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  • 8/7/2019 Ball wear and ball size distributions in tumbling ball mills_Austin

    1/8

    Pow der Tech nol ogy, 41 (1965) 279 - 286

    Ba l l Wear a n d Ba l l S i ze Di s t r i b u t io n s i n Tu m b l in g Ba l l Mi ll sL. G. AUSTINThe Pennsy lun nrn S ta t e Un iver s i t y, Un iver s it y Park , PA (US A_)R. R. KLIMPELThe Dow Chemical Company, Midhn d. MI (USA. )(Receiv ed Octaber 17.1983; ,n revised form May 15, 1984)

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    SUMMARYTh e theory o f the calcul a t ion o f the s ize

    dr s t r ibu t zon o f t he e qud tbr rum m ix tu re o fba l l s i n a ba l l m t l l i s de v eloped T he d i f f e ren -teal equat ron is so lved for wear la ws o f th eformwear ra te per bai l = r2+ Aw he re r I S ba l l radru s A = 0 gives th e Bondw e a r l a w a n d A = 1 g iue s t he Dam s w e ar l aw .M ethods o f de t e rm in ing A are i l l u s t ra t e d -Ex pe r im e n ta l da ta a re p re se n te d w h ic h showt h a t A = 0 for some cases o f we t m ul l in g ,A = I for th e two m i l l s repor ted by Dau is , andA = 2 for a case o f we t m i l l ing . Th e reason forth i s w ide d i v e rgenc e i s no t k n ow n-

    INTRODUCTIONT h e r a t e s o f b a ll w e a r m a b a l l n u l l a r e of

    u n p o r t a n c e fo r t h r e e m a j o r r e a s o n s . Fust,o n e o f t h e m a j or u n s o lv e d p r o b l e m s i n t h eo p t u n r za t i o n o f b a l l m r l l d e s i g n I S t h e ch o i ceo f t h e b a l l s i ze m t h e m i l l . T o con s t r u c t ad es i g n s i m u l a t i o n m o d e l f or a b a l l m f l [ l, 2,31 w h i c h c a n b e u s e d t o p r e d i c t o p t i m u m b a l lm i x t u r e , i t i s n e c e s s a r y t o k n o w n o t o n l y t h ee f fec t o n b r ea k ag e o f d i f fe r en t b a l l m i x t u r e s ,b u t a l s o t h e e q u i l ib r i u m b a l l n u x t u r e i n t h em i l l . Fo r w e t b a l l m i l l s , t h e eq u i l i b n u m s i zed i s t r i b u t i o n o f b a l ls i n t h e m i U i s a f u n c t i o no f t h e m a k e - u p b a l ls a d d e d t o t h e m i l l a n d t h er a t e of w ea r o f t h e b a l l s . Seco n d , eco n o m i cs t u d i e s of g n n d i n g p r o ces s e s [ S ] s h o w t h a ts t e e l l os s o f m e d i a a n d l i n e r s d u r i n g g r i n d m gi s a s u b s t a n t i a l h a c t i o n o f t h e t o t a l co s t o fg n n d m g . T h i r d , i n o r d e r f or a m i l l t o p r o d u cea t a s t e a d y o p t i m u m r a t e , i t i s d e s i r a b l e t os t a r t a n e w m i l l c h a r g e w i t h a b a l l si z eoo32-5910/85i.s3.30

    d i s t r i b u t io n c lo s e t o t h e e q u r h b r i u m m i x t u r eo f b a l l s p r o d u c e d b y n a t u r a l w e a r , w i t h a d d i -t i o n o f m a k e - u p b a l l s t o g iv e a co r r ec t eq m h b -r i u m b a l l m i x .

    P a r t i a l t r e a t m e n t s o f t h e m a t h e m a t i c s o fb a l l w ea r w er e g i v en b y D av i s [ 4 ] an d Bo n d[ 5 3. T h i s p a p e r e x t e n d s t h e i r t r e a t m e n t s a n dg r v e s s e v e r a l e x a m p l e s o f w e a r l a w s a n d b a l ls r z e d i s t n b u t i o n s d e t e r m i n e d fr o m p la n t d a t a .

    ABRASION TESTSThe a b r a s i v e n e s s o f a p a r t i c u l a r m a t e r i a l i s

    o ft e n d e t e r m i n e d b y s o m e f or m o f a n e m -p r n ca l ab r a s i o n t e s t [ S , 7 1 D r f fe r en tm a n u f a c t u r e r s h a v e d e v e lo p e d t h e i r o w n t e s t s ,a n d s o m e u s e r s a l s o h a v e d e v e lo p e d t e s t ss p ec i fi c t o t h e i r p a r t i cu l a r n eed s . A d i s cu s s i o na n d c o m p a r i s o n o f s u c h t e s t s I S b e y o n d t h es co p e o f t h i s p ap e r , e s p ec r a l ly s m ce t h e r e isl it t l e m fo r m a t i o n o n t e s t s o n t h e s a m em a t e r i a l i n d i f fe r en t ab r a s i o n t e s t e r s . At y p i ca l t e s t r s t h a t d ev e l o p ed b y B o n d [ 5]fr o m a n o n g m a l t e s t r e p o r t e d b y P. Crush . TOqu ote [ 5 ]-

