TECHNICAL NOTES 8 GRINDING R. P. Kng Copyrght R P kng 000
8. Grndng 8.. Grndng acton Industral grndng machnes used n the mneral processng ndustres are mostly of the tumblng mll type. These mlls exst n a varety of types - rod, ball, pebble autogenous and sem-autogenous. The grndng acton s nduced by relatve moton between the partcles of meda - the rods, balls or pebbles. Ths moton can be characterzed as collson wth breakage nduced prmarly by mpact or as rollng wth breakage nduced prmarly by crushng and attrton. In autogenous grndng machnes fracture of the meda partcles also occurs by both mpact (self breakage) and attrton. &ROOLVLRQ 5ROOLQJ ZLWK QLSSLQJ Fgure 8. Dfferent types of grndng acton by the grndng meda. The relatve moton of the meda s determned by the tumblng acton whch n turn s qute strongly nfluenced by the lners and lfters that are always fxed nsde the shell of the mll. Lners and lfters have two man purposes:. Lners protect the outer shell of the mll from wear - lners are renewable.. Lfters prevent slppng between the medum and slurry charge n the mll and the mll shell. Slppage wll consume energy wastefully but more mportantly t wll reduce the ablty of the mll shell to transmt energy to the tumblng charge. Ths energy s requred to cause grndng of the materal n the mll. The shape and dmensons of the lfters control the tumblng acton of the meda. The tumblng acton s dffcult to descrbe accurately but certan regons n the mll can be characterzed n terms of the basc pattern of moton of materal n the mll. The moton of an ndvdual ball n the charge s complcated n practce and t s not possble to calculate the path taken by a partcular partcle durng staton of the charge. However the general pattern of the moton of the meda can be smulated by dscrete element methods whch provde valuable nformaton about the dynamc condtons nsde the mll. &HQWULIXJDO IRUFH ) F 6KRXOGHU RI WKH ORDG ) J FRV 7XPEOLQJ Some of the terms that are often used to descrbe the moton of the meda n a tumblng mll are shown n Fgure 8. 6WDJQDQW ]RQH *UDYLWDWLRQDO IRUFH ) J &DWDUDFWLQJ,PSDFW ]RQH 7RH RI WKH ORDG 8.. Crtcal speed of rotaton Fgure 8. Meda moton n the tumblng mll. The force balance on a partcle aganst the wall s gven by 8-
Centrfugal force outward F c m p & D m (8.) & s the angular velocty, m p s the mass of any partcle (meda or charge) n the mll and D m s the dameter of the mll nsde the lners. Gravtatonal force The partcle wll reman aganst the wall f these two forces are n balance e. where s shown n Fgure 8. Thus a partcle wll separate from the wall at the pont where F g m p g (8.) F c F g cos (8.3) cos F c F g (8.4) The crtcal speed of the mll, & c, s defned as the speed at whch a sngle ball wll just reman aganst the wall for a full cycle. At the top of the cycle =0 and F c F g (8.5) m p & c D m m p g (8.6) & c g D m / (8.7) The crtcal speed s usually expressed n terms of the number of revolutons per second N c & c Œ 0.705 D / m 4.3 D / m Œ g D m / revs/sec revs/mn ( 9.8)/ ŒD / m (8.8) 8-3
The lner profle and the stckness of the pulp n the mll can have a sgnfcant effect on the actual crtcal velocty. Mlls usually operate n the range 65-8% of crtcal but values as hgh as 90% are sometmes used. A crucal parameter that defnes the performance of a mll s the energy consumpton. The power suppled to the mll s used prmarly to lft the load (medum and charge). Addtonal power s requred to keep the mll rotatng. )UDFWLRQ - G F &HQWHU RI JUDYLW\ RI WKH FKDUJH 8..3 Power drawn by ball, sem-autogenous and autogenous mlls 0 F J A smplfed pcture of the mll load s shown n Fgure 8.3 Ad ths can be used to establsh the essental features of a model for mll power. Fgure 8.3 Smplfed calculaton of the torque requred to turn a mll. The torque requred to turn the mll s gven by Torque T M c gd c T f (8.9) Where M c s the total mass of the charge n the mll and T f s the torque requred to overcome frcton. Power ŒNT (8.0) For mlls of dfferent dameter but runnng at the same fracton of crtcal speed and havng the same fractonal fllng 3RZHU Net power ŒNM c d c.n c M c d c. D 0.5 m.ld.5 m LD m D m (8.) The exponent.5 on D m has been varously reported to have values as low as.3 and as hgh as 3.0. RI FULWLFDO VSHHG Fgure 8.4 The effect of mll speed on the power drawn by a rotatng mll. 8-4
The effect of varyng mll speed on the power drawn by the mll s shown graphcally n Fgure 8.4 The speed of rotaton of the mll nfluences the power draft through two effects: the value of N and the shft n the center of gravty wth speed. The center of gravty frst starts to shft to the left as the speed of rotaton ncreases but as crtcal speed s reached the center of gravty moves towards the center of the mll as more and more of the materal s held aganst the shell throughout the cycle. Snce crtcal speed s larger at smaller rad the centrfugng layer gets thcker and thcker as the speed ncreases untl the entre charge s centrfugng and the net power draw s zero. The effect of mll charge s prmarly through the shftng of the center of gravty and the mass of the charge. As the charge ncreases the center of gravty moves nward. The power draft s more or less symmetrcal about the 50% value. A smple equaton for calculatng net power draft s 7 P.00 3 c D.5 m LK l kw (8.) K l s the loadng factor whch can be obtaned from Fgures 8.5 for the popular mll types. 3 c s the mll speed measured as a fracton of the crtcal speed. More relable models for the predcton of the power drawn by ball, sem-autogenous and fully autogenous mlls have been developed by Morrell and by Austn. (Morrell, S. Power draw of wet tumblng mlls and ts relatonshp to charge dynamcs - Part : An emprcal approach to modelng of mll power draw. Trans. Instn. Mnng. Metall. (Sect C:Mneral Processng Extr Metall.) 05, January-Aprl 996 ppc54-c6. Austn LG A mll power equaton for SAG mlls. Mnerals and Metallurgcal Processng. Feb 990 pp57-6. Loadng factor K 6 5 4 Grate dscharge Dry Grate dscharge Wet Overflow dscharge Wet 3 0 0 0 30 40 50 60 70 % loadng n the mll Fgure 8.5 Effect of mll fllng on power draft for ball mlls. The data s taken from Rexnord Process Machnery Reference Manual, Rexnord Process Machnery Dvson, Mlwaukee, 976 Gross power No load power Net power drawn by the charge (8.3) The net power s calculated from Net power KD.5 L e! c./ Watts (8.4) In equaton 8.4, D s the dameter nsde the mll lners and L e s the effectve length of the mll ncludng the concal ends.! c s the specfc gravty of the charge and. and / are factors that account for the fractonal fllng and the speed of rotaton respectvely. K s a calbraton constant that vares wth the type of dscharge. For overflow mlls K = 7.98 and for grate mlls K = 9.0. Ths dfference s ascrbed to the presence of a pool of slurry that s present on the bottom of overflow-dscharge mlls but not to the same extent n grate-dscharge mlls. Ths pool s stuated more or less symmetrcally wth respect to the axs of the mll and therefore does not draw sgnfcant power. Austn recommends K = 0.6 for overflow sem-autogenous mlls. A value of K = 9.3 makes Austn s formula agree wth Morrell s data as shown n Fgure 8.7 8-5
The no-load power accounts for all frctonal and mechancal losses n the drve system of the mll and can be calculated from No load power.68d.05 [3 c (0.667L d L)] 0.8 kw (8.5) L d s the mean length of the concal ends and s calculated as half the dfference between the center-lne length of the mll and the length of the cylndrcal secton. The geometry of a mll wth concal ends s shown n Fgure 8.6. The total volume nsde the mll s gven by V m Π4 D m L (L c L) (D t /D m ) 3 (8.6) L D t /D m The densty of the charge must account for all of the materal n the mll ncludng the meda whch may be steel balls n a ball mll, or large lumps of ore n an autogenous mll or a mxture n a sem-autogenous mll, as well as the slurry that makes up the operatng charge. Let J t be the fracton of the mll volume that s occuped by the total charge, J b the fracton of the mll volume that s occuped by steel balls and E the vodage of the balls and meda. U s the fracton of the vodage that s flled D m L c L Fgure 8.6 Geometry of a mll wth cylndrcal ends. All dmensons are nsde lners. L c = centerlne length. L = belly length. D m = mll dameter. D t = trunnon dameter. by slurry. 3 v s the volume fracton of solds n the slurry. Let V B be the volume of steel balls n the mll, V Med be the volume of autogenous meda and V S the volume of slurry. D t V B J b ( E)V m V S J t UEV m V Med (J t J b )( E)V m (8.7) The charge densty s calculated from! c V B! b V Med! m V S ( 3 v )000 V S 3 v! 0 J t (8.8) where! b s the densty of the balls and! m the densty of the meda. The effectve length of the mll s dependent on the load and s calculated from L e L.8J t ( J t ) L d L (8.9) accordng to Morrell and from 8-6
