1. The document discusses theories of comminution, which is the process of reducing solid materials to smaller sizes. 2. The oldest theory from 1867 states that work input is directly proportional to the new surface area produced by crushing. However, measurements show the energy required is much greater than needed to create the new surface area. 3. A 1885 theory states work required is proportional to the reduction in volume of particles. Neither of these theories accurately predict results from commercial crushing and grinding. 4. The author developed a third theory in 1951 stating work input is proportional to something other than new surface area or volume reduction. However, the summary does not provide details on what the third theory states.

1. BAI,I,MII,I,
1eitsnrd h tl. hundfurrAeb ccnrcntLunrpaniesandqurliqcontrolldorutodeslhroughoutdtcrofld.
I lt".l]r hc usd In (. rul,rtift lhr grindrhr||h0f all ofc Thedigitalcornterindicalesthecounldo$nendit
.l-., slrul'nffrlrcdrfLrrdrir ilr mrlu.lrncmn)fattcnti0ndurinsthistimejsnotreouired.Ajogingbltlonis
lrolidcdtopositioolhcdrun iorbedingmd urloadirg.Arlr.tnudljogispossiblefofpfeciseposilioning
StandardFeatures
. .ll ballmilk comenandard$ithooe{4 5 lb.
ch.ugc0flon hdls.ifonslmd,fcccivingtm
lUldhmd screenpm.
. lil cir-slifondru l
Models
l'ii ilqi so
220bll,Jph.60oclc
. Modell95-i1
110,'120.1th.60crcle
. lodelJ9t5I
ll0,220!.1ph,50qclc

2. F.C.BONDBICO BALL MILL
TheF.C.BondBall Mill is asmalluniversallaboratorymill usedin calculatingthe
grindabilityof all ores.
GRINDASILITY IS TIM NUMBER OFNET GRAMS OFSCREENIJNDERSZE
PRODUCEDPERREVOLUTION.
ThisBall Mill canbeusedcontinuouslyor it canbeusedfor anynumbq ofrcvolutions,
accoding to thetypeof grinddesired.Forinstance,theFrcd C.BondGrindabiliryTestsw€re
madein theBico Mill rurmingat70revolutionsperminute,with achargeof285 iron balls
rangingin sizefrom 3/4inchto 1-1l2inchin diameter,andweighingapproKimately20,125
grams.700CC ofminus 6 mesh,stagecrushed,dry feedwasusedandthecirculationload
maintain€dconstantat250%by adjustingthenumberof revolutionsfor eachgrindingperiod.
BALL CHARGE DETERMINATION
Becauseofvaxiationsin ball size,no exactnumberofballs ofeach sizecanbespecified.
Theball chargeis preparedby startingwith 285balls,consistingof approximatelyequalweights
ofvarioussizeballs.Thesesizesinclude:3/4incbrT18irrch,1inch,1-114inchand1-112
inch.....about400gramsofeach size.With 285ballsalwayspresent,someballsofone sizeare
removedard replacedwith thenextsizeof largeror smallerballs.
This is continueduntil thetotalrveightis at closeto 20,125gramsaspossible,making
thelastadjustmentwith thesmallestsizeof balls. Do notremoveall theballsof anyonesize.
Onceasuitablegrindingchargeis preparedfrom onegradeofballs, countandnotethenumber
ofballsof eachsize.
Otherbatchesofthe samegradecanbestartedusingthis count,andfirther adjustedif
necessary.It is IMPORTANT toietain the"onefor one"rutiowhile adjustingthefinal weight
to maintaintheDroDerball count.
continued...

3. OPERATINGINSTRUCTIONS
Coruectthepowetsupplytothecontolbox.Wircsarcprotudingforeasyaonnection.
Makesuleallcircuitbreake$arcof toavoidsev€reelecticalshock.
Thismachineisequipped$r'ithanautomaticcounter.Tosetthecount€r,pressthebutton
tothedesirednumb€rofrcvolutions.Theswitchhastwopositions,"RUN"fornormaluse,and
"JOG"tomovethodrumto adesiredpositiotr(foremptyns),
Toloadthedrum,"JOG"thcdrumAsninguntilit facesupwaid.Rsrnovethecoverand
gasket,Insertthesampleatrdballchargeintothedrum.Replaceth€gasketandcov€r.Setthe
numbcrofrevolutionsyoudesirconthecounterandstartthemotorbypushingthestrrtcontol.
Themachinewill stopautomaticallywhentle numberofrevolutionsisreached.Again,
us€the"JOG"buttontopositio[theopeningof thedrumupwadtowardstheceiling.Now,
remove.thecoveratrdgaskot.Agai!. 'JOG"theopeningofthedrumtowardsthereceivingpan
sothesampleandballscandischaryeilttothereccivilgpan.

4. F.c. BONDBALL MILL CEARGE
Wefird thatthequestionofcolrertbsll chargcarisesoft€uandis stillyerydebatable.
Backin 1960,Mr.Bondhimselfrecommendedthcchargcdcscribed;
43batle- 1-ll2inch(1.45)
67balls- 1-ll4iuch(1.17)
10balls- 1 inch(1.00)
7l balls- 3/4inch(.75)
94balls- 5/8inch(.61)
285BALLS Totalwt.=rDx.44.5lbs.
Mr.Bondhimsclfadmittcdtllatit isverydifrcult tospwifr agivennumberofballsof
eachweight,andin oulmostreceniconglondlncr withMr.Bon4 hal€coDtrl€ad€dachsrge
excludingthe5/8ittch(.61)balls.Becauscofthis,BicoInc.o(pcrimrntdlycameupwiththc
followingfigureforacbrrge:
25balls- 1-ll2inah(1.50)
39balls- l-1l4inch(1.25)
60balls- I itrch0.00)
68balls- 7/8inch(.875)
93balb- 3/4inc.h( .75)
285b&llr TotrtwL= apr.44.5lbs.
Weircludescveralextaballsof cachsizesothattbechargecaub€manipulat€dtoresult
in the20,125gmtotaln'it! 285ba!s.

5. BICOINc; F. C.BONDBALL.MIL,LSPAREPARTS
PLEASESPECTtrYSERIALNUMEERWEEN ORI'f,RING
(.ATAI.OG # NN-SCf,r?ITON I trS/I(G
DruD,withShaffs(PleascspeciSSA.I)
Covcr
Haagsr
TablewithLegs
PillowBlock
Asb€stosGasks,t
50IIz Dgital Comter/JoggingSwibh
60Ilz DigitslCounlF4oggingSfitclt
CouDterBrackct
Courtlr hror
PaowithSs'recaardRccaivitrgPalt
ChEgeofBaUs
ChainCuard
Chain,oncl€agth= 29inchcs
Connectinglink foi 60I{Z opaetiod
Offsctlink for50IZ opcrdion
ShaftKey
*** Sprocketsarcavsilableandquoteduponrcqucat.(S/Ned IU rcquired).
BM-I
BM-2
BM.3
BM4N
BM-9
BM-I2
BM-134
BM-138
BM-13C
BM.13D
BM.I4DO
8M.15
BM-23
BM-26
BM-294
BM.298
BM-32
n3/52
1lt5
13/6
109/50
tl.4
.3t.1
9t4
50n2
31t.5
u.4
ll.4
t/.4

6. 1.TOTURN DRUII BY
}IAND.
-_2. TURN END OF SHAET
W TH WRENCH,
(-r,ioToR c0vER BEHTND
coNT ROL PANEL)
"'lRed8$.o8it'fi'88l'J'AiPuflFo
NANDLEANDPUtLIOIiARDSYOU.
!,PESS ;HESE 4 LOWEP BU_TOS N 'IURN -O SET N LI,lBER
ABOVE EACH._ SETSWITCHON RUN.- PRESS SIART BUTTON-
f/IACHINEWILL SHUT OFF A FTER NUN4BER0F TURNSCOMPLETE.
, TO JOG N4ACHlNE' I. P.'T SWITCH ON JOG. 2. KEEP PRESSING
START BUTTON 0N AND OFF UNTIL DESIRED POSlTl0N-.'lS I
REACHED,
amzz

7. MILLWIRI}.IG DIAGRAM
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8. W]RING DIAGR.AMTOR BAI,], MII
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11. Fts. L 8+in. Hldtocone crush.tlor
t.ttidtt ctushirs ol SemnTa@ziteote
(speculat henatite). ( unboldt
Mi^iry Co., Humboldt, Michisan.)
rOMMINUTION Ih.ory is coDerred with tle relarion-
-. ship betw.eneoergt iDpr.ttaDdrbc productparticlesik
nade from s eivcn f€ed sizc. It continues to be a rich ficld
Tb. oldest therry (1867)is fiat of RrrrINoEA,LBnd it still
bas adhcr.nts. He srat.d that tbc arca of the n.w surfac€
produ€cd by crubhing or grirding is direclly proporlio.al to
tb. useful work input. Thc surfac€ area of a totr of prrticlcs
of uDiform diamct€rd is proportional to 1/d, and accord-
ing to RFnNolR thc us€ful work input pcr ton is also
proponioual to I /d. Hos'cver, tbc measuredsurfacc encrgy
of the nc*, surface ar€a produccd is only a veiy 6ma[ fr!c-
tion of thc ordcr of l/10cr0 of the energyiaput actually
required to producc tbat surface h commercial crushjna
aDd Srinding. Nearly all of thc rcquircd €d€rgy input ap-
pcars a3heat dt€r thc particLs arc brokeb.
The sccondthcory (1885)is tbat of K1cK.:H. ltal€d that
lhe work requircd is proportioDal to the rcdultioo in volume
of tbe pa iclesconccrncd.Whcrc I is the diametq of the
fecd paniclcs and p is thc diam.ter oI the product panicles,
tho reduciionratio Rr is J/p Accordiosto KICK,tbc work
idput lcquir€d pcl totr is proportioml to Iog Rrllog 2.
Siucc neiter theory agrecawith commercial qushiDg and
grindiDa r€sult!, the author devcloped thc Third Thcory in
1951-rAccording to this theory, tbe work iDput h propor-
tional to tb€ trlw crack tip length Foduced iD particle
br€aka8c,and aqualsthe work repres€ntedby thc product
mitrusthat reprcsentedby the feed. Ir particlesof similar
shape,thc @ck rjp lengthis €quivalentto the sqrare root
D.rann.n(. ArG-Chrdqs
of one-balf the surfacearea,and thc new crack tensth is
proportionarro 1I 1/i - lt/j .
For practical calculationsrhe sizein microns which 80
per cent passesis selccted as tbe criterion of particle siz€.
The <iiameterh micronswhich 80 per ccnt of tbe producl
passesis dcsipat.d as P, the siz! which 80 pcr cent of the
feed passesis dcsisnatedasF, and the work input iD kilo-
watt boursper sbortto! is ttl. The basicThird Th.o.y eqna-
to wi t0 wi
/ ? /F '. .(1)
wbere ryt is tlre work index. Tbe work index is the com-
mrnunob parameterwhich expre$esrhe resistanc€of tbe
ftat€rial to crushins and CrindiDs.Numericaly rhe work
iDdex is the kwh per shorr ton required ro feduce Ibe
mateial from theoreticallyinfinite feedsiz! to 80 pcr cenr
pasin8 100microns.equivalenrto abouto? p.r.;nl paF
sing200mcsb.Whenanythreeof rhequaDritiesin Eqoation
(l) are known, the fourrh can bc fourjd by transposingrhc
equation.Useful forms are showdin (ta) and (tb) belowj
..{la)
....0b)
Tbe sork input in Ioulesor wa {econds per gram equals
If th€ naterial is homoseneous10 sizereduction,its }l,i
value will cootinu. constanrfor all size reduction stages.
I - R.prinr.d Fton Brilirhchemical Enpineednp
- | to wi/F '
w1/F- to wil
tl to
CRUSHING&
GRINDING
CALCUTATIONS
PART I
lh. .rlehing .nd Srindinr of or.., ro.kr ri.l mii.rk k .n indusrri.l proc.$ of sr.rr inpori:nc.. Sp.ci.li!.d
enSin€.fin! khowl.d3. ir r.quir.d fo. ihe rolution of pr.ctidl probl.n! in p.nicl. ii1. r.ducrioh, rnd
codific.tio. oI rhi! kno*l.dg. hr3 h.rdl/ bciln. Th. pr...nr p.p.r i,.n rttempt to.$cmbt. r hirhty
.ond.n!.d summ.ry of th. princip.l crl.!l.tion d.thodr whi.h th. .urhor h.s found us.ful, R.f.r.nc.r 1r.
siv€. to r.ti.l.r {ith r hor..xt.niive.xpl.nrtio. .hd .:.mpl6 of ..t.ut.tion5.
by FREDC. BOND