    A flat pa dd le 3 X 1 X l/4, of SAE 4325ch r o m e- n r ck e l - m o l y b d en u m s t ee lh a r d e n e d t o 50 0 B n n e h , i s i n s e r t e d fo r o n ei n c h i n t o a r o t o r 4 5 I n c h e s i n d i a m e t e r ,w h i c h r o t a t e s o n a h o r i z o n t a l s h a f t a t 6 32t h r o u g h f a l lm g o r e p a r t i c l e s _ T w o s q u a r ei n c h e s o f p a d d l e s u r fa c e a r e e x p o se d t oa b r a s i o n , a n d t h e p a d d l e t i p , w i t h a r a d i u sof 4 .25 in . , ha s a l inea r spe ed o f 1410 f ee tp e r m i n u t e s u f fi c ie n t fo r a g o od r m p a c tb low.T h e r o t or i s e n c l os e d b y a c o n c e n t r i c d r u m1 2 i n c h e s i n d r a m e t e r a n d 4 .5 i n c h e s d e e p ,w h i ch r o t a t e s a t 7 0 r p m , o r 9 0% o f cn t i ca ls p e e d , m t h e s a m e d i r e c t io n a s t h e p a d d l e

    0 Elsevie r Sequoia/Printed in The Netherlands

  • 8/7/2019 Ball wear and ball size distributions in tumbling ball mills_Austin

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    2SO

    T h e in n e r c r r c u m f e r e n c e o f t h e d r u m 1 sl ine d wi th pe r f o r a t e d s t e e l p l a t e to f u r n i sha r ou gh su r f a c e f or c on t inu ous ly e l e va t ingt h e o r e p a r t i c l es a n d s h o w e n n g t h e mt h r o u g h t h e p a t h o f t h e r o t a t in g p a d d l e .I n ope r a t ion , s c r e e ne d pa r t i c le s pa s s ing 3/4i n c h s q u a r e a n d r e t a i n e d o n l /2 m c h s q u a r ea r e u s e d a s fe e d F o u r h u n d r e d g r a m s of314 X 112 fe e d a r e p l a c e d in the d r um ,t h e e n d c ov e r i s a t t a c h e d , a n d a b r a s i o n i sc o n t i n u e d fo r 1 5 m m u t e s , t h e n t h e d r u m i se m p t i e d , a n o t h e r 4 00 g r a m s a r e a d d e d , a n dt h e a b r a s i o n c o n t i n u e d . I n e a c h c o m p le t et e s t fou r 400 g r a m sa m p le s a r e e a c ha b r a d e d fo r 1 5 m i n u t e s T h u s t h e p a d d l e I Sa br a de d f or a t o t a l o f one hour , a f t e r wh ic hi t i s we ighe d to the t e n t h o f a m i l li g r a m .The lo s s o f we igh t i n g r a m s i s t he a b r a s ionind es A, of th e ma ter ia l_

    B a se d on a ve r a ge s o f l a r ge nu m be r s o f t e s t sc om pa r e d wi th c o ll e c t e d p l a n t e xpe r i e nc e ,B ond ga ve the f ol lowing a ve r a ge we a r lo ssformcll-_, p r e sum a b ly f or t yp ic a l s t e e l s f orb a l l d n d l i n e r s :

    Wet b a l l mi l lsBal ls .

    k g/k Wh = 0_16(A, - 0.015)1-3L m e r s - (I )

    k g/k Wh = 0_012(A, - 0 015) 3 (2 )Dr y ba l l m i l l s (g r a t e d i sc ha r ge )Bal ls .

    k g/kWh = 0_023A,- (3 )L i n e r s :kg /kWh = 0.0023Aio-5 (4 )Ta b le 1 g ive s a ve r a ge a b r a s ion ind ic e s f or a

    nu m be r o f m a t e r i a l s [ 7 ]_Th i s type o f t e s t ha s se ve r a l d r a wba c ks .

    The typ ic a l a pp e a r a nc e o f ba l l s f r om a d r ygr ind m g ba l l m i l l shows su r f a c e sc r a t c he s ,ind ic a t ing we a r by a b r a s ion [ 5 ]_ B a l l s fr omw e t g r i n d i n g o p e r a t i on s a r e s m o o t h e r b u tp i t t e d , i nd ic a t ing the r o l e o f c o r r os ion inm e ta l l o s s T h e re 1 s l i t t le d o u b t t h a t m ic r o -s u r f a c e s fo r m e d b y a b r a s i o n u n d e r m e c h a m c a ls t r e s s a r e h i g h l y r e a c t iv e u n t i l t h e c h e m i c a lb o n d s a t t h e su r f a c e h a v e b e e n s t a b i l iz e d b yr e a c t ion wi th th e g r ind ing f lu id [ 8] _ I t i se xpe c te d , t he r e f or e , t ha t m e ta l we a r r a t e s inwe t gr ind ing wou ld b e h igh ly va r i a b le c om -p a r e d w i t h a d r y a b r a s i o n t e s t , d e p e n d i n g o nth e c o r r os ive ( e le c t r oc he m ic a l ) p r op e r t i e s o fth e sys t e m [ 91 S inc e no s t a n da r d de v ia t ionswer e r epor ted for e qn s . (1) - (4) , i t 1s not