L e L( f 3 ) 0.09 f 3 J t (.03J t ) L d L( D t /D) 0.65 0.5 J t 0. 0.65 0.5 J t 4 (8.0) accordng to Austn. The functons. and / account for the effects of mll fllng and rotaton speed respectvely. However each of these factors s a functon of both mll fllng and rotaton speed. J t (& J t ) & (8.) & (.9863 c.33 c 0.497) and / 3 c [ 3 max ]exp( 9.4(3 max 3 c )) 3 max 0.954 0.35J t (8.) Net mll power kw 0 5 0 4 0 3 0 0 Morrell L = 0.5D J b = 0.04 Morrell L = 0.5D J b = 0. Morrell L = D J b = 0. Morrell L = D J b = 0.04 Austn L = 0.5D J b = 0.04 Austn L = D J b = 0.04 Austn recommends the followng forms for the factors. and /. J t (.03J t ) / 3 c 0. (8.3) 9 03 c 0 0 8 9 0 0 3 4 5 6 7 8 9 0 Mll dameter Fgure 8.7 Varaton of net power drawn by semautogenous mlls calculated usng the formulas of Austn and Morrell. Mll condtons used: J t = 0.35, E = 0.4, U =.0, 3 c = 0.7, D o = 0.D, L d = 0.5D,! B = 7800,! O = 750, 3 v = 0.459 The formulas proposed by Austn and Morrell gve substantally the same estmates of the net power that s drawn by the mll. Some typcal cases are shown n Fgure 8.8. Austn s formula gves slghtly hgher values than Morrell s for mlls that have length equal to about twce the dameter. 8..4 Power drawn by rod mlls The power drawn by a rod mll s gven by P.75D 0.33 (6.3 5.4V p )3 c kw hr/tonne of rods charged (8.4) where V p = fracton of the mll volume that s loaded wth rods. Rod mlls operate typcally at 3 c between 0.64 and 0.75 wth larger dameter mlls runnng at the lower end and smaller mlls at the upper end. Typcal rod loads are 35% to 45% of mll volume and rod bulk denstes range from about 5400 kg/m 3 to 6400 kg/m 3. (Rowlands CA and Kjos DM. Rod and ball mlls n Mular AL and Bhappu R B Edtors Mneral Processng Plant Desgn nd edton. SME Lttleton CO 980 Chapter p39) 8-7
8..5 The mpact energy spectrum n a mll The energy that s requred to break the materal n the mll comes from the rotatonal energy that s suppled by the drve motor. Ths energy s converted to knetc and potental energy of the grndng meda. The meda partcles are lfted n the ascendng porton of the mll and they fall and tumble over the charge causng mpacts that crush the ndvdual partcles of the charge. The overall delvery of energy to sustan the breakage process s consdered to be made up of a very large number of ndvdual mpact or crushng events. Each mpact event s consdered to delver a fnte amount of energy to the charge whch n turn s dstrbuted unequally to each partcle that s n the neghborhood of the mpactng meda partcles and whch can therefore receve a fracton of the energy that s dsspated n the mpact event. Not all mpacts are alke. Some wll be tremendously energetc such as the mpact caused by a steel ball fallng n free flght over several meters. Others wll result from comparatvely gentle nteracton between meda peces as they move relatve to each other wth only lttle relatve moton. It s possble to calculate the dstrbuton of mpact energes usng dscrete element methods to smulate the moton of the meda partcles ncludng all the many collsons n an operatng mll. The dstrbuton of mpact energes s called the mpact energy spectrum of the mll and ths dstrbuton functon ultmately determnes the knetcs of the commnuton process n the mll. 8..6 Knetcs of Breakage The process engneerng of mllng crcuts s ntmately lnked wth the knetc mechansms that govern the rate at whch materal s broken n a commnuton machne. In any regon of a commnuton machne the rate at whch materal of a partcular sze s beng broken s a strong functon of the amount of that sze of materal present. The rate of breakage vares wth sze. It s usual to use dscrete sze classes whch provdes an adequate approxmaton for practcal computaton. Rate of breakage of materal out of sze class = k Mm. k s called the specfc rate of breakage. It s also called the selecton functon n older texts. Note that k s a functon of the representatve sze n the class. Materal breakng out of class dstrbutes tself over all other classes accordng to the breakage functon whch s specfc to the commnuton operaton and the materal beng broken. %G S G M It s common practce to normalse the breakage functon to the lower mesh sze of the nterval so that breakage means breakage out of a sze class. If the breakage functon s descrbed by the standard form. %UHDNDJH IXQFWLRQ E LM B(x;y) K x y n ( K) x y n (8.5) 3DUWLFOH VL]H ' L ' L» ' ' M M» and Fgure 8.8 The breakage functon used for the descrpton of mllng operatons n rotatng mlls. Compare wth the equvalent dagram for crushng machnes (Fgure 5.4) 8-8
b j B(D ;D j ) B(D ; D j ) b jj 0 (8.6) The relatonshp between the cumulatve breakage functon and b j s shown n Fgure (8.8). The breakage functon has the value.0 at the lower boundary of the parent sze nterval. Ths reflects the conventon that breakage s assumed to occur only when materal leaves the parent sze nterval. The dfferences between equaton (8.6) and the equvalent expressons for crushng machnes (5.39) should be noted. K represents the fracton of fnes that are produced n a sngle fracture event. It s n general a functon of the parent sze so that K K d p d p n 3 (8.7) If normalsaton s satsfactory and f a strctly geometrc progresson s used for the mess sze then B j s a functon only of the dfference -j. For example f a root seres s used for the sze classes. D D j D ( ) j D ( ) j (8.8) b j K D D j n ( K) D D j n K D D j n ( K) D D j n (8.9) b j K ( ) n (j ) ( ) n (j ) ( K) ( ) n (j ) n (j ) K( ) n (j ) ( ) ( K)( ) n (j ) ( ) 0.44K( ) n (j ) 0.44( K)( ) n (j ) (8.30) 8. The Contnuous Mll. Industral grndng mlls always process materal contnuously so that models must smulate contnuous operaton. Sutable models are developed n ths secton. 8.. The populaton balance model for a perfectly mxed mll. The equatons that descrbe the sze reducton process n a perfectly mxed ball mll can be derved drectly from the master populaton balance equaton that was developed n Chapter. However, the equaton s smple enough to derve drectly from a smple mass balance for materal n any specfc sze class. 8-9
Defne p F = fracton feed n sze class p P = fracton of product n sze class m = fracton of mll contents n sze class M = mass of materal n the mll. W = mass flowrate through the mll. Fgure 8.9 The perfectly mxed contnuous mll A mass balance on sze class s developed by notng that the contents of the mll receves materal n sze class from the feed and from the breakage of materal n the mll that s of sze greater than sze. Materal of sze s destroyed n the mll by fracture. Wp P p P Wp F p F M M b j m j Mk m j (8.3) M b j m j k m j where =M/W s the average resdence tme of the materal n the mll. If the contents of the mll are assumed to be perfectly mxed m =p P p P p F M b j p P j k p P j p P p F M b j p P j j k for all (8.3) These can be solved by a straghtforward recurson relatonshp startng wth sze class. p F k p P p P p F b k p P k (8.33) p P 3 p F 3 b 3 k p P b 3 k p P k 3 etc. 8.. The mll wth post classfcaton. In practce the materal n the mll does not have unrestrcted ablty to leave n the outlet stream. Larger Fgure 8.0 The classfcaton mechansm at the mll dscharge end recycles larger partcles nto the body of the mll. 8-0
