small). Organic acid and base analysis, including

IPMI con ferenceD r M .L El G uindy has been appointed General Chair-m an of IPM I's 10th International Precious M etals C on-ference and E xhibition. T he m eeting is to be held at L akeTahoe, Nevada, from 9th to 12th June, 1986.The theme of the Conference will be 'InteractivePrecious-M etal T echnology-Producer to U ser'. Sixty-nine prese ntation s w ill be given by inte rna tio nal exp ertson subjects such as precious m etals in space-related in-dustries, precious m etals from natural resources, andh igh- techno logy app li ca tions.The C onference w ill m ark the official celebration ofth e te nth a nn iv ersa ry o f th e In te rn atio na l P re cio us Me ta ls

384-388.6. BOND, F.C. Wear and size distribution of grinding balls. Trans.

AIM E, vol. 153. 1943. pp. 373-384.7. WHITE, H.A. Vote of thanks to the Prentice paper. J. C hem .

M etall. M in. Soc. S. Afr., Jan. 1943. pp. 116-122.8. BOND, F.C. W ritten contribution to discussion of the Prenticepaper. J. C hem . M etall. Soc. S. Afr., Jan-Feb. 1943. pp. 131-133.9. HUKKI, R.T. Correlation between principal param eters affecting

m echanical ball wear. T ra ns. A IM E, vol. 199. 1954. pp. 642-644.1 0. T AG GA RT , A .F . Handbook of m ineral dressing. New York, JohnW iley & Sons, 1954. pp. 5-27.11. VERM EULEN, loA., and How AT, D.D., Quantitative assessment of

abrasive and impactive wear rates from ball-size distributions inrotary mills. Randburg, Council for M ineral Technology, ReportM201. May 1985.

12. BERNUTAT, P. W ear of grinding media and liner plates. Zement-Kalk Gips, no. 9. 1964. pp. 397-400.

1 3. A DA MS ON , R .J. Gold metallurgy in South Africa. Johannesburg,Chamber of M ines of South Africa, 1972. p. 45.

14. CROCKER, B.S. W ritten contribution to the paper by T.K. Pren-tice. J. C hem . M etall. M in. Soc. S. Afr., Feb. 1944. pp. 133-136.1 5. LON GMO RE , E .lo , et al. Comparison of Iow imd high discharge forb al l m il ls . Trans. Instn M in. M etall., vol. 46. 1937. pp. 562-583.

16. T AG GA RT , A .F. O p cit., p. 5-65.17. FLOoK, W . General M ining Union Corporation Ltd. Private com-m unication, 1984.18. NORMAN, T.E., and LOEB, CM. Wear tests on grinding balls.

Trans. A IM E, vol. 1983. 1949. pp. 330-360.19. NORQUIST, D.E., and MOELLER, J.E. Relative wear rates of

various diam eter grinding balls in production m ills. T ra ns . A IM E,vol. 187. 1950. pp. 712-714.

20. AuSTIN, loG., and KUM PEL, R.R. Ball wear and ball size distribu-tions in tumbling ball m ills. P ow er T ec hn ol og y, vol. 41. 1985. pp.279-286.

bioch em ical, preparative, an d assa y techn iqu es. E x-change resins and instrum entation (R eview D r F.C .Sm ith, M illipore SA , France). Polyelectrolyte fractionation processes. Industrial w ater-purification procedures, includingeffluent treatm ent (R eview : D r J .R . M illar)

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24 A PR IL 9 86 JO UR NA L O F T HE S OU TH A FR IC AN IN ST IT UTE O F M IN IN G AN D M ETA LLU RG Y