    TABLE 1Abr as i on I ndex ave r ages [ 7 J

    Mater ial Speci f ic gra -i ty *iDolom ite 2.7 0.016Sha le 2.62 O-021L.S. for cem en t 2.7 0 024Llmeston e 25 0 032Cem en t cbnker 3 15 0 071Magn esi te 30 0 078Hea vy sul f ides 3 56 0 128Copper ore 2 95 0 .147Hem at i t e 4 17 0 165Magn et i te 3 .7 0 .222Gra vel 2 68 0.283Tra p rock 2 80 0 364Gra mte 2 72 0 388Ta con ite 3.37 O-624Qua r tzl te 27 0 .775Alumina 3 .9 0 891

    poss ib l e to p l a c e e r r o r lim i t s on the i r u se _ I na dd i t ion , t he a b r a s ion t e s t doe s no t g ive in -fo r m a t i o n o n t h e b a l l w e a r la w s , s o t h e a b r a -s i on m d e x c a n n o t b e u s e d t o p r e d i c t t h ee qm l ib r m m d i s tnb u t ion o f ba l l s i z e s un le s st h e w e a r l a w i s k n o w n .

    Ano th e r type o f t e s t i nvo lve s the m e a su r e -m e n t o f we igh t l o s s o f a ba l l c ha r ge m al a bor a to r y o r p i lo t -sc a l e m i l l , un de r c ond i -t i o n s c o m p a r a b l e w i t h t h o s e e x p e ct e d i n t h ef u l l - s c a l e m i l l . Unde r som e c i r c um s ta nc e s ,da ta f r om th i s t ype o f t e s t c a n be use d to ob -t a m th e we a r la w , a s d i sc usse d be low

    BALL WEAR AND BALL SIZE DISTRIBUTIONS:T H E O R Y

    I n o r de r to so lve the p r ob le m o f c hoos ingth e be s t m ix tu r e o f m a k e -up ba l l s t o a dd to am i l l , i t i s ne c e ssa r y to c ons ide r the p r oc e ss o fw e a r a n d t h e e s t a b l is h m e n t o f a p s e u d os te a dy- s t a t e ( e qu i lib r ium ) m ix o f ba l l d i a m -e t e r s i n a m i u . T h e t r e a t m e n t b y Bo n d [l o ]m a k e s t w o m a j or a s s u m p t i o n s . (i ) t h a t t h ewe a r r a t e o f a ba l l i s p r opor t iona l t o it s S LE-f a c e a r e a , a nd ( ii ) t ha t ba i l m a k e up c ons i s t son ly o f a s ing le la r ge s iz e o f ba l l The f o r r nu la -t i o n g w e n b e l ow e x t e n d s t h i s t r e a t m e n t t oa l low f o r o the r c a se s of we a r l a ws a nd ba l la d d i t i o n s .

    I n b a l l w e a r , t h e r e i s n o p r o b l e m m d r s t i n -g u i s h i n g b e t w e e n b a l ls a n d t h e w e a r p o w d e r ,s o t h e w e a r p r o d u c t s c a n h e c o n s id e r e d

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    s i m p l y a s m a s s l o s t fr o m t h e b a l l ch a r g e .C o n s i d e r u n i t m as s o f b a l l s i n t h e mil l , con-t a i n i n g a t o t a l n u m b er o f b a l l s o f N T , w i t h acu m u l a t i v e f r ac t i o n a l n u m b er s i ze d i s t r i b u t i o no f N (r ) , r b e r g b a l l r ad i u s . L e t n r b e t h en u m b er r a t e o f ad d i t i o n o f f r e s h b a l l s p e r u n i tt i m e , w i t h a cu m u l a t i v e f r ac t i o n a l n u m b ers ize d i s t r ibu t ion o f n ( r ) . Cons ide r ba l l s o f s izer t o r + d r m t h e s t ead y - st a t e ch a r g e . A s t ead ys t a t e n u m b e r b a l a n c e o n t h i s s i ze i n t e r v a l i sn u m b er r a t e o f b a l l s en t e r i n g b y w ea r o f s i zer + d r b a l l s + n u m b er r a t e o f ad d i t i o n o fm a k e -u p b a l l s o f t h i s si ze r a n g e = n u m b er r a t eo f b a l ls o f s i ze r w ea r i n g o u t o f t h e i n t e r v a l o r n u m b er r a t e of b a l l s w ea r m g o u t o f t h es i ze i n t e r v a I ( p as s i n g t h r o u g h s i ze r ) = n u m b ero f r a t e o f ad d i t i on o f m a k e -u p b a l l s o f a l ll a r g e r r ad i i

