Ball, Tube and Rod Mills, Rose and Sullivan, Publication year 1958

Ι.Ε.J ΟΊ._. Η.

Μ.

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Κ Sυι.ι.ινΛΝ

Ί'Ιιίs l!ook tlπo\o\' S nι>w liglιt on the internal ιηeclHωics of mills. lt covers among others thc following aspects: Ball, tube, rod, ρebl>le, IHιtclι, grate-discharge, trunnion-overfiow, lιarc]inage and airswept mills; open- and closed-circuit grinding; dry and wet milling; contamination of products. Single ball in mill; cascading; cataracting; equilibrium surface of charge, etc. The power to drive a mill. Perfect crystals; heterogeneous boclies; crushing a single particle; scale effects; effects of impact velocity. Dirnensional analysis; rate of g1·inding. Effect of pulp consistency ση preferential grinding; influence of shape of grinding media; size distribution of grinding media: rate of flow through continuous mills. Surging; lifters; vib!'ation frequency spectrum: clestruction of geal'ing. Rose 's tlιeory of metal \vear; factors affecting ball wear; influence of coπosion. The l'ole of aclditives in milling.

Α

TR EA TI SE ΟΝ ΤΗΕ IN TE RN AL ME CH AN IC S OF

BALL, TU BE AN D R O D MILLS by Η. Ε. RO SE D.Sc .(Eng .) , Ph.D . AND

R.

Μ. Ε.

SU LLI VA N

B.Sc.(Eng .)

~ Ι'Π

1 9 5 8 CHE MIC AL PUB LIS HIN G CO., INC . 212 Fijt h Ave nue New York , Ν. Υ.

FOREWORD by PROFEssoR DR .-ING. RoBERT MELDAU

First published • 1958 Chemical Publisl1in g· Co., lnc. Νe1-ν York

Ι ι ι wi ιl1. onsidera ble pleasure that Ι write this Foreword Ι ιι 1 l ιoo k , written by Dr. Rose, d evoted to the analysis of Ι l ι1 i ι1((• ι· n al p rocesses of the family of mills which h ave

Ν . Υ.

I ι ι ι • l ι η ν ί ηg grinding media. I ) ι , I ose is well-known to the F achgruppe "Staubιι ι I Ιι ι i l<" of the V erein D eutsch er Inge nieure, for his lι ι Ιιι ι ·ι ·χ in Germ a ny on his researches into milling and ΙιιlιΙ 111 ι· gen eral properties of p owdered m a terials and ΙιιιΙι Ι ιι1 y knowled ge of him and of the researches which are Ιιι 111 • ι· ;ι ιτ ί d out under his direction, Ι a m sure tha t Dr. I ιι Ι ' : 1 ιιd Mr. Sulliva n are well equipped to write on the ιι l ιjι · ι · ι .

'I'IH· η d for a book d evoted to such an analysis is 1 \ ιι l ι · ιιΙ wl1en the great number of " tumbling" mills used 11 ΙΙΙ ι lιιs trΎ, their low efficien cy a nd the great resista nce Ιιiι · Ι ι 111 y have offered to a ny substantial improvement, Ι 1ι Ι II Ι·< l ιο mind. Thus, this book tackles a problem to • Ι ιιι Ι ι ιη u h thou ght has b een d evoted during at least a 1ι ιι l ιιιγ :ι nd to the solution ofwhich every advance will b e Ι ι Iι 1 ι ιι )(• . I ) ι'. I s uses, with convincing r esults, the m ethod of ι l ΙΙΙΙ ' Ι Ι Ν i ο n al analysis as a preferred approach to his problι ιι ιr :ι ιιd it a ppears tha t the statisti cal b eh aviour of fine ι 1 ιι l ι · Ι ι •Η Ι nds itself to su ch treatment, to a rema rka ble ι

Ι ι• ιιΙ .

Ί 'ο Μ Ι

P·rirι ted in t lι e Urι it e
',

k in milling terms; may the mill of Dr. Rose's

ιlι ιι ι ψ; I ΙΙ S yield ever finer products for the use of those 5

6

BALL, TUBE AND ROD MILLS

..

interested in this important subject! Ι believe th.a~ t~s book will provide improved theory and als
PREFACE

ROBERT MELDAU PoL YTECHNISCHE HocπscHULE, HANNOVER.

\ w i ιlι Ll1.c previous volumes in the same series, the object ι ι / ιlι ι : ι>r sent work is to bring together the data relating Ιιι ο n ιι· part ofthe field ofscience dealing with powdered 11 1 ι\ Ι ' Ι' i : ιl s , in this case to the internal mechanics ofthe ball, I Ι ΙI ΙΙ ι ΙΙ ι r d mills, to analyse these data in the light of the \Ι Ι Ι iι ο Ι'Η ' xperience and researches and, as far as is posΙ i ι i Ι · w itlι the present state ofknowledge ofthe subject, to ι ι ι. 1 111 ll1 I esults in a systematized form. ΛΙ Ιι · 1ti η has been given to the ball, tube and rod mill / ι ι 1 , ι ΙΙ ΙΗ : , <ιlthough the results of numerous tests on these ΙΙΙ Ι ΙΙ lι i ΙH 'R have been published, it is believed that no cοm Ι ΙΙΙ / 11 1ι Μ ίν · analysis has yet been made. Thus, fol" example, ΙIΙ IΙ ΙΙΙ Ι f.\' 11 many investigations have been carried out to Iι ιι~ν Ι lι • influence of the various parameters. defining the ι / Ι ι ι ι , ι ι · ι ristics of the mill and the charge, as far as is known Ι l ιι 1Ι ' Ιι :ιΗ previously been offered no formula for the calcuΙ ι ι Ι Ιιι ιΙ ο Ι' ιh instantaneous rate of production of specific 11ι I ΙΙΙ ' Ι 'ι· m a knowledge ofthese parameters. ln Chapter ι , lι ιι wι: νι: r, such a formula is provisionally presented. IΙ i Ν ΙΙ ι:{g sted that the formula is, at present, subject to 111 I ι ι Ιlι' f, say, ±50 % , and SO is to be regarded as giνing 1 /Ι Ι Ι Ι ψρroximation only to the desired result. Neverthel ι , i Ι iN /) lieved that the reduction of these data to such 1 I Ιι ΙΙΙιιιl :ι is a considerable advance along the road to ι l ιι ιιι κ tlι subject on a firm foundation and that the ι iiΙΙΙ Η ίπ of details, with consequent improvement in Ι ι ι ιιι . ιι : y, is largely a matter of well-conducted experi1111 11 1, Ι •: v ι though the highest accuracy is lacking in the ιι ι ι.ιll /ί) nη ul a , the work is believed to be of some ι ι ι ιι Ι ι ;ι I νalu e since the influence of nearly every variable 1.1 Iιι · ι • ιι ;ι n alysed in the light of the available data. Thus, l 11 ι 1 κ: ιιιφl , the mill operator will find information as to 7

ι

I

8

LS BAL L, TUB E J\ND ROD MIL

ball s~ze upo n t~e the prob able effect of vaήation of, say, umptιon o.f a ~11 cons er grin ding char acte risti cs and pow ~e, the d1rectω? iden ~onf e som and so h e may pred ict, with m1ll para met ers IS in whic h a cont emp lated chan ge 1n the . . likely to alte r the mill perf orm ance of Cs hani mec rnal inte the to n The r easons for restr ictio g: the mill are seve ral; these reasons bein ral desi gn and (1) Seve ral books deal ing with the gene tence, e.g. oper atio n of the tum blin g mill are in exis rung. Her r Carl Mitt ag's Hartzerkleine with the inte rnal (2) As far as is know n no book deal ing . lable processes in a mill is avai of design and oper a(3) Το have trea ted the prob lems pres ent field, wou ld the to tion addi in tion of mills, the Auth ors, have defe ated one of the objects of at mod erat e me volu ll whic h was to prod uce a sma ~tur e on the liter the i~ .gap a pric e whic h wou ld fill e mor e on wnt to n ntω 1nte the not was subj ect. It uate ly adeq ady alre part s of the field whic h are cove red. chos en since The title "Int erna l Mech~nics" has been ~ithin, the ing inat orig or in, rring only phen ome na occu 1ls such as deta s, shell of the mill have been trea ted. Thu have not ings bear and classifiers, feed m echanisms, gear ing been has ing gear of n kdow brea been cons ider ed. The since , ever how ", ging "sur with ion unct cons ider ed in conj ofte n arise from it is the Auth ors' belie f that gear failures on (surg ing) llati osci the by vibr atio ns whic h are initi ated . shell the in of' the mill char ge with a .mill has been Sim ilarl y, the pow er requ ired to driv e IS clea rly conand stud ied in d etail since the pow er dem in the mill with ge char the of n troll ed by the co'nfiguratio . mics dyna rnal inte of and is so a mat ter intro duct ion of ~ sh~rt reΝ ο apol ogy is offe red for the of comr.rnnutω~ of view of the pres ent state of know ledg e operatωns tak1ng the of e ledg solid bodies since a know

9 ied appl y tivel jΙΙ ι ιι ι • wi tl1i11 the mill cann ot be mos t effec ess for whic h proc e ofth e ledg know und kgro ΙΙiιοιιΙ : ι b Ιlιι 111 11 iR r quired. revi ewin g the • ιιιιίl;ιι:Ι y,. t~e intr~d.ucti<;>n of a chap ter grou nds that the on fied justi Is tlves addl ιιl1ιι 1 Ι ο / mill ing ify the beha viou r of a t Iι ι ~~ ιιl Hl< nces can grea tly mod so, even thou gh it is and "' 111 ι i,ι~ .turi ng com min utio n actio n is phys ical, this her ~het 11 ιι1 ιlι · Ιιιιιι ly kno~n tted from a book ι I 1 ιιι ι : ιl J' Ιηechanical, It cann ot be omi atio ns ofth e oper rnal Ιιιι Ιι /ιιιι·ρ rts to deal with the inte mill. ' ( ι. ιι ί Ιιιιιιlι give n to man y 1 •, \' Ι ι ι .ι !1 ugh final answers .can not y~t be n and appl icati on ιι l Ιlιι Ι )I ο / 1 ms enco untere~ lll. t~e d es1g that usef ullin es ed hop IS ιιl Ιlιι lι : ι/1 , ιube and rod m1ll, It r eade r. ιιl Ι ιι ι ι ιΙ Η·η ι will be sugg este d to the wor k is not a teχt this that ized real lly \\!Ι ι i I I i ι is fu pted theo ries lιιιιΙI ο iιι ι lι · s nse that wor ked out and acce in whi ch is ual man gn '" ι ιιι •Νι ' Ι1l cl, nor is it a desi ul data for usef and ulae form of Ι ;ι ι 111 Ι ι ·ι I a ollec tion ofin tere st be ι/ι Iι ιιι• ι· :ι ιid oper ator , it is hop ed that it will ator , to the oper the ιιιιl \' ιlιιι· t .the rnill designer, to read er with an ι Ιιι 11111 ι/ •ng1neer and to the gene ral ing. neer engi and "'',' ιι ι iιι mcc ha?ics ness to all the I Ιιι ιι Ιl1ors w1sh to express thei r inde bted in part icul ar and , ιιιΙiιιΙΙ w lι sc works have been cited aney whose Dev and hill Cog . Ι• 1 ~ /ι .Jol111 ross and M essrs to the U.S. ιιιl Iι . ι :ν Ό .been quo ted at grea t leng th, on of Min ituti Ι\ ι 111 11 1 111 Mιn es and to the Ame rica n Inst nission to reperr for rs, '''1' ΙιΗI Μ ιal~urgical Eng inee ous publ icati ons 1"" '''''' ι·χt ns1ve data from the vari 1111 ΙI(IΙIIII '( / jn the text. ture rs who have I /ι ιιιlοι <ιr also due to all the man ufac r nam es bein g the! , d~ta ~ "Ι'Ι'Iιι ·ι l i.ll.ust.ratio ns and ot~er IllustratiOllS for thei r ate ropn app ΙιιιlιΙ/ ι l r,I.Jf1St the e the mate rial 1 -Ν Ι Ι ΙΙΙt ·ι · nnd perm issio n to repr oduc repr oduc e Figs to "1111''''' ' · Ιο ιhe Ι.Μ.Ε. for perm issio n reas en for per~ And r esso Prof I ο Ί I, :ι. Ι Ο and 3.11, to J. Μ. Con nor Dr. to and 4.11 Fig. 1111 '"" Ι ο ι· prod uce PRE FAC E

1

BALL, TUBE AND ROD MILLS 10 and the University ofLondon for permission to reproduce Figs. 2.4 and 2.5. Finally, our best thanks are due to Mr.J. F. Hinsley and Monsieur Ρ. Fobelets, who have kindly made available the results of their researches which have extended over many years and upon whose mathematical analyses much of the treatment of the motion and configuration of the mill charge is based, for their personal assistance and advice. Finally, thanks are due to all whose writings have been consulted and to all who have contributed to the work by advice and criticism. H.E.R. PowDER SciENCE LABORATORY R.M.E.S. KING's CoLLEGE, LoNDON

CONTENTS Ι I ΙΙ

I,

1 1 '1ΈR

PAGE

Ι NTRODUCTION

17 of milliηg;

Ball mill; Ι )ι• finitioη of milliηg; Objects I ιΙIΗ· Jnill ; Rod mill; Pebble mill; Batch mills; Grate ιl1 1 ι lι:ι ι ·ge mills; Τruηηiοη σverflσw mills; Hardiηge 1111 11 ; Λ i r-swept mill ; Opeη- aηd clσsed-circuit grindiηg; I 11 y ω ill iηg ; Wet milling; Coηtaminatiση of prσducts '1

Ί ' ΗΕ ΜΟΤΙΟΝ

OF

ΤΗΕ

MILL CHARGE

35

iη mill; Mass σf balls; Iηter-relatiσηship Ι wι·ι· η cσefficieηt of frictiση aηd ηumber σf balls ; < : ι 1 ί ιt li ng; C ataractiηg; Equilibrium surface σf charge; II ιll Ι Ι':ιj ctories; Theory σf Davis; Iηterfereηce betweeη I ι ιll ί ιι t rajectory ; Theσry of νση Steiger; Magnitude

, 'i ΙΙ ).t l b all

I"

ιιl

" IIyί ng" ch a rge ΊΊ ΙΕ

aηd "rσlliηg"

POWER

ΤΟ

charge.

DRIVE

Α

MILL

69

I•' ιι: Ι.οι·s upση which pσwer deρeηds; Difficulties ίη 1\' 111 Ι ι:sts ση large mills; Tests ση small-scale mσdels · Ι lιΙ ιιι vι·stigatiση by Rose aηd Evaηs; The iηvestigatio~ ampbell aηd Coghill; The equation ofDavis · ιιl <:1ιw, I ψιιιΙiιι ιι σf Βοηd; Equatioη ofHaηcock; The effect of 1ιιιl1' 1 l 1 : ι ι·. teristics; Extensioη to a rσd mill· Method of ' ι ' 11 1 • ι ιιιl tairmaηd <>Ν Ι Ι<

ΤΗ Ε

COMMINUTION OF SOLID 109

)I) [ES

I lιι 1 1 : ιιω·e of sσlid bodies; Perfect crystals; Mosaic Ι 1l ; I fc ιerogeηeous bσdies; Discussiση of cοm ιιιiΙΙ ΙΙI ΙΙΙ ΙΙ ; Distributioη of eηergy ίη a body · duriηg ' ωshiηg a siηgle particle; Crushiηg iη a I 111 Iι II ι κ ; ,ι, Ι ' Ι ι ·ιl"; Rittiηger's Law; Kick's Law; Boηd's

ι ι

11

CONT ENTS

BALL , TUBE AND ROD MILLS

12

ΙΨΙ •: ΝD ΙΧ

l theory ; The_ory of Carey and Stairm and; Geom etrica t impac of s Effect ; effects Scale ng; crushi in rity simi1a velocity

5.

PROC ESS OF GRIN DING WITH IN MILL

ΤΗΕ

' Ι 'ιι'bles of Mill Filling Param eters

140

OTH ER CON SIDE RAT IONS RELA TING Α MILL ΤΟ ΤΗΕ CHA RAC TERI STIC S OF

178

of Prefer ential grindi ng in rod and ball mills; Effect ball of Effect ng; grindi ential prefer η pulp consis tency ο e hardn ess ο η the rate of grindi ng; Metal wear and relativ of shape of nce Influe les; partic and hardn ess of balls ; grindi ng media ; Size distrib ution of grindi ng media s; Lifter ; media ng grindi of size of D avis distrib ution R ate of flow throug h contin uous mills

7.

SUR GING AND VIBR ATIO N

ΙΝ Α

MILL 203

Surgi ng; Tests of Hinsle y; Criterion of R ose and the B1unt ; Impo rtance of the coefficient of friction of cDestru um; spectr ncy freque tion charg e; Lifter s; Vibra tion of gearin g

8.

ΜΕΤAL

WEA R

ΙΝ

221

MILL S

s; Rose's theory of metal wear; Obser vation of Engel wear; ball ing affect s Factor es; surfac of uring Armo Ana1ysis of the variou s factor s; Theor y of Hukk i; Influence of corros ion

9.

ΤΗΕ

ROL E OF ADD ITIV ES

ΙΝ

252

Α

Facto rs upon which grindi ng depen ds; Treat ment by dimen sional analys is; The influe nce of the differe nt paraof meter s on the rate of grindi ng; Equat ion for rate es surfac mill of " grindi ng; 'Άrmouring

6.

13

MILL ING

Defin ition of additi ves; Funct ion of additi ves; Wet ive; millin g; Dry millin g; Dispe rsing agents effect es ; Theories of additi ves ; Clean ing of the mill surfac ucing introd of anger D n; frictio of cient Effect on coeffi surgin g; Polar and non-p olar additi ves

236

ιι Ι iψ ; ι Index r~ 1111 • Index

253 257

LIST OF SYMBOLS /ι

I Iι

"'

I ι

11

I )i:ιn ter of particle l )in ιn ter ofball (or rod) I >i ιιn ter of mill shell Ι\ . ι Νι' of naturallogarithms ( : ι~e · !Ii ·ient of restitution Ι •: ιιι · r·~y necessary to change the surface of p owder by ΙΙιιίι amount Ι •: ιΗ · ι 'f.\Ύ necessary to change the surface of m etal ball by 1111 ί ι amount ( : 11ι · Ηi ·.ient offriction \ι tΨ I Γation due to gravity I I Μι lι ss ο[ material or Height of lifter of ball-:- H ardness ο[ I ( ι Ι ~ι ι ive hardness (Hardness Ι ΙΙ ιι · ι ί le) ll ιιι · izo ntal displacement of centre of gravity of mill ι Ιι : ι ι ·ge J ι' ι ιι · ti o nal

ball filling of mill lume of balls + Volume of space between balls Volume of mill \ Ι I · ΙΙ·;ιmeter or Specific rate of grinding Υ

\ 1ι : ΙΙ ' nme ter l , ι ΙΙ )..\ 111

fmill

\ ι i ι.ι lιι [ product per revolution : JΙΙ Ψt l ο[ rotation of shell in r.p.m. or Number of lifters ψι · ι• ι l

Γ

rotation of shell speed of rotation l 'ιιw ι • ι · ι drive the mill ι IΙΙ . ι ΙΙt iιy ofballs in ftight 1\ l ι ll Ι l 1ι·o u g hput (mass per unit time) Ι '" Iί ιι χ of element of charge from centre of rotation Ι ΙtliΙΙ Η ofmill shell or R eduction ratio ψι Ψ ίfi · surface (Surface per unit volume)

t : 11 ι iι · n l

,,

15

16

α

β γ

δ Δ ε

η

θ λ μ ν

ξ ρ

σ

φ ψ ω


Total surface Time of grindin g Time of revolut ion of mill Time for elemen t of charge to .traverse circula r path Time for elemen t of charge to traverse parabo lic path Diame ter of exit from mill Pulp consistency (volume of solid to volume of liq1.1id) Velocity of projection of an elemen t of charge or Fractional filling of solid _ Volum e of powder + Volum e of space betwee n particles Volum e ofspac e betwee n the balls Wear of a single ball Total wear of balls Ratio of circula ting load to totallo ad Angle of displacement of centre όf gravity or Angle of project ion Parame tric angle Angle of repose of charge Angle betwee n rays of the equian gular spiral or Mean density of the mill charge Semi-angle subtend ed by the charge at the centre of the mill Parame tric angle Angula r velocity of mill shell (radfsec) Angle of displacement of centre of gravity of mill charge from the vertica l Coefficient of friction of charge Absolute viscosity Kinem atic viscosity Weight ofball charge + Weight of powder charge Ratio: W eight of ball charge Density of grindin g media or Radius of curvatu re of ball path Density of powder Denotes "a function of" Α parame ter Angula r velocity of precession of the mill charge or element of charge

CHAPT ER

Ι

INTR ODU CTIO N l r Ι ΙΙ ι Ι Ιι y industr ies the final produc t, or the raw materi al 1ι 11 11 ιι · s ι~ge of the manuf acturin g process , is in powde red lιιΙΙ Ι Ι ,111 l ω conse~uen.ce the rapid and ~heap prepar ation 1,ι ι ιι ΙW< I r d matena ls 1s a matter of cons1d erable econom ic . ι ι; 1/111 1ι : 111 prepar ed 11 ο ιη cases the powde red materi al may be n a solutio from itation ι I ι 11 ι ly; fl r examp le by precip of type~ certain of ation prepar the in j Ι ΙΙ Ιι Ι ' Η wl1i h is used fine a of drying vacuum the by or dru~s, and :~ Ιιl ' jΙ ψ Ι ΙΙΙ 111 I ο Ι ι h e ma.ten al, a. process which is widely adopte d s l1ιι ι ιι : ι > ~ · 1 arat10n ofm1lk powde r, soluble coffee extract of er howev are es process Such ts. produc ι1ιι l ιιιιιl a r I ι ιι ΙΙIΙ 'ι l . : ιpplica?ili~y and in by far the grea~est numb; r of l ι ΙΙ I ΙΙ i rr: ΙI pplicati~ns t?e powde red materi als are pre1ι 1111l Iιy Lhe reduct1 0n, 1n some form of mill, of the grain 1 ι ι / Ι Ι ι .' m a terial having an initial size larger than that ι ι ΙjΙΙΙ Ι ι · ι l rn tlι e final produc t. These process cs for the reducΙΙιΙ Ι Ι ιι l Ιlιι~ 1 article size of a granul ar materi al are known Ί ιι i 11 i ιι_g" or "grind ing" and it appear s that these names Ι ι ι • ι l ιrιtcrchangeably, there being no accept ed tech111 ιΙΙΙ ιl Ι l i ll (· r· ntiatio n betwce n the two. I Ιι ι ι ·ι ·ηs ns for the grindin g of industr ial materi als are ιιΙΙ Ιιιι ι ι ιι1 s 'b ut the princip al reasons may be summa rized ιι ιιι / 1 ι Ι IΗ· Γ llowing headin gs : I ) Ί'Ιι ι· μberation of an econom ically import ant ΙιΙ : ι ι 'n al from the unrequ ired constit uents of a ΙΙ Ι ί lure. ") Ί'Ι1 · xposur e of a large surface per unit mass of ιι ι: ιι •r·ial in order to facilita te some ch emical process . 17

BAL L, TUB E AND ROD MILL S

18

redu ce the mate rial to the desired form of the final prod uct. (4) Το satisfy mark et requ irem ents.

(3)

Το

ral dressing, Exam ples of the first two classes occu r in mine desired ore the ate in whic h size redu ction is used to liber a form in to ore the ce redu to from the gang ue and also reage nts. ing leach the to ce surfa large a whic h it prese nts cinal medi y man U nder the third head ing may be classed in, izers fertil tuffs, and phar mace utica l prod ucts, foods size the falls ing head h fourt the r secticides, etc., and unde often being redu ction of mine ral ores, etc.; these mate rials hand ling, in redu ced to parti cles of mod erate size for ease ofshi ps. holds the into stori ng and loadi ng into trucks and prosuch to cted subje be to er The quan tity of powd the to rding acco ly wide s varie ction redu cesses of size l utica mace phar indu stries invol ved, for exam ple in the be can . m annu per lved indu stries the quan titite s invo of certa in meas ured in term s of a few tons, or in the case the ceme nt drugs, possibly a few poun ds; wher eas in of millions tens into indu stry the quan tities invo lved run prod uced , ng havi alone stry of tons; the British ceme nt indu ent in Cem and Portl of tons on milli in roun d figures, 12t 1955. man y For the prep arati on ofsm all quan titite s ofpo wder freis mill ball the so, even but, types of mill are avail able of tities quan st large the of ing grind the quen tly used. For st almo used is mate rial howe ver, the ball, tube or rod mill h whic mill of types exclusively, since these are the only . itude magn ired requ the of possess throu ghpu t capa city ns, and Thus , it is the extre mely wide rang e of appli catio conh whic mills the grea t indu stria l impo rtanc e, of the justi the s form h whic ly stitu te the "tum bling mill" fami . ficat ion for the prese nt work l ball mills The grea t rang e of sizes cove red by indu stria the first ln 1.2. is well exemplified by Fig. 1.1 and Fig. t 1-litre abou of mill h batc y illus tratio n is show n a labo rator in the used mill tube a n show is 1.2 capa city, whilst in Fig.

Fιο.

J.J Labor atory Batch M iJJ

Fιο.

1.2 I" aΙ"ge lndus trial Tube Mill

(Pascal Engineering Co. Ltd. )

(Ειfιιοr Alϊen & Co. Ltd.)

BALL, TUBE AND ROD MILLS 20 cement industry the tube having a diameter of about 8 ft and 1ength of about 45 ft. In Fig. 1.3 is shown a 1arge ball mill, designed for the dry. g~inding of ~imesto~e, do1omite, quartz, refractory and sιmι1ar m~ten
21

INTRODUCTION

Ι tl1is point it is perhaps of value to study the ι ι ι ιι tH · ιι clature used in connection with the mills under ι ιιι ι Η i I ration, but it must be emphasized that the 1ines of ι l ι ΙΙΙ : ι ι· ation between the types to which the names are

ΨΙΙ i iι~ ι are not very definite. Ί ' Ι ι ι: broadest c1assification

appears to be:

( I ) Ball mills. (' 1 'fub e mi1ls. ('I Rod mills. ( I Ρ bble mills.

j

Ι ' Ιι ι· ιcrm

"ball mill" is usually applied to a mill in

1 ΙΙι Ιι ;t,he gri~ding. media are bodies of spherical form Iι ι ll κ ) and ιη whιch the length of the mill is of the same ι ιιι l ι ι : ι~ tl1e diameter ofthe mill body; in rough figures the Iι 11 ι Ι I ι ι Η, say, one to three times the diameter of the mill. ll tt· " ι~b e mill': is a ~ill in which the grinding bodies ΙΙΙ I Ι i ii'Γ t Cal but lΠ whιch the length of the mill body is

11

ι ι ι Ι ι ι · i η proportion to the diameter than is the case of the Iι ι ll ι 11 i11 "; in fact the length to diameter ratio is often of

ι l ιι ιιιι l ι· ι·

ften to one.

.

Ι Ι ιι Ι 'Ο l mill is a mill in which the grinding bodies are Ι ι Ι ι I ι Ι ' ι· ds instead of balls, and, in order to avoid ι ιι ψ, lιΙΙ Ι-ζ" of the rods, the length to diameter ratio of ι1ι I ιιι i ΙΙ Η is usually within the range ofabout 1·5 to 1 and

ι" Ι

ll 11 I" '' ble mill is the name associated with a mill in Ι ι 1 lι ΙIΗ · ,ι;r.inding bodies are natural pebbles or possibly ι ι ιι ιιι Ι ιο lι s ; the general proportions of the mill being Ι ιι ι ι Ι " lι ; ιll

FIG. 1.3 Industrial Ball Mill

mill".