partcles are prevented from leavng by the dscharge grate f one s present and even n overflow dscharge mlls, the larger partcles do not readly move upward through the medum bed to the dscharge. On the other hand very small partcles move readly wth the water and are dscharged easly. Thus the dscharge end of the mll behaves as a classfer whch permts the selectve dscharge of smaller partcles and recycles the larger partcles back nto the body of the mll. Ths s llustrated schematcally n Fgure (8.0). The operaton of such a mll can be modeled as a perfectly mxed mll wth post classfcaton as shown n Fgure (8.). Applyng equaton (8.3) to the perfectly mxed mllng secton m f M b j m j j (8.34) k In ths equaton, s the effectve resdence tme n the mxed secton of the mll M W(C) (C) (8.35) where C s the rato of recrculaton rate to feed rate. A mass balance on sze class at the pont where the post classfed materal re-enters the mll gves (C)f c m (C) p F (8.36) where c s the classfcaton constant for the partcles of sze at the mll dscharge. Fgure 8. The perfectly mxed mll wth post classfcaton. The classfcaton mechansm llustrated n Fgure (8.0) s equvalent to a perfectly mxed mll wth post classfcaton. Substtutng the expresson for f nto equaton(8.3) gves, after some smplfcaton, m (C) p F M b j m j (C) j (8.37) k c If a new varable m * = m (+C) s defned, ths equaton takes a form dentcal to equaton (8.3) m p F M b j m j j (8.38) k c Ths equaton can be solved usng the same recursve procedure as that used for (8.3) startng from the largest sze. The sze dstrbuton n the product can be obtaned from the calculated values of m * usng the propertes of the classfer. 8-
p P ( c )(C)m ( c )m (8.39) Ths soluton s complcated a lttle because the modfed resdence tme s not usually known. It can be calculated only after the sze dstrbuton n the mxed secton has been evaluated. Ths requres an teratve soluton for convenent mplementaton startng wth an assumed value of to calculate m * from equaton (8.38). C s calculated from C M c (C)m M c m (8.40) and the assumed value for can be checked usng equaton 8.35 and modfed untl convergence s obtaned. The actual resdence tme can be obtaned from the load of sold n the mll and the mass flowrate of the sold through the mll or from a dynamc tracer experment. It s usually easer to trace the lqud phase than the sold but the resdence tme of the lqud wll be consderably shorter because of the holdback of the sold by the classsfcaton mechansm of the dscharge. The precse form of the classfcaton functon can be determned by measurng the sze dstrbuton of the materal n the mll contents and n the product stream c p P (C)m (8.4) The presence of post classfcaton n a mll can be detected by notng the dfference n partcle sze dstrbuton between the mll contents and the dscharged product. An example s shown n Fgure (8.) Fracton passng, % 00 90 80 70 60 50 40 30 0 0 0 Model response Product Holdup Feed 0 00 000 0000 Partcle sze, mcrons Fgure 8. Expermentally determned partcle-sze dstrbutons n the contents of the mll and the dscharged product. 8.3 Mxng Characterstcs of Operatng Mlls In practce operatng mlls do not conform partcularly well to the perfectly mxed pattern because there s consderable resstance to the transport of materal, both solds and water, longtudnally along the mll. Ths type of behavor can be modeled qute well by several perfectly mxed segment n seres wth dscharge from the last segment beng restrcted by a post classfer. The sze dstrbuton n the materal that leaves each segment can be calculated by repeated applcaton of equaton Fgure 8.3 Schematc representaton of the mll wth three perfectly mxed segments of unequal sze. Dscharge from the last segment s restrcted by a post classfer. 8-
(8.3) successvely to each mxed segment n turn. The product from the frst segment becomes the feed to the next and so on down the mll. The number of mxed sectons and ther relatve szes can be determned from the resdence-tme dstrbuton functon n the mll. Resdence dstrbuton functons have been measured n a number of operatng ball, pebble and autogenous mlls and t s not unusual to fnd that three unequal perfectly mxed segments are adequate to descrbe the measured resdence-tme dstrbuton functons. Usually the last segment s sgnfcantly larger than the other two. Ths s consstent wth the behavor of a post classfer that holds up the larger partcles at the dscharge end of the mll whch are then thrown qute far back nto the body of the mll. It has been suggested n the lterature that a further refnement to the mxed-regon model can be acheved by the use of a classfcaton acton between each par of segments, but snce t s mpossble to make ndependently verfable measurements of such nterstage classfcatons, ths refnement cannot be used effectvely. The structure of a mll wth three perfectly-mxed segments wth post classfcaton on the last s llustrated n Fgure (8.3). 7..8 6.8.4 Expermental RTD model response 6.4 6.0 Expermental RTD model response.0 5.6 Absorbance.6. tau = 5.77 tau = 0.69 tau 3 = 0.384 Slca, % 5. 4.8 4.4 4.0 tau = 4.483 tau =.653 tau 3 = 0.747 0.8 3.6 0.4 3..8 0.0 0 5 30 45 60 75 90 05 0 35 50 Tme, mnutes Fgure 8.4 Resdence tme dstrbuton functon for the water n a contnuous ball mll. The lne shows the small-small-large segment model response to ths data...4 0 0 40 60 80 00 0 40 60 80 00 0 40 Tme, mnutes Fgure 8.5 Resdence-tme dstrbuton functon for solds n a contnuously operatng ball mll. The lne s the small-small-large segment model response to the expermental data. The resdence-tme dstrbuton functon for a mll can be measured expermentally by means of a dynamc tracer test. It s consderably easer to trace the lqud phase than the sold phase but the two phases have dfferent resdence-tme dstrbutons. The water s not restrcted durng ts passage through the mll as much as the sold phase s. Consequently the hold-up of sold n the mll s greater than that of the lqud and the sold has a hgher mean resdence tme. Although the two phases have sgnfcantly dfferent mean resdence tmes, the behavor of each phase s consstent wth the threestage model that s descrbed above. Ths s demonstrated n the two measured resdence tme dstrbutons that are shown n Fgures 8.4 and 8.5. 8-3
8.4 Models for Rod Mlls The physcal arrangement of rods n the rod mll nhbts the effectve nternal mxng that s characterstc of ball mlls. The axal mxng model for the mxng pattern s more approprate than that based on the perfectly-mxed regon. When axal mxng s not too severe n the mll, an assumpton of plug flow s approprate. In that case the populaton balance model for the batch mll can be used to smulate the behavor of the rod mll wth the tme replaced by the average resdence tme n the mll whch s equal to the holdup dvded by the throughput. 8.5 The Populaton Balance Model for Autogenous Mlls Four dstnct mechansms of sze reducton have been dentfed n fully autogenous mlls: attrton, chppng, mpact fracture and self breakage. Attrton s the steady wearng away of comparatvely smooth surfaces of lumps due to frcton between the surfaces n relatve moton. Chppng occurs when aspertes are chpped off the surface of a partcle by contacts that are not suffcently vgorous to shatter the partcle. Attrton and chppng are essentally surface phenomena and are commonly lumped together and dentfed as wear processes. Impact fracture occurs when smaller partcles are npped between two large partcles durng an mpact nduced by collson or rollng moton. Self breakage occurs when a sngle partcle shatters on mpact after fallng freely n the mll. Rates of breakage and the progeny spectrum formed durng these processes dffer consderably from each other and each should be modeled separately. A ffth breakage mechansm occurs n a sem-autogenous mll when partcles are mpacted by a steel ball. Breakage and selecton functons that descrbe ths mechansm can be modeled n a manner smlar to those used for the ball mll. In practce three dstnct fracture sub-processes are modeled: wear, mpact fracture, and self breakage. Each of these produces essentally dfferent progeny sze dstrbutons and the approprate breakage functon must be used for each. The breakage functon for attrton and chppng A(x;y) can be measured n the laboratory n mlls that contan only large lumps that are abraded under condtons smlar to those found n operatng mlls. No general models for the attrton breakage functon have been developed. The breakage functon B(x;y) for the mpact fracture process can be modeled usng the same model structures that were used for ball mllng. The breakage functon for self breakage C(x;y) can be modeled usng the t 0 method that descrbes sngle-partcle mpact fracture usng the mpact energy equal to the knetc energy of the partcle mmedately before mpact. Partcles wll have a wde dstrbuton of free-fall mpact energes n a real mll and the breakage functon for self-breakage s obtaned by ntegraton over the mpact energy spectrum C(x;y) C(x;y,h) P(y,h)p(h) dh P (8.4) 0 where C(x;y, h) s the sngle partcle breakage functon for self-breakage of a partcle of sze y n free fall from heght h, P(y,h) s the probablty that a partcle of sze y wll shatter when fallng a vertcal heght h and p(h) s the dstrbuton densty for effectve drop heghts n the mll. References: Stanley, G. G. Mechansms n the autogenous mll and ther mathematcal representaton. S. Afr. Inst Mnng Metall. 75(974)77-98 Leung, K., Morrson R. D. and Whten W.J. An energy based ore-specfc model for autogenous and semautogenous grndng. Coper 87 Unversdadde Chle (987-988) pp 7-85 8-4