12. MESH
Fig.2. Expon.itidl sizedituibutio, ttlot al ote with naturul
gtuin sizebetween1OOond l5Onesll. E posutetatio Er:
0.30, P = 80 per cent pasine 4nO nicrcns. Oack lenetll
equalsCr = 24.4 cnlcc ol solid.
Fig.3. Crack lensth plot l,on Iable l-ton 80 pet cent
passi'1glN micro$ 1080 pet c.nt pdsire lm , ctohslot
all wLuesoJ e,posute ruIio Er.
Fig.4.Scalpedleed conectionplot.Scalpedleed with 80pel
cent passine 7900 niuo^s (F = 79OO).Y = 29 pet cent,
80 - Y/2 = 65.5 per cent. Slope= | t2. Catected leed
sizeFc = 12,0mthictons.
CRACKLENGTH PLOT
MESII
SCALPEDTEEOCORRECTIONPLOT
j
-l-

13. Howevcr, beterogeneoDsstructures in rock
Fo. instance,certain mat rills havc a narural grain size,aDd
thcir ryi valueswill be larg.r bclow that siz. than abovc it.
A loosely cementcd sandstoDeof 4S-meshsilic-agraiDs will
hav€ a larger ryi for a product nith broken parricles finer
than48 meshthatrfor a coarserproduct.
The etrciency of the r€ductio! machinc may also in-
fluencc thc operating work ind€x. For iostanci, a ball mill
EriDdirgan ore from 80 p.r cent-l4 meshto 80per cent-
100meehwill have a lowcr op€ratins t/t value with 1.5-in.
srindins ballstban witl oversize3-iD.balls.
A material may havc an tudacel grain sizeresultins from
some pref€rential siziry action which chatrses its natural
size distributioa. Undersiz. s.indine bals can have this
effect.
I-aboratory.determinationsof th. work index sbow the
rcsiltance to breakagc at the rize rangc tcsled, and aly
variatior in thc }'/; valDesin testsat diffcreDt product sizcs
shows that the malerial is not homogencous to size reduc-
tion. For thi! reason laboratory testr should preferably be
made at or near tbe product sizerequirediD commercial
8Imortr&
The op€ratirywork iDdexfrorn transposcdEquation (la)
can bc calculated from sizer€duction in commercial plants
to compare the plaat €fficiency wiih laboratory test results,
to compa& efrciencies of tb€ diferent plaDt siz€ reductiotr
stases, or to .ompare thc efrci€ncies of difierent plants
treatins limilar materials. The work index is particularly
vahiabl€in pr€dictingthe sizeandcapacityof Dewinstalla-
tions. Tablo IIIA in tle appendix Iists the averag€ work
ind.x values of 82 difi€reni materials.
Three PrinciDles
CommiDutiod phenomeDahaverecently beenredactedinto
tkee pitrciples,rwhich ar€ usefrnsuidesfor the considera-
tion of all crushingaDdgrindingdata.
The FiBr Priuctr'le statesthat, sinceenergyirput is neces-
sary for siz. feduction, all fccd particles of finite siu have
a certain .oar8y register,or en€rgylevel. which must be
addedto tbe enqgy iEput during crusbinaor Srindingto
obtain tbe en€rgyregisteiof the product.All statementsof
th€ energyutilizedin comminutionmust satisfythis condi
ene.gy input = en€rgy r.gistcr of product- energy
r€gisterof feed.
Th. Third Theory work index Equation (l) fo ows this
principle, with tbe energy register equal to the total sp.ciffc
energyinput io kwh per short tod. The lrork index t/i is
rhe energyreSjst.rlo 80 per c.trt pdssing100miclotrs.
If tbe enersy which has becn expendedin preparing the
fced particlesis Desl€ctedin analysioscomminutiondata.
tbe first principlehasbeeoviolated.aDdapplicationof tbe
calculated r€sult to difierent feed and product sizeseill be
The SecondPtinciple statesthat the useful.work input in
crushing and edndine is propoitioDal to the length of tb.
n€wcracktipsproduced.In ordinarycrushiosandgdnding,
rock particlesabsorbstrainenergyanda.€ deformeduDder
compressionor sbearuntil the weakestflaw in the particle
faih wjth.he formatioo of a cracktip. Tbis minutechanse
of shapecausesothercracklips to form at orberweakflaws,
and lhe particle bleaks, releasingthe bulk of tbe shaitr
energyas heat.The strainenergyrequnedto breakis pro-
porlional to fte lengtb of the crack tips formed, sincethe
additional energy requircd to €xtend the crack tips 10
b.eaka8eis suppliedby the flow of thesunoundingresidert
stEitr ene.gyto thecracktips.
Sincethe crack tip lenglh is proportionalto tbe square
root of the new su.faceareaproduced,tbe specincwork
input requiredk i.versely proportionalto the sqLrareroot
of the p.oducl particle diameterminus that of the feed
diameter,as shownin the work index Equation (1). C{ush-
iog aDd sriDdins machinesare csr€dtialtydeviccsfor the
.onversiotr of m.chanicat encrgy into strain energy inro
heat,uoder conditionswhich proftote materialbrcakag..
The en€rgy register as uscd herein r.preseDts the specific
cnergy rvhich has passed tbroush th€ material as strain
en€rcy, and iDcludes heat lossesand loss.s due to fricriolr
and other causs. It does aot correspond.to the eoergy
cont.ni of thc material.
The Third Pinciple dealswirh thc relatiooship of particle
flaws to mat.rial breakas.. A flaw is defined as atry struc-
tual weakn.ss in a particlc which rnay dcvelop iaro a dack
tp undc! straid. Flavs ar. slways pres€Dt in brittle
materials and may causc wide Erialions in thc brcakinq
sLrcD8lh!of appar€ntlysimilar pardcler.
The *cakest flaw iD a particle determines its breaking
strength in crushing and Srindi!8. It also cortrols rbe num-
ber of particlcs produced by br*kage. Particleswith the
weakest flaw8 brcak most easiiy and produce thc larsest
product particlcs. Ho*ever, they arc not necessadly easier
ro grind ro a givcn producr sizercquiriEgseveratstagesoI
brealagethan are pafljclesof rhe samesize,boseweatcsL
The Third Principle states that the weakcst flaw in a
particle der.rmines ils breakingsrrengtbbul Dot its work
index. Tbe work iodex is conkoU.d by t-beav.rsse 6aw
srructurc Lbrougbourlhc €Drre size rangc restcd.work
index variadonsar dificreni p.oduct sjze, rcsutrfrom flaw
concentratioft or shorlagesai rbosesizes,usually causcdby
natural grain siz$.
Evaluation ot Particle Size Dirt.ibution
The usual ltandard screenscaleconsistsof a seriesof
ievs with squareopeninesdifrerins by y'i. basedupon
dre 200-meshsieve oDcbiDgof 74.: microDs.There are
25,400miclors in an inch. A scrcenanalysissizedistribu-
tion of a crushedor sround productconsisr!of a tistins of
the p€r centweighrpassingor retaired on eachsi.ve in thc
Tbereis probabb a defi tr laq qbicb sorerasrberesltar
sizedi.Lriburionof crusbedor grouodproducrsrbowever,
Done of the proposed laws bas bcen generatly accepred as
Siz€distributionanalysesof crushedandsroundproducts
are commonly plotted on los-log paper with (]) the per
cenr passing as ordinatc and the parricle diaDeter (r) in
microns as absci$a. Such plots of complete samplesusuatly
sbow a fairly straisbt lioe for rbe firer particlesizeratrge
which begins to curve in tbe coarser sizes and often aD-
proachestao8trcy witb tb€ I00 per cebtpassiogIiDear rhe
rop oI tbe plot. Tbe size80 per cenrpassesmay b. Iound
from the curved portion of the plot for use in the work
index Equation (l).
Wben rbc straigbt lower porrion of lbe plo(ed line is
extctrd.dat ils slope,. ir iDrercepLerhe 100per ceorpassitrg
Iine at i.& microns. It follows a po,e. law deffned by th€
G^rEs-C^r.rD1N-ScAUfiMl.rNequation,t s,hich is
J,:r00 1.:l '" (2J
From this equationthe surfaceareaJc in cm, per gram
of cubicalplrticles of densityp, with 100per centpassing
tro microDsand sloped lo a grind linit o{ ai mrcronsn:
60.000d f/k.^ r-' I
rr ....(3)p,/.roo(t_o)L ai /' j
The grind limit ai hls lecently beenassigDedthe valu. of
0.1micron,6equivalent to 1000 Angstrom unirs. This is
abolt 200 tjmes th€ unit spacelatlice of quartz and otber
rock forning minerals.
The slop€r is often about 0.5,but may approachunity.
-3-