Ι ι 1 ι 11 IΗ' noticed that the differentiation between ball ι 11 ΙΙ t ι l Ιlι · ιu~ e ~11 arises only from the different length ι l ι ιιι 11 lt' Ι ' ι· atωs ιnvolved, and not from any difference ιι Ι Ιtιι l ι ιιιιι · Ιιlal principles. The rod mill however differs ' ' . Ι ι ι ιι ΙΙ ι ' Ι ' Iι · ιn t hat the grinding bodies are ΓOds instead of Ιιι ι 1 , w lιil ~ ι a pebble mill is a ball mill in which the ι ιι1 Ι ι ψ l ιι ιι li ·s are ofnatural stone or of ceramic material. llι 1 Ι Ι ιt 'Ν [ mill normally encountered can further be

MIL LS BAL L, TUB E AND ROD

22

of feeding the raw classified acco rdin g to the met hod l pro duc t from the fina mat eria l to, and the rem ova l of the mill. The se classes are: Bat ch mills. Gra te disc harg e mills. Tru nni on overflow mills. Air- swe pt mills. stra ted in Fig. 1.4 and The prin cipl es of these mills are illu Fig. 1.5. mills, Fig . 1.4a, the As the nam e implies, i~ the batc h ed into the mill in a load is cha rge of pow der to be gro und ess is com plet ed, is rebatc h and , afte r the grin ding proc mod e of ope rati on can mov ed in a batc h. Cle arly suc h a mod erat e sizes; say to or only be app lied to mills of sma ll abo ut 7 ft long. by eter mills of up to abo ut 7 ft diam tinu ous sup ply and con a mill of s type In the thre e othe r , and these met hod s rem ova l of the pow der is mai ntai ned s. size est are app lica ble to mills of the larg , a diap hra gm in In the grat e disc harg e mill, Fig. 1.4b rge to one end of cha the form of a grat ing confines the ball diap hra gm and the the een betw e spac the the mill and the rem ova l of the fo"r p othe r end of the mill houses a scoo fed in thro ugh a is l eria mat raw gro und mat eria l. The ofth e mill and dur ing holl ow trun nion at the entr ance end afte r whi ch it passes grin ding traverses the ball cha rge; and rem ove d by the up ed thro ugh the grat ing and is pick ugh peri phe ral ports, discharg e scoop or is disc harg ed thro men tion that scoops ar In this con nec tion , it is rele van t to the lite ratu re. In the in sometimes refe rred to as "lif ters " ter" will be confined "lif term present wor k, the use of the of mil llin er con stru cto the desc ript ion of a cert ain form ord er to pro mot e th tion, fitte d with "lif ter bar s" in be desc ribe d in a late r tum blin g of the cha rge, whi ch will section. 1.4c the raw mat eria l In the trun nion overflow mill, Fig. at one end of the mill is fed in thro ugh a holl ow trun nio n (1) (2) (3) (4)

(σ)

Batc h Mill .

ιι

Grσt~ .

ι ι I ΓJΙ)( Γ.

Surt.σc( Ot Powdv.r M~s Surtσc( Ot Bσll Mas~ . Discharge Scooρ.

(b )Grσte Dischσrg~ Mill.

S urtσc e 01 Pow drr Μσs~ SurtσceOtBa/1 Moss. Oνertlow

'~;;~~~~t$,~i~~F,..;;;j~-Trunnion. Trun nion Oνrrflow Mill . (c)

~DustHood.

r- -- ------::.-,0-

Dischσr9e Port s .

I I

I

Disc harg e Hopper. P~riph~ ral Dischargfl' Mill FιG.

1.4

(d)

24

INTRODUC TION


/

25

οι ιΙ1 ι i: ιl ιο maintain the charge at the working level. The I ι 11ι 11 ι Γ such a closed-circu it plant is shown in Fig. 1.5. Ι I ιι • 1-( t'inding in a batch mill, a grate discharge mill or 11 ιι ι ι ι11 ιι ιΙ verflow mill may be p erformed either "wet" or ι l ι " 1 l > ιl t clearly in an "air-swept" mill the grinding 1111 ι l ιι · tιrried out in the dry state. I ι ι " ΙvΙ' l" grinding the material to b e ground is fed to ι ι ι tΙ ιΙ II ίιι the form of a "pulp" containing, p erhaps, 50 % ι 1 l ιl f water. In dry grinding, however, the granular ι ι1ι ι ι , ι l is fed into the mill in a sensibly dry condition; the 11! ΙΙι ιι ί ~ι ιπe present being the few p ercent normally held

fEED HDPPER

\

I

1111 1)Owcler . Both the grate mill and the trunnion overIΙ Ι ι ιιll may be operated in either of two distinct ways, ιιι 1 11 Ι 'l'Sp ectively as ''oρ en-circuit" operation and 1 I ι ι ι I ι · ί r uit" oρeration, and, furthermore , either of Iι 1 ιt ΙΙ ' l l1 ds of operation m ay b e worked either "wet" or

ιlι

",

I ιι

1 ψι · ι1 - ircuit grinding the mill product is not subject ΙΙΙ \ ι l ~ιssifιcation, other than that which occurs witlιin I 1 ΙΙΙΙII I)Υ r eason of selective grinding of the larger parlι , . ΙΙ Η I no part of it is, returned to the mill for further 1 ι Ι ι ιιι • ι ι ι . I n closed-circu it grinding, however, the proltΙι Ι 1 ι l ι Ι κ mill is conveyed to some form of classifier, and lιι ι ιι . ιl ι · ι · ί al coarser than the required m aximum size is ι 111 ι ιι ι ι I ι the mill feed; suffiεient n ew feed material be1 ι ι l ι iΙ · I to maintain the charge within the mill at the ι ιι 1 ιΙ Ι ιι μ Ι vel. For wet grinding a hydraulic classifier is ι ι l 1 Ι ι i Ι s ι for dry grinding an air separator, normally of

ι

ELECTRIC ΕλR

"'IRICONE .. ILL

(Hardinge G_o. In c.) Fιc.

1.5

Layout of a typical closed-circuit plant

d "overflows" at the other end. In an_d the grouhnd pro u~~e grating and d~scharge scoop this case, t ereωre, are eliminated. d. h arge mill is sh own in Fig. Α variant of the grate ιsc . . . . d b th . . h' h the disch ar ge .scoop ιs elιmιnate Υ ι l .4d,_ ~η w ιc . h l d' h arge ports - with a suιtablι~ provιsωn of p enp er a ιsc . ' . "d t" hood at the exit end of the mιll._ d'fli us . ' t mill the body of whιch does not ι cr In an aιr-swep ' ·. f trunnion overflow significantly in general d~sψ~n . ro;ss~d through the mi\1 mill, a powerful stream ~ti~~~sιs rodu ced by the grindin~ and removes _the finer pa owd!r-laden air is then passc
Iι 1

ι Ιι Ηι

type, is necessary. m ethod of operation of a mill eliminl Iι ι ι · n nveyor system which was often used in earlier ι ιι 1ι l ι ) Innt to transport the product to the classifier and 1 ι 111 ι Hi zc material from the classifier back to the mill. ι Iι . ι ι · , lιowever, that such a system can only b e used ι 1 lιιll y when the m aterial undergoing treatment is ΙI Ιι Ι ι ιιιl y light, and the particles produced are sufficiently 111 ι ll , I ΙΙ ι ' ιlιe desired final product to b e entrained in the Ι ι ι ι ΙΙΙl . Furthermor e, the removal of the powder from ι Iι ι " : ι ί ι ·-swept"

26

I.


the air stream, or its utilization whilst suspended in the stream must be practicable and, for these reasons, a typical application of systems of this type is in connection with the grinding of coal for pulverized fuel firing, sincc, in this case, the fuel may be burnt directly in the air stream which carήes it. Within the classes of mills enumerated above there arc a number ofvariations; for example there occur in practicc mills in which the shell is divided into a number of chambers by means of perforated diaphragms and it is arranged that the mean diameter ofthe balls in the various chambers shall decrease towards the discharge end of the mill; such an arrangement being shown in Fig. 1.6. The reason for this distribution of ball size is that, for optimum grinding conditions, the ratio of ball diameter to particle diameter should be approximately constant. In consequence smaller balls should be used for the later stages of the grinding process, where the powder is finer, and by the adoption of a number of chambers in each ofwhich the mean ball diameter is suitably chosen an approximation is made towards the desired constancy in the ratio of the ball size to thc particle size. The problem of the optimum distribution of ball siz within a mill will be dealt with more fully in a latcr chapter, but at this point it is relevant to mention a mi\1 ίη which the segregation ofthe balls is brought about by an ingenious method; especially as the mill carries a distin tive name, even though no principles which place it out· side the classification given previously are involved. The Hardinge mill, Fig. 1. 7, uses spheres as a grinding ' agent l;>ut the body is· of cylindro-conical form and usually has a length to diame"ter ratio intermediate between thos associated with the ball mill and the tube mill. The reason for this form of construction is that it is found that, durin the operation of the mill, the largest balls accumulate a the large end of the cone and the smallest balls at the small end; there b~ing a continuous gradation of size along th cone. If then the raw material is fed in at the large end

Έ
Β

~

ι3

~

Ί::

§ ....

"'

Ρ..

s ο

Υ

l1 ι-<

28

INTRODUCTION


the mill and the ground product. removed at the sm~ll~r end, the powder in its progressιon t~rough the mιll ΙS τound by progressively small balls and ιη conseque~ce thc ~heoretical ideal of a constant ratio between ball sιz~ and article sizc during grinding is, to some exte~t, attaωed. Ρ The type of ball mill illustrated in Fig. 1.3, ιnc.oψorates a peήpheral discharge through fine screens lιωng thc cylindrical part of the mill. Heavy perforated plates pro-

29

ιl1ι 11 Ι ΙΟl xcessive. At this point it will perhaps be useful Ι ι 1\l ι ΙΙ S~ the factors upon which the choice between a ι. Ιll , . ι tιιl c or a rod mill depends. \' I Ιι ' ΙΙ η mill is used as a batch mill, the capacity of the •ιΙιllι ι l ι·:ιrly limited to thc quantity which can be handled 111 ttΙΙΙ ι ι ΙΙ y; furthermore the mill is, as far as useful work is . ·ιΙΙιΙ ttΙ Ι ' Ι , idle during the time required for loading and ιt ιι Iι 1,11 1i " Α' the machine: the load factor thus being ad1I I I .Ι r~ ted. Clearly then, there will be a considerable ιΙΙΙ 111 t lπoughput, a saving in handling costs and imΙ ι 1 ι 1l l o <ιd factor, if the mill operation is made continuous 1 l111lί Ιl f.\' ιl1 e mateΓial into the mill through one trunnion "I ι I Ι\ ί ι IΙ Iι·a wing i t ei ther through the other trunnion or Ι I ΙΙΙΙΙΙ l1 Iischarge ports a t the exi t end of the mill body. ιΙ ΙΙ ΙΙ ', l1 wever, the flow of powder through the mill is 111ι ι ι ι ttl inuous, it is necessary that the mill body is of such Iι ΙI Ι ι lt tiΊa t the powder is in the mill for a time sufficiently Ι 111 IΙtΙ ' 111 grinding to be carried to the required degree Ι Ιιιιι ΙΙΙ'SS . This, in general, demands a mill body of cοη ιιlι ι ι Ι i ι l ι· I ngth, or continuous circulation with a classifier, Ιιtιl 11 iN irι r eased length which gives rise to the "tube

I

Ιlll '

The Hardinge Mill

l ι1 Ι l tt · metallurgical industries very large tonnages have ι lιι IΙ . ιιι \l ed and, furthermore, an excess offine material ιιιιΙiι·s iΙ· ;ιble since it often complicates subsequent treat" Ι• ι ι ι j Ι Ι'Ο csses. Ι η such applications a single-stage tube 1 ιιl l ιtt ι ί ι· uitwitha productclassifier, bymeansofwhich the ιιι ιlι ιι.ι l which has reached optimum fineness is removed Ι •Ι ι 1ι ΙΙ Ι Η j) rt to the subsequent processing and the oversize

tect the screens from injury and act as a linin~ for tl1«' tumbling charge; sometimes also the fine _screen ~s f~rth~·r rotect~d by coarse screens mounted dιrectly ωsιde .ιt . this type ofmill, which is often kno\-vn as the Kruppmι~\i is of interest since it rep~esents a very ear~y type of r:ιι which with modifications, has retaιned ιts p~pu~anιy . The Krupp mill is particularly suited to the gnndι~g <~I soft materials since the rate ofwear ofthe perforated lιncι

ι ι 1111 11 ·d to the mill for further grinding, is an ο bvious ιlιιiΙΙΙΙΙ. nce continuous feed and a long mill body have I ι ι ι ι lι ι · ' p ted, however, the overall grinding efficiency of !11ι Ι Ι ι ιΙΙ 1nay b e improved by fairly simple modifications. Ι ι : ι s a lrcady b een mentioned; for optimum grinding ιιιιι i ΙΙ Ι ΙΙ ΙΙ S tl1ere is a fairly d efinite ratio of ball size to ΙΙιΙIΙΙ Iι · sizc and so the most effi cient grinding process caη ιιΙΙI l ι1 .ι ι ιaine d when a product with a large size range is ΙΙ ι I ΙΙI irι the mill. If, however, a tube mill is divided into

(Harding< Co . lnc. )

Fra. 1.7

30

INTROD UCTION


a number of compart ments. and the mean b~ll size of thc grinding media decrease s ι~ each succeedι~g compart ment; then the optimum ra~10 ~etween ball sιze and partic1e size is more nearly maιntaιned, and a better overall performa nce of the mill is achieved ; this giving ~ise to thc compart ment mill shown in Fig. 1.6. The tube mιll has thc further advantag e that, to some extent, the grinding characteristic s of the mill are under control; for examp1e, an increase in the size of the balls in the final chamber will reduce the rate of grinding of the finer fracti.ons but ~ill leave the rate of grinding of the coarser fractιon~ se~sιbly unchang ed and so the amoun~ of coarse mate~ιal ιη thι~ final product will be reduced wιthout any excessιve overall increase in fineness. The principa l field of applicati on of the rod mi~l is probably as an intermed iate stage betwe~n t~e crus~ιng planι and the ball mills, in the metallurgιcal ιndustnes. Thus, material between about 1-in. and 2-in. size may be rcduced to about - 6 mesh for feeding to the ball mills. Rod mills are, however, b eing used in closed circuit with a classifier to produce a product of 1ess than about 48-meslι . . size, but such applicati ons are unusua~. . The choice between wet and dry mιllιng ιs, ιη general, unimpor tant in small-sca le mil~i~g b~t is a major techn~cal problem when large-scale mιllωg ιη t~e metallurg~c~l industrie s is involved . On purely mechaωcal grounds ιt ι difficult to see any great diffe.r~nce i~ fundame.n~a1 principles b etween wet and dry mιllιng, ~ιnce ~ry mιllπ:ιg m.ay be regarded as wet milling wit? a f:lωd ~a:rιng the vι~co~ιty and d ensity of air, whereas wιth w~t. mιllιng the fluιd ιs a liquid. Α slight difference of con<;iι~ιon betwe~n t~e two cases however is that with dry mιllιng the fluιd (aιr) fill the ~hole of ~he mill shell not occupied by the balls, whereas with wet milling the liquid occupies only a part of the mill volum e not occupied by the solid charge. Τhiιι differenc e would be' expected to have some slight effect 011 the motion of the ball charge but the effect would b expected to be so small as to be practical ly n egligible.

31

ι lll)ί~atio n Of the literatur e ShOWS, hOWCVer, that "Ιιι11 Ι ιlιfference between the results of wet and dr Υ ι Ιlι 11 . ι ι·ι· orten '· for example th e power to . . .very Iarge II ι ιl ι I ι IH· mιll ιη whιch a given material is being wet Ι •ΙΙΙΙι Ι , ιιι ι~y b. as much as 30% less than that of a similar ι 11 ιlι y ~:· ιnd~ng_. Furtheri? -ore the ~hroughput of a cοη ΙιιιιιΙΙ Ιι ι (I ιη ι!lιs greate~ ιη wet gnnding ; probably due ,,, Ιιι /Ιι · ι: now propertιes ofa pulp in compari son with Iι. ιιι , ιΙιτιaΙ.. It appea~s, however , that the rate of ball 11 I I ιιι ι wι·nr ιs greater ιη wet milling so the decision be1 ι ιι ~ ι • Ι ι ωd dry operatio n is, in cases where the choice Ι"' ιιιιΙΙι · Ι , a matter of balancin g the economi c gain fι ·ο ιη r~duced power demand and increase d ιι ιι Ι ιι Ι•ΙΙ Iψι ι L agaωst the increase d cost of replacem ent f I I ι ι 11 : Iι tιrg and the mill liners and the Ioss of prod u~ιιι ΙlιιιΙ ιι ,ιζ the shut down period n ecessar for such reΥ 1'' • ιιιι ιιι ~ ι be m ade. is not ρerchoice this Ι Iι ι ι I ί rι ertain applicati ons, · a · ιη ιιιΙΙΙι 1! ιιιιτ , Γοr example ' the grindι'ng ofth e raw mιχ the 1111 111 ι ι Ι :ι.rΗ may be carried out either wet or dry but Ι ιιι Ιιιι ~\ ιι l t~1e cem entcan, from thenatur e ofthe material ' ιιl lιι ι.ιr· r ι 'dout dry. IΙΙΙΙ,Iιιτ _i mportant point of differenc e between wet and tι ιιι Ιlιιι ι.~· ι s_that we~ milling allows the easy introduc tion ι I 11 fΙ ll I' ;ι llVC med~a for the reductio n of the energy reΙιfιι 11 Ι ι : J)Ι' duce uωt n ew surface or for the inhibitio n of ιι ι ι ι ι ιοι1 f the fine particles . Ι t is theoretic ally possible ι ΙιιΙιιΙιlιιι· • such substanc es in gaseous or vapo ur r:ωrm · I Ιtιι Ι 1 I ι Υ ιηι·ιι b ut, ιη pr~cti~e, su~h methods have not been · r:ιact adoption , and , ιη tojustιfy wιde l successfu ιιΙiy 1 ιιllι . .. . bl I 1 "" I ,ΙΗ:ι tιca e method of ιntroducing such material S ιι

11 I

ι I ιιΙΙII ι ηg.

"" ιιlιτation. can now well be given to some of the ι ιιιι ' 11?1'1 whι~h the choice of the type of mill, or the f · d' ι ιlιι1ιl ιιl ρeratιon . of a given mill , r:ιι0 r th e gnn ιng ο a . I peris it Ι • Ιlιι ι ιnnterιal wιll depend. In the first place 1' ι1 f Ι ιιt r.est to note that the problem s encounte red · 1 11 I 1 ι Ι ιιιι·;,ιtιοη of powdere d materials are not necessarΛ~

32

INTRODUCTION


reduced when the quantity of material to be processed is small. As an example of this, in t?e preparatio~ οΙ ' materials such as Portland Cement, whiCh scll at rclatιvcly low prices, the problem is ma~nly one of o~t~ining a fi.n~l product having the required s~ze c?ar-actenstιcs .at a mιnι mum total cost per ton and, ιη th~s case, guestωns of. thι· the contamination of the matenal, dunng processιng, scarcely arise. . . In the grinding of drugs and cosmetι~ p~eparatωns, however, the question of the cost
33

ll ιιι ι. ι ιy rotech.nic materia~s

might well be ground dry, wo uld, ιη general, gιve a product having superior lt ι ι ιι Ιι · ι · ί ~ ιi cs to one which has been ground wet, but if l11 ιιι , ι!ι · πal. were very reactiv.e ~t would probably be 1 ι ιΙΙΙι lι~ι an ιnert atmosphere. Sιinιlarly, tungsten carbide ι1 11 ι ιιι the prepar~tion ο~ sintered carbide tool-tips ιιΙΙiιi i Η' gτound dry, sιnce moιsture would complicate the ιl ι ι ψιι · n t proces~es and be harmful to the final product. Ι ι Ιll11 · 1 wder pιgments would probably be ground dry ι ι ι ι Iι y Ι hi~ ~eans discoloration is minimized. The grind~ IΙΙ ιιl Ι I Ιιι ι:ηιωum for the preparation of paint is, however, Ι ι ι 11 ι , ιιηcd out wet since by this means stearic acid or ΙΙιΙΙ ιι Ιiι 'Γ antiflocculent, may be introduced. ' I ιι ι ιlly, claysforthe preparationofceramic slips would ι ιιιι . ιl > ly b e expected to be ground wet, since the initial 11 11ι llι ίιι ιι of the clay is wet and it is required in a wet form Ι •ι ιι l ΙH ι·qu e~t processi~g. Clearly,. in such a case, dry 1Ιιιι l ι ιι ι,(, whιch would ιnvolve dryιng, would be uηecο Ιιοιιιιιι ,ι nd would only be adopted in very special circumΙ 111ι ι Μ. Λη appl.ic~tion of the. ball mil1 which can scarcely l1 ι Ι , ι Ν ι' d as mιllιng but whιch should, neverιheless be ιι 11111 ι ιι ·d is the dispersion of pigment in the var~ish l ιΙΙ l1· in connection with the manufacture of paint. The •ΙΙι ι ιι t ι:n tion of solid I?aterial in the suspension is, in this ιψl1 . ιtι η, usually quιte low, gener~lly being about 10% Ι ιι l1 ι ιη c, and, although η ο appreαable reduction of the ι ι ι ι l Ιlι particles occurs, considerable ball wear is often ιιι ιιιι ιιt ιτcd . These observations are of some interest in Ιι ι ι l ' ~ι theory of the internal dynamics of the ball mill, t•Ιιι ι ιιιιι ιι lcd by one ofthe present writers, since that theory "'' 1 Ι ~ ιhat when the quantity of solid material is small in IΙΙιιι:ι ιο the volume ofthe ball charge, the particle size ι ι lιιι Ιι ο ιι might be small but that the ball wear would be ιιι ι ΙlιιΜ

ιι/ιιΙι · ί : ιb le . I ι 1111 t lιe foregoing OUtline of the problems inνolνed Ϊη l1ι 1lι • ι: ι~ο η ~f ~ mill for a given purpose and in the οpera ιιιι ιΙ I , mι lls, ιt ιs apparent that the subject is very comιlι

Ι ι ιιr· thermore,

much of the theory underlying a rigid

34


solution of any problem is in a very fluid state and, iιι conseque ncc, th e solution of a practica l problem generίιll involves tl1 e use of considcr able judgeme nt, based or previous experien ce, to temper thc conclusions from theoretic al analysis.

CHAPTER 2

Ί'Ι Ι Jl, ΜΟΤΙΟΝ

OF ΤΗΕ CHAR GE WITH IN Α MILL

I ι ι ι ι ι ι ο ι ίο η of the charge, that is the grinding media and ιlιι ιιΙ . ιι< · ι· ί al undergo ing grinding , within a mill is ofconιlι ι , ι lι l (· ιheoretical interest and practica l importan ce, ιιιιl , lι11 · tlι se r easons, has b een the subj ect ofconsid erable ιιιιl , lι y number ofworke rs, but, even so, no rigid and 'ιιιιιlι l ι·ι. ιheory, covering all the aspects ofthe dynamic s •I ι lιι ιni ll ch arge, has yet been produce d. The practica l Iιιψ ι ι ι ι . ι η of this subject clearly resides in the possibili ty .ι ι Ι ιι · 1 ι · diction of the grinding behaviou r, and other 11ι lι ι lι : ιracteristics, of a mill from the knowled ge of the ι ι ιΙΙ ι ιo ι ·ics of the elem ents of the mill charge. The ι Ιιιι ιι ι · t iι· a L interest lies in the study of the dynamic s of ι Iι ι ιι ι ι · m and in the d erivation of equ ations to define the ιιΙιι!Ι ιι ιΙ Γ th e elements of the mill charge in terms of tιιιιιi,ΙΙη :ntal qu antities such as the size and the speed of ι• ι\ ι\ ιιιι ι of the mill. Α simple example of the practica l 111 ι 11111 ι :1 η e of this informat ion is in the use of the know1 ι l ι οΓ tlιe trajector ies followed by the balls in a mill to lι Ιι ι11ιίηc the speed at which the mill must run in order ι Iι ι t t Iι ι· clescendi ng balls shall fall ση the toe of the charge, 1111 ΙΙΟI upon the millliner. The impact of the balls upon tlιι lιιΗ' t' plates can lead to u'nduly r apid wear of the I ι \Ι ι ι , : ιnd so to high mainten ance costs. Ι ιι tl1is chapter tlιe motion of the particles constitut ing ιlιι ι l ι;11·ge ofthe mill will b e consider ed; it being assumed ιlιιl ιlιιτ is no slip between the mill shell and the charge. Ιlιι ι ι · αists another type of motion, in which the charge ι ι wl1 le slips relative to the shell, which gives rise to the 35

36

Μ


phenomenon of "surging", bu~ consideration of this typ<: of motion will be deferred untιl a later chapter. Α study, from first principles, of the b~h~viou.r of a mi ll charge is much simplified if the ch<7rge ιs ιm<7gιned to bι· composed of rods, instead of balls, sιnce by t~ιs ~ea.ns thι· complication of any axial motion of th~ balls ιs ~lιmιn.at ·
τΙ ΟΝ OF ΤΗΕ CHARGE WITHIN Α MILL

l ιι tΙto ti .

ns of th_e surfaces of the rods at the point of conand free motion inside the shell Ι ιlι ιι ι ιιι a t e?. In thιs case the angle between the vertical ιιιiΙ Ιι ι · ι:adιus . vector joining the centre ofthe mill and the ι Ι ιιΙ ι ι • of gr_avιty of the pair of rods is much greater than ι Iι ΙI /ο ι· sιngle rod. But, again, equality exists between ι l ιι Ι ι~ι·.k . done to rotate. the she~l and that dissipated in Ιι 'ΙΙιtι ι <ι t the contact poιnts and ΙΠ distortion ofthe metal ιιι/ι_ι Ι'Ν. Α. further increase in the number of rods would ι ιι i ΙΙΙΙΙ · ιhιs effect until relative rotation between the rods 1 Ιι Ι

ιι ·~· ιη opposιte ~enses

I Ι

ι ιI

I

/

~-----

/

~

~

S t ιιl

11.I

---

St ι ι ι

Bolls Only. Bolls W ί thA TrocιOf Quo,.tz .

I 10

20 ι%> Fιc.

FιG.

2.1

FιG .

37

I

I

30

40

50

2.3

2.2

shell rotates about a horizontal axis with an angul:.ιι velocity η radians per second. In. such ~ ca~e the rod w~\1 lie near the lowest point of the mιll, as ιη Fιg. 2.1 and wι\1 rotate at such speed that the peripheral speed of th.e rod i the same as that of the shell. Furthermore, the dιspla<· ment ofthe radius vector joining the centre ofthe mιll aιιιl that of the rod would be such that the work done by reasoιι of the couple formed by this displacement is equal to tlι energy dissipated in the distortion of the rod and shell ΙΙ the line of contact. Ifnow two rods are placed within the mill, as in Fig. 2.'