Goldman, M and Barbery, G. Wear and chppng of coarse partcles n autogenous grndng: expermental nvestgaton and modelng. Mnerals Engneerng (988) pp67-76 Goldman, M, Barbery G. and Flament F. Modellng load and product dstrbuton n autogenous and semautogenous mlls: plot plant tests. CIM Bulletn 8 (Feb99) pp 80-86. The attrton and wear processes that occur n autogenous, sem-autogenous, and pebble mlls produce a sgnfcant fracton of the fnal producton of fne partcles n the mll. These processes are not satsfactorly descrbed by the knetc breakage models that are useful for ball and rod mlls. It s necessary to nvoke the full generalzed populaton balance model that was developed n secton.3 n order to descrbe the autogenous mllng operatons adequately. Varous fnte dfference representatons of equaton (.) are avalable but those based on the predefned sze classes as descrbed n secton are generally used. It s usual and useful to nterpret the rate of breakage process R n such a way that D R(p(x),x,F[p(x)])dx R(p P,x,F[p(x)]) (8.43) D s the rate of breakage out of the nterval. Ths conventon, whch was used to develop the smple fnte dfference models for ball and rod mlls, s useful partcularly because t s comparatvely easy to measure the rate of breakage out of a screen nterval n the laboratory and to correlate the measured rates wth the representatve sze for a sze class. Equaton (.) s ntegrated over a typcal sze class to gve û p(x) 3 P D D (x) p(x) x dx R(p,x,F [p(x)]) M j R(p j,x j,f [p(x)]) û B(x,D j ) 3 P D D P R(x) (x) p(x) a(x;x)dxdx x (8.44) p n p out where û p(x) (D )p(d ) (D )p(d ) (8.45) and û B(x;D j ) B(D ;D j ) B(D,D j ) (8.46) ûb j Fnte dfference approxmatons are requred for the remanng terms n equaton (8.44) n terms of the representatve sze for each sze class. The followng approxmatons have been used successfully p(d ) w p D D (8.47) 8-5
so that û p(x) w p D D p D D (8.48) P D D (x) p(x) x dx w p d p (8.49) P D j D j p(x) P x D D a(x;x) dx dx j p j d pj ûa j (8.50) whch mples that the products of wear and attrton must leave the sze class of the parent partcle.e., must be smaller than D j. ûa j A(D ;D j ) A(D ;D j ) (8.5) If the total regon s perfectly mxed then p = p out and equaton (8.44) becomes p p D D M j p D D ûb j 3 j ûa j d pj 3 p d p k p p j p n (8.5) Ths equaton can be smplfed somewhat by defnng the rato 3 d p (8.53) and d p 3(D D ) (8.54) p p p p n p M j ûb j j ûa j p j (8.55) p p n p M ûb j j ûa j p j j ( k ) (8.56) 8-6
Ths equaton for the well-mxed autogenous and sem-autogenous mll was frst derved by Hoyer and Austn (D I Hoyer and L G Austn. "A Smulaton Model for Autogenous Pebble Mlls" Preprnt number 85-430 SME-AIME Fall Meetng, Albuquerque Oct 985). Ths equaton can be smplfed further by makng the followng defntons (8.57) (k ) (8.58) Both N and N are zero because there s no breakage out of the smallest sze class. ( ûb j j ûa j ) Œ j (8.59) Usng the defntons of ûa j and ûb j Equaton (8.56) can be wrtten N M Œ j ( j ) j (8.60) j p p n p M Œ j p j j (8.6) Ths s the fundamental dscrete populaton balance equaton for autogenous mlls. The appearance of the term p - on the rght hand sde of equaton (8.6) should be partcularly noted. Ths term arses from the wear process that reduces the sze of large lumps n the mll charge whch s an essental feature of autogenous mllng. Equaton (8.6) can be solved recursvely n a straghtforward manner startng at p 0. However t s necessary to choose a value of 0 p 0 to ensure that the dscrete analogue of (.86) N M p (8.6) s satsfed. Usng equaton (8.6) 8-7
N M N p ( ) M N p n M N 0 p 0 M N 0 p 0 M j N p M M Πj p j j N p M j N j p j M p j j j N 0 p 0 M ( j j )p j j N p j M Πj j (8.63) whch leads to N M p ( N N )p N 0 p 0 (8.64) Thus the condton 0 p 0 0 (8.65) guarantees the consstency of the soluton. 8.5. The effect of the dscharge grate In practce t s always necessary to mantan some classfcaton acton at the mll dscharge to ensure that large pebbles do not escape from the mll. Autogenous and sem-autogenous mlls are always equpped wth a steel or rubber grate to hold back the grndng meda. The classfcaton acton of the grate can be descrbed n terms of a classfcaton functon c as was done n Secton 8... Applcaton of equaton ( 8.6) to the mll contents gves m f m M Πj m j j (8.66) where f p F C c m (8.67) and C s the fracton of the total stream that s returned to the mll by the classfcaton acton at the dscharge end. The resdence tme to be used to calculate,, and Πj n equaton (8.66) s M (C)W C (8.68) 8-8
m p F C m M Πj m j j c (8.69) Defne a new varable m (C)m (8.70) and ths equaton becomes m p F m M Πj m j j (8.7) c whch can be solved recursvely startng from = and 0 m 0 * = 0. The value of must be establshed by teratve calculaton usng equaton (8.68) after C s recovered from N C M N c (C)m M c m (8.7) The teratve calculaton starts from an assumed value of and s contnued untl the value of stablzes. The sze dstrbuton from the mll s recovered from p ( c )m (8.73) The knetc parameters k and and the breakage functons A(D ; D j ) and B(D ; D j ) must be estmated* from expermental data. 8.6 Models for the Specfc Rate of Breakage n ball mlls The utlty the knetc model for breakage depends on the avalablty of robust models for the specfc rate of breakage to descrbe specfc mllng condtons. Both functons are strong functons of the mllng envronment. Factors whch affect the rate of breakage are the mll dameter, mll speed, meda load and sze and partcle hold-up. The most mportant functonal dependence s between the specfc rate of breakage and the partcle sze and methods for the descrpton of ths functonal dependence are descrbed below. The specfc rate of breakage ncreases steadly wth partcle sze whch reflects the decreasng strength of the partcles as sze ncreases. Ths s attrbuted to the greater densty of mcroflaws n the nteror of larger partcles and to the greater lkelhood that a partcular large partcle wll contan a flaw that wll ntate fracture under the prevalng stress condtons n a mll. The decrease n partcle strength does not does not lead to an ndefnte ncrease n the specfc rate of breakage. As the partcle sze becomes sgnfcant by comparson to the sze of the smallest meda partcles, the prevalng stress levels n the mll are nsuffcent to cause fracture and the specfc rate of breakage passes through a maxmum and decreases wth further ncrease n partcle sze. Some typcal data are shown n Fgure 8.6. 8-9
8.6. The Austn model for the specfc rate of breakage. Austn represents the varaton of the specfc rate of breakage wth partcle sze by the functon k(d p ) k 0 d ṗ d p /µ (8.74) and t s usual to specfy the partcle sze n mm and the specfc rate of breakage n mn -. Fgure 8.6 Graphcal procedure for the determnaton of the parameters n Austn's selecton functon. It s useful to relate the ndvdual parameters n ths functon to specfc features of the graph of k(d p ) plotted aganst d p. A typcal plot of ths functon s shown n Fgure (8.6) and t has a maxmum at a partcle sze somewhat smaller than the parameter µ whch essentally fxes the poston of the maxmum. The maxmum occurs n the specfc rate of breakage because as the partcles get larger they are less lkely to be broken durng any typcal mpact n the mll. The specfc partcle fracture energy decreases as sze ncreases accordng to equaton (8.0) but the rate of decrease reduces as the partcle sze ncreases and eventually becomes approxmately constant for partcles larger than a few mllmeters for most ores. Consequently the partcle fracture energy ncreases at a rate approxmately proportonal to the partcle mass. Thus for a gven mpact energy larger partcles have a smaller probablty of breakng snce the energy that a partcle absorbs from the mpact must exceed ts fracture energy otherwse t wll not break. 8-0