14. Crushins or srinding io closedcircuii Foduces lessfin.s
than oped-circuit opcratiofl, and causes d to rDcreas€
Renoval of nnesbeforcrcductioDhasthe same.ffect As a
material is eJound fiDer, its value of d often apPears lo
The loA'log sizedistributionplot is donveDient.However,
the usual curvature in its upper patt itrdicates that the
actual sizedistiibution law is of the expooentialtvpc with
a variable exponen!,ralher tbao of th! po*er type witl
tbe constantexponentd.
ExDonential Size Distribution Plots
A metbod of plottios hasbc.d developedwith yiclds size
distribution lides that arc apparently quite straight for
homogeneous materials-! They follow the expon.ntial
Y =]rn-v=bte'=blt0Ax ... (4)
Ax = loeb -1oeY .... (44)
X1 reDr.s.nts lt, thc €nergy rcgistcr in kwh/toD divided
bv the ;ork index fi/i at tbe 80 pcr ccnt passingbale IiDe
where v = 20. The per cent cumulative retained v is
10o- y, 6 is th€ 100- ) interccpt,and I is the slope.On
semi-log paper I is measuredon the vertical logaritbmic
scale, and X is on tbe horizotrtal linear scale. Diasonal
slfaiqbr Iine5are drasD raoiatingfrom lhe upper lelt-!and
corn;r of lne cbarr,{hich represedreachmeshsizeor rbe
. 2"c'eens(,le andcrossthe80percenlpassingborizoDlal
baseline.fa.h diaEonallire r.lreseotsa mesbsi4 lenirg
sievewilh an openingof Pr microns,and cro$€s rbe base
line at /: lo/y'P,. The disgonal lin.s can be assisned
various Desh sizes,-withthe proper rclationshiPbetween
This plot is not as convenieDtas tbc loglog plot, but it
hasseveral advantages.The first is that the 80 per cent pass_
ins sizeP can be found with l.ss errot from P:100/v",
wbere w is rhe value of Xr at the baseline 20.
Another advantageis lhe delineation of natural or
induced grain sizes.As the sizedistribution liDe proceeds
up the chart apptoachingthe finer sievesizes,a curvedloop
to the riSbt of the indicatedslraightlide showsa grain size
deffciencv. culminating at the natual gain size where lle
loop becomesparallel10the straiShllidc.The compensaling
etain siz. excessis shor,nby the return of t1le loop to the
staisht li'e. II laboratory determinatioft of the work
index are made at th€ difierent sieve sizes, th€ los' t/t
ialues will iDcreaseas the srain deficiencv sircs approach
rbeDaruralgrain si7e,aDddecreasea( rhegraiD.rcecs sizes
where the loop relurnsto the straightline-
The natural srain size in ores usually corespotrds to tbe
unlockirg or min.ral liberationlize to $hich the ore must
be sround before concenkation.The exponentialmethod
of plouins tle size distribution furnishes a vciv good
indicarion o{ the unlocking siz. wben tbe amount of the
mine.al to b€ concentrated is 1arge. This is Particulartv
hue of iron ores,and the .xpon€ntial plots sbow clcarlv
the different unlocking properties of autoSenous and
conventionalgrinds.
Much additionalinformation can be oblainedfrom tbis
r Deot 5ia disrdbuon plot,includinScracklengtbvalues.6
l;lhe ball m;llBrindabilityreslsal 60 )oul.sinPUrPermill
revolution. tbe joules required to produce I cm of new
crack length in material of homogeneousbreakagewith
specificgravity ss is approximatelyri s8/l l.
The exDosureratio Er is relaledto b in lhe exponential
sizedistribulion Equation(6) as follows, where Er : X&/
2 - l.3ol El
tocb---1
Fj
....(4b)
Tbe data ;n Table I canbe llotted on six sheetsof single
cycleloglog pape.lo makea setof cha.lsfrom whichrhe
crack lcngth Cr of any !€8ular qushcd or ground product
can be fo'rd grapbically wbeo i$ 80 pc| ceotpassiDgsizc
P and crposD.e ratio Er arc known frotd atr €xponentral
sizE distdbution plot. On tbe first shcet tbc Cr valucs for
P = I microD ar. plotted otr tbe lcft-hand iidc, and thc
vatuesfor P : r0 microns arc plottcd on the dght-hatd
side.Each pair of points is connectcdby a straisbt linc
marked with its E value, ard intermediate lincs catr be
drawn using a logarithmic rulc. The lecond sbcet is made
by plottins values for P: 10 on the left-band sidc abd
P : 100on tbe rishnhand side, and so on for the 3€t of lix
charts, which cov.r the €ltire operating sizelangc.
T DI:E I--4'..1 Larh vd!6 lor PldiD. cr (.DFu
t6iso l0@ 10,@0 l0o,00o I
ol0 10.65 r.76 r.r7 0.495
212.0 93.0 36.6
16.39 6.60 2,47 0.915
?!?.0 n0.0 1r.r r9,t6 1,37 7.96 r.r35
t9r-9 t15.2 32.1
Surtace Ar€a C.lculations of Ground Produ.t3
Approximatc surfac€ areasin sq. cm p.r gram of cquiva-
lent cubesfor loglog size dislribution plots can be calcu-
Iated ftom Equatior (3), usins a srind limit Zi of
when the crack1cn81hC/ basbeenfou!d,'the suface of
equivalert cubesi! sq. cm p€r cc of solidsis 2 Crt.
The WTGNLRsurfaceareaSp in sq.cm/8ram is .pproxj-
mately €qual to the BL rNEair rlermeabilitysutfaceto thc
power 0.92,or S, : (rl)tr'r.
The 80 per centpassinasizeP io microDshasthe folow-
ing approximaterelationship to ibe Bun{E and W^GNER
- /2o,Jm' 3.drr rog
': sr / 6tErF
andlosP:2 los (203m/al):8.56-2.15 Ios(s,) .. .. (54)
lvork lndex from LaboBtory Test
Equationshavc been derivedfor findin8 thc work index
Pt from several types of labomtory crusbability aod
sriodability tests,Tas describedbelow.
Piecesof brok.n rock passiosa 3-in.lquar€ endret'ined
on a 2-in.squarear. mounted betweentwo opposingequ3l
30lb weight! vhich swins on wheels.when the wheels are
releasedth€ vr.islts strike simultaneouslyon oppositesides
of tbe measuredsmallestdim€osionof the rcck. The heisht
of fall is successivelyinoeased until tbe rock breaks.Tbe
imlact crusbingstr.nglh in fool-poundsper inch of rock
thickne$ is desisdated as C, and Ss is the specific eravity.
The work index is found ftom the averageof 10 brcaks,
Wi:z.sgClse
Rod Mill Gtihdability Test
...(6)
The feed is crushedto -}in., and 1250ccpackedin a
graduatedcylindcrareweiSbed,screenanalysed,andSround
dry in clos€dcircuit witb l0O per centcirculatingload iD a
l2in. dia. by 24in. long tilting rod nrill with a wave-type
lining andrevolutioncounter,rundiogat 46rpm.The erind-
ing chargeconsistsof six I.25in. dia. and lwo 1.75in.dia.
neel rods 21in. Longand weigbing 33,380grams.
In order to eqDaiizesegregationat tbe miu ends,it i!
fotatedlevelfor eisht revolutions,then lilted up 5' for oDe
.evolulion,down 5" fof anolhe.revolulion,and returned

15. to levcl lor eiSht rcvolutions cootiruously throushout €ach
grinding pcriod.
Tcsts ar€ made at all m$h sizEsfrotu 4 to 65 F.sb. At
tbe cdd ol each srindiDg pcriod thc mill is discharsed by
riltiDg downwad at 45" for 30 rciolutioos, and the product
is scr€eocdoDsi.vcs of the m$h si.zetesl.d. Thc sieYeundcr-
siz! is w€i8!.d, and frcsh unsesrcsated fced i. addcd to thc
oversiz. ro make its total wciSbt cqual to tbat of thc
t250cc origitraly charaedinto thc mi[. This is retumed
to the mill and sround for the Nnb.r of i€vol tions
calcu.latcdto sive a circulating load equal to the weient of
tho ncw fced addcd. Thc stinding period cycl€8 are coD'
tinucd until thc nct srams of sicve undersize produccd pet
t€volutioD rcrchcs equilibrium atd reversd its dire€tio! of
iocrcasc or dccrca3.. Ther the undelsiz! product ard cir-
c lating load are scrccn analy!€d, aod tbc averagc of the
Iast thrce net Stams pd revolution (G/p) is tbe rod mil
Sriodabilily.
Wlere F is thc sizc i! nicrons which 80 pcr ceat of the
oe* rod mill fe€d passcs,and Pr is thc oFning of the sievc
siz. lestcd in rnicrons. theD the rod mill wo.k index ryi is
calculatcd from tbe fonowing revised (1960) cquation :
/10 l0
wi:62ltP,or3 y tcry)o6'b ) |
-^--/--l
.,.,(n
vr vrl
Tbis r/i valucshouldconformwltb the motor outprit
Dow.r to ao av.ragcoverflowrod mill of 8ft iolerior
diamercrErindins$.1 in op.n circuit.FordrysdndiDgthc
work inputsbouldbemultipliedby 1.30.wlrcrc D isthenil
diamet.r iftide thc liDingh fe€t,tbc work input shouldbe
multiplicdby (6/D)o'
Ball Mi Gtindabiliry T.'t
Th. standardfccd is prepar€dby stagcdushios to all
passinsa 6 mesbsicve,but fiocr fe€d cao bc uscdwh€n
nccssary. It i! screcnanalys.d and Packedby shakiDsi!
a 1om-ccgraduatedcylinder, and the weisbtof 700cc is
placedin thcmill andsrounddryat250pcrcentcirculatins
load.Thenill is l2ir. x l2io. wilh toundcdcor.ers,ald
a shootblioios exc€ptfor a 4in. x 8in. handholedoor
for cbarging.It hasa rcvolutioncounleraadrunsat?0rpm
The erindingcharseconshtsof 285iroo ballsweiShing
20,125grams.It consislsof about43 1.45_in.balls,67l.l7'
ia.bslls,l0 l-in. balls,710.75-id.bals,and9406f in balls
with a slculat€d surfac! at@ of 842sq.in.
Tcstsarcmadcai all si.vc sizesbelow28mcsh.Alt€r the
tust gdodirs pcriodof lm r€volutions,tbc mill i! dudped,
th. ball cbarseis screeoedout, and tbe 700cc of matcrial
is sde€Dcdon si€vesof thc mcshsiz. testod,with coafict
protcctingsicv.s if ncc.ssary.Tbc undersizcis rci8b€d,
atrdf.erh unscgresatcdf..d is add€dto thc ovcrsizcto bring
its wcightbact to that of the original charS!.Thctr it is
rcturdedon to the bals in the mill and SrouDdfor the
numb€rof revolutiods€alculatedto goduce a 250p€r ccnt
circulatinsload, dumpedaod te:croeo.d. Thc Dumb.r of
revolutionsrcquiredis calcrllatcdAom th. rcsults of thc
pr.viouspcriodto ptoduccsievcuod4sizccqualto I /3.5 of
the total charscio thc mill.
Tho g ndiog p.riod cyclcsarc contiNcd utrtil thq oct
grams of sicve und€rsizeproduccd pcr mill rcvolution
reach.scquilibdum ard rcv.tscs it! dhcction of incrcas.
or dlcrcas.. Th.d the uddc$izc product and circulAting
load arc a.recnanalyscd,and thc avcragrof thc last tbrce
trctgradsperrevolu.ion(Cbp) is tbc bal rnin grindability.
WbcnF is the sizein microm which 80lcI c.nt of thc
ncwballlDi fe.d pass€s,Pis thc micro$ which80pcl c.at
of the last cycle si.vc undcrsizcpfoduc! pass.s,ad Pr
is thc openingitr micronsof the si€vcsizc&stcd,thcn thc
ball mill *ork iodex ryt is cdculatcd from thc folowirs
!.vis€d (1960).quation:
/r0 l0
wi _ 44.SttpJa.t'/ lcbeoBLl--7=- -.1 ....(8' wf /r/
Th! averag€valuc of P at 100 m€h is 114 microF, rt
150 m.sh i. is 76 microns, at m0 mesh ii is 50, ad at
325 mlsh it is 26.7. These v&lues of P arc to b€ uLd in
Equatiotr (8) wher P cannot be found from sizadkEibutioD
The ryi yaluc from Equalion (8) should codorn with
the motor output powcr to atr average overflo{, bal mill
of 8ft intcdor diameter SrindiDg w€t iu closcd circuil Fot
dry srindins tbc work input should no.maly be muliblicd
by 1.30.However.bal coatinSard packiug can incrcasc
thc work input in dry Sxindilg.
WbereD is the mill diamet€rimid. the lining jn ft, lh.
wort inpul sbouldbemultiplicd by (8/D)".