I ι ι μ,ι· Ι y precluded and, in this respect, the charge beι lιn ost as a solid body. Ιl ιι ι · o ιτectness ofthis view is supported by the work of Ιιι 'ι / Ιlι · present authors who, with a co-worker Rose ιι l Ι •, . ιιιs, ω has shown that, for all other factors r~main ιι ' ψι : ιΙ , the power to drive a mill increases with the 1Ιιιι of tl1 . ratιo (D jd); that is for increasing numbers of ιl l ιιιc l s (ιη the. exp~riments balls were used). This rela1 11 Ι 1 ψ , sl1own 111 F1g. 2.3 indicates, at once that the ι ι ι l ιι l ' lli·' 11t . h of ·the centre of gravity of th e c'h arge 111ι ι w ιι ~η 1ncreasing number of balls, which is in ι ι ι ι l ιι n · wιth the reasoning given above. T his figure ι ι

38

D RO D MI LL S BA LL , TU BE AN

of the tio n in the dis pla cem ent fricalso shows tha t the va ria of t ien ffic coe less wh en the cen tre of gra vit y is mu ch hig h, wh ich ind ica tes tha t, for is es fac sur wh en tio n be tw een the g of the ch arg e is effective this con dit ion , the loc kin olv ed. bu t a few ele me nts are inv of the mi ll is so low tha t th If the spe ed of rot ati on ted , the n ele rat ion ma y be neg lec e will effects of cen tri pe tal acc ch tre of gra vit y of the arg the dis pla cem ent of the cen possible lim itin g con dit ion s is o inc rea se un til eit he r of tw ing : be s ion dit rea ch ed ; these con to ma inforce at the shell, necessary of th (Ι) Th e tan ge nti al tre cen ss ma the tai n the dis pla cem ent of the fri cti on al force. Wh cn cha rge , becomes eq ua l to wh ole ch arg e slips bac k. this sta te is rea ch ed the wh en , cem ent , θ, rea che s 30° (2) Th e ang le of dis pla n fall the s in eac h lay er since the centres of the rodof rod s collapses. on ver tic al lines, the he ap eta l e, since the effects of cen trip nts In fac t this is a triv ial cas me ele affect the mo tio n of the acc ele rat ion pro fou nd ly tio n ο/' speeds of rot ati on the mo w giv cn t of the cha rge , bu t for slo tha to s ate xim mill ap pro will the ch arg e in a pra cti cal s) s mo tio n the balls (or rod mill, by case (2) abo ve. In thi he oft ll cen tric wi th the she tra ve l on cir cu lar arcs, con is rea ch ed , aft er wh ich th y y ilit tab th un til the po int of ins is inc lin ed at ab ou t 30 ° to ίιι ich wh e, fac n tio roll do wn the sur mo pa ral lel layers. Th is nt ho riz on tal , in a series of en thr ou gh the tra nsp are tak sho wn on a ph oto gra ph . 2.4. It will be no tic ed tha t ιι en d of a mo de l mi ll in Fig s the mi dd le of the ch argr, ard sm all "v ort ex " exists tow the balls no lon ger roll down on ati rot of eds spe r pro · At hig he bu t, at a cer tai n po int , are the sur fac e of the cha rge rea fte r describe approximatelg the th s jec ted int o spa ce an d ain me eti ng the ba ll mass; ag ore bef ths to b pa rab oli c pa ar pe ap in Fig. 2.5 . Th ere ba ll pa ths be ing as sho wn me s for these two types na no un ive rsa lly ad op ted the evi den ce ap pe ars to be ί t bu , rge cha ing ' mo tio n of the the first typ e an d "ca tar act for g" din sca fav ou r of "ca

Μ

AR GE Τ ΙΟΝ OF ΤΗΕ CH WI TH IN Α MI LL

(Dr.

y ο[ London) J. Μ. Connor and the Uniυωit Fιc.

(Dr. J.

Μ.

of London) Connor and Ιlιe University Fισ.

2.5

39

2.4

~ IΙΙ ' Ι ' Ι

Ν

OF

ΤΗΕ

CHARGE WITHIN Α MILL

41


40 for the second type. These names will be adopted for thc present work. As the speed of rotation of the mill is increased thc partic1es are projected with progressively greater ve1ocitics until the theoretical trajectory for a particle, which is in fact 1ying against the mill shell, wou1d fall wholly outsidι· the shell. Clearly, since the partic1e cannot pass through the shell, it wou1d 1ie against the shell throughout the cyclι· and so be carried around continuously with the mill shell. This condition is known as centrifuging and the speed ο(' rotation at which it occurs, for the outermost 1ayer ο(' particles, is known as the "critica1 speed" of the mi\1. This speed, the critica1 speed, is of considerab1e importan ι· in mill techno1ogy since, for example, other factors beinf.( equa1, the equal performance of two mills of different siz ·ιι demands that the ratio of the actua1 running speed to thι· critica1 speed should be the same for the two mills. An expression for the critica1 speed of a mill will now bι· derived. By reference to Fig. 2.6, it is easi1y seen that if thι· trajectory is not to fall inside the shell, the radius of curv<ι· ture of the path, ρ, must be greater than R, that is thaιι that of the mill she11. Furthermore, this must be true fοι• from wherever the trajectory might start. From the dynamics of a partic1e it follows that for tl11· trajectory,

Fιc.

Ι l ιιι Γι' m the conditions 'fi d IΙΙιιιι ('>. 1) into (2.2). speci e ρ~ R, so, substituting

-R

~

V3 gV cos

α

vz

v

gcos

χ' = cos α χ" = ο} y' = V sin α- gt y" = g A1so, from e1ementary mathematics, the curvature, ρ, of any curve is given by {1 + (dy fdx)2}3/2 ρ= d 2yfdx2 which by substitution (for examp1e, see Todhunter's jerential Calculus, p. 174) gives (y'2 + χ'2) 3/2 ρ = x'y" -y'x"

2.6

ιι1

cos

α

V2

α~ -

gR

.... (2.3)

lιιιι ιιι Η α ::Ι> 1·0 so the limitin cond' . . . '' Ι ΙΙ ιιι ι iR to g It!On IS α=Ο wh1ch cor-

DU·

V2jgR ιι ι

1 i ιι (~

Ιι ιι ιιι

V=

which

ωR

'

ω2

= 1 = gfR

ω= y(gfR)

.... (2.4)

-

---~~-

--

-~--

-----·

ΜΟΤΙΟΝ OF ΤΗΕ CHARGE WITHIN Α MILL

42


When the diameter of the ball is taken into considera· tion, the radius of the outermost ball path is giyen by -!(D-d) which, when substituted for R in equation (2.4) , gives

and this ση expansion by the binomial theorem leads to ω = y(2g/D ) . (1 +0·5d/D)

.. . . (2.r.)

If, as is usually the case in practice, the value of d/D iH less than about 1/2 0, the second bracketed term may, i11 general, be taken as unity. In this treatment ω is the angular velocity of precessioιι of the mill charge and, because of slip between the mi\1 shell and the charge, this is not necessari1y equal to η, thc• angular ve1ocity of the mill shell. If, as is often the casι~, this slip is assumed to be negligible, then ω= η and thcs equations may be cast into the slightly more convenicnι practical form Ν, = (60/2π)y(2g/D) = 76·6y(l/D)

....

(2.β)

where D is the mill diameter ίη feet and Ν, is the criti :ι\ speed of the mill in r.p.m. The above formula is based on the assumption thιιl there is no slip between the ball charge and mill shell anιl, to allow a margin of error, it has been common practiι: to increase the coefficient in the equation by as much :ιΗ 20%. That relative slip between the charge and shell is of ίnι· portance in this connection is supported by the work οΙ Rose and Evans
-~ -------~----

43

IIΙ ~ed that centrifuging in a mill with porcelain balls and ιc ι ,Υ oc~~rred when the actua1 speed exceeded the ca1cul , Ιιι Ι d h cntιca1 va1ue by about 5 οιο ι to 10 οιο· ι It ιs . qu estιon. ' Ί, > .' owever, wh~~her with modern liners maintained

ι e a~onab1e condιtωn

ω= y{2g/(D-d)}

-

the increasing of the va1

f h

' ι ιc·fficιent of equation (2 6) by 20 οι . ue ο t e ι lc·Nirab1e. · ι ο ιs necessary or I ~ttentio_n will now be turned to a more detailed analysis 'ι ι ιe motωn. of t~e. ball charge. Ι η the first p1ace, ο η the ; ο ιι_η~~ ο~ 1 ,~ νι~ wι

1 . e consιdered, even though, as shown 1ater

s~mplιcι~y, the ana1ysis first developed by

Ι ιΗΙ ιη _certaιn r~spects, an over-simplification.

' . of · 1 a poιnt Ρ ' Fig · 2 · 7' at whι'ch th e proJ. ectωn : ::: partιc e occurs a~d let this. point be at a distance r from 1 . e~tre of the mιll. In thιs connection the " . t f I'' c~ι ·ctιon" may be considered to be the point at whf~~ t~e I ι ι' 1 ι; Itory of the partic1e is influenced by gravity and is not Υ controlled by the packing of the ,Ι Ι ι ι ι ι • re. surroun d'ιng .onsιder

Ι

11

I•ι' .r _ proJectιon . . to occur, it is necessary that the radius of ; ΙΙ ι vLιtu[ιe

of the path under the influence of gravity should Mma er than the ra?ius of curvature of the initial 'I ι ι ιιlarι{ath. An expressωn for the radius of cίιrvature of :ι ' \Γfr~mu~~f~h gravity is given by equations (2 .2) and cc

:

r ~ Ι ι ι • IΙ ,

Ι •' ι ·οn:ι th~ geometry,

ι·: ; :ιcΨLβιο~ -α

Ι 'Ιιi Ν

α

=

ω2rf g

ωr,

- _, (2. 7)

however, it is c1ear that the line of

Α mduhst be at right angles to the radius vector

an

ence r

Icι'

α)

for the limiting case and since V = cos

ο

V2f (g cos

=

(gfω 2 )

cos

β

.... (2.8)

quation is satisfied when point Α is distance gf 2 centre of the mill, a.nd furthermore, since t~e

ι· ιhe

Μ ΟΊ' CΟΝ

44

BALL, TUBE

AND ROD MILLS

OF ΤΗΕ CHARGE WITHIN Α MILL

l11 ι ι ι , is the radius of the Davis circle and n is in r.p.m. ιi11, in this treatment, slip is usually neglected and so 11 1" ι ·Ν ι · rΊ ts the speed of the mill in r. p.m. Ιlι ι ιι ·::ιjectori es ofthe ball paths may, on the assumpι ι ιιι Ιlι rι ι ιhe ball paths are parabolic, be easily plotted. Ιl ιιι , l ιy ιhe integration ofequations (2.1) and taking the ιι o fΊhe co-ordinate system at the point of proj ection. χ = Vt cos α } ττ · α - ι gt 2 • • • • (2. 10) Υ = Υ t sιn 2

FΙG.

FIG. 2.7

. nd from the theorem of gc<~· angle ΟΡΑ is a nght a!l~~e a circle is a1so a right anglc, !ι metry that the a~gle wιt ιη ~· h projection takes plac Ι follows that the lιne fro~ w /~c 2. the "Davis circle". arc of a circle of radιus g ω ' . an . . his leads to 1he expressιon By a simple substιtutωn t (2.< ) rc = 0·408/n2 • • • •

45

11111 ι l i ιninating

2.8

t between these equations gx2 y = xtanα- V .... (2.11 ) 2 2 cos 2 α '''''" wl1ich the parabolic trajectories may be plotted. Ι οιτ important than a know1edge of the form of the 1 ΙjΙ 1 ιor· i es is a know1edge of the form of the surface in Ιι ι / ι Ll1e parabo1ic traj ectories terminate. Ι • ιοιι1 Fig. 2.8, it is clear, from requirements of

eηteri~g aηy ~η,

MI LL S TU BE AN D RO D

ouηt

BA LL

of materia1 quaηtity 1eavtηg ' 46 qua1 to the b am the t tha t oss a y, coηtiηuit stbo1eice tra jec tor ies caηηο cr ' mu q'q ι Ρ Ρ ηt me ηu1are1e ce the ρara ·η the sam e anηu1ar e1eme~t as it aηd so, siη iηed ?Υ pωηts: ~face bR c is defers iOη ~f thc tra jec tor y mu st termtηa~: b . d by the iηt ect r IS ~h it origiηates. ~hus, the ere wh r; ius tanl~c1e ofr ad suc h as R, wh tch are .ο a c ori iηates. Α represeη~atiνt:. h WI~ y tor arabo1ic tra jec the traJeCtbory. dgby the . η . as. . tiO padius at wh ich R r oiηt, suc h as ' m ay be . the oiηt of proJec o1 a wlth of the pa rab .Y't ues, of h Ρ va1giη fhe ori η ο pω the of the s ge reιι::.erred to these co-or.diη atc ou1 th e circ1e of rad ius r th~ c Cir the f ο . oη Th e equati is . α -- ο ... . (2.12) 2ry Slη χ2 +y 2 + 2rx siη α-

determiηatiOΡη οΙ

ο taiηe

χ aηd tafkil:t~rsectfoη

by (2 .11 ). On 1a is giveη ara bo12 ressωn . exp the ), (2. Piη hΊ) oft iOηm (2· 1 uat .eq the d aη fro Υ 1ηg tut sti sub

g2x4 4V2 cos4 is

obtaiηed;

~ + (sec2 - {; sin V2 cos α α

· h fro m w h IC

g2 ( 4V2 cos2 No w fro m

α

χ2 - Ο

α- V2 ~~s α)χ2

α)x-grcosα + V2-- Ο

) x2-(gtaη

equatioη

(2. 7), α

ing . the immediate1y pre ced wh ich sub sti. tut e d Ιη 1eads to ο χ2 -Χ tan α= --ο;3 4r cos α

aηd

Ο

an d a1so

V2 = gr cos

Th e

=

so1utioηs ο f

t h'IS equatioη are χ

= 4r siη

α

χ

=

cos 2 α

equatioιι,

Ο

... . (2.13)

ιI

~Ι Ο Ί' Ι ΟΝ

ΤΗΕ

CH AR GE

WΙτΗΙΝ Α

in the pr ese o wh ils t the fou rth , <ιrcat zer ι 'i'1 ΙΙ ιΙΙ),ιιιrιi.~ Ll1 req uir ed . sti tut ed Ι lι Ι'ΙΙ Ι/1ίs value for χ is sub t, rl rι f Η ΙΙ'Ι t/) la, it is fouηd tha 1

I

MI LL

47

cle iη fou r ral , a pa rab ola Cuts a cir qu art ic lr·;ι r Lhat, in geneηt the of ts roo case, thr ee OF

ι •Ιιιt . ιι1C/,

equatioη

giveη

by

iη

equatioη for

the

2 . . .. (2.14) y = -4rsiη α cos α t the essary to plo pu rpo ses it is ηοt ηecint οb be y rι r ι ·, Γι t" most the ma R po reps:reseηtative cto ry, /ιι/ ιι rιι/ι·ι/ Ιr·<~ ι.rι· me tri cal ly thu ι the mill an d let Ο be the ceηtre ofcir Ι rι Ι•' ί,ις. 2.8 let po intRP cu lar pa th of Q represeηt the ι Ιι1 1rιrιι: •nt ric circle 2ω , repregf ius rad S is the arc , of ι/ιι / ι ι/Ι. Th e arc OP t a ho rijec Pro . urs occ projectioη tre Q cen ' rι Ι rr ι~· ι/1 1iη e at wh ich th t the circle at Qaηd wi Th e rr rιl 1/ Ι l1rough Ρ to cuius R. at cle cir t the ι lι ι ι ι/ ιι· an arc ofaradpoiηtQPon, tothecusurface iη which the ιιι ιιι/ Κ is the n epe ati ng the balls me et the ch arg e. R eηt rad ii, lea ds to the fer dif mbe r of cirςles, of ΙI ,,ι ι ι ι ιί ι ·ι·rιu I sur fac e bR C. is sim ple : ι ·ι rι· j)I"Oof of this coηstructionthe α, an d len gth QR is giveη by I Ί ι ι· I ng th PQ = 2r sin -yR)2}1!2 QR = {(xQ -xR )2 + (yQ 1/ rι ~ ι·v 'Γ, xQ = 2rsiη α; YQ = Ο 11 ·".ιι = 4r siη α cos2 α; YR = - 4r siη 2 α cos α 111

2

ι/ ι rrι.r"

coηstruction

+ (4r sin2 α cos α)2}1!2 {(2r siη α+ 4r sin α cos2 α)2 fic ati on leads to n exp an sio n an d sim pli QR = 2rs in α ati on shi p used d thi s is pre cis ely the rel an QR = PQ , Iιι Ιιιι n. tio uc str on ι l1 ι· k fro m the cen tre of 1 '/ιι: curve ab is a cir cu lar arc struc

ι ι/{ ι /ιίι /1

48

tc· face wh ich does ηοt oscula the mil l aηd represeηts a sur S, aηd, iη coηsequeηce, fronι ί ιι wit h the segmeηt of circle OP pla ce. Th e rad ius of this arc wh ich ηο projectioη takes ηl{ oti the mill cha rge aηd, deη fixed by the ma gni tud e of es the expressioηs the rad ius by r;, Davis giv ... . (2.15) r;/R = Κ

app rox im ate ly; J filliηg of the mill.

thi s

beiηg, iη

expressioη,

Ν

the

OF

0- 4α) τ = τ(36360 c / 11 ,

i / ι ι· lim

Ι ' ιι l ιy

· par abo lic pat h is e reqωred to tr?-νerse the τ-

... . (2.l(i

-0· 02 4+ 0·3 9y (7- 10 J)

κ=

wh ere

49 MI LL ΤΗΕ CH AR GE WΙτΗΙΝ Α e τ [ι Ιlι ιι , Ί ',, iH l /n aηd thebytim ' c> or the ball to cover the eη giv is 11 ι ι I ι I ι• I );) tl1 ~ ~ Ο ' ΙΊ

RO D MIL LS BA LL , TU BE AN D

1 '

2 siη α cos α

fractioηal

t the mill

tha thaη zer o, it follows Siηce Κ caηηot be less ereηce bι~ erf iηt t abo ut Ο· 7 wit hou filliηg caηηot exc eed vis sta t ιι Da lly tua Ac takiηg pla ce. tweeη the par ticl es ur, r; should ereηce sha ll ηοt occ tha t, iη ord er tha t iηterf iη revolu ed 28/n 2 , wh ere n is the spe ηοt be less thaη 0·2

per second. wh ich is iη "fl igh t" caη br Th e fractioη of the cha rge s. cal cul ate d fairly sim ply thu (2.14) , giveη, fro m equatioηs is 2.8 . Th e aηgle β iη Fig by η2αcosα-rcosα)

-= - -πn

=

( 1/r)(4rsi

(4 cos3 Fro m var iou s siη β aηd

so,

cos

α-3

α)

tites trigoηometrical ideηti

r = JR 2;r ;2

= cos 3α,

cos

bu t

3α

= cos (18 0- 3α)

-β), siηce siη β=cos (90

cos

(90-β)

= cos

90-β =

or

(180-3α)

180-3α

-β= 90-3α

or

aηgle

4α.

... . (2.17)

par tic le whilst i11 t~ughbyichtheis α+ 90 + 3α- 90 n

passed Now, the the par abo lic pat h is α+ 9

+β

ηd to be 54 ο 44' Ι ιιι/ Ι I ΙiΗ aηgl~ wil l be fou . b th d a ηd 1e aηg 11 , Ιι Ν ι; f this e a OVe expressioηs, l t lS fou η

Iι I

Ι/

wh

· r.p .m. , the tim e per revolu If the spe ed of the Inill is n

Τ

1Ί, = 0·3/n c = 0·3 9/n; 1 . ηth · of ηti spe Ιιιιι , Ι /ι tim e e Circu ar pat h ' as a fJractωη ι is 1e cyc the of • ι ιι Ιιιιι Ί'

'

this leads to

· · · · (2.18)

. ed tha t th 11 ιι ιιw ί t is ass um η oft he cha rge ma y tω mο e . th by d ι .. 11 ι ι ι·c·s nte par tic le at the rad ius mot~ofol of the ιι / ι ι ι ι;' η of. the ch~ rge essary to fi d t h e aηg1e , It IS ηec η rad ius . ι ι ΙΙ Ι Ι 'Ν[) ηdΙηg to the

tioηs

siηβ

χ

V cos α

· = 1/n ' •

0·39 0 56 =5 6 % ... . (2.19) 1- 0·3 = ' f b tioη is giveη by / ι ιι, ( / ι η um er ο cyc1es p er revo1u τ,; τ= 1/0·7 = 1·45 · · · · (2.20) . . · Ι 'I t firs a as t of " ι ~, lt foll ows tha e . 1 approximatωη , 56 % ι ι ιι ι ί ι η iss p eηtiηth' Circu ar p ath aηd a Ι so t h at the d mill sh 11 11 1 , ι I · tweeη the the cur ves dP a bR C is aηh b iede u occ a are ιl ι",, ο Γ the ~ e cha rge wh eη the mill ι . ι ι ι· ·s t. It also fol lo! s tha {' η 1aη. average, every ball ι ι ι I Ι 'Ν aηo the r 1·45 tim es p er rev o utiOη of the Inill.

WΙτΗΙΝ Α MIL L ΜΟΤΙΟΝ OF ΤΗΕ CHA RGE

MIL LS BAL L, TUB E AND ROD

50

, is ope n to criticisn1 The analysis of Davis, just outl ined the fric tion al cha re, mpl exa on a num ber of grou nds . For the effects of intc racteristics of the cha rge are igno red, cha rge in a givcn the actions betw een the elem ents of een adja cen t traj e betw e enc rfer inte the and traj ecto ry etal forces acti ng on trip cen the tories are neg lect ed, as are the equ atio ns involv d the part icle s. Fur ther mor e, some of wid e app roxi mat ions . app ear to incl ude a num ber ofra ther atte mpt to deri ve ιι ous Eve n so this wor k was the first seri , whi lst form ing a good rati ona l theo ry of the ball mill and also pav ed the way for basis for mill calc ulat ions , it had late r studies. the abo ve trea tme nt The first imp orta nt mod ific atio n to ted out that th poin who 5); ger( Stei was sug gest ed by von alon g the ascending hyp othe sis of free flight of the part icle lid since the con · inva is bra nch of the para boli c traj ecto ry traj ecto ry result ιι any g alon tinu ous pro ject ion of mat eria l Thu s for the ascending in con tact betw een the part icle s. g the arc is con stan t ; bra nch ofth e par abo la the spee d alon ht of particles. flig not vari able as is the case with free tha t the intrinsi c On refe rrin g to Fig. 2.9, it is seen al acce lera tion ί radi equ atio n to the nor mal μ is, since the of the grav itati ona l ent pon com mal nor the pro vide d by acce lera tion , give n by V2 fρ

whe re ρ poi nt Ρ.

= g cos

α

FIG. 2.9

Λlso from equ atio n (2.21),

V2

dα

g cos

V 2 sin α dα

dy = Ιι ο ι n

g

whi ch 2

V2

- dsfdα = g cos α

••..

(2.21)

y-y o = V g

V2

or from whi ch

V2 g

dα

= - cos

α

ds = - dx

-ds

α

lιtt l sinc e dy= ds sin α,

ecto ry at tl1 is the radi us of curv atur e of the traj

Thu s

5}

g

cos

α

[log cos α]α

αο

cosα

1og cos αο

. . . . (2.23)

equ atio n (2.22), l•'or sum mit χ= χ, and α,= Ο, so from V2 χ., -χο = g-αο

(2.24)

52

MI L L ΜΟ τΙΟΝ O F ΤΗΕ CHAR GE WITH IN Α


and from equati on (2.23) ,

V2 y, - yo = - g log sec α 0

.... (2.25)

The surfac e from which the projec tion of the partic lcH occurs is unalte red by this treatm ent and is, as beforι·, 2 given by a circle of radius gf 2 ω drawn in the way pn·· themselves are con· ories traject The ned. explai y viousl sidera bly altere d, howev er, since for the same initial conditions, the horizo ntal veloci ty of a partic le, at the summ it , is greate r under the assum ption of von Steige r, and so thc partic le travels a greater distan ce horizo ntally . Also tl1 r. height of the summ it above the point of projec tion is dif~ ferent in the two cases. Furth ermore, since from contin uity consid eration s, a trajec tory on termin ation must cut thr. surfac e of the ball charge at the same radius as that a ι which it comm ences, it follows that the equili brium surface of the ball charge is modif ied. These differences arι• clearly broug ht out by the diagra m of Fig. 2.10 in whi Ιι traject ories startin g at the same point are plotte d ίιι accord ance with the two treatm ents and the corres pondin H surfaces of the charge are shown . As alread y stated , the motio n of a mill charge is considera bly more compl icated :than the previo us treatm entH would imply , and so far the variou s propo sed treatment κ are not compl etely satisfa ctory. One of the most compr ehensi ve analyses, and that upοιι which most of the remai nder of this chapte r is based, ί κ that develo ped by Barth (6) and, indep enden tly ofBar th, by Hinsle y and Fobele ts. (7) * , was d evelop!'cl * This work, which h as not previously been publishedwho were appar·· over a number of years by M essrs. Hinsley and Fobelets, ently unawar e of the prior work of Barth . their mater irι l Messrs. H insley and Fobelets h ave very kindly m ade all on the baκl• written, was apter ch this of uch m and authors availabl e to the of the presc: rΙI of their m a terial, before the work of Barth came to the notice · writers. indebtedness Ι ιι The present authors are pleased to acknow ledge their tter. a m this in Fobelets and M essrs. Hinsley

53

Iη ιhe first place the equili brium surface of the mill ve Ιι . ΙΙ 'g~ is de ter~i_n ed and for this purpo se the effecti r) powde plus (balls ιιι · Ιlι 1ent of f~1ctωn * of the c? arge n frictio of ent i coeffic d eral1ze gen Α ιιιιι Ν t be establ1shed. of heap l conica large a g formin by ined determ ΙΙΙΙι y be

1 1

CuRνε ΡΑ - PARAθOLIC τ RAJEC TORV.

ΡΒ- τ R λ JE C TORY O rVoN SτειcΕR

Bc- Ρ••••οιιc

τ RAJεcτo•v .

·

FIG. 2. 10

Ιlι ι · ιηill ~harg e on a flat surfac e, as slιown in Fig. 2.11; the lιι • 'Ι ' be1ng formed by shovel ling the mater ial ση to the the coefficie nt . I 11. th is oveι·all coefficie nt there are compon ents such as ; a ''plough -

ιι llι,:ι •! ι οη between th_e ba ll a nd the surface of the powd er particle of the ba ll through ';' Ι : ~~ ιοι· representιng the force req ~ired to move the tip a factor which, for Ι '' ""' lace layers of the_ powder partιcle ιη q uestion, anddisplace ment factor ·'" I of a _better expressιon, can be called a mecha nical of th~ Ιlιι ιιι rιgnιtud e of wh ic~ is determi ned largely by the mode of packίng

hich are rubbed 1'' 11 IJι• l ., and th e coeffic_ιe n toffrict ion between the p articlesw e compon ents 'I ΙΙΙΙΗ tοηe another dur~ngd ιsplacement. T he m agnitud eofthes l; for eχ Ι ι l ι·Ι~· rmιned by physιca l cha racteris tics of the powder materia of ploughi ng of ιιι ιιι l ι , ~Υ the h ard ness, whιch wιll determi ne the degree Ι"''"' οί contact of the ball through the surface of a p article.

54

ΜΟ ΤΙΟΝ OF ΤΗΕ CHARGE WITHIN Α MILL


apex of the cone and allowing the stable angle of r~posc Iι ι become established. The angle of repose, γ, ιs thι · ιι measured and the coefficient of friction λ is given by λ= tanγ

55

<) η 1ι ι

equating the forces acting on the particle the follow•, •xpression is obtained:

λιιι '/ 2

cos

( β- α)+ λmg

sin

β+ mrη2

sin

(β- α)

= mg cos β

.... (2.2(1)

. • . . (2.27)

Let the upper surface of the mill charge be given b">:' tlιι · curve Α-Β of Fig. 2.12 and take the centre of the mιll : ι the origin; al~o take the co-ordinate axes as shown. Let tlι

Fισ .