The graphcal procedure s mplemented as follows: ) Extend the ntal straght lne porton of the data curve as straght lne A havng equaton k k 0 d ṗ (8.75) ) k 0 s evaluated at the ntersecton of the straght lne wth the ordnate d p = mm.. s equal to the slope of the lne. K 0 s often represented by the symbol S whch s called the selecton functon at mm. k ) Evaluate ratos such as c/d to form 0 d ṗ at a number of partcle szes as shown. k(d p ) Plot k 0 d ṗ k(d p ) aganst the partcle sze as shown as lne C n Fgure (8.74). The slope of the resultng lne s equal to because, accordng to equaton (8.74), k 0 d a p k(d p ) d p /µ (8.76) Parameter µ can be evaluated n one of two ways. Construct lne B, parallel to lne A and passng through a pont f whch has ordnates equal to 0.5e. Ths lne ntersects the data curve at abscssa value µ as shown. Alternatvely the sze at whch k(d p ) s a maxmum s gven by dk 0 (d p ) 0 (8.77) dd p whch mples that d pmax. (8.78) 8.6. Scale-up of the Austn selecton functon When developng scale-up rules for the specfc rate of breakage, t s necessary to dstngush between parameters that are materal specfc and those that depend on the materal that s to be mlled and also on the geometrcal scale of the mll that s to be used. The parameters. and n the Austn model for specfc rate of breakage are usually assumed to be materal specfc only whle k 0 and µ depend on the geometrcal scale. The graphcal constructon that s outlned n the prevous secton reveals the role that each parameter n the Austn selecton functon plays n determnng the specfc rate of breakage n a ball mll. It s not dffcult to determne the values of these parameters from data obtaned n a batch mllng experment n the laboratory. They can also be 8-
estmated usng standard parameter estmaton technques from the sze dstrbutons n samples taken from the feed and dscharge streams of an operatng mll. In order to use ths model for smulaton of other mlls t s necessary to use scale-up laws that descrbe how these parameters vary wth varatons of the mll sze and the envronment nsde the mll. The domnant varables are the mll dameter D m and the sze of the balls that make up the meda d b. These varables together determne the average mpact energy n the mll and both have a sgnfcant nfluence on the value of the constant k 0 n equaton (8.74). Because the specfc rate of breakage s essentally a knetc parameter, t obvously ncreases wth the number of mpacts that occur per second per unt volume n the mll. Geometrcally smlar mlls havng the same fractonal fllng by meda and rotatng at the same fracton of crtcal speed 3 c wll produce nearly dentcal mpact frequences per unt volume. The mpact frequency per unt volume should vary at a rate proportonal to the speed of rotaton. The varaton of mpact frequency wth mll fllng s rather more complex and purely emprcal scale-up rules must be appled. The effect of ntersttal fllng s also modeled emprcally to reflect the fact that not all of the slurry remans n the regon of the tumblng meda where energetc mpacts occur. A pool of slurry can accumulate at the toe of the charge for example and ths s largely devod of mpact that cause breakage. As the ntersttal fll fracton approaches.0 the mpacts are ncreasngly cushoned by excess slurry between meda partcles. The scale-up law for parameter k 0 s k 0 k 0T D m D mt N.3 6.6J T 6.6J.3 3 c 0. 3 ct 0. exp[5.7(3 ct 0.94)] exp[5.7(3 c 0.94)] exp[ c(u U T )] (8.79) The subscrpt T n ths equaton refers to the varable determned under the test condtons for whch the parameters are estmated and the correspondng varable wthout ths subscrpt refers to the large scale mll that must be smulated. The varables J and U are defne by the load when the mll s statonary as shown n Fgure 8.7. The meda ball sze also nfluences the specfc rate of breakage. Smaller ball szes produce less energetc mpacts and each mpact nfluences fewer partcles n the mmedate vcnty of the mpact pont between any two balls. The actve zone n the slurry between balls s obvously smaller wth smaller balls. Smaller balls are also less effcent at nppng larger partcles. Offsettng these effects that tend to decrease the specfc rate of breakage as ball sze decreases s the ncreased frequency of mpact that results from the ncreased number of smaller balls n the mll. The number of balls per unt volume vares as /d 3. The net result of these competng effects s revealed by experment whch show that the specfc rate of breakage scales as /d n where n s approxmately equal to whle the partcle sze at maxmum specfc rate of breakage ncreases n drect proporton to the ball sze. The parameter µ whch D m J = Fracton of mll volume flled by balls U = Fracton of meda nterstces flled by slurry. Fgure 8.7 The geometrcal constants for mll scale-up are defned by the load condtons when the mll s statonary. defnes the sze at whch the specfc rate of breakage s a maxmum ncreases wth ball sze rased to a power close to and also shows a power dependence on the mll dameter. Operatng mlls always contan a dstrbuton of ball szes and to accommodate ths the scale-up s weghted n proporton to the mass of balls n each sze. 8-
k(d p ) k 0 d ṗ M k m k d T d k d p µ k N 0 (8.80) where d k s the representatve dameter of the kth ball sze class and m k s the mass fracton of the ball sze class n the mll charge. µ k s gven by µ k µ kt D m D mt N dk d T N 3 (8.8) Recommended values for the constants are N 0 =.0, N = 0.5, N = 0., N 3 =.0 and c =.3. Large sze ball mlls have been observed to operate neffcently so that k 0 should be scaled by the factor (3.8/D m ) 0. when the dameter of ths type of mll exceeds 3.8 meters. Austn LG Menacho JM and Pearchy F A general model for sem-autogenous and autogenous mllng. APCOM87. Proc 0 th Intnl Symp on the Applcaton of Computers and Mathematcs n the Mneral Industres. Vol SAIMM 987 pp 07-6 8.6.3 The Herbst-Fuerstenau model for the specfc rate of breakage. The varaton of the specfc rate of breakage and of the breakage functon wth mllng condtons can be substantally accounted for by descrbng the varaton of these functons wth the specfc power nput nto the mll and examnng how the specfc power nput vares wth mll sze and wth desgn and operatng condtons. Ths ndrect method s partcularly useful for scalng up the functons from batch scale and plot scale testwork. Ths s called the energyspecfc scale-up procedure and was developed ntally by Herbst and Fuerstenau. (Int Jnl Mneral Processng 7 (980) -3) The specfc rate of breakage out of sze class s proportonal to the net specfc power nput to the mll charge. k S E P M (8.8) where P s the net power drawn by the mll exclusve of the no-load power that s requred to overcome mechancal and frctonal losses n the mll. M s the mass of the charge n the mll excludng the meda. S E s called the energy-specfc selecton functon for partcles n sze class. The energy-specfc breakage rate s commonly reported n tonnes/kwhr. The essental feature of the Herbst-Fuerstenau model s that S E s a functon of the materal only and does not vary wth mllng condtons nor wth mll sze. Ths assumpton s equvalent to the postulate that the amount of breakage that occurs nsde the mll s proportonal to the amount of energy that has been absorbed by the materal that s beng mlled and t s mmateral how the energy s actually mparted to the partcles or at what rate. Thus the method can be used to scale up the breakage rates over broad operatng ranges. In fact t has been found from expermental observatons that S E s a functon of the ball meda sze dstrbuton n the mll. It should be determned n a laboratory experment usng the same ball sze dstrbuton as the full-scale mll that s to be modeled or smulated. 8-3
It s usually convenent to use equaton (8.8) n the form k S E P W (8.83) where W s the mass flowrate of solds through the mll. The product k can be used n equatons (8.3), (8.38) and (8.6) and nether the mean resdence tme of the solds n the mll nor the specfc rate of breakage k need be known explctly. Ths elmnates the need for complex emprcal scale-up rules such as equaton 8.79. The rato P/W, whch s measured n kwhr/tonne s the net specfc power consumpton n the mll. The varaton of energy-specfc breakage rate wth partcle sze s gven by 0 3 0 0 s * S E b a 3 0-0 - 3 d * 0 3 4 5 6 0 3 4 5 6 0 3 3 4 p Partcle sze mcrons d p Fgure 8.8 Graphcal procedure for the determnaton of the parameters n the Herbst-Fuerstenau energyspecfc rate of breakage. 8-4