16. Hadercv e Gtindability Ratins
WhcreItd rcpr.s.nts theHardgrov! grindability ratina,r'
thc equivalcntw€t edndinswork index is found from I
capacilycaDbe estiluted from lhe crushcrsetting,.ccentric
throw and work index of the material
The product siz€sof jaw crusher!aad primary Srraroly
crusberswilh steep crushiDs conesare coDrrollcd priDcipally
by the opetr side settins of tbe crusher. Wb.rc O$ is rhe
opeDsides€ttingof thc crusherin incbcsat the bottom of
the crushingchamber,the 80 pcr ceDrpassingsizcP of tho
crusherproduciin micronsis calculatedfton E4uarion(10).
I i!. cquals25.400microns.
p : (2s,4M)(osr)(0.042i+ 0.40) ....00)
The product siz.s of conecrushers,with their flat clush-
ing conesand ielatively higb spccds,arc conrroltedpriDci-
patly by tbe close sidc setting. Wherc Crs is the ctose side
settingof the cone crusherin incbcaat thc bottom of tbe
crushins chamber, as commonly d.temircd by passiog
a pie.c of lead though the crusher, sod Ecc is tte eccenrric
tbrow in inches at the bottorh of thc crushing cor€, thc
product sizeP is found from
(25,400)(C$) (7,s..) (0.027r + 0.70)
.(e)
Crurhing
Crushingis acc.mplishedby contactwith metalor other
surlacermaintaineditr a ffxedpositionor in a risidly con-
straincdmotion path, althoughmany dusbcrs bavesafety
f€atu€s which allow teleaseurder exc€sriv.pressure.This
is contrasted$ith erinding,which is accomplhhedby the
frcc rnotionitr responseto sravityand other forcesof
unconnectcdmediasuchas rods,balls,rock piec€saEd
pcbblca.
Frecmcdiagrindinshssscvcralinhcrentadvantagesover
fixed nedh crushing,andasreductionmachinlry increases
in sizc and strengthlargcr particlcsbccomcamonableto
Srindingwbichcouldforncrly b€rcducedonly by crushing.
Cas€sin point aie tlo devclopm€ntof largeperiphcraldis-
chargerod miUsand autogcDoussrindios mills. Hos,ever,
tl€ comnercial Foductio! of pa icl.s larger than about
* in. i! sti a crushinspfocese.
Gushing is uually doft &y itr s€veralstageswith small
r€ducdon rstios ranging fron 3 to 6 in each stage.The
mschin$ us€d includcr gtratory sushers, jaw crulhcrs
Ooth singlcand doubl€togglc),$uhiDs.rolls, and impact
clushcrs,hammcriills or pulvcrators.It is doncwith botb
3c.€€ncdand natural f€cds,in staSesvith scrccnsbctwceD
cachstagcto rcmovcund.rsiz€,asw.1l as in opencircuit
and in closedcircuii witb lcrccls.
Exc€ssivomoistu€, ffn$, or both,id the feed can cause
packing in the cr$her, r$ulting in a deqeasein capacily,
incrcaseio powerdmw4 andincreascin the croshingplanr
work index.This isusuallyrcmedicdby screeningout more
ffncsaheadof the crush€r.
Crushcrmoror sizcsare usuauylimitedto protectrhe
crushersaSainstbr.aka8e.For thesamereasanuncrushable
piecesof metallre usuallyremovedfromthefeedmagnetic-
ally, or thc crusheris designedto openup and let tben
CtutherPtoductSizes
Thecrusberproductsizewhicb80pcf centpassesat full
....ol)(1Ecc- 2css
If the matcrial is very slabby, tbe vstu. of p mav bc
somewhatlara€r than tbai irdicated by Equatioru (10)and
(ll). These equationsare useful wh.n scrccu analys€soI
lhe crusher products are not available.
Scalpetl Feed to Crushett
The Third Theory equations rcquire a '.Datur.al" feeal
containioStho Daturalfinesproducedin the pr€viousreduc-
tior stagcs.When fioes are removed from tbe f€ed, the
relationshipbetwe€nF aod P is alrered.In most crushing
installations wbere fines smauer rhan the crushcr dischargo
opening are removed from the feed by screenhg, thc work
input per ton of original feed is not matcrislly decreascd,
exceptas the removal of finesprev.ns rb€ abnorrnalcon-
dition of packidgin the clusher.It hasb€eDfound satisfac-
tory to djsregardthe scalpiDgoperatioD,andto co$idcr th6
feed to the screen or grizzly as equivalenrf.ed ro the
crusher.This is pleferablein most cascswb€re the Brizzty
separalingsizeaDdhou.ly tonnaserhrough are nor kno*n
H,){ever, in some insianceswhere much of the fineg
i,ren removedthe cor.ectjonfor scalpedfeeomusr o€

17. '
mad.. This is done empirically by using that incrcascd
normal f€ed sizcFc which is equivalcntio s,ork input per
ton to the 80 pcr cent passing size F of lhc scalped f€ed.
Theper centpassingsiz€dhtribution tin. oflhe scalp.dfe.d
is ploltcd oD logJog paper. A liDe wilh thc normal slope
of 1:2 is drawn through the E0pe. ccdt passingpoint a to
ih intersectionr. with the size wbich 5 pcr cent of tbe
scalpedfccd passes;a parallel line is drawn through the
point with co-ordinatesF, (80 - Ycl2), and its tute6ection
with the 80per ceritpassingline givesthc valueof Fc.
WheEpieccsall of onc diamcler of.J micronsar€ fed to
a crushcr the equilal.nt 80 p€r c€nt lassing sizc F. b lhat
of a Third Thcory siz. disrribution liac with an exposur€
mtio E of 0.05and the sarn. crack leaStbas the panicles
fed.' The crack IeDSthin cmlcc is Ct = 173| la. T],neFc
valucs arc lisred in Table II. When the fced codsists of
parliclcs of several different diameters d withou! fines, the
equivalentcorrcclcdfeed sizeFc can b€ computcdas the
wcigbtcd averas. of thc ditraent sizesd,
r^rLF rr-l{urldr r0 F cdr Pstut si:.-.il,ro,pm(r4 dr d u!(mr
itr cm/c.-ltl'/r, r,
-
slots- 0.11609V,'
ton sround should bc mrdtiplied by (D /8)trr to find the work
index flom Equation (la).
FinenestAdjustnenl
Expcrietrce bas shosE that .xtremcly finc g!indi!8
requiresadditionaletrergyirput bcyoDdthat iDdicatedby
tbe work index Equation (1). Thcrc arc scvqralpo$ible
rcasonsfor ihis, including:
(a) The ball size! cuslomarily employcd arc too large
for €xtremely fiDe particlcr.
(b) In finc gridiDs thc closcd-circuit classification is
usually .ith.r iDefrcictrt or absent.
(c) In dry srindiog of fitre particlcs th€ amount of ball
coating ranses fron iDcipient 10 complct.. Ball
coaling cushionsrhe metal contact! and d.cr.ases
sriodins efficiency.
(d) In wet sdnding a thick viscous pulp can caule ball
cusbiooing and d€crcasegrindiDs efrciency.
(e) The production of particlcs sma.Id than thc grind
limit ti of 0.1 rnicron involves breakas. across thc
uDflawed space lattice, aDd requir.s seveml times as
mucb enerSyiDput as thc customary brcakag€ alonS
planesof latticedisplaccrDcnt-6CoNidcrableamounts
of traos-srind-liDit particlc! may bc produc.d in
ultra-fin€grinding.
Wh.n tbe prodoctsizcP is Icssthan 70micronsthe work
input I/ as calculatedfron laboratory testsis nultiplicd
by tbe followiog empirical adjustmcotfactor ,t! which
P + 10.3
L145P
The fine product lactor rl was dcrived for the frne dry
srindins of c€mentclinker,andappliesto dry srinalinsdown
to
"
valuesof 15 or less-For wet fine gdndirg,,{r should
havc a maximum value of 5-
Convers€ly,thc plant opcratinswork index ftorn Equa-
tion (la) sbouldbe dividcd by 11 and by (8/D)tr,for dtuect
compariobsirh tbe laboratorywork index lti.
Prcper Grindins M.dia Sizes
Th€ siz€ of tbe griDding rn.dia is one of the pridcipal
factors afecting th€ efrciency and €apaciry of tumbling-
typc gridios milh. It is bcstdetermir€dfor any particolsr
iDstallation by lengrhy comparative plant tesb with carcfully
kept records. Howevcr, a mcthod of calculating the propcr
sizes,basedupon coffcct theoreticalprincipl$ and tcstcd
by experi€nce, can bc very l€lpful, particutarly in n€w
The gcneral principle of selection should be that th€
proper sizeof th. make-up srindins n.dia is the siz. which
will just brcak the larsestfeedparticlcs.If tbe mediais too
larse,the Dumberof breakinsconlactswill be.cduccd,and
ihe €x&emefinesmad€ by eachconiacrwill beincr€ased.
If rhe mediais too smal, therewill be yastedconlactsof s
foic. insufrcientto brcak the particle!contacted.ln eitber
casethe srinding efficiencywil be reduced,but the rse of
undersircmedia is usually more harmful lhan the use of
Let , = make-upball, rod, or pebblediameterin inches;
F = sizein microns80p.r ceotof the newfeedpass€s;
7i
-
work iodexat thc fecdsizeF;
C3 = f.action of mill critical specd;
sc = specific8rality of materialbeing ground;
D = mill diamere.in fect insideliners;
K = an empirical.xpericnceconslant.
In o.dinary ball mill opcrationf-in. steelballswill€fiec-
lively crind averasesiliceousore with 80 per cent passing
lmm. of *ilh F:1000microns or about 16 mesh.It
Grinding
Corcction lor Feed and Products
Closed-circuitgrinding and complex grinding circuits
which includecoDcentrationand separation€quipmentare
best analysedby consid.dng each circuit as an inlegraled
unit. In closed-circuitgrinding the unit consislsof tbe mill
and the classifier,with a singlefeed to the mill or classifier
and a singleclassifierundersizcproduct.Calculationsfrom
the mill discbar8eand circulatiDgload are usually unsatis-
factory becausethe harderfraction of the materialaccuhu-
Iates.and the circulatingload hasan unknown hiSherq,ork
index than the new feed.If lhe closedcircuit includ.s con-
centratingequipment such as magneticseparatorswhich
reject a taiUng,the product of the ItindinS circuit is a cal-
culatedcompositcof the classifierunde.sizeand the separa-
tor tailirg, which shouldalwaysbe screenlnalysed for this
Calculationsinvolvjns urnatural feed from which part
or all oi the nneshave been removedshould be avoided
lvhenevercalculationsof inlesraled ci.cuits can be sub-
slituted.However. the empirical mcthodsdescribedunde.
''ScalpedFeed to Crushere"can be usedwhen necesa.y
when Er.naingre r arc made In o .mcll. .01 .n rous.
pilot-plantmill of diafreterD, tbe grosspowerinprt per