2.11

Fισ.

mill shell rotate with an angular velocity η and .also assunιι• that the particles of the charge describe circular aιη around the origin with this velocity. Ν ow any particle, Ρ, of mass m, ο η this surface, at ιι distance r from the origin, will be in equilibrium under tlι influence of three forces: (1) The radial "centripetal" force: mrη 2

l ttl ιi ply this equation by 1fmη2 and rearrange.

( •/ ιι ~ sin

β+r

cos

"

β

=

cos

α+r

gfη 2

cos

sin β sin α) r sin β cos

β-

ιιιι I ιh en substitute χ= r cos α, lι y iιι β which then gives λ(gfη2+χ

(2) The weight: mg (3) Frictional forces acting betweent11epaι:!_icle and iι neighbours.

2.12

I l ιι\VCVCr, cot

cot

β=

y = r sin

β+y) = gfη2

dyjdx,

cot

α,

α+

r cos

-

β-χ+y

λ(gfη2

sin

α

and also divide cot β

SO

dyfdx(gfη 2 +y- λχ)

β

+y)

-χ

.

56

Μ Ο τΙ ΟΝ OF ΤΗΕ CHAR GE WITHIN Α MILL


57

Now let z= (gfη2 +y), which, on substitution, gives, (z-λx)(dzfdx)

=

-λz-χ

dz λz+χ dx = λχ-z This is the standard form of homogeneous equation ο Ι ' first order and may be solved by putting z = kx, wherc λ is a function of χ. This, on integration, gives, log χ λ tan-1 k-t log (1 +k 2) +log C z Jx 2+z2 λ tan-1 χ -log ~

~:::-----Ιλ•Ο·25

λ•Ο-5

b

α

z x2+z2 tan- 1 log - c Now ifwe write: x2+z 2 =R 2, tan- 1 (zfx) =ψ, the exprcli·

or

sιon

χ

R =

CeλΨ

(2.28)

is obtained; this expression being the equation of aιι equiangular spiral. . . . Also since R2 =χ2 + z2, ιt follows that the ongιn of tl11· spiral (the pole) where R=O, . must ?e situat~d at χ = Ο,, y = - gfη2; that is on the vertιcal axιs at a dιstance gf'r! ~ above the centre of the mill. (lt should be noted that thιH distance is also the diameter of the circle, the Davis circl<·, defining the surface at whic~ p~ojection occurs.) T?e co· efficient C is a parameter whιch ιs related, but not sιmply , to the degree of filling of the rnill. . The configurations of the surface of t~e. charge wι~Ιι various values of speed coefficient of fnctιon and mι 11 filling, as calculated on this basis, are shown in ~ig. 2.1 :ι Examination of this figure shows that, for a gιven ~ιll filling and coefficient offriction, the surf~ce. con~guratι ιι of the charge is not greatly altered by vanatιo.n~ ιη ( Nf Ν,) the ratio of the running speed to the cntιcal speccl . Similarly, for a given value of the ratio Nf Nc and a giv ·ι ι

c Fι G.

2.13

ιιι • Ιrί .ient offriction, the slope of the surface does not alter ι ι ι · .ι t ly with the rnill filling; furthermore, the line defining 1Iι ι ιωrface is almost straight. ln fact the average slope of Ι Iι ι • ιωrface increases very slightly with d ecreasing filling. Ί Ίι c variable which has the greatest influence on the 1 ι ι ιι Ι iguration of the surface of the charge is clearly the ι Ιιι· l llc.ient of friction and, all other variables remaining

1

58


unaltere d, the slope of the surface increases rapidly wit Ι ι increasin g coefficient of friction. It is desirable that the meaning ofthis surface should l>c· borne clearly in mind. lf the derivatio n of the equation ί studied it will be seen that the equation represents the su ι· face upon which a particle is in equilibri um under tiH· influence ofthe centripe tal force, based on the assumptioιι that the particle is traversin g a circular path around tlιc· centre of the mill with the same angular velocity as tiH· mill shell, ofthe weight ofthe particle and ofthe friction:ιl force between the particle and the underlyi ng surfa τ . Now, clearly, in a real mill the balls which come up fr 111 the body of the ball mass must either be projecte d or muHt roll down to the toe of the charge and so, in cascadinH, this mathem atical surface cannot represen t thefree surfaα• of the ball charge. lt is probable , however, that a surfaα· of approxim ately this form exists a few ball diametcι ·H below the free surface; that is, there exists an equilibriuιιι surface down which the cascadin g ball charge rolls. ThuH, such a free surface cannot exist with cascadin g moti ιι , although it is probable that οήe does, over a portion of tlιc· charge, in the case of cataracti ng motion. It is clear from the previous discussion that, ifprojectioιι of the balls is to take place, the charge must cross the Davi circle. Now, the equilibri um surface, as defined by tlιι• equiang ular spiral, can fall either below or above the DaviH circle; as shown in Fig. 2.14. When the equilibri um suι· face falls below the Davis circle, the particles, as , th(Ύ emerge from the main mass of the ball charge, pile ιφ until the Davis. circle is reached; as shown in Fig. 2. J!"> . This is clearly possible since in the mass of balYs lyiιι H above the main equilibri um surface but below the D avi circle, there exists an infinite number of equilibri um suι· faces such as ab. The inner boundin g curve for this mass of balls is :ι circle concentr ic with the mill shell and tangent to tlιc· equilibri um surface. The position of this circular arc anιl the equilibri um surface is, of course, fixed by the conditioιι

ΜΟτΙΟΝ OF ΤΗΕ CHARGE WITHIN Α MILL

59

Ι I Ι . Ι t ιhe "rolling" mass of balls and the "flying" mass are

Ιιι~ c · t l1er equal to. J, ~he static filling of the mill. Since, in

' ase, the prOJCCtωn takes place from the Davis circle lιι · nstr_ucti?n fo~ the terminat ion of the trajectories: ν 1 :η. e~rl1er, 1s val1d. ln the second case, in which the projectio n I ιιι Ι ΙI ιbnum surface _falls above the D avis circle, reached. is surface ΙIΙΙι 'Ν not occur unt1l the equilibri um ΙIΙ Ι

Fι c .

2.14

Fιc .

2. 15

Ί 11ι 1 ι· ρ ΙΌjection does not now occur at the Davis circle

' Ιlι Ι ι ιn~vious construc tion must be modified. 11 ι · asoning very sim~lar ~ο that used previously, it can I Ιι lι own _ that the te.rminatiOn of a paraboli c trajector y, ΙΙΙ ΙΙΙΙΙΙ ·nCing at a pωnt Ρ' above the Davis circle is ob1 ΙΙΙι c ·ι Ι w~en, ση Fig. 2.16, distance P'Q.: = Q:R 'and the /'ι 1 ΙΙΙ / C ' zs on the D avis circle. Thus the construc tion is ιιιΙI : ιι· to the pre~ious except that point Q: is on the Davis ' 11 Ι ι· whereas pωnt Q is on the mill circle. So for either

-----

--

--

60

~



61

case the point of termination of the trajectories may IH determined by simple graphical methods. lt is interesting to have a knowledge of the proportioιι of the charge which is in "flight", and of that whiclι i "rolling"; also to see how these quantities vary with tlιc• speed of rotation of the mill and with the coefficient οΙ friction of the charge material. Α simple construction ί(ιι this purpose will now be described; this construction bcίιιι.ι due to Fobelets. (Β)

Fισ.

Fισ.

2.16

CASE 1. The point of projection on the Dav~ circle: (a) As in Fig. 2.17, draw the mill shell circle and t\ιι• Davis circle. (b) With compasses determine position of point \{ ; the termination of the trajectory originating at 1'. (Use point Q on mill circle as centre.) (c) From the pole C (ση the Davis circle), draw radiιι vectors at 10° intervals; taking vector CR as datuι11 . (d) From C set out along the successive rays a lengtlι Ca = CR. e-λ(π/lΒΟ)δ, where CR is the length '\{

( ι ·)

( Ι ')

(~)

2.17

measured_from the figure, δ is the angle, in degrees, fr
62



this Step may be dispensed With and pointS ι I ι joined by a straight line. There is then some los~ οΙ accuracy but this is not generally of great practι<· ιl importance.) (h) Determine the area SbaRPCS. This when exprcsHc·cl as a fraction of the area of the mill circle giv s ./ , the static mill filling. (i) Determine the ratio of areas PCSbP and PbaR I' ; this ratio being the ratio of the "flying" charg to the "rolling" charge. When the equiangular spiral 'enters the Davis circle tlιι · treatment is slightly more complicated and demands sοιιιι · trial and error. Thus, 1

CAsE 2. The equiangular spiral enters the Davis circl (a) As for Case 1. (b) Assume a position for the point of projection, I'', and using the construction of Fig. 2.16, determiιιι· point R; the termination of a parabolic traject ι· which starts at Ρ'. (c) and (d) As for Case 1. . . (e) J oin points such as "a" and thc equιangular sjnrιιl should pass reasonably close to Ρ'. If it does not,. st:ιι· t with a new point Ρ' and repeat the constructιon. (f) to (i); As for Case 1. The basis of this construction is as follows. lf the flight trajectories are parabolic, then the hoι· i zontal velocity χ' is constant and given by χ'

=

ωr

cos

α

... . (2 .2!1)

where α is the angle of projection and the other symboiN have the same meaning as before. Also the time of flight, ΤΡ, is given by ΤΡ

= xfx' = xfωr

cos

α

. . . . (2.30)

63

l)uring this time, however, the rotation of the mill will charge at radius r,

Ι ~ι i ι1 g, from the depth ofthe mass ofthe ι c·olumn of balls of length ωrΤΡ. ιι

'1'11is is the quantity of balls projected in time Tp, from Nlrip of width dr, and this may be written dq1 . Then dq1 = = l •'ωm

ωr(xfωr

xfcos

α)

α

α

= PS

(2.32 )

from equations (2.31) and (2.32) q1 =

f

PS dr

Ι ιc· ι· the integration is carried out along Ρ. This integral is, however, equal to llι•, ι ι r bPCS and hence the construction.

l1 to

(2.3 1)

Fig. 2.17, however, it is evident that

xfcos Ί'hen,

cos

the curve from the area ofthe

lthough there is still an element of approximation in 111 iNtreatment, it appears to be superior to that of Davis in ι l ι 1ι the effects of frictional characteristics of the charge ι ι ι • included and the computation of the "flying" charge ι l ι ΙΙ 'Ιi not involve reference to some mean radius. Ο η the Ι ι ι Ν i Η of this treatment it is possible to study the conditions l ι ιι taracting or cascading in some detail and this will I IΙIW be done. . IΙ will be noticed that, so far, no rigid definition of cas' ~ι l ing and cataracting has been given. ln fact such a ι \ 1 l i nίtion is probably impossible since it now appears that ι lιι ι ·' are not two types of motion of the charge involved, I Ι ΙΙΙ a single type of motion in which, in certain cases, some 1\1ω i ls ofthe motion are not apparent. For the present purΙ•ΙΗ Ι ' it is probably sufficient to name the motion "cataract1ΙΙ " when an appreciable ga:(> exists between the parabolic 11 ι j • tory of the innermost particle and the upper surface 1•l t \ι rolling charge. Cascading then corresponds to the ιι ιι 1 t i η in which this gap is so small as to give, in general, ιlιι appearance of being non-existent.

64

~~ O'I'JON OF ΤΗΕ CHARGE WITHIN Α MILL


ι ι 1l ~;ιρ it follows that ι ΙIΙΙ ιιιίll filling for which

This difference is c1ear1y shown in Fig. 2.18, which art• the trajectories for a mίll runnίng with a definite chaη:~ι· at a definite speed but with different fillings . ln Fig. 2.18:ι , with the hίgher filling, the space is very small a~d wh ιι allowance ίs made for the finite size ofballs the m1ll woulcl appear to be casca~ing. In the sec~nd ίllu~tration therc ί ιι a definite space wh1ch, even allow1ng for 1nterference b ·tween the balls in the different trajectorίes, would not b<• obliterated, and hence the mill wou1d be described :ιιι cataracting. For the present purpose it will be assum ιl

65

these va1ues are the maxi1num cataracting will be apparent.

ΊΆΒLΕ

2.1

Maximum static mill filling for cataracting to occur Ν Ν,.

Area of circle

ιnill

( ίη2)

Area giving "rolling" charge (in2)

Area giving "flying" charge (in2 )

Jmax

(%)

Ratio Ratio Μ, Mr Μ,

Μι

-ι

ι

11•11 οι.

ιι

75

11'1

111

ΙΗι

0·75 () ·()

11

IHi

0·75

O·!J

(b )

Fω.

2.18

that when this gap is 1ess than 0·05 of the diameter of tiH· mill the charge will appear to be cascading; this gap bein κ cho;en because, in view of the finite size of the ball in :• rea1 mill, it is improbab1e that a gap_ smaller than tl1ι H· figure wou1d be discerned. By the p1ottlng of a number
3· 14 3·14 3·14 3·14

ο

ο

ο

ο

0·50 0·81 1·28

0·15 0·50 0·9

20·5 41·0 69·8

0·23 0·38 0·42

1·0 0·77 0·62 0·58

3·14 3·14 3·14

0·46 0·90 1·26

0·15 0·5 0·9

19·0 45·0 69·1

0·25 0·36 0·42

0·75 0·64 0·58

3· 14 3·14 3· 14

0·38 0·82 1·15

0·15 0·50 0·9

17·1 41·5 65·0

0·28 0·38 0·42

0·72 0·62 0·58

IΙ ί :-; seen that a fourfo1d νariation in the coefficient of Ιι11 Ι iι η produces, for a given speed of rotation, but a ιιι ,li ι· ib1e variation in the maximum mill filling which will ι" ι ιιιίι cataracting. Thus, it is probably safe to say that ιlιι ι · ι· sults, shown graphically in Fig. 2. 19, are app1icable Ιιι , ιιι y practica1 mill with an accuracy sufficient for any ιιιι 1 ιιι ;ιl purpose. The genera1 va1idity of these resu1ts has ι1ιι iv d an amount of indirect support in the following ιl

.

ΊΊtι· photographs ofthe motion ofthe ball charge, pub11 Ιιι · ι l by Rose and Evans, <9 1 have been questioned ση ΙI ι ι A'•·ounds that they show η ο cataracting. Α study of Ιlιι orίgina1 artic1e shows, however, that the three cases ιιι .! = 0·25, J=0·5 and J=0·75, with N fNc=0·56 in ι ιιlι ase. From Tab1e. 2.1, however, it is seen that each 1ι l Ιlι se va1ues of J is above the maximum for which

66


ΜΟ Ί'[QΝ OF ΤΗΕ CHARGE WITHIN Α MILL

cataracting will be apparent, and so the results shown ίιι the photographs are to be expected. The introduction of a coefficient of friction of consttι ιι ι value, as in the extended theory, is not completely adι·· quate, however, since a study of Fig. 2.13b suggests tl1 : ιι the average angle of slope of the surface of the chaΓ!{« ' should decrease with increasing filling, whereas tiH•

Ι ι Ι ι ι ιιι ιι , λ, would bring about an even better agreement

l ιι ι 1\Ψι :ιι tl1eory an~ practice. Unfortunately, the introducΙiιιιΙ ιι / Ν ιι h a vanable quantity very much complicates

ιlιι ιι ι. ΙΙIΙ ·matical analysis. \ 1Ιιι , ι1 1 and very important, use to which 'ι111 Ι ι ιι ι · ιι ns can be placed is to ensure that

1·0

I

I

Ο·

·-

81-- -

)(-

λ

• O·S

λ=

1·0

λ=

2·0

j

--, ~ 0-6 z

Pol< .

1/

:::;

.J

t'

ι: -'

I

Ο· 4

.J

i

v v

CASCADING

RεcιοΝ .

0 ·2

ο

0.2

--

CAT ARACTΙNG REGION

Ο ·δ

0 ·4

Ο·θ

Ι ·Ο

(t-y'Nc)

FΙG.

2.19

measurements of Ros e and Evans(9) show that the slορι· increases with mill filling. This difference is easily xplained since the charge consists of a limited number οι · balls of finite size and, in consequence, the energy di ιι sipaied in friction in the charge increases with the size οι· the charge. This has an effect equivalent to an incre;asc in the coefficient of friction of the charge; and so brings aboιιl an increase in the slope of the surface of the charge, in tlιr same way as is shown in Fig. 2.13a. Thus, it appears th :ιt the int.r oduction of a variable value for the coefficient οΙ '

67

FIG.

2.20

the foregoing the condition

68


that the charge shall not slip ση the milllining is fulfill '(\ . For this purpose, (1) Use the previous constructi ons to de~ermine ~11ι ' configura tion of the "rolling" charge ιη the mι\1 ; shown shaded in Fig. 2.20a. . , (2) Determin e the position of the centre of gravιty, (,, of this area. . 2 3 Through G plot the radial acceler~tιon ω r!: ( ) d' 11 and the gravitatio nal acceleratιon g vc.rl!· ra llιa ~Note. radial force = MrGω2 and vertι :ι\ ~~rc~· = Mg, ~ο these vectors are proportio nal to thι• forces.) (4 ) Obtain the resultant of thes~ vectors. . Draw a straight line connectιng the centre of mιll ~) (S) to the point Ρ, where the resultant cuts the mι\1 shell, and determine the angle. ε. . . , . (6) Compare tan ε with the coefficιent offnctιon λ. ~οι safety λ must exceed tan ε by a reasonabl e margΙΙι. The last condition is clear from Fig. 2.20b, ~ince thι· resultant has radial and tangential compo.nents a~ showln . The tangential componen t tends to gi":e nse to s~φ a: t ~~: surface and slip will, in fact, occur ιf the ratιo ο .t ι tangential force ~ο. the radial force exceeds the coefficιcn t of friction: that ιs ιf tan ε> λ. I Ι.

2

. 3.

4 5. 6. · 7. Β.

9.

REFERENC ES VANS D. Ε. Proc. Instn mech. Engrs,Lond., 1956, 170, RosE, . Ε ., and E ' 773. . f 1 R Η Ε and ΕνΑΝS D. Ε. Loc. cιt., r e . . . GOSE, . w'' Lecture 'to the Fachausschuss fίir Staubtechnιk of tlιι· RUNDER, · Ι V.D.I. Dec. 1955. 19 61 250 Trans. Amer. Jnst. min. (metall.) Engrs, 19 , , · W Ε D AVIS, · • VoN STEIGER R. Zement , 1929, 18, I \06. B ARTH w. 'τech. Mech. Thermo-dynam., 1930, ι, 321. . . ' HINSLEY, J . F ., an d FoBELETS ' Ρ. Private Communιcatιon to tlιr authors. FoBELETS, Ρ. Procέdέs Henricot, 1955 . . RosE, Η. Ε., and ΕνΑΝS, D. Ε. Loc. cιt., ref. Ι,

Η

CHAPTER 3

ι ιιι ·: ι

WER REQUI RED MILL

ΤΟ

DRIVE

Α

I ι ι ι 111 ιw τ required to drive a tumbling mill is of interest Ιιι ι ι I ι Ι ι ι ι Ι ι designer and to the mill operator: to the former ι Ι ι . ι Μ iΗ f design for the determina tion of the necessary 1

111 tl1

ι 111 1 , ,1ll ιΙΙ ιι lιΙΙΙι ·

l1

Ι ιΙ κ.

elements of the machine; and to the latter beother factors being equal, the most economic al is that which demands a minimum power for

I l1ι • ρ wer required to drive a mill depends, to some ι ι 1 ι ι Ι ο ιι every one of the physical dimension s defining 1!1 ι 111 i 11 l1ell and,ball charge and ο η many of those definιι ιlιc • 1 roperties of the powder charge. Thus the number Ιι l ι 11 i • ι bl s involved is very large. Since even a moderatel y ι ΙΙ ΙΙΙ\ ι lι · ι · theory for the internal dynamics ofthe ball mill, 111 · 1 iι · Ιι all these variables are given due importanc e, has ι 1ι Ι I ιι · ι · 11 propound ed, the calculatio n of the power requireΙIIι IΙl of a mill, from theoretica } COnsiderations, Cannot be ιιΙιι l ι , ,' imilarly, owing to the great number ofvariable s, 1111 ι ΙιΙΙtρlete experimen tal investigat ion of the power deΙιt ΙΙtιi Η f ball, tube and rod mills has been made, the ιιιΙιιΙΙιιt of work required for an analysis of all of the ιι Ιlι l ·s being prohibitiv e. l11 olι~ιbly one of the most comprehe nsive experimen tal Ιιι ι Ιt li ι:( tions of the power demands of the ball or tube ιιΙΙIΙ iM that carried out by one ofthe present authors, with 1 1 1 ι w rker, Rose and Evans, ω by use of small-scal e ΙΙΙΙιΙiι • I Η; the test apparatus being that shown in Fig. 3.1. l\ 1 ιι . ι· [ small-scale models, a very large number of tests ιιΙ.Ι Ι ι· carried out in a reasonabl e time and, furthermo re, ο Ι ι I \)lcs WruCh WOU}d be difficu}t tO Change Oll a large69

70

scale test may easily be studied. Νο valid major objection to the use of small-scale models exists siηce by use of thι~ method of dimeηsioηal aηalysis the results may be geηeral ized to be applicable to mills of aηy size. The valicl geηeralizatioη of the results of the model test demaηds that there should be a complete dyηamical sirnilarity bctweeη the systems, aηd, iη the case of the ball rnill, th · pressures iη the bed ofmaterial iη the mill caηηot be madι·

ter of ball, d, the deηsity, ρ, of the ball, the volume ι ιιpiecΓby tlϊe ch~rge (iηcludiηg voids), expressed as a Ι ι ι ι ti η of the total mill volume, J, the speed of rotatioη, ι , Ι l1 c acceleratioη due to gravity, g, the coefficieηt of ι ι Ι i ιutiΌη of the material of the balls aηd mill, e. The ΙΙΙι~νι· r, Ρ, would also be expected to depeηd upoη the Iι ιl l owi ηg characteristics of the powder; the represeηtative ιiΙ , Ι ΙΙ Η~ter of the particles, b, the energy ηecessary to bring ιlιtιι ι l uηit increase 1η the specific surface of the powder, Ι •, ι ι1d the volume, V, occupied by the powder charge ι Ι ι ι ι Ι ι ι Jiηg voids), expressed as a fractioη of the volume Iιι Ι wι· η the balls iη the mill. Furthermore, the power, Ρ, ι ι ι ι I ι be expected to depeηd upoη the effective kiηematic •ι ιιΝ i ιy of the mixture of powder aηd fluid, ν, the effective •ι, ιι ~ ί ι y of the mixture, σ, aηd, iη the case of wet milling, Iι\ Ι Ι ι · ratio of the volume of the solid material to the ιι lιιιιι of the liquid iη the pulp, U. Fiηally, wheη the ιιlι ι ior ofthe mill is fitted with lifters it would be expected ι Iιιι Ι ι l1 power, Ρ, would depeηd ο η the ηumber oflifters, n, ι ι ι ι I ι φοη the height of the lifters, h. l ' lttι s, the power caη be expressed symbolically:

ιiΙ. ι ιn 111

I'

FιG.

3.1

Apparatus for Tests on small-scale mills

strictly similiar iη a large aηd a small mill. Ενeη so the satis· factoΓy results obtained by use of model tests suggest that errors arisiηg from this source are small. Ιη the pres 111 chapter the questioη of the power to drive a mill will bι· approached from the poiηt ofview ofthese riιodel tests aηιl the results ofpΓeνious workers studied by com~arisoη wiιlι these data. The power, Ρ, to drive the mill would be expected to eter, D, thι· depeηd upoη the leηgth ofthe mill, L, the d/

71

Τ ΗΕ POWER REQ,UIRED ΤΟ DRIVE Α MILL


ι/>(L,

D, d, ρ, J, Ν, g,j, e, b, Ε, V, ν, σ, U, h, n)

(3.1)

Ιι ι ΙΙ ' φ deηotes some fuηctioη of each of the quaηtities ΙΙ iι ί ιι the bracket. Ι ' Ιιι • ιι from the well-kηowη priηciples of dimeηsioηal 111 ιl yH is it is easy to derive the expressioη iη terms of •IΙΙΙΙΙ ' Ιι ~ ίοηless groups,

(ιι n'ι :ιρ)

=

Φ{ (~), (~), (~), (niτ2)' (i), (D3~2ρ)' (n~N ), (~),

(J), (j), (e), (V), (U) , (n)} .... (3 .2 )

111 ι ι ι/> deηotes some fuηctioη of each of the dimeηsioη ι οιιps upoη

the R.H.S. of the

equatioη.

72

I' Ιffl, PO WER REQ.UIRED ΤΟ DRIVE Α MILL


Theoretically, the dimensionless groups can be combined in any way and it is the function of experiment t determine these relationshipso ln the present work, however, it has been found that, in most cases, the functions of the dimensionless groups may be multiplied togethcr and the results of the experimental analysis have been arranged in this wayo On this basis equation (3ο2) can be written:

(n5~3ρ)

=

Φι(ntz) ο Φz(~) οφ3(J) οΦ4(~) οφ5(n) χ Φιz(~ 3; 2 ρ) 0 Φι3( U) ο Φι4(D:Ν)

(3ο3)

ln the work of these investigators, attention has becn confined to dry grinding and, in this case, the groups (νfD 2 N) and ( U) are eliminatedo The first variable to be studied was the effect of thc length of the mill on the power requirements: that is, th ο determination of Φ 2 (L/D)o The experimental results of' Rose and Evans fall on a series of straight lines passin~ through the origin and so it follows that Φ 2 (LfD) is ;ι linear functiono Thus, ο ο

•

•

(3.4)

The critical speed of a mill has already been discussecl in Chapter 2 and it has been shown there, by equation (205), that the critical speed is given by (3.5) ω, = y(2g/D) which, by transposition, gives

gfD =

ΚΝ, 2

ο ο

roup (g/DN2) is proportional to the square of the actual speed ofrotabe written:

Ιlιι ι .~ι .iο oΓthe crit~cal sl?eed to 1 ''" ΙIΙωι the relatωnshφ may

ο ο ο ο

(3ο8)

IΙ

ΙI ι ι Ι ί1 Ι ~ ws that in order that dynamical similarity, as Ιιιl ι Ιιι ~ group, shall hold between a number of mills of ι l ll ι ιι ιιl ι li ameters, it is necessary that they shall all run ι Ι ΙIΙι Μ~ Ι ιnc fraction of their respective critical speedso Ι I Ι Ι i ιι. ~ cordance with the findings of Gow Campbell

ιιιι I ι :11 Ι ι ί 11 <2> and other workerso

χ Φ6(~) οφ,(f) oΦs(e) οΦ9(i) οΦιοα) oΦn(V)

(LjD) = K(LfD)

ιι ΙIΙΙ\ Ιlι '

73

..