ln S E /S E ln d p /d p ln d p /d p á (8.84) whch represents the logarthm of the energy-specfc breakage rate as a power seres of the logarthm of the partcle sze. Usually two terms n the seres are suffcent to descrbe the varaton n suffcent detal for most purposes so that the functon becomes ln S E /S E ln d p /d p ln d p /d p (8.85) A typcal plot of ths functon s shown n Fgure (8.9). The parameters n ths functonal form can be obtaned from expermental data usng the followng smple graphcal procedure. ) A sutable reference sze d p s chosen to gve the reference value S E at pont a on the curve. d p s usually taken as the top sze for the problem on hand but that s not essental. ) The turnng pont on the curve s located mdway between ponts a and b and the coordnates of the turnng pont are (d p*,s * ). ) The parameters n the selecton functon are related to the coordnates of the turnng pont by ln S /S E ln d p /d p (8.86) and ln S /S E ln d p /d p (8.87) These expressons are obtaned by dfferentatng equaton (8.85) and settng the result to zero. 0 ln d p /d p (8.88) Ths must be solved smultaneously wth ln S /S E ln d p /d p ln d p /d p (8.89) whch yelds equatons (8.86) and (8.87). Ths model for the varaton of specfc rate of breakage wth partcle sze s useful to model the effect of ball sze dstrbuton n the mll. The parameter determnes the sharpness of the maxmum n the plot of S E aganst the partcle sze as shown n Fgure 8.8. must always be a negatve number otherwse the graph wll have a mnmum rather than a maxmum. Large numercal values of make the peak n the curve sharper and consequently the rate of breakage falls off rapdly at the smaller partcle szes. Conversely smaller values of make the peak flatter and breakage rates are 8-5
mantaned even at comparatvely small szes. Smaller meda are assocated wth smaller values of. determnes the partcle sze at maxmum specfc breakage rate through the expresson ln d p d p (8.90) The use of the energy-specfc selecton functon to scale up ball mll operatons s descrbed n the followng papers. Herbst JA, Lo YC, and Rajaman K, Populaton balance model Predctons of the Performance of Large-Dameter Mlls. Mnerals and Metallurgcal Engneerng, May 986, pp4-0 Lo YC, and Herbst JA, Cosderaton of Ball Sze Effects n the Populaton Balance Approach to Mll Scale-Up. In Advances n Mneral Processng. P Somasudaran Ed. Socety of Mnng Engneers Inc. Lttleton, 986 pp33-47. Lo Yc and Herbst JA., Analyss of the Performance of Large-Dameter Mlls at Bouganvlle usng the populaton balance approach. Mnerals and Metallurgcal Processng, Nov 988 pp-6. 8.6.4 Specfc rate of breakage from the mpact energy spectrum. The selecton functon can be calculated from the fundamental breakage characterstcs of the partcles. The selecton functon s essentally a measure of the lkelhood that a partcle wll be broken durng a specfc mpact event. In order for a partcle to be selected for breakage durng the event t must be nvolved wth the event (e t must be n the mpact zone between two meda partcles) and t must receve a suffcently large fracton of the event mpact energy so that ts fracture energy s exceeded. Integratng over all the mpacts n the mll gves Specfc rate of breakage p(e)w(d P p,e) P(E,d P p )p(e)de de (8.9) The followng assumptons were used for each of the terms n equaton (8.9). Partton of energy among partcles nvolved n the mpact 0 0 p(e) 0.376 (e 0.). (8.9) p(e) = 0.5α Eexp(-αE) + 0.5αexp(-αE) α = 0.000 α =.000 00 Mass nvolved n the mpact from sngle mpact measurements on beds of partcles. w(d p,e) kd / p E 0.4 kg (8.93) Dstrbuton of mpact energes n the mll based on DEM smulatons p(e). Eexp(. E) ( ). exp(. E)(8.94) The probablty of breakage P(E,d p ) s log normal wth Selecton functon - arbtrary unts E 50 56 d p J/kg (8.95) 0-0- 00 0 0 Partcle sze Fgure 8.9 Specfc rate of breakage calculated from equaton (8.9) (plotted ponts) compared to the Austn functon equaton (8.96) (sold lne). mm 8-6
These expressons were substtuted nto equaton (8.9) and the result s compared wth the standard Austn functon n Fgure (8.0). k 0.4d 0.5 p (d p /0).5 (8.96) 8.7 Models for the Specfc Rate of Breakage n Autogenous and Sem-Autogenous Mlls The essental features of an autogenous or semautogenous mll are shown n Fgure 8.0 The most sgnfcant dfference between ball mllng and autogenous mllng s the presence of consderably larger partcles of ore n the charge. These are added n the mll feed and act as grndng meda. Consequently the average densty of the meda partcles s consderably less than n the ball mll and ths results n lower values for the specfc rate of breakage when compared to ball or rod mlls. The average densty of the load s proportonately less also and as a result autogenous mlls can be bult wth sgnfcantly larger dameters. D m Post classfcaton Self breakage L c Pebble dscharge D t Equatons (8.7) defne the sze dstrbuton n the charge of an autogenous or sem-autogenous mll usng the populaton balance model. The sze dstrbuton n the mll dscharge can be calculated from the sze dstrbuton of the charge usng equaton (8.73). In order to use these equatons t s necessary to be able to calculate the selecton functons and breakage functons for the separate breakage mechansms that occur n the Slurry dscharge L Fgure 8.0 Schematc sketch of autogenous mll. mll namely attrton, chppng mpact fracture and self breakage. Chppng and attrton occur on the surface of the partcle and the nner core of the partcle s not affected. Partcles are subject to attrton through frcton between partcles and also between partcles and the walls of the mll. Partcles are chpped through mpact aganst other partcles or aganst the walls of the mlls. The sze dstrbuton of the progeny from attrton s somewhat fner than that from chppng but these two subprocesses are normally lumped together. Goldman et. al. (988, 99) and Austn et. al (987, 98 ) have measured wear rates n comparatvely small test mlls n batch mode and all the data can be descrbed by a wear model of the type d û p (8.97) 8-7
Austn et. al. favor û = 0 whle Goldman et. al. (99) found û = 0.37 n a.75 m dameter plot plant mll and û = n a 0.75 m laboratory mll (Goldman et. al. 988). The surface specfc wear rate depends on the mllng envronment decreasng as the proporton of fnes ncreases n the mll charge and ncreasng wth mll load and mll dameter. Careful experments have shown that the surface specfc wear rate on a partcular partcle decreases for several mnutes after t has been ntroduced nto the mll envronment because ts ntal rough surface s subject to chppng and attrton that decreases as the partcle becomes rounded. Because of ths phenomenon the populaton balance equaton should also allow for a dstrbuton of sojourn tmes of the partcles n the mll. Ths level of fne detal s not justfed at the level of modelng that s descrbed here. Percentage passng (log scale) 00 P A Attrton products No breakage here so ths part of the curve defnes the attrton breakage functon. D A No progeny n ths regon Resdue of the worn feed partcles Relatve progeny sze D/Df (log scale) Fgure 8. Schematc of a typcal progeny sze dstrbuton from a 0 mnute tumblng test..0 The value of the specfc attrton rate s ore specfc and should be measured expermentally for the ore. An attrton method has been developed at Julus Kruttschntt Mneral Research Center and s descrbed n ther 5 th annversary volume. A sample of the materal n the 38 mm to 50 mm sze range s tumbled for 0 mnutes. The sze dstrbuton of the charge after ths tme s determned by screenng as s typcally bmodal as shown n Fgure 8..Estmates of the specfc attrton rate and the attrton breakage functon can be easly estmated from ths graph. From equaton (.04) dm dt Œ! s x û 3m (8.98) x û Snce the partcles do not change much n sze durng the test, x and m can be assumed constant. dm dt 3 m(0) d û f (8.99) where d f s the geometrc mean sze of the orgnal feed partcles and smple ntegraton gves m(t) m(0) 3 m(0) d û f (8.00) and P A 00 m(0) m(t) m(0) 3ût d û f (8.0) Snce the plateau n Fgure 8. Usually ncludes the pont D/D f = 0., P A s closely related to the t a parameter that s quoted n the JKMRC attrton test result. The fracton of the orgnal partcles that has degraded by attrton durng the test s equal to P A, the heght of the plateau on the cumulatve sze dstrbuton curve. Thus 8-8