18. follows theor€tically,r aod i! coDfirmed by experiencc, that
2-in. balh are suitabl. fo! 4-mrn feed- 3-in. balls for 9-mm
particl$, .tc. Th. batl size should vary as the squarc root
of the paniclc sizeto be broken.
From tbeo.etical considerationssthe proper make-up size
a of steelor castiron batlsis found fron
largediamcte.mills draw rnorepower with largeball sizes.
Ball rationins. whicb is tbe regular addition of dcfilite
propofiioosof ballsof djfferentsizes,may be usedwhena
is intermediatebetweetrtwo commercialbausizcs,or wher!
an unusualsizedistributionof the fccdrequAesthe addition
of some smallerballs with lhose of the calculrtcd sizc,.
For Rod MilLs
From theoreticalcolsidelarions,the proper diamcter B
of mak€-upsteelgrinding rods i! found from:
- /rI / ssr.t/ir
": "/ rmc"/r-l . ...0 2)
Th€ empiricalcoastaot.K is found by expericDceto be350
for wei sriDdirs and 335for dry srindiDs.
The comrncrcialsizc ncar.st to t h ordinarily sclect.d
for ibe make-upball siz€.However, when B is lessthan
I id., it E|aybe cconomicalto sclecta largcr ball sizefor
the3ercasons:(1)thecostp€r ton of the smalerballsis
increased;(2)lcsswearis obtainedfrom th€smallerballs
before they are di$harsed fron th€ mill; (3) the smaller
balls may plue thc gratesof diapbrasmdischargemills; (4)
synholildEqudonNo'
o
4P
Yt.
n
4,3. l,
n,+n
,,
n,r+
Ftu D'dridiBri.nr hcror.
Rod hnl
-R'
rdiu$m.dr rrcror
Bd onl Rr rdiBh.nt rrcror.
rbiD. {ifac. ao jn rq cn/srh.
E$o!.ntirl !ir. dndbudon o.6ddi.
FLbggincb.imDldc'ulhin'nmsrb'
&r 6r dturdiB rord/100
Tour cack ldrlh</d.
P.r.d'
-dshr
orcon6,B'o
'
rad.
cyda Fi. m'nnc inrR. '.njsr6chr
on.
P.r edr or dnchlrF rasine riz. ,.
F.dtl. th6w df c;shq in i.cb6.
F.ri rzFnikrcci0 Fenrpekr
lqrivrrmr 3iz .. &,rp.! f..n.
PdcmrolD4f.!dDaliJa!i'P-
16 tuj.e!$rF !; hil n.- red.
lra{d3rctr eirdbilily
Kilovat! s to! ol bdk,
Kilof,d! id tm of rcd!.
l43rh ol !n incio! in tei.
Pq.4lpuiainhilldjtch'ie'
o@id.cruh.r&ni4inb.hB.
Producr!i!F n...o 30tsr@t pn*i
si.v.o!ai4inmjclot'FA-cah&bj]ny,
Fd.rroo
'.p
or nilr
'o
rocr di:.b!re
Ps dr of ov6ir rrsid !iz. P.
opti-uE ro.t nill r.dldjon n.it.
Pdcot'ai4i!claliid@rRproducl'
Fflcdon or nill varud. b.low dischrrce
work indcr: kwhjon
'o
30 Fr @r-
wort rdd: rwbhod ro D@dud 5;..
when the reductionratio R/ = r/P is lessthan 8, the cal-
culatedvalueofB shouldbc increasedby 1in.
For Pebble and Rockled MilLr
Pebblesfcd to pebble mills, and the ruck fed to auto-
genous rnils wbere tbe large piecessrind the smaller
particles, are selected10have the sameweight as steel balls
suitablefor the sameseNice.WbenB is tbepropermake-up
b8ll sizcaccordinsto Equation (12),theDthe properlebble
or srindingrock sizeof speific gravitySs is A x 0.8/58)+.
Size Distribution oJ Gdndiry Media
All types of grinding mcdia commonly wear down to
sizEssufncientlysmall to discharsefrom tbe mill with the
materialbeins ground.However,in someiod mills broken
and worn rods are removedmanually.
It hasbeendelerminedtha! a film of metalof udit thick-
nes is worn from any siz€ball in a mill in the samesrind-
ing time. If the weighl loss is periodically replaced as
make-upbalh of sizea, th. ball chaigereachesac equili-
brium sizedisributioo which extendsdown to almost the
ball sizedischargedfrom the mill. Tbis equilibriun sizedis-
tributioDfollows the equation
^ et l--wig
ro0r lm cr v,
.,.. (13)
.... (14)
whe.e ) is the percentageof the lotal equilibrium cbarge
passinsany size r. Equatior (14) presumablyholds for
srindins rods and pebblesasweil as balis.
In order to obtainconsislentperfolmancein wet srirding
mills, the initial media charseshouldbe madeup from the
severalsizesavailableto besimilar to lhe equilibriumcharse
defined by Equatioo (14). This cao.be approximatedby
drawing on loglog paper the srraight per ccnt passinA
lide witb a slopeof 3.8tbrough 100per centpassingsize8.
The initial chargecorDpositionis detclminedby mdking the
poids midway betwe€nthe ball or rod sizesto be used.
ff ball rationirs i! to be used,tbe initial chargcshould be
proportionedbetweenthe two ball sizesfed.
In commercialrod mills no rod siz€ssmallerthan 2+in.
should be used in tbe initial charge.In commercial ball
milk (he minimLm si/e used is commonly I iD.
In dry-grindingmilh the metal wearrate is so much less
than in wet milh tbat two yearsor more may b. required
to rcach equilibrium,aod the initial charg€caDbe propor-
tioned lo fit the mill feed without referenceto the equili-
The weisht. volume, and surfacearea of steel balls or
rodsof diamel€rt inchescanbe foudd from Tabte III.
r,s,

19. Fip. l. Fouteen
86-ht ba nius
istallei! at So then
Petu CopPer CotP.'s
corcentotor, foque'
palo, P.tu, griu.Iing
30,ffiO tons ol coP'
PART I I
Milt TlPc Rod
Mcdia: lb/kl9h 021 rods
LiniDg: lb/twh O.oAi
realor the m6tal savine!resultinSfrorn the usc of hard
altoy mil lirings or mcdia arc usualy Srcater id drf
Srindingttan in wet griDditrg,
Ab.arion Tcat
Abrasion t€its to indicatc m€tal wcar ar6 madc a3
folows: o 4O0slamsof -l +
'-in,
paniclcaof thernrttrirl
to be Estedar; Dlaccdin a rotAtiry drutn for 15nin Th'
drun sbow€rsthc rockthousb a rapidy rotatingiop.llcr
containedwithin it. The imPcller lotat€d at 632tpm and
consistsofa3x 1x +i[. stcclpaddl€with2 !q.in.crPoscd
to wcar, machiDedof SAB 4325 st€cl hardeD€dto 50O
Brircll. The gram! of weislt lost bv thc pad{ c after im'
pactinglour Juccessive400-8rambatchcsof rcck for 15min.
iach ii called the abrasionirdcx and dcsienat.d$ ,ti
Thecombinedproductof thefour ls_min.pcrioal!i! sctccD
aoalysed;it av€ra8es80per centpassins13'250micron$
Paddlesof sDecialallov si.els and cbetirom csrl bc
r.sEdwith a sdtrdardabrasiveFaterialto detcrmiDcr€la-
tiv€ wcarraqistanceof the metals.
Ta av.raeaAi valuesof sometvpical materialsfron
125testsartlsted iD Table IIA in tho lpp€ndix. A ftm
cofielation with actualwcsr rat€sin €rushcrssDdeEiodhg
mils hasDotbecdmadeasyet; however,it is apparcutthat
theabrasionindexvadationssho$nin TableIIA ar! mucb
sreatcr than tle wriations to be expccLd iD tho ts.tal
wearratesin cornmercialmachines.It is obscrvedtbat anv
correlationb.twe€ntbework index andtl€ abrasioEhdei
is vc.y slight
Prelimirary indications ate that ir wet srndins thc
pourds of bals p€r kwh equals(li + 1) dividcdbv a
- 9-
Bd
0.14 balls
0.020
Th. nctal wesr io drv SritrdiDgaveragesaboul oo€'
."ve*l tnal ot wlt ItiDdiDStbe samernatedal Howcver'
-r""
* U" mat.rials Srouod dry are sofi'r aid less
Ii-#iu. rl* tlo'. gr-'a q'ct Batlcoatiosio drv srioding
can reduc. tbe metal w€al still turlher'
Wcff over taf of tl. nelal wear i! wct SriDdiDgreulLs
ftom conosion,or dissolutjoofrom lbe'ctivc nasetrt metal
l"J""es iont.uattv bcids producedin tbe mill For tb's
t'ffi"F*,"F,*"+H'[+"t:r#':#.1"ji{;T|i"#tTJ
CRUSHTNG AND
GRINDTNG GALGUTATIONS
Dl FRED C. gOND
Th. concludinsp.rt of thi. rrtlcl. i5 conc.rn.d with r nuhb'r of h'toR rfi'di" th' gindina
J"..'i ."iiriiit..t
"q"lpn.nt.
such r5 th. rhdion or nill volufr. occ!pi'd bv th' trind-
i;;-;;:;.,.h. qu.ntni; or rods dd b.rk b b. 6nt.in.d in . mirr, {"r or mill
'nd
;;"ndin!;.diun,.id mlll tP..d in le/ms of th. critic.l tP..d. Th. .ff.cti of hill dhh't'r' or
;ownw;rd rlltt.sc ol b.tl .hrrg.. rrtio ov.Eiz.:f"d. uPonP'rtormrn"
'nd
Powcr
'on{hts
don rre d'(us.d, oP.n-.ncuit multitliatio^ frcloB rrc liv.n ror.onrcrtlng clor'o{rrc!E
work v.t{.s to the oP.n_circuit vrlu.r .nd. 6n.llt. .o'r.ctiq r.cio6
'r'
lN'n ror worx
i'a.i *rirtion' *hicli .rit. wh.i l.boBtory srind.birit/ ind ioP.<t cruthinl t'ris 8iv'
diF.r.nt wort Ind.x v.lu.t.
M6t3l Wcar
r f,ETAL $.ar is us'rallylbo s'-.ond largeqt6itr8leiten
lVl
"i
**'r. is conventiooalItindins, aod io wcr
iii'-ai'g i*i"l"ti*' il mav lpproacb or
'ven "cerd
thc
DOWCrCOSL
' wlcralwcaris commonlv.xpre$edin lb/ton susbcdor
-iiii.
rl"".ii". i*r"toDs ;; fcedaod Eoduct siz€sald
L workindcxarecliminalcdbverprcsliDsm'tal coDsumP'
Go ,, nltWl" iucludios rcj€€tcdwom parb: it can be
ott i". i.on tu pourds of metal coDsumedpcr toD
crusbedor eroud and thekwh/ion
Tbe poundsof nelat botbwom awav-abdscrappeda
*.'i-"Le tuti""
-itt
tte abEsiv€ncssof tbc oreandlhc
"trnsi6o
t."itta""o ot rt" IDetalTh€ averagiv'lues froE
"-irie"
rt-to of mils witb ordharv miU--liriDg'srDd
eFindingEcdia,grindingw.t a largprangEof siliceousor€s'
i r,iLTr:!.1*';5;&".df"':i'i