(3ο6)

lf now Ν is the actual speed of rotation, it follows th< ι ο ο ο ο (30 7)

' tionship between the P?Wer group ( PfD6N2ρ) ΙΙΙΙI [ IΙ ι: gr~up (Ν,/ Ν), oaso determιned by experiment, is I Ι',',' 11 Ι!ι fιιgο 3_.~, and ιt ιs seeon that for speeds less than ιιιι ΙI Ιtl ι11 cntιcal, the relatωnship Φι' (Ν,/ Ν) can be ,, 1 , ι ~~~ · 1 by Φι'(Ν,JΝ') = K 2 (N,jN) 2 ο ο ο ο (3ο9) ll 11

11 · 1<

Ι ιιιι 1\~ = 3°1 3 for lifter mills and 3°66 for mills without ι ι, yYithin this r~nge these relationships can be used ι I 11 ι ΙI y ι η the oequ atωr:s for the determination of power 1• ψΙΙΙ 'Ι ' ι l ιο dnve a ~ιll. The deviation frοήι linearity

I IΙ ι

'' "' ιι '' Υ

theose cuivesoιs exactly what would be predicted since, ideally when the IΙ ΙΙ IΙ· Ι ιιι ll charge is ~er:trifug~ng, the centie ~f gravity of 1111 1l1 11·g would coιncιde wιth the axis of rotation and Ι I Ιι ι ιιιw •r required to drive the mill would be zeroo l11 ι IΙ t' ry, the centrifuging of the charge takes place in Ι 1 1 1 :ι ι ιd each layer form~ at a de~nite speedo Ι η pracΙιι ι , l ωw ver, the p:ocess ιs complιcated by interaction I ι ι I \ t ' Ι ' II tl1e balls, lιfters, etco, and centrifuging occurs ιΙΙΙΙΙΙ ΙΗ' less gra~u~lly over ~ wide range of speeds, this Iι ΙΙIΙιι ' ι the devιatωn from lιnearity, shown in the figureo 111 ι • (lι se curves are plotted on logarithmic scales the 1 111 \'Ι'Ν H l1ou~d be asym~totic to the negative branch of the 1 Ι , : ιnd ιη fact the lιnes show this trendo ,Ι ιιtι: ι·, ~s has already been mentioned, the critical speed 1 ι ι Ι\ τιοη for the occurrence of many phenomena in a IΙ ιι ΙΙΙ

ΙIΗ' retιcal consιderation

75


ΤΗΕ POWER REQ.U IRED ΤΟ DRIVE Α MILL

mill, it is perhaps of interest to mention that ~uring th<· course of tests at various speeds, the configuratιon of th balls within the shell was observed through a perspex enιL

ιι l ι ·ι ιation exceeded the critical value Ν" by a very small ιιιι ο uηt . For a smooth mill, cataracting of the charge ι 1 ιHτally commenced at speeds of about 0 ·8Ν" while, 1 lιι • ι1 lifters were fitted to the shell, cataracting cοm ιιιι 11 · d at about 0·6Nc. This is roughly in accord with the ι ι 1ιlts of Chapter 2, illustrated in Fig. 2.19, that if a mill \ ι ι\ ι a 30% to 40 % filling is taken as repres entative, the

74

s.ooo

Smo!,hJII.i

I

ι.

I

ί

1,000

I

500

I

I

1./

I I

I

I

I

/ /1~ Mill with lifters.

7

\•ο

I

I

I

1/

1·0

I

/

oj_ 5

I

5

10

I Fισ.

100 50 SMOOTH MILL Nc/ N 10 5 MILL WITH LIFτERS Nc/N

1·0

FIG.

500

1,000

5Ο

ΙόΟ

3.2

cap and in the case of the smooth mill, centrifuging οι · the Όute; layer of the balls was not observed at speeds UJΙ to 1 ·2Ν" the highest speed reached during the tests, bιιt when effective lifters were fitted to the shell, the οutι·ι layers of b alls commenced to ((entrifuge when the spcrιl

3.3

lιΙιιιιl(lary between cataracting and Ι " Ι '( I [ rotation is about Ο· 7 N c.

cascading is when the

\'Vitl1 a smooth mill surging of the entire charge was ιιιιιι·Ιimes observed for speeds below about 0·4Nc. ! 1ο ι η ing now to a study of the effect of the density of the ι· ιΙιιι Iing media upon the power required to drive a mill it l ιι llιι ws that since the mass of grinding bodies has a ιlι ltιιil ' configuration, as suggested by Fig. 3.3, then the Ιιιιφιι • ιο drive the mill will be r9J20rtional to the weight ιι l Ι Iι ι · harge; that is, without owder pro ortional to the ι \1 ι ι i ι y of the material of' the al1s or rods. The correctness

ι ~ -~- • .ι~-~-- --

76

' Ι ' 11 I~ P OWER REQ.UIRED ΤΟ D RIVE Α M I LL


of this view is established by Fig. 3.4a, based on the work of Rose and Evans, ( 3) in which the measured power is plotted against power calculated on the basis of this assu mption for charges of glass, steel and lead balls; th. · relative densities of these materials being given in Tab]ι:

10.000

5.000

~ ~

~J(jlι

"' Σ :ε:

<( α::

1.)

1.000

:ε:

u I

v·

α::

~

ο

500

ι

I / 4:Πooth LίD'=

"-

ο

:;'

"'"'

"'ο

mill; • steel balls;

100

100

500

,_/

ο

1·0; D/d = 11 ·0; J = 0·5. lead balls; + glass balls

1.000

CALCULA rED POWER - CM

5.000

TABLE 3.1 Materia1

7"

..

I

"""-

v

10,000

GRAMMES PER SEC .

77

Specific gravity

Glass

2·65

Stee1

7·8

L ead

1 J.4

ιιιιι ι · onfi guration as it has when balls only 111 ιl ι , Ll1 configuration is unaltered by the

are present, inclusion of ιl ι ι ι '' ιwιl red material, then it follows that the torque to ι !ι • ι t l1 c mill will be proportional to the total weight of the ι I ι ι ι ~ι, ι · i ι1 the mill. Thus it would be expected that, at least ι ι Ιιι·~ ι approximation, the power to drive a mill conι ιι ιιιιι f', R powder can be calculated from the equation 'lψlιι ι l>l to a mill containing balls only, provided that ιlιι ι l ι · ιι Η ity of the balls, in the latter case, is increased by an ιωΙΙιι ιι t sufficient to include the effects of the additional ι l ι ι to the powder charge. If this point of view is ιιι ι ιι l ι ψ Ι ι ·ιΙ , it follows that the mill charged with powder lιι ιιι ι ιΗ·s a special case of the mill containing only a ball 1 l1ι1ι ι-ι ι·, and effects arising from the particle size and other 1 I ι 11 ι ιι: t ·ristics of the powder charge may be expressed by ι '''ι ι •ι · ι i ns to the equations applicable to the mill without I '"~ ι Ι ι· ι· . On this basis, for a mill containing a powder 1

Iι

11

ι-ι ι :

(a)

I 'ι ιΙνι • ι· input

FIG. 3.4

(Τ

rque due to ball space charge + Torque due to

I' wder) χ Speed χ Proportionality factor

3 .1. Ι t is a t once seen tha t an extremely good linear r lationship holds between these variables and so it must b<· concluded that the suggested relationship is valid. If now the material undergoing grinding occupies th · spaces between the balls of the ball charge, it follows that the centre of gravity of the powder mass will coincide witlι that of the ball mass. Thus, if the mill charge retains th ·

Κ χ

(Wt. of ball charge + Wt. of powder charge)

Ί ' Ιι : ιι these terms are simply additive is supported by ι l ιι ι · ιιι ν s of Fig. 3.4b in which are plotted data given by ι Ί ΙΙ \ Ι ιί ΙΙ and Devaney,< 4 J for the power required to drive a ι ιιι l ιιιίΙΙ , in which the rods are tubular and the effective ι l ι ι ι i ι y of the rods is varied by "filling" the tubes with


' Ι ' 11 Ι•; POWE R REQ.UIRED ΤΟ DRIVE Α MILL

various mateήals. For these tests the mill was grinding chert, for which σc::::. 3·0, and, since the void space between the rods is about 15 % of the mill filling, it follows that, if the simple addition is valid, the power is given by thc expression of the form y = mx+c

ll ι11 tl1 functional relationship Φ 10 ( σ/ρ) is given by ιlιι ιιιιιltiplicative factor (1 +0·4σ/ρ) . I IΙΙ ' ν, lidity of this relationship has been further eχ ι ιι 11 ι ΙΙ ' Ι I l>y tests carried out using powders consisting of ι ιι ι ι ι , iΙ' η powder and silica sand; the density of these 111 ι ι ι ι t: I ~ bcing given in Tab1e 3.2.

78

Furthermore, the power is given by

79

TABLE 3.2

Ρ= Κ(Ο·Ο15 χ 3·0+0·85ρ)

and it is seen that the curve given coincides very closely t this expression. Ι· 5 ------r------τ----,-----,

Material Iron Emery Silica sand

Density (gfcm3) 7·9 . 95 2·7

ιr:

~

I ι 111' sentative ι Ι ι ι ι ιτduction to I IΟ) :ιbove, are

ο

Q.

w

V>

ιr:

ο

:τ:

ο EFFECτiVE

DENSIH OF RODS .

(b) FIG. 3.4

If now ξ is the ratio of ~,hL power tQ drive the mill charged with balls and powder to the power to drive th<: same mill witfi a a Ι cfiarge only t hen:

.

Wt. of ball charge χ Wt. of powder charge Wt of ball charge ξ = lf also it is assumed that the porosity of the ball charg · and of the powder is 0·4 in each case, then: ξ

(1-0 ·4)ρ+0·4(1-0·4)σ (1-0·4)ρ

= (1

+0·4σ/ρ)

1~ ο,-4 /· 2,"1 /-χ β, ~ { 7 14 1 + ο ι4 ,χ2ι (;./~,& ~ i ) 13

( .... (3.10)

data from both smooth and lifter mills, a consistent basis by the use of equation shown in Fig. 3.4c. Ι ι~ Ι! Ι ιηίnation of these data shows a scatter over a range ι ιl ιlιοιιι 5 % with no consistencies in the deviations. Thus, ι 'Ι ψι· ω·s that these differences arise from random errors ιιιιl ΙΙ() the functiona1 relationship Φ 10 (σ/ρ), given by ι ιιιιιιiοη (3.10), again appears to be satisfactory. It is ι l 11 1ωssible that some of the scatter arises from the actual 1ιι ιιι ι Μ i ι y of the powder and ball charge differing from the ι ιιιιι ·cl value of 40%, but such deviations are scarcely ι ιlι ιι l :ιbl e . Thus it is concluded that the function φ 10 (σfρ) ιιι ι y ι , · replaced by (1 +0·4σ/ρ), with ah accuracy suffiι ι ι ι ι ι· r practical purposes. The remaining functional ι ι l ι Ιιionsl1ips cannot be represented by simple algebraic ι Ι Ι ι 't 'HS i OllS and SO are best presented by graphs. Ι ιι ι Ι ι first place it follows from Fig. 3.4a that, since 111 ιtι · ι· i <ι ls having a wide range of values of coefficient of ι ι 11 ι ιιl ίο η are used and yet the points suffer no significant ι ~t ιΙι · ι·, the effect of the coefficient of restitution on the l "'~νι · ι · rcquirements ofthe mill is very small. Thus, at least Iι 11 ι I ι'l\ tical purposes, the function Φs (e) may be neglected.

80

' Ι ' ΙΙ Ε P OWER REQ.UIRED ΤΟ DRIVE Α MILL


This result would be expected since, although the coefficient of restitution of the metals would affect the conditions of rebound when clean solid particles make contact, the effects of changes in the coefficient of restitution of th · metals are, in the case of a ball in a mill, masked by th energy dissipating capacities of the loose bed of balls anr\ JSO r - - - - . - - - , . - - - - - , - -

--,----,---·-τ-~--,

~

w

"'>-

81

'• Ιιιιιι.ι l area of_the charge and the radius of the couple 1ι ι ι ι , Iιο ι h of whιch depend ο η the mill filling J. Ο η the Iι 1 ι , ι · ι l1 se assumptions it is a simple matter to calculate ιlιι ι ι ΙΙ 'ψι in terms of the mill filling and, when this is ιlιΙΙΙΙ , ι Ι Ι " theoretical" line ofFig. 3.5 is obtained. This ι1ιι ΙIΙιιι l f treatment is a gross over-simplification how' ι ι , in c, as explained in Chapter 2, the curve 'repre' ι ι ι ι ι ' ι h surface <;>f ~he charge depe.nds ο η the filling and Ι ι Ι ι I ΙΙ ιι ; ιl cl1aractenstιcs of the mateπal. Furthermore, the Iι ι ll1 '"' • not mathematical points and so there is mutual

~Joor--~~~-+~~-+---+-~~~~J-4-~ z υ

Smooth mill.

Σ ~2\0,L--~---~--~--~--_L_ _J_J___j

1 :

I

i

.;?: Ι ι ~J~Γ'--,Γ--4---~--~--~--~~~

.

ό

Th.ιoret ical Curνe. Ε χ ρ~rimental Curνιz.

8 -Smooth Mills. @ - Lifter Mills.

Lifter mill; n

J

= 6.

200 r~--~2S~-~S0~-~71~-~,~00--~,2LI_ _ _I\LO_L~ι7S SPEED-R .P.M.

χ

Quanz.

8

Emery.

Ο

Iron.

(c) FIG.

~.4

powder upon which the ball falls and by powder betwecιι t.h e contact surfaces. If, on referring to Fig. 3.3, it is assumed that the curv<· representing the free surface ofthe charge is a straight lin<· and a~so that the angle θ is unaltered by variation of thι· quantιty of the charge, then it is clear that the power ιο drive the mill is proportional to the product of the cross-

--i---__;~---\--1 0·6

0·8

1·0

FIG. 3.5 ι ιl1 ι 1 \· ι ·cnce between the balls. There is also interference Iιι Ι w ~η the balls and lifters, etc., and, when the relation1ι ι iHdetermined by experiment, a vastly different curve ι ι . :ιls? shown in. Fig. 3.5, is obtained. The importanc~ ι ι l Ι lι ι ~ dιfference wιll bec?me apparent later, when various Ιι ι ψ lι Γι d f~πmulae, whιch. have been proposed for the ι ιιιwΙτ requιrements of a mιll, are discussed. 11 l1 a~ already .been explained in Chapter 2, in which I Iι ι ι ηοtιοη of a mιll charge is discussed that the coefficient ι ι l ι ι ·ί ι: ιί οη between the ball surfaces ~nd the number of l ι . ι ii H ίη the charge are, from the point of view of motion of ι Ι ι , ι ι'Α' , not independent variables. For this reason the

82


functions Φ 4 ( d/D ) and Φ 7 ( f) must be considered toge tiΊt: ι·. Ι η Fig. 3.6a are shown the results of tests on a smo t Ι ι mill containin g clean steel balls and in Fig. 3.6b of ι IH

a mill with lifters. Examinat ion of these curves showN that, for the practical case of (D/d) greater than 20, tl11· value of the function does not differ significantly froι11 unity. For values of (D fd) appreciab ly less than 20, how· ση

ΤΗΕ POWER REQ.UIRE D ΤΟ DRIVE Α MILL

83

ι·ν r, the value of the function is not independ ent of the

of the mill. to test the effects of variation of the coefficient order Ιη ι ι f ' Γriction tests have been carried out in which: fιlling

( 1) The balls and mill are cleaned and tested dry. (2) The balls and mill are cleaned and slightly oiled. (3) The balls and mill are cleaned and treated with molybden um disulphide . (4·) The balls and mill are contamin ated with a trace of powder. (5) Α full powder charge is used. ' l'he coefficient of friction between surfaces treated witlι ιιιolybden um disulphide is about 0·05 as against 0·1 to 0·2 Iι Ι Ι' i1ed surfaces and 0·3 for dry steel on steel. The coeffiΙ Ιι · ηt of friction between the surface contamin ated with ι ιιιwder is unknown but is probably very high, say 0·5 lι t l·O. Ι n. order to investigat e the effect of small quantities of ( ΙΙtwder, quartz powder from the sample passing through ι ' 00-mesh sieve was added to the ball charge under ΙΙινι·stigation and no~mal tests carried out. The quantity ιι l ι > wder used for th1s purpose was 5·0 % of that required tιι fiiL the void spaces between the balls. This quantity of ι Ιι , ιι·gc was sufficiently small to ensure that, for all pracΙΙι . ι! purposes, the change in torque, from the correspon dIΙ ψ, <:nse of the mill charged with clean dry balls only, was ι lιι ι· ι the change in the coefficient of friction and not to Ι Iιι i η reased densi ty of the charge . As a resul t of these Ιι IH, no significan t difference was found between the ιι ιΙit Ν for the cases (1), (2) and (3) above, and the cοm l ιΙ ιιι ·< l Γesults for these tests can be presented by the full l1ιιι • ofΊ?ig. 3.7, this curve, ofcourse, correspon ding to that Ι ιι ι (. / ) = 0·5 on Fig. 3.6. It is probable that this drooping ι lιοlι': ι teristic arises from the comparati ve freedom of ι ι ιΙ , ι Ι ίο η of the balls when the coefficient of friction is low ιΙΙιl wl1 cn the number of contact points between the sur1 11 ι · ί s small; as is bound to be the case when (D / d) is small.

84

'Ι' ΙΙ Ι•; PO WER REQ.UIRED ΤΟ DRIVE Α MILL


The effect of adding a small quantity of powde~ ix shown by the broken line on Fig. 3. 7, this curve beωg. adequate for all values of (J) and, furthermore, .a study οΙ Fig. 3.6 suggests that this curve is adequate fo~ l1fte~ mi_lls. Thus, it m ay be concluded that ~or smooth. ~11ls gnnd1n A" materials having a low coeffiCient
111 11 ι lι:ιι · r d with iron shot or glass balls, has "slumping" ιι ΙΙiιι ι l ι :ιr e been observed during dry milling. Ιlιιι , ι1 cxplanation ofthis phenomenon can be offered Ι ι11Ι I i ρr bable that it is in some way bound up with a 11 Ι ι Ι iο ι ι ί η the frictional characteristics of the mill charge. Ι Ιιι ι τsul ts of tests carried ou t to determine the effect of ιlιι ιιιιιηl> r of lifters, n, on the power requirements to

J • Q-75 ,

.......

,Ι

1·0

0 -8

-~;.::-::

~

J·σ s-

v /

I\ 11 J· 0 ·25 ....-

Bo lls Onl y. StΙZc l Bolls With Α Trocc Of Quortz

I

I ο

\ Ο.

20 . D (Χ!) Fιc .

30.

-Ι• Ο

~~=ι

I li

ο

,I

D;d • 20

t?f± ι I

0 ·7

-

ι

40.

~

ιj

S tΙZcl

---

............_

(')

ιι

0·2

85

5 IO(D/d) 15 for(IJ.j)<20 Muιιiρly j1) &y Ί 5 ι I

20

25

-

50 ,

2

4 N umb εr Ο Ι

3.7

Fιc.

It has been suggested by that for val~,es of tlι,c ; Fatio (Dfd) greater than about 80 the ball charge slump~ the curve of Fig. 3. 7 falls for values of the ratιo an d ' thus ' . also state d t h. at t h' α . greater than this figure. It 1S .~s.eHecl ι particularly marked in wet mίlling, but, s1~ce 1t 1s rall1c· ι· unusual to use balls .so small that the rat.ω of (D fd) c ceeds 80, it is probable that this effect 1s not of gn·:ιt practical importance. In view of this comment the present authors havc, l1y use of the small-scale apparatus, subseq~ently extend.c·cl the range ofFig. 3.7 to (D jd) :::200 and, 1n no case, wιtlι

6 L if t c:r s - n .

Β

10

3.8

Bond, <5>

ιlι

rp

• ι :ι nιill, that is to establish 5 (n), are plotted in Fig. 'ΙΊΗ : Γunction 5(n) is influenced by a large number of ι ιιι , ιl> l ι·s to a small extent, and the curves presented in ΙlιΙ Ι ψ;ι ι τ· are the lines best representing the whole of

\11

Ι lι1 ι ιl ιιιa.

rp

This method of presentation is adopted since it l1Ι ι Ι lι a t the errors introduced by the use of a single line, 111 ι 1ι ι Ι :ι many, are not significant but that the simplificaι 111ι oiJtained is worthwhile. 1 1' ι ο ι η the figure it can be seen that, provided the number •I lιll ι ι·s is greater than about 6 and the ratio of (Djd) Ι

86

'1'1 1 Ε POWER REQ.UIRED ΤΟ DRIVE Α MILL


exceeds about 20 the function has a constant va1ue οΙ' unity. Thus it fo1lows that, provided t11ere a.re suffici nt 1ifters to effective1y key the charge to the mιll shell: tl1c· number of 1ifters is unimportant. For a number of 1ιftcι·H 1ess than 6, but with (D fd) 1ess than 20, there is sοηιι· dependence upon the va1ue of mill filling as shown by thc· branching of the curve to the 1eft of th~ figure. For prac··. tica1 purposes the va1ue of the functιon, for va1ues οΙ (D fd) 1ess than 20, can be ca1c~1ated by m~1tip1ying tlιι· re1evant va1ue read from the maιn curve ofFιg. 3.8 by thc• 1·0

0 ·8

-

.

@

-® ι--

οο6

~

'ΦΟ4

""Θ::.

e - D;d=20 @ - DlιJ = 10

Q-4

ι ο

Rot

0 ·1

ιο

Fιa .

I

ι Ο·

3.9

appropriate correction factor read off from the graρlι inset in that figure. The effect ofthe height ofthe 1ifters is given by the fuιιι · tion φ 6 (h/D) and the resu1ts of tests t? de~e~mine this fun.c· tion are given in Fig. 3.9, from whιch ιt ιs seen t~at Ιοι mills in which (D fd) is greater than about 20, the heιght οΙ the 1ifter has 1itt1e effect on power requirements. For t Ιιι• 1ess usua1 case, in which (D fd) is 1ess than 20, there js ι considerab1e deviation for these 1imited cases, but η straight-1ine interpo1ation between the t~o curves ~ιl Fig. 3.9 is probab1y adequate. T?ese conc1~sιons a!e val~cl on1y for 1ifters of reasonab1e heιght, that ιs, for 1ιfters ιιι which h is not greater than, say, 0·2D and not 1ess th:ιιι about d/2. If h exceeds 0·2D the 1ifters undu1y interfi·ι·r

87

ιιlι ι Ι ι ~: motion of the ball charge, whi1e if h is much 1ess

ιΙι ιιι

ιlιι 11

rl/2 the balls ride over the 1ifters and so 1ocking to

Ιι ι · ΙΙ d.oes not occur. This at once suggests that "wave"

Iι Ι ι- Ι :ιp", and other such 1iners cannot act as effectiv~

lιll ι ι , .

I'I 11:. · .ιrect of the partic1e size on the power requirements

ιι l ι . ι ιι ιll wou1d be expected to be small, provided that the I' ιι ιιι· Ι c~s we~e not so small that powerful aggregation takes

Ι'''' ι · ..lf ,t,hιs were to occur, the charge cou1d become I' ll y- lιl{~ and the J?Otion of the ball charge would be ι ι ι ιι ιι s ly ιmpeded; wιth a consequent effect on the power ιlι ι ι ι . ω ds ofthe mill. l ' lι c : results of tests, carried out on smooth and lifter ιιιι ΙΙ Ν, .to ~etermine the form of the function φ 9 (Dfb) are ι ι· ι ι ιη Fιg. 3.10; these tests being such that the va1ue of ι l ιι μ; r· up (Dfd) exceeded 19, (J) varied from 0·1 to 0·75 ιιιι l Ιη the case ofthe lifter mill, (hfD) =0·1 and (n) =6 · Ι ι INt the group (D fb) was varied from 77 to 740· thi~ ι ι11 <: f values ofthe group (Dfd) largely representin'g the ι ι Ι ι a1 range cσyered by the opera ting condi tions of a ι ι ι.ιι·. ~umber of mιlls. The points relating to different test ι ιιΙΙι l rtιοηs .are seen to suffer a considerab1e amount of I ι ι ( τ. Thιs sc.atter pro?ably arises from the interdepend1 ιι.ι · c :. f the varιables whιch has been discussed earlier, and I ι ι I.t .that, un1ess ~ very great complication in the form ι ι / . ιc l dιtJOnal correc~ιon factors is introduced, this must be ι ι ι · ι: ι?ted. For .practιcal purposes such complication is not ιιι tιl ι d, and, ιη the present work, mean curves have been ι lι , ι w η through the points ofFig. 3.10. When this is done Ι l ιι · r: ιι rves for the two cases are seen to be identical and I' Ι ι· the fu?ctional relationship Φ 9 (D jb) shown in Fig. 3.11. I ) ·creasιng the va1ue of the (D fb) ra tio in a given mill is "' ι • ιι to decreas.e the power necessary to drive the mill. This ιι ι . ι y be ex~1aιned on the basis that the 1arger particles I 11:11·ate adJ.aCent balls to a greater extent than do the ιιι . ιll r partιcles, and, therefore, increase the dilatation of Ιl ιι · h~rge . The centre of gravity of the combined charge is Ιlιιι s dιsplaced towards the vertica1 through the axis of

I''

Ί ' ιι ι~ PO WER REQ.U IRED ΤΟ DRIV E Α MILL

89

ι ιι l 1ιlο ι1 nd so the torqu e is decre ased. In the small mills ι ι ι ι / Ι ο ι· ιhese tests, the prese nce of very fine powd er, such Ιlιιl Ι lιι; ratio (D j b) great ly exceeded 800, result ed in ι I' 1ι ,1ι i n of the powd er from the balls, bu t this only . The powd er formed ' ' ' 1 ΙΙ ιι·ι·c! at speeds in excess of O·BNc wall of the shell, rical cylind the d ι ι ιιr ι ι ί ιωοus layer aroun decre ased by a mill the drive to red requi ιιιι l ιlι · power Ι ιι ι • .ι mount when this occur red. This pheno mena would ι ιιιι / ι, ιl>/ y not occur with large indus trial mills, but data is

L-+--+--+--t--ι-ι R

with ΙΙΙιl ι ι v:ιil able ση this point . The quest ion is boun d up ofBOO jb) (D of ratio a and le partic ofthe size 1/ 1ι . ι/ ιs lute

ι.__-4----+--+-~---:-"~·t-liι....,-ι---ι ~

(/

~

.._.f!ll',

.. , " · ·

L-+--+--+-----ir--τ---ι ~ ~

s

,,ιι , •Ι

I ·δ

)·4 ·2

ο

200

400

600

(%)

800

ι οοο

FIG. 3. 11

ι ι ιι ι ί/ls used in the tests corres ponds to a charg e consisting ιι l ι •: ιι· ιicles of0·003-in. avera ge diam eter. Generally, howt in the case of very ' 1 ι ι ·, it would be expec ted that, excep

. --L~-~~~-~~-~~~-~6-~όo ~L Sll~W HlOOWS (qja)'ψ ..:.

e>ό

CD

6

,.._

ό

SlliW 'ι13ΗΙ1 (qja)'ψ

"'

ό

ό

IΙΙΙ Ι ' Rrind ing, this effect is unim porta nt. er Ί 'lι roughout the series of tests, the value of the powd ; unity at ant Ι ι ll i ιι ,ς- group (V) has mainl y been held const is e charg er powd the ΙIΙΙ being the condi tion that Ιlιι·οr tically just sufficient to fill the void space betwee en ι Ιιι• l>alls. In fact, the powd er increases the total volum of e of ιlιι l1arge and a layer ofbal ls may occur at the surfacwith ed unfill is which en ιlιι· ·harge, the void space betwe ιιιι wder. This restri ction on the value ofthe param eter (V)

jl

•• , •• ,

' Ι ' Ιf Ε POWER REQ.UIRED ΤΟ DRIVE Α MILL

90


has been made because, in practice, less powder than ί ~ necessary to fill the void space is rarely used since a paι ι of the ball charge is then inoperative in the grinding pr cess; even though the power necessary to maintain tiH· centre of gravity of the inoperative portion of the ball charge in the displaced position must still be suppli cl. Conversely, if the powder is excessive, choking of the mill occurs, and a portion of the powder charge is remote fr0111 the ball charge at any instant and so cannot be effectiv ly ground. Thus the decision to restrict the value of the parnmeter (V) to unity is based upon the practical importan ι • of this case and the necessity, in view of the large numbι· ι · of variables involved in the problem, to liinit the amount of experimental work. Although this restriction is useful from the point ofvi w of limiting the number of variables to be investigated, it imposes the restriction that the results are strictly appli ι~ able only to batch mills and not to the industrially iιη portant grate mills and overflow mills. Thus the results ο Ι the present investigation, which has been carried out Γο ι · the case of ( V) = 1·0, are strictly applicable only to a bat Ι ι mill. The results have, however, been extend ed direct1y ιο the cases ofthe trunnion overflow mill and by inferenc to the grate discharge mill. Consider now the extension of t IH· work to these cases; the mills together with a batch mill have already been illustrated in Fig. 1.4. With a batch Inill, as previous1y suggested, the centrc οΙ gravity of the ball charge and powder charge sensi])l coincide and so the density correction ( 1 + 0·4σ/ ρ) prι · vious1y deduced is applicable. In the grate mill the powder surface probably slο μι · from the in1et end of the mill to the discharge end in ιlιι · manner indicated in Figs. 1.4b and 1.4d. In this case tlιι · moment ofthe powder charge about the axis ofrotation ιιl the mill is probab1y much the same as in the ball Inill ; tlιι · greater moment at one end of the mill being rou g lιl y ba1anced by a smaller moment at the other end. Thus, ιlι ι · method of the present work would be expected to lι r

ψι ι lιι ·: ιblc

91

to a grate mill with an accuracy sufficient for

11111 ι ιιιιη:>οses .