    T h e n u m b e r o f b a l ls w e a r m g o u t o f t h ei n t e r v a l i n t i m e d f m c l u d es a l l b a l l s b e t w een ra n d r + dr w h e r e d r 1s d e f in e d b y-(4xrp,) dr

    = ( r a t e o f w ea r o f each b a l l ) d t= f (r) d t

    w h e r e f (r) i s m a s s p e r u n i t t i m e . T h e n u m b e rr a t e o f b a ll s w e a n n g o u t i s t h u s

    =N cWr) f(r)-~T dr 47 i+p , (5)E q u a t i n g t o t h e t ot a l n u m b e r r a t e of a d d i t i onof ma ke-up o f a l l l a rge r s izes , n , [ l - n(r)] ,nT[l --n(r)] =NT

    givingcwr) = nTpb4m [1-nn(r)3r2

    dr NT f (r ) (6)T h i s i s t h e b as i c d r f f e r en t i a l eq u a t i o n d e f i n m gt h e d i s t n b u t r o n o f b a l l s =es N (r ) , w i t hboun da ry cond i t ions o f N( r ,,) = 0 an diWIll,, ) = 1, wh ere r,,.,,,, is th e m i n i m u m s i z eo f b a l l w h i ch can ex i s t m t h e m i l l an d r,,,,, ist h e m a x i m u r n s iz e o f b a J l a d d e d . T h e e q u a -t ion u - n p l i ci t l y d e fi n es t h e r e l a t i o n b e t w eenb a l l a d d i t i on r a t e n r a n d t h e w e a r r a t e . E x p e r -i m e n t a l m e a s u r e m e n t s o f N(r ) , NT a n d n T fo ra k n o w n ad d i t i o n o f b a l l s n ( r ) e n a b l e s f (r) t ob e ca l cu l a t ed .

    The difference n m as s r a t e s o f b a l l s en t e r -i n g a n d l e a vi n g t h e e l e m e n t b y w ea r i s t h eco n v e r s i o n o f m a s s t o p o w d er , w h i ch a ts t ead y s t a t e eq u a l s t h e ad d i t i o n o f m a s s o fm a k e - u p b a l l s T h u s

    r a t e o f co n v e r s i o n o f m a s s t o p o w d er i n t h ee l e m e n t 4 4= n r[ l --h (r ) ] -(r + &)3& --m3pP ,3 3 1= n,[l - n(r )] pb4mr2 drClear ly , a l so

    (7 )r a t e o f co n v e r s i o n o f m a s s t o p o w d er i n t h e

    e l e m e n td.Mr) dr= NTf @) 7 (8)

    E q u a t i n g eq n s . ( 7 ) an d ( 8) m v es an a l t e r n a t i v eder ivat ion o f eqn . (6) .

    T h e r e l a t i on b e t w e e n c u m u l a t i v e m a s s fr a c -t ion M(r) i n t h e c h a r g e a n d n u m b e r fr a c t i o nN(r) isdM(r ) = ~&&,~&~(r )Al so , m a s s a n d n u m b e r fr a c t i o n i n t h e m a k e -u p a r e r e l a t e d b y

    w h e r e m (r ) i s t h e cu m u l a t i v e m a s s fr ac t i o n o fb a l l s le s s t h a n s i ze r i n t h e m a k e -u p a n d m r i st h e m a s s r a t e o f m a k e - u p (p e r u n i t m a s s o fba l l s ). The se con ver t eqn _ (6) tow(r) = m ,K4xp, [l-n(r)]?d r f(r)w h er e , s i n ce g n ,n p , = mTJ( l / r3 )dm(r ) ,

    n-lax 1K= _f 7 a m (r ) (9 )mlnA co n v en i en t m e t h o d o f an a l y s i s i s t o

    a s s u m e t h a t t h e v a r i a t i on o f f (r) w i t h r c a n b ea p p r o x i m a t e d b y a p o w e r f u n c t i o n r2+ 4wezra te= f ( r ) =Kp b 4 m2 * Porf(r) = (pb4m2)(Kr) (10)w h er e f I c an b e p o s i t i ve o r n eg a t i v e . S i n ceWe= ratI2 1s -d(47+&/3)/dt , Kpb 4irr 2+* =(-4npb /3)dr3/dt = -4m pbr2d r/dt a n d t h e w e a r

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    d z s t c n c e p e r u n i t t i m e ( -dr /dt) is K T* (for A =0 t h e w e a r d i st a n c e p e r u n i t t i m e = K). E q u a -t ion (6a) becomes

    AI(r) = (m,K/K) i [l - n (r)]r3 -a d r (11)