20. Dumberwhich ransesfrom 6 to 9, atrdIid€r w€ar iEabout
onc-cishth of bal wear. For wet rcd mills the divisor
rang$ from 4 to 6, and liner w.ar is aboutone-s€vmthof
rod w.ar. Wear in dry srindins t about onc-scvlrih of
wetBrindins.Thc .veragcmetalwearin cnshirs is rougbly
comparabl.to the liner wear in wet srioding.
Volume of Grinding CharSe
Tbc ftactior /p of thc total iot€rior mill volum€occupied
by llc srirdins charsc can be foutrd by Equation (1t,
whereD istheinsidcdiaractcraDalO b thc €rtical distaocc
do$n from th€iDlidetop of theIni[ to thc lcvcled griDditrg
charS!: O should bc the &vcrag€of m€asu.crDentsat the
contr. and borh .d3 0f thc mi[.
vp= 1.r3- r.26AP ....0t
speedis about 77 pd cent of criticat: for wct rod milh ii h
about 70 per ceDt of critical. Somewhat slower spe.ds arc
oftenfound to b6 moree4onomical.
Dry-grindins mills atrd pebble mills usualty operare at
about lbc same sleed! as wct mith! nith the maxiDum less
lirnited thaD iD wet mills-
Laboratory sludies hav€ showo that or thc iising side of
th6 mil each circular row of gritrdirg ba s slips dowqward
tilb lespectto the nexto!te. row it restsupon,tbuscausiog
somc grindina in tbis portion of the mi . A similar dippase
is Dot obs€rvcd in rod Eills.
At spccdsfaster than 60 per ccnt of critical the srnaller
balls or rods in e Sdnding charac tcDd to concsnEatc qritb
the pulp nea! ih. IiDiDg of a cylin&icrl mill, ard lhe larg.r
media lre displaccd roward the ccrE€ of th€ cbargc. Advan-
tagc is t.kea of ftis iD ball mill6 to movc the smalld balls
loward th€ dischargc end of thc milt by spiral liftcrs Aailina
towerd the mill diacharge; or by rnaking thc mil sh€
sli8hdy conicd, with the smallest diamctcr at thc dischargc
Smal-diameter mils arc cornlnodly opcratd at somc-
what highcr fractions of tbeir critical sp.lds than are lars€
mills, indicatirg tlat the proper mill sp.ed h itrtermediare
between a constant fraction of criticsl spcedand a coDsrant
pcdpheml speed. An approximat€ empirical cqurioD for
tbo maximum pra€ticalrpm of werbal mills,designaredas
No = 51- At)losD . . .. (20)
Effed of Mill Diam6t€.s
At a consiantvolumefraction t/p themassof rhegrind-
ilg cbarSevarics as D'. At a coostantfraction Cr of the
critical sped the pcripheralspeedvariesas v5. It follo*s
rhat tbc pow€r input to a conventionaltumblios mill
theor.ticallyvariesastheinterior diametcrto tbe2.5Dower.
Meaqurementsbav. shownrharlbcwer6n. srirdi;g mitl
capacityvaries as D'.. Thc diamcterexpoDeatincreaslr
sliSltly in mils opcratedunder bish impact condirions.
Theoretically, iho maximum expon.nt under reductioo
ertirelyby inpact is 3.0.
Mealulemdrts haye aho sbown that tbc power inpur
expooentactually vadcs as Dar. The decreasefrom tbe
theoreticBl2.5cr(poDentprobablyresult!ftom energyfrom
tbefalling bals or rods beiss tsansferredbackro th€ mifl
she on its do$n-8oingside.Theactualdjametercxponent
per lon of grindins nedia is 0.4insr€adof th€ thcoretical
0.5.
Tho difideDce betweenthe two observeddiameterex-
ponentsof 2.6aDd2.4is 0.2,yJbichistheexpon.ntd€filing
themechanicaladvantag.of lars€diametermilb. Mechani-
cal emciencyircreascsasthc inrerior mil diarneterto rhe
0.2pow.r, aBdtbe kwh/ton requiredto sdnd d.creasesin
the samcratio. Sitrcethe standards'ork iDdcxis basedon
rDillsof I ft int€rior dhmeter,rhccomputedkwh/ron (I/)
byEqDatior(1)for anymin of D fr interiordiametershould
bemultiplied by (8/D)'t
Powe. p€r Ton of c.indlng Media
Thepowerinputrequiredia tumblinsmitlsis calcutated
from tbe power requircdpcr ton of srindins mediauuder
tbe mill operating coDditioDs.It vari.s wirh the ftaction
Zp of themill volumeoccupiedby thesrirdins charge,tb€
fractio! Cr of the critical spccd,rod tbc iDrerior mill
Equatioo(21)sivcs-(r/, th. mil inputkW pcr ron of
nelvgrindingrods in cotrvenlionalwetSrirdingovdflow
lod mills.
Kvr=Dt4(6.3-s.4vdcs . .. .(21)
An accumulalionof brokenrods in thc mill can reduce
tbe actualpower d.awn by asmuchasl0 per ceDt.
Thoweighiof thesrindiDgcbrs€ catrbecalculatcdftod
D, Izp, atrdthc insideleDstbof the mi[.
Loos€roudd balls wilhout b!€akagcweigh290lb/cu, ft,
looserods 3901band silica pbbles 1001b.Measuemcnts
of thc weisht of steelba s contain.d in a cubic foot. bor
are usually lccsthan 2901bbecau. of thc incr€ascdvoid
sprce at th€ sid€sof the box, and tbc w€ielt coatained
decreas.sastbc ball siz€idocas$. For accurat! ocasurc-
mentsthe smallcstdimeDsioEof the contaiocrlhouLl bc at
least 20 times that of tbe largest ball. Brok€n balls caD
reduccth. spccificweightto 2801b/cu.ft, and b.okcr rcds
to 3401b.Thc spedficweisht of both ball add md charges
caDalsobc rcducedby a contentof hard particlesof the
material bciog ground.
Tons of new balls 7b cotrtaio€dio a min ar. found
approximatelyfrom Equation(tO, atrdlons of rew rods
?/ from EquatioD(17).
Tt = VpD'L16.8
....00
....07)
.... (19)
A charseof Brindirs ballscontaiDsapproximately40
per cenl of void space,aDdrods in lir€ar contactcontaiu
20 per cent voids. Broker orc coDtainsapproximately40
p€rceatvoidspace,andweighs100lb/cu.ft multipliedby
its specificsravity ovcr 2.65.At 80pcr ccnt soli& or morc
rhevoidsiD a grinding chars€of stcelballs caDcortaiE 14
pcr centof thebatl wcightiD ore multiplicd by S8/2.65.A
rod chargowith therodsin liaearcontactcanco aiD7 per
cen. of (hc rod *eigbt. Howcv€r,botb bals and rods arc
commonlyfor€edapart in the mill atrdmay coDtainmore
orethaotlese miEim m amounts.
Tbe p€r ccDtsolidscontainedin th€ pulp within a wct
Brindins mill is ordiaarily more tban that mtering and
Icaviry tb€mill, siDcethcflater fows tbroughthc mill fastcr
tlar the h€avierparticl.s of ore. This diferenceis io-
creasedin 8ra.eandpedpheraldischars€milb-
Criticel Mill speed
Tho theoreticalcritical spcedNc h rpm b the speedat
rhich a particleof Dodiameteragainstthe mil linins witb
no slippasos?ouldcentrifus!. It is found from Fauation
(18),c,hercD is th€intcriormill diamctcritr fcct.
Nc = 76.61fd ... 08)
The ftactioD Cr of the critical speed rcprcscnted by alry
bil rpm is found from Equatiotr(19).
Cr=0.01305x rpmx y'b
Tumblins mills are usually d$ign.d to rcvolve at a con-
stsnt fractioD of th€ critical speed.Incrcascd specdiocr€ases
the mill .apacity ard power draught,but alsoincreasesthe
metal w€ar and mainteDaDcecostper ton Sround.For con-
ventionalwct grindiogball mills morethan 8 ft in diameter,
*ilh peripherallifters anda ball chargeof morethan 30 per
centof the ioteriormi1lvolume.the maximumDracticalmill
- 10-

21. 4
f,
I
-r
! CLASSIFIERPERFORI{ANCL
,r/ i i.. -----------TLoi-
* E Ei: ovERs,zE
- %cuMoN
E[=i:
:
-
3i<'
- [ -'" --as
S 3l vAR,Al,oNs-PLor
; ;*
W.7Oa |<rh/SHoRTlON
Fig. 2.
TOP: Classifiet perlorma ce plot.
Ov.tsize per ceat .am. ok plott?d against und.6ize
D.t ce Dasinz. InrcB..tior showt E3 p.r .enl
ildsifi.t Zfrci.icr anl t6s nkrcas pa iaE eze.
BOTTOM: Coftection lor wolk indet variations.
Gtindabiliry TestFWi = 14.4 at 14 mesh.
Wi: 1O.4a, 2$ nesh.
Gind in ball ni| lrcn 80 pet cent pasins 1m0
miooAs to 80p?t cehtpassinelN nicrons.'lit =
10.9,Wit:r4.4i lrcn Esuatioi (42)W=7.O8kwh
Equation (22)sives.Krr, the mil irpur kW per ton of
srindiEgballsfor coaventionalw.t-gdnditrgball mills using
makc-upballs lar8cr than about oDe-eighticthof thc mill
Kt tb * 2.8DLt(3.2- 3v p) Cdr - o.1|Ztrw, .... Qz)
For dry-grinding smtc discbargcmils Xn, should bc
multipliedby 1.08.
If Od r€prcscnt!&c vcrtical distaDccin feet ftom thc
insidetop of thc mil to th. lowastdischarsepoint, then
' Vpd= t.r3- r.26AdlD ....Q3)
wheret/pd is llc fractior of theintdior mill volumebelow
dischargclcvcl. For a full lowlev€l gtat discbary. Vpd
equal!0.029.For welgrindins srate|nd low-leycldiscbargp
. - l, . O.4O- VDdI
m'rls.rnuruprynlrD ny
l, + 2l l.
SlumD Coffection
Lars+diametq ball mills fed with small rnak&up balls
losepower becauscof €xce+livcdownwardslippagcof the
bal chalsc or the siDSsid. of tbe'milt, and this lo$ of
powd inult d€creas.!thomin capacity.An cmpiricslslump
subtractionquantity Sr is computedfor wet overnowand
satc ball rnills by Bqu&tion(24),to bc subFactd from the
K)r, vdue of Equation(22).No subEactionis madcfor vtct
ball milts with , < 8 or for dry ball millswith D < 10.For
dry mils with D > 10thc slumpquantityto be subtrastcd
from Xr, is tbec-quartersof Sr asfoundbyF4uatiotr(2t,
whcr€, is thedianeter of tbc make-upballsf€d in inch.s.
"":(i#-t- ,.,.Q4)
Efi€ct of Reduction Ratios
The reducrionratio Rr is the ratio of the sizeof the rew
mill fced to that of tbe final prod ct, or F/P Witb scalp.d
Iccd, FclP should bc used.Rod mills arc pa.ticularly s€$i-
tive to u avourablercductionratios,atrdif F/P is smatlcr
than about 12 or larser thatr abouf 20 the twh/tor (,'/)
required fo! grindingincreases.If Rto represenhtbe opti-
mum rod oill reduction ratio, its approximatevalue is
(25)
Whcntheactualreductionlatio.Rf is muchsmal.r or
largerthanRro, thework input P from E4uation(l) 3hould
bo multipliedby thecmpirical adjBtmcnt factor 1,, wher.
A'=r+2(Rt- R0)'1300 ....(26)
Sinc€pcdpheraldischargcrod mills havclos, reduction
ratios,thcy Dormallyr.quirc anincrcasciD P.
Ba[ rdills arc lcssEeruitiveto changesin reductionratio
ttad rod mills. Ho$cvcr, whm R/ b.lomes le$ thar about
3,particularlyin 1befDe srirdins of concentsat$,7 frorn
Bquation(l) shouldbe nultiplicd by the cmpiric{l adjult-
mcntfactorlt, where
zX(R!- 1.351+ 2.&
--' 20(Rr- 1.35)
Free Verticll Orcillatlon
A body id fre€ v€rtical oscilation fals from it! highast
posilion under tbe influcncc of gravity and tu stopp.d
lorcibly by atr equal deccleraiion;it is thctr acc.l.ratcd
upwad at tb! samelate andr.acbcstbasamchiSh.stpori-
tioDasbcfor.. It dcscdb€sa sinplc barmonicmotion ovcl a
vertical distanccof ,fi inchcswith .prr cycl.! pcr minut!,
....(28)
The critical frequencyof vibrating bodic! is calculst d
fiom Equation(28)i as tbe cpn (oi rpro) dccrc*cs below
thc critical, th€ tcndcncyto mai sio frcc vdtlcal orcilla-
tion caNesthc amplitudci to incr.aic.
Thopowero€ccasaryto maiDtainfrc€ vcrticd oicillation
is direcdyproportionalto thc pcriod of os.ilatiod (l/cpn)
aodto y'n. Wbere-Kvois the kilowattsn!.cssaryto mair.
tain a ton massin frec vcrti.al oscillation.thctr
....Q'
... (29)
Th€ vertical componctrt of thc motioD of a particle of no
diamc.€r a8ainst thc lining of a bal mill at griticd spccd
colrEspondsto frce v.rtical oscillation*ith h: l2D, znd
Ef,ect of Overiize F€ed
Fced particlq which are too lars. for thc grinding ballt
or rods10brcak are sradually worn down ir the mill r,itl
a considerablelossof srindif,geficieocy.If rhemill product
Kvo = 328lcpn= l.tt li
Rto=8+5LlD
- 11'