Ι ιι Ιlι case of the overflow mill, in order that the proι lιιι Ι N lι a ll overflow through the hollow trunnion, it is ιιι ι ι NH: ιry that the genera1 leve1 ofthe powder surface must ι ι ιι ι ι 'Ν Ι > nd to about a 45- 50 % filling of the mill Then Ιιι ιι tl1e ball filling is about 45-50% the overflow Inill • ''' ιι :φ nds to a batch Inill, and so the power input shou1d I ιι ι lιι ·ι· tly calculable from the present work. For small

1·4

~r----

I

lron Powd~r -p·7·8

~ ........

~

Silicα Powd~r- p:2·6:"

Not~!

0·2

Α 'Stondαrd' Mill ls Α Mill ln Which Th~ Powd~r is Just Suffici~nt Το

IΤο Fill Th~ Sίoc~ B~ιwv~n ~h~ Bαll;--

ι

"

ο

0·1

0.3

0 ·2 Boll Fι c .

Charg~

0 ·4

0 ·5

-J .

3.1 2

ι ιlιιι·s οΓ ball ~harge, however, a correction factor is ιιι ι ~ ·:-;~ary an? th1s has been determined by experiment and Ι ~·, ι V('l1 Ι η F1g. 3.12 .. Ι η the use ~f this factor the power is ι ιlι · 11l a~ed on the bas1s that the m1ll has a ball charge equal ιιι ι Ιι :ιι In the actual overflow Inill and the normal correcι ι ι ιι ι lί>Ι· pow~er. charge is introduced. This calculated power Ι Ιlιι · n multφlied by a factor, corresponding to the value ιι l Ιlι ' actual ball charge; the factor being read from Fig. I I ' . Th~ shape of the curve of Fig. 3.12 would be ex1" ι Ι ·d, Since for very small ball charges the powder

BALL, TUBE AND ROD MILLS 92 standing above the ball charge 1eve1 contributes a rel ative1y 1arge moment, and so increases the power requirrιl over that ca1cu1ated on the basis of a batch mill. F ι intermediate fillings, however, the contribution to tiH· moment by this additiona1 powder is small, but there iN, in addition, a 1oss ofmoment due to this additional powdι· ι· disp1acing the centre of gravity of the re1ative1y heavy ba ll charge towards the centre of rotation. Strict1y, there shou1d be a correction curve for ea lι va1ue ofthe density ofthe charge materia1, but, in fact, t11t' correction is small and when regard is paid to the overall accuracy which can be expected in ca1cu1ations of tl1i. type, interpo1ation between the curves of Fig. 3.12 is ;ιll that is justified. The parameter Ε is the characteristic of the powcl<'l' which is a measure of its resistance to grinding. Thus tlH· quantity (EfD3N2ρ ) wou1d not be expected to affect tiH· instantaneous power input to the mill, a1though it woulιl be expected that the rate of grinding is dependent upon i t. The question of rate of grinding will, however, be treatcιl in a 1ater chapter. The tests carried out with different materia1s in the mill , in order to determine the effect of the density of tlιι · powder, show no significant variations which can l>c ' attributed to the difference in the va1ue of the param tι'l' Ε between the materials. Thus it appears that in connt('• tion with the power input to the mill, the dimensionlι·NH group ( EfD3N2ρ) is not a significant variable, and κc1 Φ 12 (ΕfD 3Ν2ρ) is e1iminated from the equation. ln order to compute the power required to drive a mill it is necessary to inc1ude the values of all the various r ·lc· vant factors previously discussed and on the basis of tlιi H foregoing work, the equation for the power becomes

(n5~3 ρ) =(ι+ 0 ~4 σ)·(~)·Φι'(~)·Φ3(J).Φ4(~)·Φσ(ιι) χ Φ 6 (i) .φg(i) -Φ11(V)

.. . . (3.1

Ι)

ΤΗΕ POWER REQ.UIRED ΤΟ DRIVE Α MILL

IH

Ι 'c'

93

4> 1'(NcfN) is read from Fig. 3.2,

Φ3(J) is Φ 4 ( dfD)

ιιιι l

read from Fig. 3.5, is read from Fig. 3.6 or Fig. 3.7 as required by mill parameters, Φ 5 ( n) is read from Fig. 3.8, Φ 6 (hjD ) is read from Fig. 3.9, Φ 9 ( b JD ) is read from Fig. 3.11, (/>n( V) is read from Fig. 3.12 for a trunnion overflow mill or is taken as unity for a batch mill or gra te discharge mi11.

app1ied to the computaof 1arge industria1 mills I ι . ι I' ι on the published data given by Taggart, {6) and th~ ι 1 111 ts of these ca1cu1ations are shown in Fig. 3.13. From Ι lι l .ωrv~ it is seen that the points ι:ιre scattered a1ong good Ι ι .ι ι ις ht 1ωes, even though the ratω of the horse-power to ι I ι ι vι· the 1argest mill to that required to drive the mode1 ι ι ι l ' the order of 30,000 to 1·0. Thus it appears that there systematic errors in the resu1ts of the mode1 tests. 11 c· π Ι ιι connection with the scatter ofthe points, it is perhaps ιι l ~ 11terest to mention that the data quoted by Taggart, \ Ιιι ~ hav~ been apparent1y obtained by circu1ating a •Ι ψ~ tιonnaιre to a large number of mill owners are for ι l ιι· Γι llowing reasons, subject to some uncertaincles. ' 'I ' Ι ι · equation (3.11) has been Ιιιι ι ι ο Γ the power requirements

The power quoted is the power input at the motor terminals. Furthermore, this figure is often estimated; or even the nominal installed power is quoted. (2) The overall efficiency from the motor terminals to the mill body is unknown and can vary from mill to mill. For purposes of ca1culations an overall efficiency of 85 % has been assumed for mills with concentric drives, and an overall efficiency of 75 % for mills with girth ring drives . (3) Th~ figures for diameter and length are quoted on varιous bases; for example some are the nominal dimensions of the mill and others are inside new (Ι)

ΤΗΕ POWER REQ,UIR ED ΤΟ DRIVE Α MILL


94 610

HJRENCI 600

ctischargc: mil1s . .......,

Graι~

HOL[GεR

~

v

L_

SUIY-IΙNE ο

v [{;'

Ιt ENGELS

....

v

~~Η ENGH~ L MAGNA

L.

ο

!? ιοο i' 2Ι

V O ο

LL·

v/ F~NOH D 'ίERDE

lL/

F'

NIHD

lEROE

1 • "'-...

Overfiow trunnion di scharge:

millιι .

~--·ο

I ~OLI

v/

HI.&.MI

ιi/

·'/

//

UrAH__,) ARTHUR

}'

~ ιs ο ~

iL:=

v

iz ·; that is to machines absorbin g even several thousand lιoι·se-power. The method of calcu1.ati on develope d in this wo rk will be demonst rated by applicat ion to the data, ι ι • l a ting to a large mill, publishe d by Carey and , Ί;ι irmand.<7) Λ lifter mill of6 ft diameter , revolvin g at 18·7 r.p.m., is ι ι i rιding coal. The body of the mill is divided into two ι·ι : ιίοηs : one section, 7 ·5 ft long, being charged with 5 tons ι ι ! ' RLcel balls, of 2 in. average diameter , and the other sec111)11, 15ft. long, being charged with 6 tons of "Cy1peb s" ι ι l ' a in. average diameter. l•'or the section 7·5 ft long and assuming 40% voids lιι · Lween the balls and in the coal, the above data lead to:

~/

L v-v

ο

ο

v

ιιιodels may be applied with confiden ce to the calculati on ιι l ' Lhe power input to industria l mills of even the largest

/

~

ο

ROAD

~/ο

~/ 10

000

'"

JOO

'"

)00

OβSER I/E O POWιR ΤΟ DRΙ V E

)10 • ΟΟ GRAH BA.LL

'"

500

510

600

650

100

HΙlLS-HORSEPOWER

~~so--~•oo~7..,.o-,,~oo-,>~so--~Joo~~JΙo~~•oon-~..~O-Ι~ OBSER VfD POW[R

95

Volume of balls = 38 cu. ft ιιιcl Volume of chamber = π/4 χ 62 χ 7·5 = 212 cu. ft (J) = 0·18 Ί'hus Ι •' ι om equation (3 .5), the critical speed for both sections I :11. r.p.m. Ν w from the appropri ate graphs,

= 9·2, Φ3(J) = 0·65, Φ 4 (Dfd) = 1·0, Φ5 ( n) = 1·0, and Φ 6 (h/D) = 1·0

(/> ι'(Νc!Ν)

ΤΟ DkΙVE QνΕ~.ΙιΟW βλιL HΙLLS-HORSEPOWER

Fro. 3.13

liners. Since the power input, Ρ, is related to tiH· diameter D by the expressio n Ρ = KD 2 ·5, the uιι certainti es in the diameter of the mill can produ · ι· consider able inaccura cy in the calculate d valuc οΙ ' the power input. (4) The density of the material being ground is nol known precisely. In view of these uncertain ties, it is believed that tl11· agreeme nt between the calculate d and measur~d pow rH is as close as can reasonab ly be expected , and It may \)ι • conclude d that the results of this work upon small-scalι·

ι ιιιι I since the particle diameter, b, is probably less than

t

ι ιlιοιιt Ί'Ιιeη (,

in., Φ 9 (D fb) = 1·0. from the equation

~3 ρ)

(ι+ ;~σ) . (Ε)·Φι'(~)·Φ3( J).φ4(i) 0

χ φ5(n) ·Φ6(~) .φg(~) 0·4 χ. 1·4) χ (7·5) 6 χ9·2χ 0·605 χ l·Ο 79 1·0 χ 1·0

( 1+ χ

96 Ρ

or

= 820,000 ft pdlfsec = 49·5 h.p.

I 111

If, for the section 15 ft long, it is assumed that thι : "Cylpebs" pack with a voidage of 40%, then (J) =Ο· J :1 and, by similar calculations Ρ= 66 h.p.

From the published data, the measured power to tl1ι·. motor terminals is 103 k W, and so the power demand ο Ι 86 k W by the millleads to a combined efficiency of mοι ι· and transmission of 83%, which is reasonable. Thus tlιc· agreement is as close as car.ι be expected . Considerati on of equatron (3 .11) wιll show that for speeds 1ess than 80% of the critical,

D5Ν3ρ.(1+ ο:σ) · (~)·(~γ.φ 3 (J)

so

Ρ = =

tl1is CaSe, Ρ=

KD Z·o

(3 .14)

lιΙι Ιι i ~ a widely accepted relationship for the power

lo

(1.

mil1.

I ι ι Ι . ι ι · ι, it has been found by Gow, Guggenheim , Campj,, 11 111 ι Coghill,
ι

IIJ

I"

ractice, the power demand is given more near1y lationshi Ρ Ρ= KD2·6 .... (3.15)

I~ 4 '/d) φ(

,. Ο Ι'!

σ.e

'''"''~'"'"'/

/

(%) Ο·ό75

11

ι

Kfy'D

D5. (K/D) 3/ 2 KDH

j ll

11 1

2

.... (3.12)

the remaining groups of the equation being neg1ected fι ι· the time being. Now, for a given mill filling and a ~iven. material bein l-ζ ground, Φ 3 (J) and (1 +~·~σfρ) are ιnvanant, . and for : ι constant fraction ofthe cntιcal speed Φ 1 (Ν,fΝ) ιs constanι. Furthermor e, since the mill runs at a given fraction οΙ ' its critical speed, it follows from equation (3 .6) that Ν=

I

Ιψlι l

Iι

Thus the power to drive the who1e mill = 49·5 + 66·0 = 115·5 h.p. = 86kW

p =

97

'Ι ' ΙΙ Ι ~ POWER REQ.UIRED ΤΟ DRIVE Α MILL


. . . . (3. ι :Ι)

This equation is true for complete similarity betwcι · ιι 1engtlι and diameter of the mill but, for a mill of constaιιf length, it follows that (L/D) oc (1/D)

ο

10 Fω.

(%)

15

20

25

3.14

Fig. 3.7, however, it would be expected thq.t varies approximat ely as (D fd)O·l and in Fig. 3.14 lι o wn the curve (Dfd)0·075 plotted ση the same axes qs Ι ιlιι ι l o tted curve of Fig. 3.7. It is seen that the fit is ι ι ι ο ι ι ι b1e. Ί'Ιιιι s it follows that in tests in which the ball diameter is ιιιιιι lnιιt the ratio (D fd) varies direct1y as D, and so the ιιιι w ι ι· ιο the mill, which varies with Φ 4 (Dfd), will vary as Ι ιιι ι : ιpproximately. Thus, it follows that, if the ball diaιιιι Ιι · ι · is maintained constant, the equation (3. 14) must be ιιιΙΙIΙiρlied by DO·l, which brings about agreement with ι Ι(Ιι . ιtίοη (3 .15).

Ι •' ι ·οιn ·l• ι ( f) jrl)

98

ΤΗ Ε POWER REQ.UIRED ΤΟ DRIVE Α MILL


One of the earliest attempts to derive, from theoreti :ι l considerations, a formula for the power to drive a mill i~ that of Davis, (lO) who starting from the basis of the b :ι 11 trajectories discussed in the previous chapter, calculated tiH· striking velocity and so the kinetic energy of the elem ni H ofthe charge in a giveri trajectory and, by integration, tlιc · kinetic energy to the whole charge; this kinetic energy clearly being supplied to the mill from the external sour

99

Ί ' Ιι equation (3. 16) is not directly comparab1e with ι ψιnιίο_η (3.11), since for a given composition of charge, Ιl1ι w ιght of the charge, W, cannot be varied indeι ιι ιι ι l 'ntly ofthe diameter ofthe mill and similarity main1 ι i ιιc ·d at the same time. Thus, the functional relationship

•I• 1(.1) of the previous work is absorbed in the form of ι ψι :ιιίοη (3.17). Furthermore, the equation of Davis ι I,., i ν d for the conditions of maximum efficiency and

I

is so

· general. equation developed by Bond(ll) for mills dryling is

1ι ο ι

η

1 ι1ι1 =4~------~-----4-------+------~~~~~

·~

I'

1·26{D0·4 χ

~c (6· 16- 5·75J) -

0·1

χ 210(N,/N-0·6)-l}

~N3J-------+------+-----~~~--~------I

.... (3.18)

I

Q 1-

2

~------~----~~~---+-------+------~

c(

(( ~~----~~----~------+-----~~-----1 ο

6

2 L~

nqth-

FιG.

β

F~~t .

3.15

On this basis the following formula for the power input to the shell, when working under conditions ofbest theoreti :ι l efficiency, was obtained: (l-K3)

(l-K5)

Ρ= WR3 /2{ 0·004467 (l-Κ2)1 Ι 8 -0·003700( 1 -Κ2 ) 318

( 1-Κ7) }

+ 0·000880 ( 1 _ Κ2 ) 518 where the value of mate equation: Κ=

Κ

....

(3.1β)

is given by the following approxi -

-0·024+0·39(7-lQJ)I /2 .... (3 .1 7)

Ιν l ιc · Ι ·c Ρ is the power draft in kilowatts. per ton of grinding Ι ι ι ιii Η. The equations of both Davis and Bond are based ιι ιιl y on. the mill diameter, tht: mill filling and the speed ιιl ι·otatωn express_ed in ~~rms of the critical speed; the lιιιιη s of t~e equatωns _ansι~g from the empirical expresNII 111 ~ definιng the relatωnshφs between several variables. ιιιι · ' the weight of the ball charge enters both these ι tj!Ι :ιιions directly, it follows that the power is proportional Ιιι Ιlι length ofthe mill; a conclusion which is in accord

1vi 111 the equation of Rose and Evans. (12) Ι l1as been suggested by Gow, Guggenheim Campbell ΙΙΙ c l Coghill,
(0·5L-l)K+1

.... (3. 19)

L is the length ofthe mill and Κ is a constant having ιlιι · value of0·9 for mills oflength less than 5 ft and 0·85 for ιΙιiΙΙ · of length greater than 5 ft. This function is plotted in Ι •' ί ι.(. 3. ~5, and it is seen that. for long mills the power ι c: ?s1derably smaller than ιs demanded by a linear

1 Ιιιτ

Ιι11ι ιωη.

)

BALL, T U BE AND ROD MILLS 100 The higher power requirements of a short mill woul I, p erhaps, be expected, since the end plates of the mill migl11 cause a "piling-up" of the charge at the ends of the mill . Such an effect would, however, be expected to be η 1-ζ · ligible in a long mill. Thus, this deviation for a long mill iM surprising and it is believed that this formula should bc· accepted with caution. Hancock
' Ι ' ΙΙΕ POWER REQUIRED ΤΟ DRIVE Α MILL

Iι ι

101

ι ι Ι ι 11 ~S

Ι' = == ==

K 1D 5 N 3 χ D~

2 χ Φ 2 (J) χ t χ ρ( 1 +0-4~)

Κ 2 Νχ{D 3 χΦ 3 (J)}χLχδ 0·0005 lNC3Lδ

(3.2 1)

Ιl ιιι Ν, ·ι he general form of ι ι ι ιηn nt b~t the value Ψlιc ·: ι·s to be lll error.

where C is the length in feet of the chord defined by tl~e · surface of the mill charge, L is the length of the mill ίιι feet, δ is the mean specific gravity of the charge and ιι ί the speed ofrotation in r.p.m. This formula approximatc·H to the formula of Rose and Evans insofar that the tcr ιιι ρ ( 1 + Ο-4σ jρ) is proportiona1 to thc mean density ofthe pulι) , and so to δ of cquation (3.20), and in that the powcr ί proportional to the length of the mill. Furthermore, the quantity C3 may bc written:

the two equations is in close of the coefficient (0 ·00078)

'(C/D)3

ByExpHim~nt . ?. '\

C3 = D 3(C/D )3 but (CjD ) is the sinc ofthe semi-angle, Δ, subtended by tlιι · surfacc of the chargc at the centre of the mill and so C3 = D3sin3

0·2

0·4

Δ

Examination of Fig. 3.16 shows, however, that for tlιι · practically useful range, of up to 50% filling, the curvc• ofsin3 Δ and Φ 3 ( J) practically coincide. Thus C3 = D 3 χ

ο

Φ 3(J)

Also from equation (3.6) it follows that for constaιΊcΎ in the ratio of running speed to critical speed

n oc 1/y D Then, on making these substitutions, equation (3.1] )

Ρ= D5Ν3ρ χ Φι'( ~) χ Φ3 ( J) χ ( 1 +0-4~) χ (t)

Fισ.

J

Ο·δ

0·8

1·0

3.16

J ι'rH· values of_ mill filling greater than 50%, however, ΙI ιι : <ιgreement ιs not good but, in practice, high mill Ιιllιιι s are_ perhaps not so frequently used as the lower , ι\ ιι ι~s so ιt ~ay concluded that, provided that the ι ιι Ι ' Ι"(' t val:ιe ιs _assιgn_ed to the coefficient, th.e equation of 11 , ι 11 ock, ιη. spιte ~f ιts simpler form, is, over a practical ι ι ΙΙΙ Ι{ ofvanables, ιη good agreement with those of other Ι\ΙΙΙ Ι· ker~ . Thus this formula includes, approximately, the ι \ιι,ιι. ·nsιonless group Φ 3 (J) of the previous work.

?e

I lιe dependence of the power demand of a mill on the ι ι 11 ure of the pulp does not appear to have received a great

\ ι

102


deal of study, and, in genera l, observ ations on this matt ι· take the form of genera l statem ents. For examp le, T aggart05) states that, other things being equal, wet millin~ require s 60-90 % of the power for dry milling . This vicw is confirm ed by calcula tions of the presen t author s, basccl on figures for two mills of about 8 ft diamet er and 45 1ί long kindly made availab le by Messrs. Edgar Allen & C . ln this case the power as calcula ted by the formul ae ι· Rose and Evans was in extrem ely good agreem ent witlι the observ ed power require ments of the mill when dry grindin g (within 5 %). Howev er, the power calcula ted οιι the basis of dry-gr inding formul ae with modifi ed chargc• density was found to be about 20% higher than the actu~ιl deman d of the mill when grindin g wet. It would be expect ed that a fine "smoo th" fluid pulι> would reduce the power deman d, since such a pulp woulcl act as a lubrica nt to the mill charge and so reduce thc· equilib rium displac ement of the centre of gravity of thc· charge . This reduct ion of the power require ment of tl1c· mill is probab ly bound up with the phenom enon οι· "slump ing" ofthe charge , mentio ned earlier. More viscou ~ pulps would, by increas ing the displac ement ofthe charg<', lead to increas ed power consum ption. With increasin !-ζ "tackin ess" of the pulp, a reduct ion of the power requircments would be expect ed, since, in such a case the baliH might adhere so well to the shell that a part of the chargc· is spread over the shell and, thus, is balanc ed about tl1c· axis of rotatio n. This decreas e in power deman d witlι pulps of increas ing thickne ss is in accord with thι · observations of Gow, Gugge nheim, Coghil l and Camρ bell (16) but these worker s did not carry the tests to tlιc· poi~t where the balls adhere d to the shell of the mill. ThiH effect is, howev er, also observ ed in mills which are used for the dispers ion of pigmen ts in varnish , and similar dutic11, and so is very real. From what has already been said about the interactioιι betwee n the coefficient of friction and the magni tude ι · the ball charge , it would be expect ed that the effect of th

ΊΊ. l Ε POWE R REQ.U IRED ΤΟ DRIVE Α MILL

103

ι Ιι ι ιιι c: ίη pulp conditi ons would be relativ ely more imιιιιιl ι ιιιι in mills for which the charge is small. This conι Ιιι Ιιιι ι in fact follows from the observ ations of Gow ι ' '' ι· n lιeim, Campb ell and Coghill
ιr

~ 1 · 5r---t---+"""--7L_+-----;;;~-----1

ο α.

ω

V> α:

;Ξ

1· 0

ο

r---i-7'-'------+---:τ"'---+--+--~

0·2

0 ·4

Ο · ι!

0 ·8

1· 0

Puιp CoNSΙSHNCY

FIG. 3.17

Ι ιι νiew of the lack of data on this subj ect, it would 'Ι'ΙΗ ': ιr that the best correct ions for pulp charac teristic s Ι I ΙΙ ι · Ι ι an be made is to assume that, for very thin pulps ιι ι ι l ιι ι derate values of J and (D jd) the power required is ι ιll % f that for dry milling , and then to use a linear interιιιι l .ιti η for less liquid pulps. For Iarger mill fillings or ι ι • . ιl ι • ι· values of (D fd) the reduct ion in power would be ι Ιιaη for the previou s case and the multip licatio n Iι Ιιι Ιο ι· slιould. b~ m~dified accord ingly. The genera l corι ι ι Ι ι ι •ss of this vιew ιs suppor ted by Fig. 3.17 iη which the

104

ΤΗ Ε POWER REQ.UIRED ΤΟ DRIVE Α MILL


results of tests by Gow, Guggenheim, Campbell and Colζ hill, (16) are plotted. Although these ρoints are small iιι number and rather closely spaced they give lines whi Ιι when extrapolated to zero pulp density give 0·22 and 0·1·5 horse-power as the powers to drive the mills without pulι ι charge ; these figures being in the ratio 2·05 to 1·0. R eference to equation (3.9), however, shows that, ;ι /1 other variables being maintained constant, the ratio of tlιι · powers is proportional to the ratio of the square of thι · speed; in this case (0·70/0·50) 2 , that is in the ratio 1·96 to 1·0, which is in good agreement with the figure 2·05 to 1·0. Furthermore, for a mill running at 70 % of the criti : ι I speed the ratio of the power when grinding a pulp of 50 % density to the power when dry grinding is about 0·6 to 1·0: which is in accord with the previous general statemcn t that when wet grinding the power d emand is about 60 % of that when dry grinding. Thus it appears that, ν ιγ roughly, the linear interpolation previously suggested ili adequate. The possibility of calculation of the power required to drive a mill from the knowledge ofthe horizonta1 displa ι· ment of the mill charge has been developed by Carey a nιl Stairmand. (17) From the principles discussed earlier in this work, it iN easily shown that Ρ= WN!fι, ( 1 / 3·14)

105

Ιι ιιwη in Fig. 3.18. These curves are stated to be applicable Ιιι ιιιills having diameters ranging from a few inches to f ' l 'c ι feet and, ΪΠ practice, tO giνe reaSOΠably reliable I ι ι ιιl ιs .

.nalysis of this method in the light of the previous disshows that the effects of the weight of the charge, ιιιιl so Φ ( 1 +0 · 4σf ρ ) , are inc1uded. Also the effects of

ι ι ιr Μ ι οη

-- -

Smooth

Μiιιs .

- - Ltfttr Mills

0 ·20

ο

-40

0 ·60

D-BO

ι-ο

tyN,

. . . . (3. 22)

where Ρ is the power input to the shell in kilowatts, tιι maintain the displacement of the charge, W is the wei g lιt , in tons, of the grinding media plus powder, Ν is the mill speed in r.p.m. and Η1, is the horizontal displacement ίιι feet of the centre of gravity of the charge. Thus, the pοwι· ι input may be calculated provided th e valιι e of Ηι. can Ιιι · obtained. For this purpose Carey and Stairmand furnish a gra ιιlι which gives the value ofthe ratio H1,/D (D is the mill di:ι meter), in terms ofthe mill filling and the ratio ofthe sp · ι·ιl ofrotation ofthe mill to the critical speed; this graph bcirι ~

FIG.

3. 18

•/• ι '(Nf Nc) , Φ 3(J) and of lifters are included, and so it ιιιι ιι·, rs that only variables which have been shown to be ι ι l ι · omparatively minor importance are omitted. Thus, it 1 · ιιιι l d be expected that this treatment would have an Ιι ι ιιι·acy

sufficient for most practical purposes.