    I f t h e m ak e - u p I S m d e f m i t e s i ze s o f b a l l s of~1 , r2, - -, rk , --- r m, an d i f t h e s i ze s a r eo r d e r e d r m a v =r,>r,...>r,> _ >r,>r m m , a n d m k 1 s t h e w e r g h t fr a c t i o n o f m a k e -u pof s ize rL, eqn . (9 ) becomesK = zn z , / rk 3 (94I;Sl m r l a r l y , 1 2~ I S t h e n u m b er f r ac t i o n o f b a l ls o fsize rk , w h e r e n h = (m k /rk 3)/K T h e nn (r) = I:+n

    m--l +nk , _ +n k r, 1 (12 )

    a n d m T fo llows f rom eqn s . (9a) an d (12) us in gM(r,) = 1 . I n g en e r a l , t h e ch o i ce o f m a k e -u pb a l l s i ze d l s t r l b u t m n n ( r ) t o g i ve t h e ap p r o acht o a d es i r ed M(r) i s a t r i a l -an d -e r r o r ca l cu l a -t i on w h r c h r e q u ~e s a k n o w l e d g e o f K a n d A

    F o r a sz ngle sEe of ma k e-up bal l n(r) = 0fo r r < rm a r a n d

    M(r) = [ ,2 :),1 (r4 -- - rm m 4 -A)

    Si n ce M(r,,,) = 1, K = 1/r , , ,r3,(4 - A)IL~ ~,

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    2of d i sc r e t e s i z e s, no t a c on t inu um , se pa r a t ioni n t o t h e v a r i o u s h a c t i o n s i s s t r q h t f o r w a r da nd th e we igh t l o s s o f e a c h s i z e c a n be de te r -m m e d . E x a m p l e s o f t h e t h r e e m e t h o d s a r egiven be low

    EXPERIMENTAL DATA

    Da vis [ 4 ] ga ve da t a f r om a Ha r dm gec on ic a l we t m i l l a nd a 6 f t dia. by 8 f t d r y ba l lm i l l wh ic h ind ic a t e d A = 1 , c o r r e spon dm g toa we a r l a w o f we a r r a t e 0~ ba l l we igh t , d 3 r _Us ing m e th od 2 , Lor e nz e t t i [ ll ] ha s f oundtha t e qn . ( 10 ) un th A = 0 i s a good a s sum p-t ion f or we t m i l l ing a n d he g ive s K va lue s r a n g-mg f rom 3.8 pm /h for a re la t ive ly sof t i ronor e to 15 4 pm /h f or a n e x t r e m e ly ha r dcopper ore , for Armco Moly-Cop ba l ls .

    O n t h e o t h e r h a n d w e h a v e a n a l yz e d d a t af rom 2 w e t i n d u s t r i a l b a l l m i l l w h e r e m a k e -u pwa s of a s ingle s ize of ba l l Table 2 g ives th em i l l ing c ond i t ions . F igu r e 1 shows thec um u la t ive we igh t f r a c t ion o f ba l l s ve r su s ba l ld i a m e t e r , a s m e a s u r e d b y e m p t y i n g t h e m i l lc o n t e n t s a n d co u n t i n g a n d s i zi n g b a l ls u s i n gc a l ipe r s Two d i ff e r e n t s r z e s o f m a k e -up ba l l swe r e t e s t e d . I t a pp e a r s tha t A = 2 in th i s c a se ,s m c e t h e d a t a a g r e e d r e a s on a b l y w e l l w i t h t h eform of eqn. (14) wi th A = 2 , th a t i s , wearr a t e in m m /h a r 4_ The va lue s o f K ob ta ine df r om e qn . ( 13 ) u s ing th i s va lue o f A we r e(7 6) (10m6) m m - h -i for t h e e q u d i b r i u m b a l lc ha r ge ongm a t ing f r om 100 m m m a k e -upba lls an d (12.3) (10m6) mm - h -i for th ato r ig ina t ing f r om 75 m m m a ke -up ba l l s . I ts h o u l d b e n o t e d t h a t i f t h e s m a l le r b a l lsf or m e d by we a r we r e so ft e r be c a use o f t hel o ss of a h a r d e n e d o u t e r l a ye r t h e n t h e ywou ld wea r fas te r , g iving A < 0 .TABLE 2B a l l w e a r d a t a o n 4 3 m d ia . b y 5 m lon g w e t o v e r -f lo w m r l l g rm d m g a b r a s iv e in o rg a n ic m a te r i a lType o f ba l l s tee l , spec if ic

    g rav i ty 8 5 ,600B r i n e 11 h a r d n e s s

    D ia m e te r o f m a k e -u p b a l l , m m 100 75Media we igh t , t 110 110Dai ly add i t ion , t 2.0 2.7D a i ly th r o u g h p u t , t 1600 1480S te e l p e r t o n th ro u g h p u t , k g /t 1 25 1.82Steel loss, kg/kWh 0 068 0.098

    ocz

    001 I I I I ,llll10 10 40 m 100

    aall owmerer mmF ig . 1. C u m u la t iv e b a l l s i z e d i s t r i b u t io n a t s t e a d ysta t e (see Tab le 2 )_