22. is coarsc,the loss in milt capacity can b. quite large; ir
dereasesas the proporlior of the lotal work cotrsumedin
fineerindirg increases.The decreasein capacitycaulcalby
overeizeerirding mediais aot aspronouncedasthat caused
When Fo reprcscDtsthe maximum 80 pe. cdt passing
feed siz. in microns wbich does not appreciably decreas€
th. erinding efrcicocy (is not oversire) nith tle ball or rod
siz! calculatedfrom Equation (12) or (13), aDd the rrork
index is 13, Fo is about 4000foi ordinary ball mils and
30,000for rod mills.
Forba mirsro : ao,my'(l3VD4 . .. .(30)
Forrodmillsfo = 30,000,/(3)twi ....131
The work input Il/ for oversiz! feed (F> Fo) is calculated
w-
tho samelinear or logarithmic gsph shect.Thc poiqr *herc
tle two lin6 cross h thc partirg size, aod the pcr ceot
passiDgat tbc cro3sirg poiDt is the per cetrt cmciency of the
Per€€nta8e Circulating Load
In .losed-circuit reductioD the per c€nr circularing load
(100 C0 is 100tnnes the ratio of the weisht of separator
oversizc returning to the rcductioD machire io tbe weieht
of the new feed enterins tbe circuit h the same time
P : microns80 p€r ceoto[ scparatorundcrsizepasses.
Fp = per cent of new fecd passiDssiz, P.
Dp : per cent of iDacbine discharse passirg sizeP.
Rp : per ceDtof separator oversizcpassingsizcp.
When tbe new feed enters the re.luction machinc
Cr:(80-Ddl(Dp*Rp
Whcnthenewfeedenterstheseparator
ct:(8o-Fp)l(Dp_Rp)
The quaDtitiesrequircdin EquarioDs(37)aDd(38)caDbe
deteimioedfron loglog or exponenrialplots of complere
size distribution aDallses. However, rh€ sdeen adalyses
evailablemay not permi! detclminatior of thesequanrities,
and C/ must be calculatedfrom the old equationslrbascd
on the perceDtagespassing?00meshor any otie. availahlc
If . represents the per cent of tbe classifi.r fine producr
passing 200 mesh. r is the per cenr of rhe classifier coarse
product passing ahesame sizc, and m is tbe pcr ceot of thc
mill dischargcpassiDgthe samesize,thcn whcn tbc Dcwf.ed
ente$ the erinding mill
ct = (c- m)l(m _ t) . .. . (39)
When tho new feed enters rhe classifier, and ld r.pteseors
tbo per centof tlc newfeedpassine200rncsh,thcD
ct=(c-t")t(n-t) . ... (()}
Ary other suitablesc.eensizemay be subsrituted.How-
ev.r, tbe Cl values calculaledfrom dn.r.nt scleer siz.s
usuallyshow a wide variatioo.and tbe meshsizeat whicb
the circulating load is calculated should be sp.aiff€d.
Per Cent new Classifler Feed to Closod-circuit Mlll
When a rod mill in open circuit dischargesro a cls$ifi.r
in closed circuit with a bal mill, rh. per ceDt of thc rod
mill discharse which enl€rs the ball miI as scalp€dncw feed
,- fc" or rodmi discharyel[% pn , lm-% Enl
'*-l passiogparrirgsi.. I I r0o
-r
tmc/ I
..., (41)
Closed-circuit Versus ODen-circqit Eell Mill
Grinding
The listedwork indexvaluesapply to ball rnills srindiDs
w€t in closedcircuit. For dry sriDdiDsitr closedcircuit, the
workinputZ shouldbemultipliedby 1.30.
The conversion_toopen-circuitSrinding,eilherwet or
dry, is dodeby multiplyinstbe closed-circuitwork input
,y by aEopen-cilcuitmultiplicatior factor. Itr tbis coDDec-
tion the circulatirg load Cl shoutdbcuoity or grcater.
This factor varieswjth rhe refercnc€pe! ccnt passilg!
or tbe per centpassiogat whichtbc opcr- and closcd-
circuitsrindinsafecompared.For iosratrce,iJrhe.efereoce
percentpassingis95,tbernuttiplicationfactorwil bemuch
largertbatrat 80p€rceDtpassitrg.ff it is requiredth.t 95
pe. ceotof tbeproductpasles200meshtbechangefron
closedcircuitto opencircui!wil requirca mu€hgreater
increasein powerthanif ir isrequiredthar80pe.centpass
200nrcsh,or aayotherspecifiedmeshsize.
(F - FolFoTRr+(Wi-lto wi 10wil I
L/ P /F I L
.... (32
Grind Difrerential
Tbe gdnd differeDiial Gd evaluates the diferen@ h the
particlc sizcsof the concentrateand tailing when griDdisg
for rnineral unlockiDg and cotrcentration by flotatioD or by
cravity. An inqeased grind difierential can b€ of mal'or
importancein suchsrindils cjrcuits.
WhcreP is tbe 80 per centpassidssizein micronsof tle
fe6daoconcenr.ation,Cp is tbe 80 pe! centpassingsizeof
tho concentrate,and Cw is the per ceDtweight qf the coo-
cenrratedividedby rbef4d, the griod difierentkF is
P (150- Cr) - 50cp
.... (33)
P(5o-Cwr+50Cp
A typical copperflotatiotr plant has a sriDd differeDtial
of 1.34.Any chansein thc sinding circuit $fhichwould iD-
dcase tbe Srind diferential *ould resott iu relarively
coarsersrindins of lhe g3trgue.and should favourably afiect
costs,recoveryand gladc.rr
WbcE makibg rwo grinds to 80 per ceot passio8a giveo
product sizeP, witb the tust srind diferential c4 larger
than thc secoDdsrind diferential GdL the relative IbechaDi,
cal efrciencyof the filst srind to the secondis:
Rer.En:./r + f91', l) - l9,r, l) ....,r,a Gdr.l 1l Gd,+ tJ -
ff the recessionfaclor R/ is tb€ number of standardy',
scr€eoscalcspac€sbetweensizeP andsizeCp,then
RJ: (toEP- losCd ltoc,/1
N
P:cPQ)l
Tbc g nd difereolial cd can be calculated from the
rec$sioDfactor Rl aDdthc p.r cent weigbt of the conceu-
tral€ C}? by substituting the value of P ftoln Equation (3O
inlo Bquatioo (33).
Th€Schuhmannslopedoflhe heavierconcentrat.is com-
modly greatertban thatof the lighter taitings.
Clairifler Performance
In closed"circuitreductiontbe efficiencyaDdtlc s€paraG
ins size,cut poitrt or lartins sizeof tbe classifieror screen
are important. The efrciency is commoDly expr€ssedas
I0{ minus the per ceDtof finishednalerial or "unders" in
the ove.sizeleturned to the rnill or crusher,a! a cenau
The paldns sizeh defioedas the sizeat which the per
cent"unders" in tbe separatorovenize equalsthe per cent
"overs" in the separatorlndersize.The separatorlnder-
sizeper cent passing,and the separatorove.sizeper ceil
cumulalivo relained on, are plorl€d as smooth cu.ves on
-i2-