ΊΊι power requi~ed to dri~e a rod mill does not appear

Ιι ι Ι ι : ινe b een extensιvely studιed. As a first approximation , ιΙ wo uld be expected that the value of the function

•/• ι ιι(σ/ρ) would be different to that for a ball mill but

106

BALL, TUBE AND RσD MILLS

ΊΊ Ι 1': PσWER REQ,UIRED ΤΟ DRIVE Α MILL

that the values σf the remaiηiηg fuηctiσηs wσuld ησt bι· seriσusly altered. Siηce the "vσidage" iη a bed σf circular rσds, packed in the tightest array, is abσut 10 % (0·095) aηd assumiηg th · vσidage σf the pσwder tσ be 40 it fσllσws that

ι Ιιι ιιιt · Ι y high cσsts for the tarriage σf replacemeηt balls · ιιι ιlιι· ιι s. σf such lumps σf ore be justified ση ecσησmic ι ι ι ιιιΙΙΙ IH· sιηcc the ~r~ηdiηg efficieηcy is adversely affected ιlιι 1.' lι y aηd the mιllιηg.cσst correspσηdiηgly iηcreased . Ιlιι · ιιs of such ηση-cιrcular σr ηση-spherical materials ιιιιlιl l ιad tσ a "lockiηg" σfthe elemeηts σfthe charge ιι 11l ιι ι aη iηcreased displacemeηt σf the ceηtre σf ι ι 1\ tt y [ the charge, with a relatively iηcreased power ι 1 ιιι 111 I. The pσwer demaηd wσuld, furthermσre be mσdi IIΙΙI lι y. ιh~ tightηess σf packiηg σf the griηdiηg bσdies, ltΙΙ l1 Ι Η ~ιffereηt tσ that fσr balls, aηd by the deηsity σf tlιι 111 . ι τιal. The pσwer tσ drive mills charged with such ιιι 111 ΙΙ;ιls dσes ησt appear tσ have beeη studied hσwever ΙtιΙi ι wiιlι. the preseηt state σfkησwledge, little ~σre tha~ ι ""Ι l1 ι·s ιιmates σf the pσwer calculated ση the basis σf 11 I III'(~VΪOUS equatiσηs mσdified ση the liηeS Suggested, 1 111 Η ' ιnade. IIΙJ • fσregσiηg work deals eηtirely with the pσwer ιιι 1Ι ΙΙΜ: tιΎ tσ keep the charge withiη the mill shell iη ΙΙΙιιlιι~ιι. Clearly, iη σrder tσ determiηe the pσwer tσ be ιι ι ι ι ι lιt ~d by the driviηg mσtσr, it is ηecessary tσ add tσ this I ι ΙΙΙ ι .thc pσwer ~σst iη the beariηgs su ppσrtiηg the mill Ιιι 11 , ιιι th~ beanηgs su~pσrtiηg the iηtermediate speed ΙιιΙΙ :ιηd ιη the reductωη geariηg. The estimatiση σf ιlιι ι I sses, is, h?wever, a ~atter σf geηeral eηgiηeeriηg IΙΙιιwlι·dge, aηd ιs ησt specιfic tσ the tumbliηg mill aηd ~ ~~~ l1 ιn a tters are specifically excluded frσm the p;eseηt

%,

Φω ( ρ ~)

_ ( 1-Ο·l ) ρ+0 · 1(1-0 ·4) σ

=

(1-Ο·l)ρ

l+0 · 067σf ρ

.... (3.23)

Alsσ, fσr a giveη value σf mill filliηg (J) the lσad σf rσds iη the mill will exceed that σf balls iη the ratiσ σf thι· vσlume σf rσds tσ the vσlume σf balls, that is,

1 - 0·1 = 1·5 1 - 0·4 Thus, the pσwer tσ drive a rσd mill, withσut a charge, will be giveη apprσximately by

PR = l·5PB

pσwd ι·

.... (3.21)

where ΡR is the pσwer tσ drive the rσd mill aηd ΡΒ is tl1ι· pσwer tσ drive the cσrrespσηdiηg ball mill, as calculat ·cl frσm the fσregσiηg fσrmulae. It is prσbable, hσwever, that the actiση σf the pσwdιτ tσ dilate the charge, aηd sσ tσ reduce the hσήzσηtal diHplacemeηt σf the ceηtre σf gravity σf the charge, wσuld bι· mσre marked with rσds thaη with balls. This wσuld le
107

ιιΙ I ,

REFERENCES I (ΙΙΜ ι ι, , Η . Ε ., and EVANS, D. Ε., Proc. Instn mech. En:grs Lond

17 .

Ι :ιιw, Α . Μ., CoGHILL, W. Η. , and CAMPBELL, Α. Β . (ιιιetall.) Engrs, T ech. Pub. , Νο. 326. 1 1ΙΙΝ Ι t 1 Η. Ε., and EVANS, D. Ε .. Loc. cit., ref. Ι .

Ι lιΙΙιι.ιι ιι, W. Η., and DενΑΝΕΥ, F. D.

'

.,

1956 170

'

'

Amer. Inst. min.

Ball Mill Grinding, U.S. Bur. of T ech. R ep., Νο. 581, 1937. ι llιι Νο, F. C. Proc. Instn mech. Engrs, Lond., 1956, 170, 800. ιι I ΛΙΙUΑRΤ, Α. F. Handbook of Mineral Dressing (New York), 1945. I: ιιι~v, W . F.,. an~ STAIRMAND, C . J. "Recent Developments in Mιneral Dressιng , Instn Min. Metall., Symposium, 1952. Mιncs,

108


Μ., CAMPBELL, Α. Β. , and CoGHILL, W. 11 . 8 . Gow' Α . Μ ., GU GGENHEIM . , 87 51· 112 24 Tran>. Amer. Inst. mzn. (metall. ) Engrs , Α' Ί. t ' . . (metall) Ειιψ . ι 9. FAHRENWALD, Α. W ., a nd Lεε, Ε. Ε . mer. ns. mzn. . . • T ech Paper, Νο. 375 . 19 19 61 256 10 DΑνιs, ·Ε. W . Trans. Amer. Inst . ιnin. (ιnetall. ) Engrs, , ' · 11 . BOND, F. c. Proc. Instn mech. Egrs, Lond., 1956, 170, 800 12: Rosε, Η . Ε . , and ΕνΑΝS, D. ΕΜ. LCoc. czt., rLeLf. Α 1. Β CoGHILL W 11 . 13. Gow, Α. Μ. , GucGENHEIM, ., ΑΜΡΒΕ , . ., ' ·

CHAPTER

ι Ι Ν 'ΓΗΕ

14. Η~~"σ~~: :{:·:.· Trans. Amer. Inst., min. (metall. ) Engrs, 1934, 112, 77 • 15. TAGGART, Α. F. Loc. czt ., r ef. 6 · d C GHILL W. 11 . 16. Gow, Α. Μ., GuooεNHEIM, Μ ., CAMPBELL, Α. Β. , an ο , Loc. cit., ref. 8. · f 7 17. CAREY, W . F. , and SτAIRMAND, C. J. Loc. czt ., r e ·

4

COMMINUTION OF SOLID BODIES

Ι ι ι ι ιιι ·οbl

ms associated with comminution of solid bodies and complex. In spite ofthe large amount ι / IΙΙ\ΨΗ Ιi ation which has been undertaken, the present ι ι Ι ι ιιι · knowledge is such that there is no theory by means ,f 11 ιιίι · Ιι the behaviour during fracture of even a single Ι ΙΙ ι 11 Ι ι • under the simplest _Rossible mode of loading can ι ι ιι ι · li ted in reasonable detail or with moderate

ιι ι ιΙιιιιιι:rοus

Ι

ΙΙΙΙΙι , Ι( Ύ .

Ιιι ιl ιι:

absence of an adequate theory applicable to even ase, it is clearly unreasonable to hope that the l ιι Ιι.ι ίοιιr of a comparatively complicated system, such as ι 111 11 , ί ιι which numerous p articles are involved and the ιι" ~ι lι • ο [ loading gives rise to both impact and abrasive 111ιιl ί ι Ι Α' of the p article, can be predicted from theoretical • 11 11 Ιιl< · ι · a ιions. Because of this absence of adequate theoιι ι l 1 , ι\ l
ιιιψl<:

109

-

110

~-

-~~

---

proper ties which spriπg from the crysta1 structu re aπcl associated factors, such as crystal lite size aπd arraπgemcιιt of the crystallites withiπ the macro crystal or powd • ι· particl e. Such materi als are homogeπeous. Materi als like iroπ ore, however, will geπerally be composed of particl es of almost pure iroπ oxide, iπtermixι:ιl with particl es of silica. Thus, metal ores iπ geπeral, co<ιl ls. F ι· aπd cemeπt c1iπker are heterogeπeous materia sounιl is there later, ed discuss be reasoπs which will t differcn have al materi of types two these that evideπce fractur e charac teristic s. The liπe of demarcatioπ betwc 11 the two types ofmate rial is ποt sharp, howev er; for althouglι a heterogeπeous materi al is fairly obvious, homogeπeoιιN materi als have iπterπal defects which iπtroduce some ι · Πieeharactenstics of hetero eπeous materia ls. Coπsider, iπ t e rst p ace, t e acture ο a heter·oor ir 11 geπeous particl e; for examp le, a cube of graπite grairι the at m mediu iπg cemeπt the , geπeral iπ Siπce, ore. bouπ'Crarιes ιπ suc a maten a may e coπsιClered to Έaνι· a ower streπgth thaπ the graiπs themse ves the fracturι • Wtl , !Π the maiπ, OCCUf by failure of the graiπ bouπdarit'N , ter a while the particl e will be brokeπ dowπ iπto tl11· reachι·ιl coπstitueπt graiπs. Wheπ this stage has beeπ by fr~H'· about t brough be oπly caπ further size reductioπ ίt stated, as er, howev Siπce lves. themse graiπs ture of the is probab le that the crystals of such materi als are stroπgι · r tlιi H thaπ the materi al at the bouπdary, griπdiπg beyoπd parιi Ιπ stage. earlier the iπ thaπ lt poiπt is more difficu t l~ι· to ratioπs coπside these applied has gω Hίitti • cular, griπdiπg process. Thus, a graph of fiπeπess attaiπed agaiπst the expeπcli ture of work πecessary to attaiπ the fiπeπess would be ι· pected to have a form somew hat as showπ in Fig. 4.1 :ι . Clearly , iπ the case of metals, it would be highly improlι able that all the graiπ bouπdaries would fail before aπy of the crystals failed. Micro- photog raphs, by Felix a11ιl Geiger, (2 ) of samples of steel which have failed undt~ r· brittle conditi ons show clearly that the fractur e is pro_pιι-

1

111


Enorgy lnput . Hεtrrogrnrous ΜσtιηΌι. α.

j~tl En
Enorgy lnput. R
c. FιG.

4.1

ιι ιιιι · αys tals would occur simultaπeously with the failure ιι l ι Ι ιι • l)ounda ry and th~s w~uld elimin ate the sharp corner Ι ι • ιι11 t l1 curye and so gιve nse to a non-liπear relatioπship. I ιι ΙIΙ ι· rushιng of a homogeπeous materi al howev er the • ι ι would be differeπt. If the crystal lattic'e were pe~fect ιιι ι Ι Ιlι plateπs of the press aπd the polishe d faces of the

112

crystal were absolutely plaηe aηd parallel, theη theor ti · cally the stress would be uηiform througho ut the parti clι ·. Presuma bly, wheη the stress at which failure occurrr
1

!

ΟΝ


ΤΗ Ε COMMIN UTION OF SOLID BODIES

113

. : ι η apprecia ble fractioη of the cross-sectioηal area 11/ ιl ιι · ι >:ιr ticle. They may arise from defects iη the growth of ι Iι ι '. Ι ystal, t~ermal str~sses, shock loading, etc. Defects ι1Ι Ι l ιι Ν ιι ature, ιfpreseηt ιη aηy apprecia ble ηumber are of ΙιΙψιι Ι' Ι ηηce iη coηηectioη with fracture siηce clearly' they ' ' ι ιιιι 11. ι~ tc p 1aηes of weakηess, similar to the graiη bοuηΙIΙ ι ~.·~ ι ι~ a .heterogeηeoι:s p~;ticle, aηd so would give rise ι 11 1 ιγιηdι.ηg charactenstιc curve as for a heterogeηeous ιιιι/1 ι ' Ι ;ι l , Fιg. 4.1a. Ι ι 1 Ιlι ~?ird c~a~,s may be placed :-v~at, for brevi~y, may l ιι ι . ιllι: ι mos~nc defects. '!he ?asιc ιdea uηderlyιηg this I ΙΙ ΙΙιl or de ect IS tfiat there lS evιdeηce that material s are Ιιιιιlι ιι Ι of elemeηtary "bricks" haviηg a size of the order ιι / ι lιnndred lattice dimeηsioηs. Thus, for examp1e / Ι Ι l ι (•I) sugge~ts. that the micelle, which represen ts th~ 1Iι ΙΙΙΙ ' Ι Ι L ' ΗΎ uηιt ιη coal aηd coke, has a size of about 35111 : llumphr eys-Owe n<5 > states that miηeral salt (NaCI) Iι ι ιιι lemeηtary structure of about 1500 Α. One of the 6 Ι ΙΙ Ι ' ΙΙ ι w~i ters, Rose, < > ~a.s showη that. curves of both the Ιιι ll ι l nsιty and reflectιvιty of graphιte have a discοη Ιi ιιιι ί ι y, which probably iηdicates a chaηge ofstruct ure or 1 1l1 ~ 11 in the surface coηdi tioη of the primary parti~les l ιι ΙιιΙΙΙΙ g' the aggregat e, at a particle size ofabout 500 Α. Iι ΙΙ ()W the cr stal st~ucture is "P.erfect"' the elemeηtary ιη aη ordered WaY. · the adjective IΙΙIΙ ι .~ :ιre arraηg ηg that the arrayΌfthe "bricks" iηdicati merely ι" ι·ι: f ΙΙ Ιi ιι1~ Ι ~τ d ~?d.not,~hat there are ηο defects, of the type 1ι I ι~d as mιcro defects above, iη the crystalla ttice. If. I II 1 Iι ι ι1νι • νιτ? ! he crystal is "mosaic ", the elemeηta:ry: block~ be 11 1 ιιοι ιη aη ordered array a,nd such a crystal would the corres1 Ι ΙΙΨ iι : d to be weaker iη brittle fracture thaη I ΙΙΙΙΙι l i ΙΙ g crystal with ordered array. Α crystal structure of ιl ιι ιι :ι ιure wo:uld agaiη suggest that griηdiηg would Iιι 1 11 111 · more dιfficult wheη the particle size reached that 1l ΙI ιι·. Ιnosaic bloc~. ~tartiηg from a large particle size, the ι• ι ΙΙι ι iΙ ng charac~enstιc curve for a mosaic crystal would be ι ι ιι · ι ΌΙ ·d to be ~η ~hree parts, as showη iη Fig. 4.1 c. Ι η 1ιι ιιι Ιιι;c, the gnηdιηg of material s to below the mosaic ι 1ι ι ι

Ι

114

Ο Ν Τ ΗΕ COMMINUτiON OF SOLID BODIES


block size is rarely carried out and so evidence as to ιlιι behaviour of particles in this size range is lacking. It is known that, as grinding proceeds, there is an ίιι creasing degree of aggregation of the particles, whi ι makes it increasingly difficult to separate the primaιγ particles of the aggregate as, for example, in the prepar: ι tion of dispersions of the solid material in a liquid vehi l1·. The underlying reasons for this are not clearly understoolι mechanisms which would lead to this result are operativι ·; for example the reduced area around the contact poiιιt between a pair of balls, within which crushing must tak

/.

IιiΙΙ 111 IΙιι1

I

115

the fineI wi ιι lead to a decreased rate of grinding as Ιιιπ ases. Another possibility is the carrying out of a ι ~:ιr ticle by the air expelled when the falling ball

'/''"' ι:ι :l1es the stationary one, or even welding together

οι ι l1 ι fin st particles by the blow would tend to produce a I ι ι ι ιιf ' thc type shown. Thus, there is certainly no ground lιιι l;ιting, withoutfurt her evidence, that the curvature of ιl11 Ιi ι ιι • constitutes proof of increasing strength of the Iι 11 ι Ι ι Iι·s with decreasing size. Again, the existence of fine 111 ι ι ι · ι ί~ι l, along with the coarser particles, may exert a ι 1ι Ιιiο ιι ίηg effect during milling; the fine material interlι 11ιι!i wi th the mechanical action of the ball on the

,_

"'

!~--------~~--------~

...

ι:!

c ~ c

---~ -~------~~-------Timc Scaιc . FJG . 4.2

material being ground by forming a resilient in which the larger particles become embedded:* Ι lι ιl this mechanism is ofimportan ce is supported by the ιιl ι ι • ι ν tion that the rate of grinding in a batch mill is in' 11 1~1 ·d when the charge is removed from the mill, the Ιιιιι · ιι ί ved out, and the oversize material remaining again 11 ι ι ι Ι ' Ι Ι d to the mill. Ι •:ί t l1cr of these explanation s forms a more practical eχ ιι l , ιιι : ι ι ίση for the decrease in the rate of size-reducti on of a ι tιιwι l ·r with increased milling time than does that based hypothesis that the mechanical strength of the ιιιι ι Ι ι 11 ι.ι ι ι · ι· ί al increases with a decrease in particle size . ι ι l1is point it is interesting to consider the distribution ιι l ι l11: energy supplied to a particle during crushing. Ί' IJi s cnergy would be distributed in:

ι ι

t:ι ll ine

ι1111ι 1i

116

ΟΝ ΤΗΕ COMMINUTION OF SOLID BODIES


( 1) Elastic deformation of the particle. (2) Plastic deformation of the particle.

(3) Gliding or slip on atomic planes, twinning and p Η sibly ot.her lattice re-arrangements within ιJιt • mineral. (4) Increased surface energy of the body.

1 11 1 ι ι •, (7)

117

who in 1867 stated, apparently axiomatically,

ι I ι 11 '' ιlι energy necessary for crushing is directly prοpοr Ιiιιιι ι ι l ι the new surface produced". Continual reference ι ιι ιlιί Μ 'law" is found in the Jiterature SO it is important Iι ι Ι i li'i i ιnplications should be carefully considered. I 'ι ι ι ·ι · Ly from the point of view of an energy balance, this

wou ld hold provided: ermanent lattice mονι · rήents, wfiicli wιΙΓ6e smalΠor ar , 5'ΠΉ1 e materials, anιl ~ο Incre~sed surfac~ energy, i'S"absoroea.. llri:ι:.!KJhe crus lι · !E,g; process and does not reap2ear as heat. The remaincln ofthe energy supplie , however, will reapp"ear as heat anιl this difference at once gives rise to the idea of the enerAY which is "necessary" for crushing and that which iM "sufficient". Thus, it is essential that sufficient encrgy should be supplied to bring about enou,g_h elastic deform : ι tion for fracture to occur, and no less guantit;y will suffiι· · When fracture occurs 1 however, this energy, less that cοι· responding to the permanent Iattice rearrang_ements anιl to the increased surface ener.gy~ ears as eat and ί ι is this difference which is measured when a mill is run in : ι calorimeter. Since, however, the energy for e astic, aιιιl possioly that for a part of the plastic, deformation ί essential iffracture is to take place, the use ofthe absorbι · ιl energy as a standard of comparison is grossly misleadiιΙ f.\ as to the efficiency of a crushing process. As an exampl οΙ this discrepancy, it has been observed tlιat the encιι.\ supplied in free crushing is about five times that calculatι : ιl on the basis of surface energy, whilst for a ball mill ιlιι• value of the ratio is often between 500 and 5000. When a bed ofparticles is crushed the energy expendι·ιl in friction between the particles must be supplied, bιι ι , again, this ultimately reappears as heat and performs :ι Η 1 usefu1 wo.rk in grinding. Thus, friction in the bed ol particles will, again, lower the apparent "efficiency" Γ : ι crushing process. Turning now to the so-called ''laws of crushing" whiι· lι have been propounded, the earliest of these is that of Ri ι The

energ~orreseondin

to

( I) The whole of the energy considered is absorbed as (' 1 )

Hurface energy. 'L'Lιe energy per unit surface is independent of ρ article size.

Ι lιι l : ιw could not hold when strain energy and plastic ιiΙ l ο ι ι η tion energy is included. It cannot therefore be 1 1ιι τ t d to apply to the whole comminution process or to ι Ιιι ι'l! ι nminution of ductile materials. It would be expected

ι" I Η ' <ιpplicable

to the comminution of brittle materials

lιι ιι the energy expended in elastic deformation is de' Iι ι ι Ι ι·c L from the energy input to the particle. , \ί 11 , for spherical particles the specific surface, in area ιιιι ι p r unit mass, is given by

S = 6Jσd

.... (4.1)

Ι ιι ι·ι~ σ is the density ofthe materia1 ofthe particlt; and d ι Ι ι ι I )itrticle aiameter, It fol ows t at tlie increase in specific 11ι I:H; · during grinding is given by

S 1 -S0 =

(6/σ)(l/d 1

-l/d0 )

= S0 (R-l)

.... (4.2) .... (4.2a)

Ι ιι · ι·ι : R ( = d0 (d 1 ) is the reduction ratio. IΙ 11 w W is the energy necessary for the creation of one ι ι ι ι i ι nf' new surface the ratios of the energies necessary to ι ι ι I ι Η: unit mass of a given material through the reduction ι ι ιl os R 1 and R 2 is given by

WRΙ

RI-1

WR2

= R2-l

.... (4.3)

118



and ifthe reduction ratios are so large, in comparison unity, that it may be neglected, then WRι f WRz

=

Rι /R z

.... (4· .3ι ι )

Furthermore, the energy necessary to reduce unit of material from size d0 to size d1 is given by Wι

= =

(6/σ)

W(1fdo -1/dι )

W(So-Sι)

witlι

mn NM

.... (4Λ) .... (4.4:ι)

These expressions, the use of which involves a know ledge of the results of sieve analyses or of the specific sιιι· face determination, are easily applied and are widely usc ·ιl for the control of fine grinding operations. The law of Rittinger appears to have remained unch:ιl 1enged unti1 1885 when Kick propounded the "Gcsc·t ι. der proportionalen Widerstande", but since that timc.: ιι controversy has raged, concerning the validity of the two laws, which has not yet been resolved. The "law" as formulated by Kick is: (1) The quantities ofwork necessary in order to prodιιc · c · equivalent changes in shape in two geometricall ι similar bodies of the same material are in the rat ίο of the volumes or the weights of the bodies. (2) Bodies of given shape or form, in order to undcrι.~o a definite given subdivision or change in shaρι•, require an expenditure ofwork equal to the prodιιι · t of the weight and a quantity which is the quantit y of work necessary for unit weight of the same sh ;ψι· to undergo the same deformation or subdivision. (3) The forces necessary for corresponding changcs ίιι shape of two geometrically similar bodies of tiΙΙ· same material are in the ratio of the correspondiιικ cross-sections or the surfaces of the bodies. Actually the second statement has precisely the saηιι• information content as the first but is stated in an altern:ι ti.ve form.

Ιι

btained experimental

I

veήfication

119

of this law

Ι ι " ι 11 ι s ιs ο η the crushing of simple ο bj ects and, by Ι Ι'\11 iιι κ the law ta size reduction generally, was appar-

' ιιtl

.ι l>l c to refute*the law ofRittinger.

Ι ιι ιl{e case or

ittinger's law, an expression for the

11 ι Ιι i ι η ut in order to bring about any specified reduction ι 11 11 ι ΙΗιy be deduced. Thus, Kick's law states that the ssary to reduce a body ofunit size to a number of 1 ι ιl ιιι· I' ιιtιι Ιι ·s of, say, half the size is constant. Each of these Ι Ι Ιιtιι lι·s, however, requires only one-eighth (!-3) of the ι 11 1 ι ΗΥ f the original body to reduce their size to one-half, lιιιt , . ι Ν ιh re are now 8 (23) ofthem, the energy to reduce

.. 3 .ι

~

-~

.-----τ-----.--:-;-:-;--;----;---=""

2 1----+----~.r

::>

~~ I ~--~~---+----~--~ .,c

.....

ο

δ

4

2 Rεducιion

Raιio-

β

R

FIG. 4.3

Ιιι ι • i H iιt to one-half their size is equal to that required to ι 1iι ι ι · ι: ιhe original particle to particles of one-half of the 11 ο Ι' the oήginal body. Thus, to reduce α unit mass of a ιιι ι ιtι · ι · ίal to a given fraction ofthe ori inal size re uires the , ι ιι • ιι ιl.ίture of a constant amount of energy; regardless of tl11 ιι ι ί ιi al size of 12article. l ' l ιi s can be illustrated graphically as in Fig. 4.3. ι 'Ι ι . ιι· Ι y, if the reduction ratio R is defined as d0 fd 1 , when ιlιι ιt ι ίι lerial has been reduced to one-half the original Ι ι ι I α = 2 and one unit of energy has been supplied. Το ι ιι l ι ι <' · tlιe size to one-half again changes the reduction ι ι Ι ο f'rom 2 to 4, but again one unit of energy is required; ιι ι ι I ΝΟ ο η for further subdivision.

120


οΝ ΤΗΕ COMMINUTION OF SOLID BODIES

The relationship between W and R for such a case i. clearly W = KlogR .... (4.5) but this relationship can be obtained formally a3 follows:

I Iι ι • f'orcgoing analysis brings out a number of interestι 11 111 ιi ιι ts in connection with the fields of applicability of ι l1ι ι ι wo laws. In the first place Kick's law states that the ι 111 ι •,y ι bring about a specified change in the size of a 1ι 11 ι 1ι 1rιss of mateήal is constant and, furthermore, since 1l1ι Ι IΙ 'rιίη energy to induce a certain critical stress in a 1111 ι ιιι : ι ss of material is also constant, it would ap ear that Ιlιι ι ιι 1 ·ι· ·s involved in breaka e in accordance wιt ιc s I 1 . ι ι ·c· boa orces an furthermore, after fracture the ι ι I fl.Y~lιoula appear as heat. τ us, ick's law would be 1ιι ι ι ι ·ιl to ho1d or particles for which the surface energy 111 ~·, li ible; that is for large particles during coarse ι 11 lιiιι g. ι iιιι ι ν rse1y, Rittinger's law, which states that the energy Ι Ι I ι 111 1'(·d for fracture is proportionaJ tO the lleW SUrface Ι ιιι ΙΙΙΙ ' < I , would be expected to hold for very fine grinding ι ι 1ι •, ί 11 this case, the surface is large and the surface ι ιιι ι \Υ requirements could swamp the elastic energy 11 ι ι 111 ι ·c·ments. In fact there is evidence that, in practice, ι ι ι , ι ι ι: rushing is roughly in agreement with Kick's law, ι11 Ι 'Ι'ftS fine grinding is more nearly in accord with I ι ι ί ιι g r's law. At this point the implication of these two I '"' ί ιι relation to the physical process of crushing will be ιιι ιι . ί ι red. , Ίιιι·ka}(9) has pointed out that although Rittinger's law ι 1 Ι " ι·~ι:nts a logical interpretation of the process of fοrma ΙiιιιΙ οΓ new surface, the manner in which the process is ΙιΙΙΙ .ι t d is disregarded. ln technical size reduction, the 1ι ιι ι i ι ; l s are crushed by impact, pressure or abrasion. The 111 ι .ιι energy of the resulting newly formed surface is • 11 ιι ι ·c l at the expense of the elastic deformation energy of ι Ι ι ι Ι ι dy; the balance of the energy reappearing as sound ι 1111 l ιcat . Smekal has also pointed out that particles caη ιι ιιΙ I>• considered as being ofhomogeneous structure, and 11 1 Ιιί g( ΙΟ) has shown clearly how the influence ofthe microι ι ιιι· ture affects the validity of the laws of crushing. The Ιιιψοrιaηce of inhomogeneity in the particle is shown by ιlιι • nιc t that it is generally accepted that the calculated

(Ι/Κ) dW =

Increase dR in the reduction ratio expressed as a fraction of the initial value of R

= (R+dR-R)/R = dR/R or, on integration, W = Klog R

Spqcίlic

Fω.