    I n t h i s t e s t i t a p p e a r e d t h a t t h e w e a r o fl a r g e r b a ll s w a s m u c h fa s t e r w r t h r e s p e c t t os m a l le r b a l ls t h a n p r e d i c t e d b y t h e B o n d e x -p r e s s ion , l e a d ing to a m u c h f la t t e r ba l l s i z ed i s t r i b u t i on . I t a l s o a p p e a r e d t h a t w e a r r a t e sw e r e fa s t e r i n t h e e q u i li b r i u m c h a r g e o fsm a l l e r ba l l s . F o r e xa m ple , t he lm e a r we a rr a t e o f 75 m m ba l l s wa s 11 pm /h f or t hel a r ge r ba l l m ix a n d 17 pm /h f or t he sm a l l e r ,c o r r e spon d ing to we a r r a t e s o f 1 .6 a n d 2 6 gp e r b a l l p e r h o u r . T h i s i s p e r h a p s d u e t o t h eg r e a te r num be r o f ba l l -ba l l c o l li s ions p e r un it u n e f or t h e s m a l le r b a l l m r x T h e t o t a l n u m -b e r o f b a l l s p e r u n i t m a s s of cha rge , NT, ISg ive n by1 4-A dmax'-d-dmml-A~Pb"&= 1-_a d 4-A-d 4-A (15m a x m l nFor A = 2 , the ra t io of NT for d,,, = 75 mmto NT for d,,, = 100 m m , a s sum m g d,, =12 mm, is 1.7: th e wear ra tio o f 2.6 t o 1.5 is= 17 also.

    The m e th od o f f ol lowm g ba l l we a r w i tht im e i s i l lu s t r a t e d by th e f ol lowm g da ta . Two

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    h u n d r e d a n d t w e n t y b a l ls (1 t o i l) w e r e t a g ge d (l-5/100):24 = 0.6 X 10m 3_ Sm ce th is dat aby d r i l l ing a 6 3 m m d ia . ho le in to e a c h ba i l f e llows the B ond we a r la w , th e kg /kWh o fa n d fl l lm g t he ho le w i th low m e l t ing po in t T a b le 3 c a n b e c o m p a r e d w i t h t h e v a l u ea l low Using eqn . ( lo) , in t egra t ion of p r e d ic t e d f or qua r t z i t e f r om Ta b le 1 , t ha t I S ,--d(47rr3p,/3)/dt = dp, ,4rr + p gives 0 062 wit h O-C20 kg/kWh.Ve r m e u le n e t a l [ 12 ] ha ve a l so r e por t e d

    m a r ke d ba i l we a r t e s t s on 60 m m dr a . ba l l s i na 2.4 m X 2.4 m (8 ft X 8 f t ) r u bbe r -l ine d ba l lm i l l r u nn ing a ; 85% of c n t i c a l spe e d w i th aba l l c ha r ge o f 22 ton ( m e t r i c ), g r ind ing a bou t65C t per da y of an ore . Sam ples of 10 of th em a r k e d b a l ls w e r e w e ig h e d a t v a r i o u s t i m ei n t e r v a l s c o r r e s p o n d i n g t o k n o w n t o n n a g e smil led . Equ a t ion (16) for A = 0 or 1 can bep u t a s

    1-A = (1 - 4)H1-4 A# I.0

    r 0 (16)h-l 5 = Ktr 0 A=10w h e r e r. IS t he in i t i a l s i z e o f t he ba l l a t t u net = 0 , wh ic h wa s 50 m m . F or a c on t inu ousover f low disch ar ge mil l of 4 m id . by 4 8 mlong , we t g r m dm g a ha r d go ld o r e a t ove r1GOO t pe r da y ( se e Ta b le 3 ), t he va r r a t ion o fth e r a d ius o f ba l l s a s a f unc t ion o f g r ind in gt im e i s shown in P ig 2 . I t i s c l e a r t ha t t h i sr e su l t i s c ons i s t e n t w i th A = 0 . t he lo s s o f OSof ba l l r a d ius r e p r e se n t s a l o s s o f ha l f of t heba l l we igh t_ a nd g ive s a we a r r a t e K of a bou t19 pm /h This va lu e in eqn . (13) g ives ap r e d i c t e d m m l m u m fr a c t i o n a l w e a r r a t e o fth e to ta l cha rge of mr = 1 .44 X 10m3 f rac t ionp e r h o u r , a s s u m m g t h a t ~ r m m , k m a , J 4 < 1.0.H o w e v er _ t h e a c t u a l f r a c t io n a l w e a r r a t e w a s

    TABLE 3B a l l w e a r J a t a on 4 m d i a by 4 8 m l ong we t ove r -r low m l !1 g r I nd I nggo ld o r eTyue o r b a l l s t e e l , sp e c i fi c g ra \ 1 ty5i5 Brmel l ha rd nessD ia m e te r o f m a k e -u p b a l l . m m 1 00Dal ly add l t lon , t 1 .5Da l!y th r ough put . t 1100 - 1200S te e l p e r t o n th r o u g h p u t , k g /t 1 3Stee l loss, hg/kWh 0.062