23. Som. apFoximatc opcD-.ircuitmultiplication factoft for
vario'rs r.ferencc percentag.spassingare lisred below in
TableIV. Theyfollow an exponeniialfunction.
TASLE IV
50
60
'lo
80
90
92
95
98
1.035
t.05
t.l0
1.20
1.4'0
1.46
1.57
1.70
Correctionfor Work IndexVariations
WhenaDor. hasmturalgrainsizes,it sometimeshapp€rs
tbatlaboratoryrcd mil andballmi gdndabilitytesisand
impact crustiDgteEtssive dificr€Dtwork index (7r1 values
Appendix
on th. same samplc at difer.nt producr lizes. Tbis corE-
plicates the calculatior of the work iDput rcquir€d (p) in
kwl/ton from Equation (l), which is !s.n with rbc dcsir.d
capa.ity to fnd thc required rDoror and mil siz!.
Wh€o this bappens thc various work iddci values ar.
plotted vcnically as orditra!$ on lo8-lo8 pap€r against tbc
80 per cent passing size as abscissdc,and the poin& arc
connccted by stlaight linca which arc cxtcrided horizoDralty
to the edg.s of rb€ pap.r. Tbe 80 pcl cear pa.gsiDgsizcs ol
Sriodability l.st ploducts are fourd by divjding rbc r.r.cn
opeDin$ Pl in Eicrons by log 20 (1.301),or by the valucs
siv€n folowing Equation (8) for fine bal miI tcsts. Thc
80 pcr cent passing sizo is 1.5 iD. for rhe impac! cruhitrg
The work indcx valuesat tl. dcsignat.d 80 pc! cerr !ass-
ing product size P a l fced sizc F arc found from ihe
plot and dcsiguated as tri and ,t/i. Thctr thc work irpul
ry is fouDd from F,quation (42) as givon below:
_. lO W t0 wi, , 10wit - wt,,
*-@- -J--
-
....tort
syEb.F rn E{ur.ion No.t
x"
,6
,d
,N
k
,,
R.d nil nr drutuar rerd.
Brll birr i' rdBdot f.cror.
rlrim.udr4 116 in e, h/{b.
r$oo.nri:r $E dBdbnuon ad6ddL
Pd sr qmr.rN lordroo.
Middr 30E ar otdol16frr. b!q.
rmr h.k iddh<n/c.
!ruliono|diltc!i.i.d!Dej.
O@ sid. corhr 3.triu b D.h6.
Ps dr *.irhr olcoDHhk D fd_
P.r ar Furi,r b .r!.16s 6no ,rcd(r
cyrA ro ohur.in fG vdial ofirrrion.
P4 ed ol dncnu.. !ain! u P,
Ex'ffilj'|olqDccrrtio'
E4uj'd.oltEofrlBd|ed.
Pq 60r ol-!.r fe! puiq rir,.
l-d c9nJ.peiq j' Eiu r* r..d
lan nijr sindrbjliry a-c, &. ar8/
R.d oi'r gild.bir y A-c! @ s@s/
arindi4mni..j.mfutrnt.
s.buhDrm miftd 100d q. ole.
I4idt o, bin uEnd in fct.
!q !.'t' P!.ti.. b @l]l dn:hq.,
9e hu bilitr€d b En.
!i.E oFlt8 i! E 6od-A,{ sidlb0rry,
Rdlton rdr tdra 6!n6trt Dd
F_61 or o6i:.JlBi4 rizo P.
9t -!Ero
r.d EjI nihdid nd.,
!u.!. |40 4r 6/ttu,
l.ll r m! dbs.cdo! ouurxv.
& 6 pg arp. w|@.udre
Pa @ p|!n$_b cl.t i6a co!ru DtbdEr,
FEdd of Dix $rltd b.ro' d&cbrs.
wo'k iutsr h trvb F rron @!.
wd( rndd: lwlhd
worr brl: twnhd o Eldd.i:a
P.i 6r @d.ds F.lr.n d uv riz..
6rots or li:. dndb!!i@ I&
wort bcd r6E DIo. r Dodud riz
".worr. I!d.: r@ iro' ri fd dF F,
nr.
n.*. ts
iq,
", "
'b $r: erF rh ds6 d .F
b. que r F4u.u@ (a, o3), (3e, Dd raa)re bor d4r'yh by .b. ruuoi
TABLE llr_^ h.do! I'd'r Td RTIB
.T',*ji l'}l'"#*,
,to"
^
of l in' y I j! ,rnicl.' Tb. Eod
l3lo,?s
i
rirr"r rr,zso-
- 13_

24. is coarsc,tbe loss in mill capacity can be quitc largc; ir
decrcasesas thE proporlion of the totll work consumedin
fine srindingiocrcascs.The decreaseir capacitycaus€dby
ove.sizegriodinemediais not aspronounc.d ss that caused
When Fa represeotsthe maximum 80 per cent passirs
feed sizein rnidols which doesnot appreciablydeqease
tbc grirdiDg efrcicncy (is not oversiE) with the ball or rod
sizecalculatedfrom Equation (12) or (13), and thc work
index is 13, Fo is about 4000for ordidary ba mills and
30,000for rod mills.
porballmitbro = a0o0y'057fr .,.. G0)
ForrodmillsFo:30,m0 /G)m ..,.(3t)
The work input 7 for oversizc feed (F > Fo) is calculated
,,, lgw _ gwtl IR .(wi_rt6_Fofof
" :
L,/F
^/FJ l------------o-- )
.. .,(32)
Grind Difrerential
The grind differentialCd eaaluatcsthe difcrdce iD the
particle sizesof tbe conceDtrate and lailing when grindiDg
for min€ralunlockingand concenbationby nobtion or by
gravity. Ao increased srind diferential can be of major
importancein suchgrindins circuits.
wlere P is the 80 per cen!passiDssizeiD micronsof the
fecdto concentration,Cp js tbe 80 pcr cent passingsiz€of
lhe concenrare,and C19is the per centweighr of the con-
ceatrate divided by thc fc.d, tb. grind difierentialu is
P (lJo - Cw)- 50 Cp
.... (31)
P(50-C,)+50CP
A iypical copper flolation plant has a eritrd differeDtial
of 1.34.ADy chaDsein tle srindins circuit which would in-
creasc tle sdnd difier.ntial would result in rclalively
coaner sdoding of tbe 8atr8ue,aDdshould favourably a6ect
costs,recovelyand grade.ll
Wbeo makiog ttro griod! ro 80 per ceol passinga eiv€n
product siz€ P, witl tbe first grind difierential cdr lals€r
thaDtb€ lecond Srind differ.ntial Gd,, tbe relative mcchatri-
cal efrciencyof the first grjDdto the secoDdis:
t------;;,----;: ;--
Rcr.Etr-J,-(#)-(Fl . tt4)
ff the recessionfactor Rl is the numbq oI slardard y'z
scre€nscalespacesbetweensizeP and sizeCp, tlten
Rt = (losP - loscdllos/2
nt
P:cPQ)1
Lhesamelirear or logaritlmic Sraph sheet.The point wb.re
ihe two lioes crolc is the parting sizc, and th. per cent
passingat the crossiDApoint is the per centemciencyof the
Percentage Ci.culating Load
ID closed{ilcuit reduction the per ccDr cicularira lord
(100 c/) is 10Otimes the latio of the wcighl of separaror
oveftize returning to the reductionmachinero rhe weight
of tbe new f€ed €Dtering rhe circuit io the same time
P : microDs 80 pcr ceDtof scparator undcrsizc passes.
Fp : per ccnt of new fced passingsizc P.
Dp : pe! centof machincdischarscpassinssiz! p.
Rp = p.r centof scpararorovcrsizepassinssizeP.
when the new feed eDtersthc reduction machine
CI=180-Dp)l(Dp-Rp)
rflLen the trewfeedeDterstheseparator
ct=(8o-Fdl(,p_Rpt
Tbe qDaDritiesr€quiredia Eauations(3, and (38)cau!c
determined from log-1og o. exponential plo& of completc
size distribution analyses. However, the s$een analyses
availablemay not permit det.rmioationof rhesequantiries,
and C/ must be calculatedfrom the old equationltrbas.d
on the percentagespassing 2-00mesh or any otner availablc
If . represents the per cenr of the classifier fine producr
passing200 m$b, r is tle p.r cenr of the classifiercoarse
product passiDgtbe samesize,and u is the pcr crnt of ttc
mill discharg. passioglbe samesize,thctr 1,hcn the ncw fc€d
enterstbe srinding milt
CI=(c-'1:.)l@-t) .. . . (39)
when thc new fecd €oterNtbc classifier,andt rep.escnts
aheper cent of the new feed palsing 200 mesh, then
cr: (c_ l(n _ t) .. . . (40)
Aoy othersuitablescrcensizemaybc substitutcal.How,
ever.tbe C/ valuescolcularedftom ditrerenrscr€.osizes
usuallyshowa widc vadatioD,and the ma3hsizeat which
thccirculatiDsloadiscalculatedshouldbespecified.
Per Cent new Clersille. Feedto Closed-<lrcuit Mill
Whena rod rnill iD opencircuit discbarscsto a classificr
in closedcncuirsith a ball mill. tb. DerceDtof th. rod
mill dischars€whichentcrstheball mifas lcalDednewfeld
,- loo orrod mindischarsell%En , tm % Eff.l
'*-L passinspaningsL" I L lm
-r - 100cr I
.... (41)
Closed.circuit Versus ODen-circuit Ball Mill
Grinding
The listedwork index valuesapplyto bau rni s Sridditrg
wet in closedcircuit. For dry gritrditrgi! closedcircuit, th.
rvorkinputtYshouldbcmuiiiplicdby 1.30.
Th€ conversion'toopen,circuitgdnding,cirberwct or
dry, is don€ by multiplyitrs the closBd-circuir$ork input
Irl by an open-cncuitmultiplicationfactor.In tbis comec-
liofl ihe circulatingload Cl shouldbeuniry or greatcr.
This factorvarieswitb the rcferencepcr cedtpassing,
or the per cent pasibSar whicbthe opcd-atrdcloscd-
ci-cuitgrindidgarecompared.For iostaoce,if thereferelce
ler centpassingis95,1hemultiplicationfactorwill b. Eucb
targerthan at 80 p€r centpalsing.If it is rcquir€d thar 95
percentof theproductp:sses200meshthe chanaefrom
clos.dcircuitto opencircuirwill requirea muchgreaicr
incrcasejn powertbanjf it isrequiredthat80percentpass
lO0incsh,or anyothersDecifiedmesbsjz€.
The sriDd ditr rential Gd caD b. calculated from tbe
r€c€ssionfactor Rl and the per cent weisht of tbe coocen-
trateCw by substitutinstbe value of P from Equation (36)
ioto Equation (33).
Tbc Schbmann slope
'
of tbeheavierconc.Dtrateis com-
monly gr€aicrthanthat of the lightertailings.
Classifi€r Pe.formance
Io closed-circuitr.duction th€ eficieocy and the separan
itrg size,cut point or partingsizeof the classifieror sdeen
are impo(ant. Tte efficiencyis commonty expressedas
100minus the per centof fiDishedmaterial or "uDders"in
lbe oversia return€dto the mill or crusber,aL a cqraro
Thc parlinS sizeis definedas tbe sizeat wbich the per
cent"urders'' in the separato.oversizeequak the per cenr
"overs" itr the separatofundersize.The separalorundei-
sizepe. cent passingiand the separaroroversizeper cent
cumulativeretainedon, are plotted as smooth cu.ves on
-.,2,

25. -
br T9E
of Mrbrtrl'
cluijon $ouid b. ucd i.
"Dt
it't .h. .wrry @rr hdd vd!.! L,Ed tuE ro ,!*r6c idrrulioE, |hs d! indMdut vdliod bd*& 6{cld! i! rry
r19 2.69
r6,30
9 r.76
27 2,66 r0.r3
5 2,5.
1 2.1r
16.16
7 2.73
9 3.00
1 2.15
rJ.r3
19 2.36
id
z's4
P. ltrnEr. j.r,rD!.rr di aulb.t.runtrktnd4 Bzttin.
!Krn, FrIMtd.
' Bo^D. F. a. lllni, Th.ory ol Caanian. oa', /.t u r Itud., tetl,
tlrt 4a1t Mhin. EBtLnl^t, }1[1y 1952.
'Bom, F. c. 'ThG PriDcirls ol corhirution ,
E 'l'inqer4 of CoDmiDurid I , A.IM.F. T..l'.
'8.!D, F. C. 'ConFE.rim o' rh. Thi'd Tn.ory .
'BoND, l. C. 'sundnd G.irdrbil
sie s.l€.lo!! Minins En.in..ri4. Mry
r9s3". ,r.r.M.E. T,@,. !953. tll. 592.
! BoRrM, B. B. "Mcr$ri!3 rh. RdaliE Abrasiv.n.s , Pr .Id Oufr,
Ind Gftdidg cr,c,rhioon. cdadian vnht
107,r66tz ttn'. t9s1,
"DoNo. F. c. M!rh.6rh! or C(fq rnd
'I^@tI Edttboo ot Mhtd D4rnir. se, 6, r. i6 ('r.y)
^sTM
' T^ao^Nt. EnAb..k ot Mk?o, Dr.sr", s.c. 19, F. r02 @il.y)
- 14-