P A 0t A d û f P A 300ût d û p t A 300 60 m û (8.0) The specfc rate of breakage due to mpact follows the pattern observed n ball mlls and the Austn method can be used to model the specfc rate of breakage due to mpact fracture provded that due allowance s made for the lower densty of the medum partcles. In sem-autogenous mlls due allowance must be made for the presence of steel balls as well as autogenous meda. These effects are modeled by the Austn scale-up procedure by ncludng the autogenous meda n the meda sze classes n equaton (8.79). However the lower densty of the meda partcles must be allowed for and n equaton (8.80) both m k and µ k must be scaled by the rato! k /! s where! k s the densty of meda partcles n meda sze class k and! s s the densty of the balls n the test mll. The sharp decrease n the specfc rate of breakage that s evdent for partcles that are too large to be properly npped durng an mpact event s especally mportant n autogenous and sem-autogenous mlls because the coarse feed supples many partcles n ths sze range. An ntermedate sze range exsts n the autogenous mll n whch the partcles are too large to suffer mpact breakage but are too small to suffer self breakage. Partcles n ths sze range can accumulate n the mll because they nether break nor are they dscharged unless approprate ports are provded n the dscharge grate. Ths s the phenomenon of crtcal sze buld up. The phenomenon of self breakage s completely absent n ball mlls but t plays an mportant role n autogenous mllng. The larger the partcle and the greater ts heght Specfc rate of breakage (log scale) Regon Meda - meda m pact Regon Regon 3 Self breakage Crtcal sze buld up Partcle sze (log scale) Fgure 8. Schematc representaton of the specfc rate of breakage n SAG an FAG mlls. of fall n the mll, the larger ts probablty of self breakage on mpact. Partcles smaller than 0 mm or so have neglgble breakage probabltes and consequently very low values of the specfc rate of self breakage. Specfc rates of breakage by mpact fracture and self breakage are addtve and the varaton of k wth partcle sze over the entre sze range s shown schematcally n Fgure 8.. The overall model s obtaned from equatons 8.74 and 5.8. k k 0 d ṗ d p /µ drop frequency G ln E/E 50 E (8.03) In equaton (8.03) E s the average knetc energy per unt mass of a lump of sze d p when t mpacts the lner or the charge after beng released at the top of the mll durng tumblng. Ths s calculated as the potental energy of the partcle at a fracton of the nsde dameter of the mll. E 50 s the medan partcle fracture energy of a lump of sze d p. For larger lumps ths s ndependent of sze as gven by equaton (5.0) but s materal specfc. The drop frequency s calculated from the asumpton that each lump wll be dropped once per revoluton of the mll. 8-9
and drop frequency = 0.705 3 c /D m ½ from equaton (5.50). E fgd m J (8.04) 8.8 Models for the Breakage Functons n Autogenous and Sem-Autogenous Mlls The breakage functon for mpact and self breakage s determned prmarly by the mpact energy level and the t 0 method s used as the model as descrbed n Secton 5.4. For self breakage, the mpact energy s a functon of the lump sze and the heght of the drop and ths s calculated as the potental energy of the lump at ½ the nsde dameter of the mll. For mpact breakage the average energy s calculated as the net specfc power nput to the mll charge. The breakage functon for the products of attrton and chppng can be obtaned from the sze dstrbuton of the products of the sngle batch attrton test because there s no sgnfcant rebreakage of the attrton products durng the test. Ths breakage functon s modeled usng a smple logarthmc dstrbuton A(x,D A ) x D A A (8.05) D A s the largest fragment formed by attrton whch can be obtaned from the measured attrton progeny sze dstrbuton as shown n Fgure 8.. A s the slope of the straght lne porton of the curve n the attrton product regon. The breakage functon for mpact fracture s modeled usng the same models as for ball mlls. The breakage functon for self breakage can be determned by droppng ndvdual lumps of ore and determnng the sze dstrbuton of the products. 8.9 Mll Power and Mll Selecton. 8.9. The Bond method The correlaton between materal toughness and power requred n the commnuton machne s expressed by the emprcal Bond equaton. The work done n reducng a mass of materal from representatve sze d 80 F to representatve sze d 80 P s gven by the Bond equaton (5.7) P o K d P ½ 80 d F ½ 80 kwhr/ton (8.06) A reference condton s the hypothetcal reducton of ton of materal from a very large sze to a representatve sze of 00 mcrons. Ths reference energy s called the work ndex of the materal WI. WI K 0 K (00) ½ 0 < K 0WI (8.07) P o 0WI d P ½ 80 d F ½ (8.08) 80 8-30
n these equatons d 80 must be specfed n mcrons. The representatve sze s conventonally taken as the 80% passng sze. WI can be determned from a standard laboratory test procedure. The Bond equaton can be used for crushers, rod and ball mlls. WI s usually dfferent for these three operatons and must be measured separately. The standard laboratory test for the measurement of the Bond work ndex was desgned to produce an ndex that would correctly predct the power requred by a wet overflow dscharge ball mll of.44 m dameter that operates n closed crcut wth a classfer at 50% crculatng load. D 80 F and d 80 P n equaton (8.08) refer to the feed to the crcut as a whole and the product from the classfer. The work ndex for the ore as measured usng the standard laboratory method must be adjusted to account for varous operatng condtons before applyng t to calculate the energy requrements of an ndustral mll that dffer from ths standard. Ths s done by multplyng the measured work ndex by a seres d F 80 W tonnes/hr Ball mll Open crcut P d 80 F d 80 W tonnes/hr Classfer MF d 80 Ball mll d 80 MP d P 80.5 W tonnes/hr Closed crcut Fgure 8.3 Applcaton of Bond work ndex for calculatng power requred for open and closed mllng crcuts of effcency factors to account for dfferences between the actual mllng operaton and the standard condtons aganst whch the work ndex was orgnally calbrated. (Rowland C A Usng the Bond work ndex to measure the operatng commnuton effcency. Mnerals & Metallurgcal Processng 5(998)3-36) The effcency factors are: EF : Factor to apply for fne grndng n closed crcut n ball mlls =.3. EF : Open crcut factor to account for the smaller sze reducton that s observed across the mll tself (open crcut condton) as opposed to the sze reducton obtaned across the closed crcut. If two ball mll crcuts, one open and the other closed, as shown n Fgure 8.3. If both crcuts produce the same d 80 P the power requred for the closed crcut s gven drectly by the Bond formula P CC 0WI (d P 80 )½ (d F 80 )½ (8.09) and the power requred by the open crcut s gven by Table 5. Bond work ndex effcency factor for wet open crcut mllng. Reference % passng 50% 60% 70% 80% 90% 9% 95% 98% Open crcut effcency factor EF.035.05.0.0.40.46.57.70 8-3
P OC 0 EF WI 80 )½ (d F (d P 80 )½ (8.0) wth EF =. If the two crcuts are requred to match ther product passng sze at a dfferent % passng say d 90P,the factor EF wll have the dfferent value as gven n Table 5. EF 3 : Factor for varaton n mll dameter. Larger mlls are assumed to utlze power somewhat more effcently than smaller mlls. Ths factor s calculated from EF 3.44 D m 0. for D m <3.8 0.94 for D m 3.8 (8.) EF 4 : Oversze feed factor. The optmal d 80 F sze for a ball mll that grnds materal havng a work ndex of WI kw hr/tonne s gven by F O 4 4.3 WI ½ mm (8.) When the feed has a sze dstrbuton coarser than the optmum, the mll must draw more power to acheve the desred product sze. The approprate effcency factor s gven by EF 4 (WI 7)(d F 80 F O ) R r F O (8.3) where R r s the reducton rato d F 80 d P 80 EF 5 : The work ndex n a ball mll ncreases when the reducton rato decreases below 3 and the effcency factor s gven by EF 5 0.03 R r.35 (8.4) To sze a ball mll or rod mll that must process W tons/hr t s necessary to calculate the mll power requred from P P o W (8.5) and fnd a mll from the manufacturers catalogue that can accept that power. If manufacturers' data s not avalable, equatons (8.) or (8.4) can be used to select a sutable mll. Ths wll not produce a unque desgn for the mll because many combnatons of D m and L wll satsfy these equatons and the aspect rato of the mll must be chosen to ensure that the mll wll provde suffcent resdence tme or specfc power nput to produce the requred product sze dstrbuton as calculated from equaton (8.3) or (8.73). It s mportant to note that the geometry of the mll wll 8-3