    TIME IN MILL daysFig. 2 C h a n g e o f b a l l d im e n s io n w i th t im e i n *mill,r , , = 50 mm (see Tab le 3 )

    II3 rz _ = 1-E A = 1.0r 0 r0IAL = 1 = -Ktr0 3ln(mlmdw h e r e m i s t he m a ss o f ba l l s a t t im e t , w i ththe e f fe c t ive g r ind ing t im e be ing p r opor -t iona l t o the a m oun t o f o r e m i l l e d Ana lys iso f t he r r da t a sh ows tha t A = 0 f it s t he r e su l t sin a s im i l a r m a n ne r to tha t o f F ig . 2, s e e F ig3 , bu t w i th t im e a bou t 15 t im e s a s long ,giving K = 0.7 p m / h _

    Ve r m e u le n et al c la im e d tha t t he f itof d a ta to A = 1 .0 wa s as good as th a to f A = 0 , ba se d on a c om pa r i son o f co r r e l a -t ion coe ff xie n t s , a n d the r e f or e r t wa s no t pos

    0770 so 100 15 0 ZOO 25 0TONS MILLED =lO-=

    F ig _ 3 . C h a n g e in b a l l d i a m e te r w i th to n s m i l l ed(650 t/d), r 0 = 30 mm , d at a of Ve r m e u l e n e t a l [12]

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    7/8

    2 8

    s ib l e to d l s t ingu l sh be twe e n A = 0 a n d A =1 .0 . Howe ve r , t he y a ppe a r e d +a ha ve - pe r -f or m e d th e ba lys i s c m ( r + e r r o r ) i n one c a sea n d ln [ r 3 + e r r o r ] i n the o th e r , wh ic h i s c l e a r -ly no t s t a t i s t i c a l ly c o r r e c t . Eq ua t ion ( 16 ) c a nb e p u t a sr / r0 = 1 - z = fl(t ) A=0

    r/r0 = exp (-Kct) = f2(t) A=1The su ms of squ ar es of [ (r / ro) (expt ) - ( r / r , )mod ei lZ a r e 0 .00053 for A = 0 an d mor e th ando ub le th is at 0.00113 for A = 1, clear lyind ic a t ing the be t t e r f i t :>i t he m ode l vv l thA = 0 M or e p r e c i se s t a t i s t i c a l a n a lys i s wou ldr e q u i r e r e p h c a t e d t e s t d a t a .

    The va lue o f K for A = 0 gives a p r e d ic t e dm m i m u m f r a c t lo n a l w e a r r a t e o f t h e t o t ic ha r ge o f 0.9 X 10m 4 f r a c t ion pe r hou r , whe r e -a s the a c tu a l m e a n va lue w a s a bou t 1 .5 X 1W4f r a c t ion pe r hou r . Me a su r e m e n t s o f t he ba l l sa n d c o a r s e m e t a l f r a g m e n t s d i s ch a r g e d fr o mth e m i l l w i th ba d ly wor n d i sc ha r ge g -r a t e s ga vea va lue o f a bou t U 4 X lo - f r a c t ion pe r hour ,t h u s g i vi n g a c o m p a r i s o n o f m i n i m u m w e a Tr a t e o f 1 .1 X 10m 4 f r a c t ion pe d hou r f or t h edir ec t expe i im en sa l da ta t o 0 .9 X 10e4pr e d lc t e d & o ,n the K vz lue . Th e m e a nm e a su r e d s i z e o f d i sc ha r ge m a te r i a l w a s sve na s a n e q u i v a le n t d i a m e t e r o f 2 8 m m .

    As a n e x a m p l e o f t h e t h i r d m e t h o d q f d e -t e r m inm g ba l l we a r , t yp lc a l da t a fr om a p i lo t -sca le mil l might be as fo llows. A cha rge of380 kg of 60.8 mm (2 in) balls a;ld 120 kg of25.4 mm (1 in) ba l ls wa s tu mb led for 48 h .The lo s s in we igh t o f 50 8 m m bJ l s wa s10 kg an d th a t of 25.4 mm wa s 7 .9 k g. Ther e l a t i on b e t w e e n w e i gh t l o ss a n d ch a n g e i nba l l r a d ius i s

    = 1 - f ra . c t lona l weigh t lossThis gives r/r0 = 0.991 for the 50 8 mm ba l lsan d 0 .978 for th e 25 4 mm . Eq ua t ion (16)e v e s t h e r e l a t i on b e t w e e n r/r0 f or two s i z e s a s1 - (rllrolY-* ro20 I-A =lO A+11 - (r2/ro2) - A To1 (17)ln(r,,/r,)/ln(r,,/r,) = 1-O A=1The d a ta g -we th e fo l lowing va lue s o f t he l e f t -ha nd s ide of th is equ a t ion - 1 .2 for A = 0 ,2 5 for A = 1, 5.0 for A = 2. Th u s th e r esu lts

    a r e c o n s is t e n t u n t h A = 0 a n d t h e h n e a r w e a rra te i s 11.2 P m/h.

    DISCUSSION AND CONCLUSIONS

    The s i tua t ion i s c l e a r ly c onf use d The da tagiven by Davis [4] g ive A = 1 , wh erea s Loren -ze t t i [ l l] rep or t s A = 0; th e Verme ulen e t a l[ 12 ] da t a a n d one o f O-Z da ta s e t s f or a l a r ge1ni l .l a lso give A = 0 . The oth er da ta se t g ivesA = 2_ Disc uss ions w i th p l a n t sup e r in t e nd e n t sm th e S ou th t i c a n go ld f ie lds ( fo r e xa m p le ,R e f . 13) sugge s t t ha t i ndu s t r i a l e xpe r i e nc e i st h a t e q u i h b r i u m b a l l c h a r g e s d o n o t c on t a ma s b ig a d i c t ion o f ba l l s ne a r t o the m a ke - upba l l s i z e a s p r e d ic t e d by the B ond we a r l a w ,A = 0 I t I S poss ib l e tha t a n inc ons i s t e n c ye x is t s b e t w e e n t h e t a g g m g m e t h o d o f d e t e r -m i n i n g b a l l w e a r a n d t h e m e t h o d o f d u m p m go r s a m p h n g t h e b a l l c h a r g e . T h e w e a r p r e d i ct e df r om the t a gge d ba l l t e s t fo r t he Ve r m e u le ne t a l . d a t a w a s i n r o u g h a c co r d u r l t h t h er e qu i r e d m a k e -up I f a l lowa n c e wa s m a d e fo rd i sc ha r ge o f 28 m m e qu iva le n t d i a m e te rm a t e r i a l . Howe ve r , f or t he da ta s e t o f F ig . 2 ,t h e p r e d i c t e d m i n i m u m w e a r r a t e w a s m o r eth a n tw ic e tha t a c tua l ly m e a su r e d _ To r e so lveth i s p r ob le m , i t wou ld be a n a dva n ta ge i fi n d u s t n a l e x p e r i en c e o n d u m p e d c h a r g e s w e r eo p e n l y r e p o r t e d a l o n g w i t h t h e g r i n d m g c o nd l t lons l e a dm g to the ba l l s i z e d i s tnb r l t l on .T h e t l s a t m e n t o f t h e u n s t e a d y s t a t e c h a n g ef r om a s t a r t ing ba l l s i z e d i s tnb u t lon to th ee qu l l lbnu m ba l l s i z e d i s t r ibu t ion lv ll l bet r e a t e d m a l a t e r p u b l i ca t i o n

    R EF ER ENC ES

    L. G Aust in , R_ R. Kl impel , P T. Luck ie an dR. S C. Rogers, S r m u Z a t i o n o f G ri n d r n g Cw c u r t sfor D es i g n , Design and I n s t a l l a t i on o f Com m i nu -t ion Circu I t i , SME, Denver , Colora do, 1982,301 - 321, ch ap 19L G Aus t i n and R R Khm be l . m J Her bs t ( ed Mod e l i n g f o r S ca l e -Up o f Tu m b l i n g Ba l l Mil Ls ,Con tr ol 84. SME, Den ver , CO, 1984.L G. Aus tm , R R Kl lm p e l a nd P . T Luc k ie , T h eProcess En g i n eer i n g o f S u e Red u c t i o n : Bo l lMz l Z i n g , AIME, Denver , CO, 1984 ( in pre ss)E. E Davis, ZYan s AIME. 61 (1919) 2 5 0F. C_ Bond , Met a l Wea r in C ru s h i n g a n d Gr i n d r n g ,AIChE Ann ua l Meet ing, Hou ston , TX, 1963, F5 4

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    28 6

    6 _A L Stern, Cfiem En g , 69 (1962) 1297 Commmufron. V. C Marshal! (ed ), I Chem.

    Er,~'_,London,19'?4_S I J Lln and S Nldlr, i%fot Scr Eng.. 39 (1979)

    1939 K A NataraJan, S C. Riemer and I. Iwasakl,

    Corros~uc end Eroswe Wear zn &fagnetrc To c o n i t eG r r n d m g 7 AIlME Annual Meeting, preprmt 83-4,1983

    10 F. C Bond. ~llrnrng Eng., 10 (1958) 592.

    11 J J. Lorenzettl, Ba l l S r ze Drs t ra b u t ro n -f ro mC o mp u t e rS i m u l a t i o n t o Pro d u c t , 3rd Symp. onGrindmg, Armco Chzle, S_A.M.I.. Vu i a de1 Mar,1980.

    12 L A. Vermeu!en, D_ D. Howat and C. L_ M_Gough, J S _4 f i I n s t . Mm Me t a l l. . August 1983,189.

    13 D. D. Hcwat and G. J. Wright, Th e Po t en t ra l r t y o fHeat -IZeoted Bal l s zn the Grm dm g o f Gold Ores.Mlntek (South Afruza) Report MlOl, 1983.