determne the power nput not the tonnage through the mll. The power consumed wll gve a certan amount of sze reducton and the output sze wll be a functon of the tonnage. By contrast a crusher wll produce a fxed sze reducton rato and the power wll vary to match the tonnage subject to the maxmum power avalable from the motor that s nstalled on the crusher. 8.0 The Batch Mll 8.0. Batch grndng of homogeneous solds The commnuton propertes of materal are often determned n the laboratory by followng the sze dstrbuton durng mllng n a batch mll. In such a test, the mll s charged wth a gven mass of feed materal and the mll s operated wthout contnuous dscharge and feed. The sze dstrbuton n the charge n a batch mll changes contnuously wth tme. The mass balance must be wrtten for each sze class. For the top sze M dm dt k Mm (8.6) It s usually adequate to assume that k does not vary wth tme and ths equaton can be easly ntegrated to gve m m (0)e k t (8.7) Ths soluton plots as a straght lne on log-lnear coords. For the next sze down: M dm dt k Mm k Mm b (8.8) dm dt k m k b m k b m (0)e k t (8.9) Ths s a frst order lnear dfferental equaton whch has a soluton of the form: m Ae k t Be k t (8.0) where A and B are constants that must be determned from the form of the dfferental equaton and from the ntal condtons as follows. Frst the proposed soluton s dfferentated dm dt Ak e k t Bk e k t (8.) These expressons for w and ts dervatve are substtuted nto the orgnal dfferental equaton Ak e k t Bk e k t k Ae k t k Be k t k b m (0)e k t (8.) whch smplfes to 8-33
( k B k B)e k t k b m (0)e k t (8.3) so that B s gven by B k b m (0) k k (8.4) A must be evaluated from the ntal condton. m (0) A B (8.5) whch gves A m (0) B (8.6) and m m (0)e k t B(e k t e k t ) m (0)e k t k b m (0) (e kt k k e k t ) (8.7) The soluton can be contnued n ths way to develop the soluton from sze to sze. The soluton s tedous but not mpossble. It s better to develop a soluton that works automatcally for all szes and whch s especally suted to computer methods of soluton. Note frstly that a general soluton can take two forms: m M. j e t j m M j m j (0) j (8.8) The coeffcents. j are not functons of tme but are functons of the ntal condtons and the coeffcents j are not functons of the ntal condtons but they vary wth the tme. The coeffcents. j can be developed through some recurson relatonshps as follows: The dfferental equaton that descrbes the varaton of each of the sze classes s dm dt k m M b j m j (8.9) j The general soluton s now substtuted nto ths equaton M j. j e t k M. j e t M b j M j j j l. jl e k l t (8.30) Re-arrangng and collectng terms 8-34
The order of the double summaton must now be reversed M. j (k )e t j M j Mj b j. jl e k l t (8.3) l M. l (k k l )e k l t l M l M b j. jl e k l t (8.3) jl The change n the lmts on the double summaton should be noted partcularly, The regon over whch the double summaton operates must not change as the order of summaton s swtched. Now the terms are collected M l. l (k k l ) M l jl b j. jl e k l t 0 (8.33) The coeffcent of each exponental must be zero f the summaton s to be zero for every value of t. whch provdes the value of each value of. except for... l. l (k k l ) M b j. jl (8.34) jl k k l M b j. jl for >l (8.35) jl. can be obtaned from the ntal condton m (0) M. j. M. j (8.36) j j All the coeffcents can be solved by recurson startng from =.. m (0) M. j (8.37) j. m (0) (8.38). b k k k. etc. Ths recurson s most useful n the form. j c j a j wth j< and c (8.39) 8-35
. l c l a l c l k k l M b j c jl a l jl (8.40) k k l M b j c jl jl. a m (0) M c j a j (8.4) Then the c l 's are ndependent of both the tme and the ntal condtons and they can be calculated once and for all from a knowledge of the specfc breakage rate constants and the breakage functon. j 8.0. Batch grndng of heterogeneous solds The batch commnuton equaton for heterogeneous solds s dp j dt j S j p j l K l kk l S kl b jkl p kl (8.4) In equaton (8.4),j,k and l ndex the parent partcle composton, the parent sze, the progeny composton and the K K progeny sze respectvely. l and l are the left and rght hand boundares of regon R n the Andrews-Mka b dagram for parent partcles n sze class l. jkl s the dscretzed verson of the functon b(g,dg,d). In practce t s convenent to decouple the sze reducton process from the lberaton process. Ths can be done by usng the condtonal breakage functons b jkl b j,kl b,jkl (8.43) b where j,kl b s the fracton of materal breakng from class k,l that reports to sze class j.,jkl s the condtonal transfer coeffcent from grade class k to grade class gven that the partcle transfers from sze class l to sze class j. b,jkl s usually represented as an Andrews-Mka dagram. We refer to b j,kl as the sze breakage functon and b,jkl as the Andrews-Mka coeffcents. Equaton (8.43, 8.43) s completely general and does not depend on the assumpton of random fracture. b j,kl and b,jkl are condtonal dstrbutons and must satsfy the condtons M N b j,kl (8.44) jl and M b,jkl (8.45) 8-36
Typcal examples of the dscretzed Andrews-Mka dagram are shown n Fgures 8.4 and 8.5. It s mportant to realze that these represent just two of the many dscrete Andrews-Mka dagrams that are requred to characterze any.63.63 6.00 6.00.3.3 8.000 8.000 5.657 5.657 4.000 4.000.88.88.000.000.44.44.000.000.707.707.5000.5000.3536.3536.500.500.768.768.50.50.8839E-0.8839E-0.650E-0.650E-0.449E-0.449E-0 Partcle composton %Vol 0% 0% 0-0 0-0 0-0 0-0 0-30 0-30 30-40 30-40 40-50 40-50 50-60 50-60 60-70 60-70 70-80 70-80 80-90 80-90 90-00 90-00 00% 00% Relatve partcle sze Fgure 8.4 Internal structure of a typcal Andrews-Mka dagram showng both the feeder and attanable regons. The feeder regon s ndcated by the shaded bars n the upper half of the dagram and the attanable regon s ndcated by the unshaded bars n the lower half of the dagram. The heght of each bar n the feeder regon represents the condtonal mult-component breakage functon b 4,0kl where k and l represent any parent bar n the feeder regon. The heght of each bar n the attanable regon represents the value of a m,n 4 0. partcular ore. Fgures and 3 show a dscretzaton over 9 sze classes and grade classes whch requres 9 = 8 separate Andrews-Mka dagrams, one for each possble combnaton of k and l. In general a theoretcal model of the Andrews-Mka dagram s requred to generate the approprate matrces whch can be stored before the soluton to the batch commnuton equaton s generated. Approprate models for the Andrews-Mka dagram are dscussed n Chapter 3. 8-37
A soluton to the heterogeneous batch commnuton equaton can be generated by explotng the lnearty of the dfferental equatons to generate the general soluton j p j l. jkl e S kl t (8.46) k The coeffcents n equaton (8.46) are related to the selecton and breakage functons and to the ntal condtons usng the followng recurson relatonshps. jmj 0 f gm (8.47) j. jj p j (0) l. jkl (8.48) k.63.63 6.00 6.00.3.3 8.000 8.000 5.657 5.657 4.000 4.000.88.88.000.000.44.44.000.000.707.707.5000.5000.3536.3536.500.500.768.768.50.50.8839E-0.8839E-0.650E-0.650E-0.449E-0.449E-0 Partcle composton %Vol 0% 0% 0-0 0-0 0-0 0-0 0-30 0-30 30-40 30-40 40-50 40-50 50-60 50-60 60-70 60-70 70-80 70-80 80-90 80-90 90-00 90-00 00% 00% Relatve partcle sze Fgure 8.5 Internal structure of a typcal Andrews-Mka dagram showng both the feeder and attanable regons. The feeder regon s ndcated by the shaded bars n the upper half of the dagram and the attanable regon s ndcated by the unshaded bars n the lower half of the dagram. The heght of each bar n the feeder regon represents the condtonal mult-component breakage functon b 7,0kl where k and l represent any parent bar n the feeder regon. The heght of each bar n the attanable regon represents the value of a m,n 7 0. 8-38
. jmn j ln k S kl b jkl. klmn S j S mn (8.49) Note that the summatons n equaton (8.49) run over the feeder regons and not over the attanable regons. Ths soluton to equaton (8.4) s based on the usual conventon that breakage mples that all progeny leave the sze S class of the parent partcle. The pathologcal case j S mn s occasonally encountered n practce. When t occurs, t s usually handled by makng a slght adjustment to the parameters that defne the relatonshp between the specfc rate of breakage and the partcle sze to assure that no two values of are exactly equal. S j Equaton (8.46) represents a complete and convenent soluton to the dscrete verson of the batch commnuton equaton wth lberaton and ths soluton produces the sze dstrbuton as well as the lberaton dstrbuton as a functon of the tme of grndng. C:\My fles\wp8docs\notes\met600\module 6\Grndng.wpd January 8-390, 999