.... (4.())

Surlacc .

4.4

Alternatively, since R=d0 fd1 or R=S0 fS 1 , equation (4· ./ι) may be written: W = Κ log (d0 /d 1 ) •.•• (4.6:ι) or W = Κ log (S1 /S0 ) •... (4 . 6 \ι ) The essential difference between the law of Rittinger aι11Ι that of Kick is that according to Rittinger the energy ι·c · quired to crush a particle depends upon the initial si~.c· of the particle, whereas the law of Kick states that tlιr energy required depends only upon the reduction ratio, The relationships b etween energy input and total surfaι· r, for various values of initial fineness, according to the law of Rittinger and Kick are shown in Fig. 4.4.

121

122


chemical bond stren th of a material is of the order ο/ Ο-1 οοο times t e actua mechanica strengt οΤ tlιι · matena . -urtfi'e'rmore itΊs generally 'Delieved Π1:ιt -τ e aiscr~pancy is due to points of inhomogeneity iΙΙ the crystal structure; the elastic energy for fracture b ci ΙΙ Η concentrated at these points due to notch effects. tπ It is based on the analyNi~ of actual size-reduction machinery rather than 011 theoretical consideration of fracture. According to tl1i ~ the,ory, termed by Bond the "Third Theory of Co~mi~ιι tion", the effective work done in crushing and gnnd111 ~ resalts primarily in the formation of crack tips and it iH directly proportional to the total crack length for~ed. Ιιι equidimensioned particles the total crack length IS stat<·ιl to be equivalent to the square root of one-half the surfa<·r area ofthe particle. The total work input represented by ι ι given weight of a crushed or ground product is t~ereίl rι · inversely proportional to the square root of the diamcιn of the particles of the product.

Ο Ν Τ ΗΕ COMMINUTION OF SOLID BODIES ΙΙ

Jιj!

123

represents the work required to reduce, from a F microns to a product with 80% Ρ microns a unit weight of material, then

f, ι ι l w iιlι 80 % passing

I' ,_ "' Α"

(4.7) Ι ιι ι ι · W . is the "work index" or the kilowatt-hours per Ιιιι ι ιτq u'i red to reduce from a theoretically infinite

I ' 1ι I iι · l • size Ι •: ψ ι : t tion 11ι l ι ι Jι'

to 80 % passing 100 microns. (4. 7) can be transposed so that when the work W; is known the work input W required by any feed and product Ρ can be found. Thus,

W= W;

y'F - y'P 100 y'F ·γp

(4.8)

11 Ν represents the reduction ratio, or F JΡ, then

y'F y'R y'F-y'P = \IR-1 W;

w

y'R

y'P

= vR- 1 Ίοο

.... (4.9)

Ί ' Ιι ι· mesh size in microns, of the sieve through which /11) ' 1 ~, ο Γ th e mat~rial passes is selected as the criterion of Ι ι ι ι\ :ιnd product sizes since this is a figure which is readily tt lιtι iωιble by plotting the results ofscreen analyses. lt IS ι l , ιιιιιι:cl to be more reproducible than the size of mesh ιlιι ιι ιι !-ζlι which 100 % of the sample passes, but ot~er ι ιι Ι <·r·s use the sieve which passes 100 % ofthe sample w1th

cess. l1 as been pointed out by Walker, Lewis, McAdams ιιιι l illil and<Ι3) and more recently by Coulson and ι ι ι lι : ιrdson (14) that the various laws of comminution result ιι ι • ιι ·rgy relationships which can be derived from the

Ι ι ι Η ιι

Ιι

ι

fll 'ι ·~sion

dWJdd = -Cd"

.... (4.10)

124

Ο Ν ΤΗΕ COMMINUTION OF SOLID BODIES

B A LL , T UBE AN D ROD MILLS

which states that the change in diameter, and hen 11 1 specific surface, brought about by a unit of energy, iH . ι function of the particle diameter. If the index n is put equal to -2, and the equaιioιι integrated between the limits d0 and d1, tlιe expression

W = C(l/d1 -l /d0 )

. . . . (4·. I I)

is obtained; this expression being identical with equ al i ο 1ι (4.4), and so is Rittinger's law. Kick's law may be derived by putting n equal to - J ίιι equation (4.10) and integrating between the same linιiι as above, whereupon the expression . . . . . (4.1 J :ι)

which is identical with equation (4.6a), is obtained. In the same way, insertion of n = -3/2 gives dWfdd = - Cd-31 2

which when integrated between the same limits as ρι·ι · viously, gives . . . . (4. 1 '~) or 2C d0 l /2 - d1 I /2 .... (4.] ι :ι ) dol /2 W = dιΙ/2 . Bond's law thus represents a condition between tl1o ι • corresponding to the laws of Kick and Rittinger in n· l i Ι tion to the energy n ecessary for crushing through a g ίνι • ιι reduction ratio. It has been pointed out by Dobie< 15) that if the en c ι·l-( • WR necessary to crush a material through a redu ιίοιι ratio R is written in the form. . . . . (4. 1:i)

where d0 is the initial size of the material, then the Rittinger and Kick are contained therein.

law~ ιιl

125

Ί ' Ιιιι s,

WR = bfd0

(Rittinger's law)

ιιιιl

WR = α

(Kick's law)

Ιιιι ιlι ι· nnore

WR = bfy'd0

(Bond's law)

there is a reasonable amount of evidence that r's law applies moderately well fo_r fine grinding ιι1ιl Μ iΙι e Bond's law is deduced from the results of ιΙιιιιιι • ι· us test on real mills, it appears that, for fine grindΙιι , i Ι is necessary to have a law in which the work input, 11 1 , Ι ο bring about a given change in the reduction ratio, lιι ιι ιlcl be a function of the fineness of the material and, ΙιιιΙIΙΙ · ι ·mοre, WR should increase with increasing fineness. I Ι ι ι ι lιi H basis Carey and Stairmand(l6) have suggested that, I 11 ι ΙΙ 'Ι l 1' tO proνide a criterion for the performance of ιιι IIΙιι l{ machinery, a crushing coefficient which increases ΙΙiι lineness and is related to the size distribution of the I' ι ι Ι iι · l ·s produced by the crushing is required. In this case ιl ιι ιτ ushing coefficient is d efined as the energy cor" 1 ιο ιιdίηg to a given reduction ratio, and so co.ι;τesponds ι ι ι 11' 11 bove. Ιιι ol'der to obtain such a "crushing coefficient", Carey ΙΙιιl Kιait'mand made use of the similarity in frequency ι I Ι ι i l>ιι tions of crushed products and so developed the ιιι 111 iρle of "associated energy". Ί'Ιι • characteristic grading produced by free crushing is I• Ι 1111' l a "natural" for the substance, and it was found ΙIΙ ι ιl il' the largest particle of a natural be crushed and ιι lιlι · ι l to the remainder of the distribution, the resulting ι ιιli ιι is again a "natural". By summating the energy ι ι ι I'' i ι· d to crush from one natural grading to another ιιι . ιllι· r one, it is possible, by laboratory experiments, to ι~lιΙ,ι ί η the energy required for free crushing between 11 ιi ΙΙι ·;ιl distributions of given maxima sizes. Thus, for ιΙιφlc, ifitis assumed that a "natural" with a maximum Ι ι ι ol' 1 metre represents zero associated energy, and it is lιιΙΙΙΙΙI ιhat the energy required for successive crushings to :Ί ιΗ~

ι ι Ι ι ι ί ιι ~

126

ΟΝ ΤΗΕ COMMINUτJON OF SOLID BODIES


produce a "natura1 " ofmaxim umsize 1 cmis 0·5 kWh /tοιι , and that a further addition of 0·2 kWhfton produ <'Μ ι "natura1 " wi1h a maximu m size of 1 mm; then t~e e?<'Ιϊι,~ associated with a natural of 1 cm maxιmum sιze IS Ο ι kWhfton and the energy associated with a natura1 of 1 ηιιιι maximu m size is Ο· 7 k Whfton. In order to derive va1ues for the associated energy, ΙΊ ι ·ι crushing was effected by crushing between horizoιιt ιl p1atens with a 1imited reductio n ratio; a 1oad versus d_ Ιlι· ι tion curve being used in order to compute the energy 111JHΙI to the crusher. By use of the figures so obtained for tlιι associated energy, a figure for the mechani cal efficien y cιl any grinding process may be obtained from the relatioιι ship. Mechani cal efficiency of grinding Energy associated with product - Energy associated with Energy used by the mill

-

Γc ~ c·c I

In the absence of estab1ished laws of comminu tion, tlιι methods of Bond and of Carey and Stairman d proviclι · 1 means of deriving a _crit~rion of performa nce for m~Jiiιι ~ machine ry. The denvatωn of values for the assocι:ιtι · cl energy or work index is tedious and the use of siC'Vι analyses for the size frequenc y determin ations limits tlιι fineness of grinding to which the method can be appliι · ιl , but, nevertheless, these methods appear to form usι·/Ίιl bases for the design of crushing machine ry. Turning now to the efforts which have been madι· tcι establish the validity of the varioίιs laws of crushing Ιι experim ent, it must be S!ated at O~Ce t!ιat much of I Jι. work must be treated w1th reserve. ,The reason for tlιι Λ caution is that the πieasurement of both the energy inJHΙt to the crushing system and of the fineness characteristic Λ of the product are problem s of the utmost difficulty. • Conside r first the difficulties involved in the determiιι 1 tion of the fineness of the crushed material . The us · οΙ

127

extrapol ation ofthe grading curves into the and, since very high specific surface is assoι i ι Lcd with the smaller sizes, small errors in the extrapol a1ι οιι will give rise to large errors in the calculate d surface this 111 ιhe product. Theoreti cal difficulties also arise in surface external ifthe even , example for Thus, nection. ι1 ιι ι ΙΙΙι ιld be determin ed with absolute accuracy , the existence 111 internal surface within a particle of any real substanc e ι ι ιι ds, to some extent, to reduce the particle to an aggreιt · ofsmalle r particles . Thus it would appear that the surI Ι ιι · · which should be measure d is the external surface lus Ι l ιι · surface of an crac s wlϊ1c are su ae~ wι e fo Ι / ι ι · olies1ve forces, between the two boun aries of the . 1 ι . ι k, to be ~ffectively zero. t first sight it would appear that a better approxιma Ιi ιι ι ι to this true surface would be ο btained by the use ιι / Lhe adsorpti on method of surface measure ment. UnI Ιιι l ιιnately, however , the theory of the adsorpti on method ι ill in an incompl ete state and in conseque nce no more ι ι l i :_ιnce may be placed upon the adsorpti on method, as an 11c · 1ι rate measure of surface, than any other. Even if this 1ι • not the case, however , there still remains the guestion , ιι l wlιe ther the adsorbed molecules wed e ο en narrow 'IF- .. ..,.1.:-:-. ~SCSrsurface ,W.:~ I I ιc: ks · and 50 whether ad sorptlOll ffiCt oas 1 Ι ιiι: ι ιs not strictly free surface in the present sense. Thus, 11 ι ιι us t be admitted that no known methods exist, for the ι l 1 Ι ι : ι' mination of the sιze or ~ecific surface of the smaller l ιι ιι • Ιί ns, which are free from criticism. Ί'Ιιc method of solution of tne particles , adopted by 1\ 1 ιι · ι ίη(l7) and by Gross and Zimmerly<ιs) is also open t_o ι ι 11 iι• i sm since, for example , it is known that a crystal 1s ι Ι Η ι ι lved at different rates ο η faces parallel to and per1'' ιι ι lί ular to the lattice planes. Hence, a figure for the ιιι Ι ι ι c area based on the average rate ofsolutio n, without ιi Ι t• ι· ιain k~OWledge that the particleS are geometri cally Ι ιιι l l : ιr, is unreliab le. supplied Ί ' ιιrning now to the measure ment ofthe ener ted that apprecia e once at Ι ιι Ι Iι • crus ιng process. Ι t wi ιι · νcs involves ι ιl >sieve range

I

128

129


I Ν' ΤΗΕ COMMINUτiON OF SOLID BODIES

the energy to which reference is made in the equation~ 1ιl comminution is t e energy usefully expended in the fr: ι< tιon of the partic e and not tlίat which is in the crushiιι B system as a wfiole. onsiaer, for examp1e, the measuι 'Ι ' ment ofthe energy transferred to a single partic1e by a dι·cψ weight. The initial potential energy of the weight may Ιιι · determined with considerable accuracy and, by the usι~ cιl modern laboratory methods, the height of rebound of ι IΙ Ι ' hammer may also be determined. Thus, the energy givcn ιιιι by the weight may be determined with adequate accura<' . The energy given up by the hammer is, however, in ΙJΗΙ ι converted to strain energy, and thence to heat, in part ιιΝc · ιl to increase the surface energy of the powder, in parι ιιι give kinetic energy to some ofthe particle, in part to sοιιιιcl energy and in part to "pelletiz~ some of ~he prodιιcι. 11 , however, this heat energy coul<Γbe deteπruned, then, rtN ιι first approximation, the energy used to fracture ι lι1 particle could be regarded as the difference between ι Ιιι energy given up by the weight and the heat ene~gy .' c· jected. It is easily shown, however, th~t the determii~a~ιoιι of this quantity of heat is extremely d1fficult. Thus, 1f ι ι ι assumed that the hammer weighs 10 lb and falls throuι.~lι , say, 3 ft, and also that the metal masses (hamη:er, anvίl , etc.) which will be heated weigh 20 lb, then a s1mplc ι::ιl culation shows that, if all the potential energy were οιι verted to heat, the average temperature rise woιιld lιc about 0·05Ψ. Thus it is almost impossib1e to make a lιι· ιι balance with worthwhile accuracy. For this reason, ι lιc analysis of the results of crushing tests are usually b~Ν ι · ιl ση the assumption that the whole of the energy lost by ι Ιι,ι hammer is expended in fracture, but such an assumpt10ιι Ι clearly of doubtful validity. Α further complication is that it is possible thaι ι Ιιι geometry of the syst~m and the rate. of loadiD:g of ι I ιι particle have, for a g1ven energy dens1ty, some ωfluι· ιιc c on the fineness of the product obtained . With the drop-hammer apparatus, the energy, althoιιl(lι available in the hammer, is apparently not effectivc~ ly

ι ι ιιι lc'l't' d to the particle. Thus it would appear that the ιιι ι μ, y ι r·ansfer process is, or can be, a controlling factor.

ll ιi N vi w is supported to some extent by the statement Iι 11.111 , ιι Ο) that the Hadsel mill, in which ore is lifted and I ι 11 11 ι I I' pped ο η to steel plates, has a power consumption •I 11 I' kWhjton which "compares favourably with ballιιι 11 1 onsumption for feed of i- in."

'

Νι

ο

ο

ο

v

ο

ο

L

/

/

v

/

v

40 30 20 10 Nft Enι;:rqy lnρut.- Kqm .Cm./Gιn. Fισ.

L

50.

·4.5

Ί'ι :Μ ts upon quartz, by use of a drop hammer, carried

ΙΙΙII l>y Gross and Zimmerly
using a drop-weight apparatus

ιιι i l ar to that of Gross and Zimmerly, also carried out ι ι ι ι lιing tests, bu~ in this case the air permeability method

----

130

.

~--

ι

- - - -



was used for the determination of the specific surfa1· ι ·. These workers also found a constant relationship betw<ΨΙΙ work input and surface produced for glass, fluorite a ιιιl calcite, Fig. 4.6a, but, for salt, which is a softer mineral, .ι

l ιιιι ι ι l spectrum" to explain similar results obtained for ι Iι ι ι i rιding of sa1 t in a ball mill. These resul ts are also ιι

131

ιι · ι · ord ance with the work of Smekal(9) mentioned

I'' ι \ i oι ι s !y.

Ι ι ι o ι·der to carry these investigations into still finer size ι ιιι ι ·ιι , Kwong and Piret< 21 > made use of an adsorption

e

~

u

~400~----~~~74------+-~L_-i u

.. ι

''ι Ιιιιίψι e for specific surface measurement, with ethane as ιlιι . ιι Ι Μ rbent. For the harder minerals, they found an Ιιιι ι ι . ι Ν · in the energy input per unit increase in surface 111 ι Ι Ι οι· increasing fineness (Fig. 4. 7) but claimed that ι Ι ι ιι· ν aled no discrepancy with their previous work

u ~

~200~--~~--~~------+------i

έ2ooor------r-----+~L-~

.

~

~

z

Ν·

Ε

u ι

ο

40

20

Nct Enεrqy ιnput-

CIO Kgm.Cm.j Cm.

~ΙΟΟΟΓ---~~-----+-----4 "'"

80

~

α

~

z

&

ο.

u

~

300 200 ιοο Ν• ι Energy Jnput -Kgm.Cm./CCm.

Ε

u

..

FIG.

ι

~250~----~~---4------+------i

"'.." ~

8 -

Natural .

z

0 -

Synιoεtic.

ο

40

20

Νεt Enerqy lnput-

CIO Kgm .Cm./Gm.

80

b. Fι G .

I

4. 6

non-linear relationship (Fig. 4.6b) was found. This rcΝιιlι indicates a decrease in the amount of surface formecl, IH I unit input of energy, for the larger values of specific ΝΙΙΙ face. Plastic deformation of the sa1t crystals was suggcslc•cl as the reason for this. The work ofHίittigω is ofinter ·st iιι this connection, since he suggests the presence of' ι

4.7

ιι ιι ι.~ ρ r meability techniques, since the results were eχ

Ιιιιι l ι ·< l into much finer size ranges. This work also shows ι l ι ι I ον ' r the size range studied, the ratio of area by ad•Ι ιι ι ι iο η to the area by permeability is fairly constant at ι i ΙΙΙΙΙ I 1·9-2·0. Work was a1so carried out on the slow

ιιι ιιψι · ·ssion crushing of particles, and in this case, the

ιι ι I input was measured by computing the area under

ιlι ι ι ι ιr νe obtained by plotting the displacement of the • 111 lιing surfaces against the applied load. The results Ι ' Η · ~·.8) show clearly that the ratio of new surface to ι ιι ι ι ι • y is greater for slow compression crushing than for

Ι ιιψ . ι ι · ι crushing. This deviation between the results was

1

1ιi , Ι in

d tentatively by changes in the usage of energy in

ι 1 1 ιι . ι Ι d drops of the ball of the impact crusher.

132

ιΙΝ ΤΗΕ COMMINUTIO N OF SOLID BODIES


If it is assurp.ed that the average stress concentratioιι ί 11 is a function ofthe energy input for each ιl1 ·ι 'J ι ο the all, and that a critical stress concentrati 11 ι ne~essary for fracture to οι:;cψ, then initialiy, lll the c!Ι'(ψ we1ght crusher, the stress concentration in the bed is 11ί ι.~ l1 , and therefore the energy is used efficiently. However, wiΙiι ~essive drops the effective energy input per drop ιlι creases by 40-50 °0 due to the frictionallosses between ι I Ι ι par IC es, an so t e average stress concentration, and (' ΟΙΙ sequently the num - er ot fractures, decreases. th.Ξ_material

3000r------ ------.------ ------·

Ι ι ΙΙ Ιιι ι ι ·

was caused by high local stress concentrations at

" I ι Ι ι ι · l y Γew points; the e~astic deformation energy, which ι ι ψι~ι · ; ιι ·s

mostly as heat, IS thus small compared with the

ιιιl ι ιι ι · Ψ?duced. When fracture occurs at high energy

""'

ι ιιtΙ·atωηs,

however, fracture starts at a number of high level of en_exg;χ; ιΙΙIΙ ΙIΙοη, andsince t~e fracture at most points will / ιιιιι ι · ι :ι Ι. only a small _d1stance before meeting a further 111 ι ιl rI'~Cture a relatively high..:eerce!ili}g e of the elastic ι ιιι ι 'y_ w1ll rea pear as heat. The effiCiency ol the process ι•• ι ι ι Ι ~. ;~ ιnce more .:eωnts are at a

•

ιi

~00

"'

έ

"''u

~

Ε

u

~2000~-----------1----τ7~----~ υ

ι

,;..

..

Sιnglr Cryιtαlι .

Multiplc Croinι.

•

·;;:

u

. I

• ο-

t'"200

"'"

ι

. ..

::>

ο

~

~ 1000~------or~~~----------~

n. 100

z

~ 11'1

"'~ .

~

u

. )<

ο

100

Enrrgy lnput- Kgm .Cm./θnι.

200

z

~

ο

200 400 600 Enrrgy Drnιity- Kgm.Cm./Gm.

FIG. 4.8 FIG. 4.9

The work of Axelson and Piret(20) is of particular in tι · ι ι · ι in the study of crushing, since these workers have invι· ιl gated the slow crushing of single polished and unpo liHiι ιl quartz crystals . By us e of a technique for energy ίιφιι Ι measurement similar to that described above, it was ((Ηιιιιl that the necessary stress for fracture lay between 9780 : ιιιιl 500 kg/cm 2 , the natural crystals tending to have thc Ι : ιι Ι{ ΙΙ stress for fracture . It was observed that although thι: Ν ιιι face created is greater the greater the energy concentt':ιtiιιι (kg cm fg), the surface createa _eer unit energ!J.'inf.J.ut de ι·ι· ιι or Increas1ng energy concentratιon, as Shown in Fi . ·f e lίypoth esis used in order to explain these results is ι ι when fracture occurred at low energy concentrations, ι Ι ι

133

134


linear relationships were found to exist betwee!l cnι · ι Κ)' input and new surface area for Cumberland Barylι 'H, North Wales limestone and Durham anhydrite. ιι adsorption technique, with krypton gas as the aciHoι bate, was also used to measure the total surface, bιι ι "" evidence of a constant ratio between the surfac ' Iι\ adsorption and permeability was found; in fact the r:ι ι iι ι was found to decrease with increasing fineness . The ~uestion of elleting_, of the roduct mentioιιι • ιl ~ier, is of some iήterest since t e researclies of Βοwι1ι•ιι aή τ~> sho.;t:hat impacted "sίιr.faces may bc οιιιι Sfiongiy wel eCΓ'tO'g'Ctner. I'i: ιs eviaent that i this' h.~l)ι · ι t ere will e a consideraple Ι ο ,ut surtace · with οιι · quent increase in particle size. .. Res-eafclies oy :M"acκenz!e and Milne<23) show thaι ιlιι X-ray diffraction pattern of muscovitep ecomes mor ιlil :fίi'Se as grinding proceeds until, after 8 hours, no s lι ; ιιιι pattern remains. After further grinding the pattern ι ι · appears but has a different form to the original. This rι·Ν ιιl suggests the possibility of re-crystallizati on of tlιι· ι matenals an , a1tho·u gh the evidence is not conc1usivι· , ιt ιs suggestive that the same ,ε2 enomenon mi ht occuι· iιι the fine rnilling ot other materia s. In fne worκ whιcfi fias 6een dιs cussed, in this chaptι· ι·, Ιt has been assumed that there is a similarity in the cruslιiιι attern of materials comminuted by slow crushing antl 1 blows of. various }!l<;ιgnitudes. owever, none of t1ιι workers appears to have attempted to establish eχρι·ι i mentally the truth of the assu·m ption. This raiscs ι Ι ι following question: If a given input of energy to a part ίι Iι produces a given increase in surface and assuming, Ι ι'' example, that an input of twice the energy produces twiι ι the surface increase, then are the size frequency curvcN Ιιιι the two products similar, with the scale of particlc 11i1. changed, or is there no relationship between the produι : t , other than a two-to-one ratio of the increase in finencNH ι' This question has been studied by η en (24) J1o ever, who has s~q~:,ι. tl.le.t the cumulative curves of tlι

Ν ΤΗΕ COMMINUTIO N OF SOLID BODIES

I'' ιιι lιι

135

ιs obtaiμf..ςl, by g,ushi!.lgM,<,L~~callysirnilar bodies 'ι ι ι ΙΙ' ιl1ogonally affiiιe, ~ aήa"Lnence ιt fi5l ows a : ' ( I) The size distributions for the materials are geometrically similar. (' .) :rhe specific surfaces. of a series of products vary ιnversely as the absCissae of the cumulative curve ΓΙ r any specified value of the ordinate. ' Λ Ν • rie~ of product curves for various porcelain cubes, ιlι · t •rmιned by Andreason, are given in Fig. 4.10, and it ιoor-------,-,-----rr-------,

80

.. ....... 60 ..., Ν

c :;,

.

.,. σ

..

c 40

...u...

α.

20

Pr ~ ssurq lmρact

ο

Crushinq. Crushinq.

4·0 Grain Slzq- Mm. FIG. 4-.10

6·0

136


is seen at once that, for all practical purposes, thesc : ιι ι · orthogonally affine. These curves are also representa tί ι of curves obtained by tests on feldspar cubes, gl : ι H cylinders, etc., and so it may be stated that there is sοιιιι experimental proof that geometrical similarity holds ίιι the fracture pattern. The present authors, Rose ;ιιΙΙI Sullivan, have, in the experiments on the crushing of Ι ιι r spheres, also found that the product curves are, roug lιl , orthogonally affine and, in this respect, confirm the wοι lι of Andreasen.

I ι

I I

ι_------------ -

ldealized.

Real .

Fracture Patterns According to Andreasen. Fιa . 4 . 11

Andreasen has also put forward a logical explan a tioιι for the existence of such similarity. Thus, in Fig. 4.1 Jι ι i shown the pattern obtained by the actual fracture οΙ ' 1 cube and in Fig. 4.11 b, an idealized fracture pattern wltiι Ι ι can be deduced from it. Ifit is assumed that the prin iιι ι ΙI plane of failure is parallel to one of the diagonal pl a ιιι • 1 then it follows that the grain size of unity comprises 50 ΆΊ, by volume of the material, the grain size of-!- compι· i N ι ' 25 % , of t comprises 12-!- % and so ο η. Ι t is also clear t l1 11 the weight cumu1ative curve for such a product j M 1 straight 1ine passing through the origin. Thus, t l~ι · ι ι appears to be a fundamental r eason for the type of ιιι• ι obtained by Andreasen. His work is also interesting in tlι.ιl the treatment leads, at once, to Kick's law, and Andre ίιN rll

( Ν Τ ΗΕ COMMINUTION OF SOLID BODIES

1 1ιl . ιiι1 · th at the c1oser a

137

roximation to Rittin er's law

lιt1 Ι ι} obtaιnea<:Ίq.r,~ ne ,g:ιn 1~ anses rom a oss

ι 1! ι 11 1 ι ncy of th e rmll wfl en _ri~In hner ~artιc es. \ιι interesting pωnt wnic emerges f~ e anaiYs'i'; Qr \ι ιι Iι·ι· ηs n's drop-weight tests is that, for the larger cubes, Ι lιι ιl{ rcement between the product size obtained for ΙΙΙι1 · 1·n .rgy input, by slow crushing in a press and by drop ι ι i\ Ι ιt, ιs very close. For the smaller cubes however the 11ι ιιι Iι ι ·t size from slow crushing is in very close agree~en t 1 ι Ιl1 Ι lιat ca1culate~ from the s1ow crushing of the large 1ιιiΙΙ 'Η nd hence wιth that obtained by impact crushing ιιl Ι IΗ : large cubes, but the product obtained by the impact ι ι 11 Iι i ι ιg of t~e small cubes is far too coarse, although the 1ιιι ν ι 'Η are stιll orthogonally affine. Since Andreasen used ι Ιι κ ι · and Mϋller(27) andΌthers int~ Ιlι Ι ' 1Jossible existence of a general law of size distrib~tion lιιι 11 wdered materials a1so suggests the existence of a Ιιιιιι l : ιmeηtal mechanism of comminution but "laws" of 1:ι~ · ι'< l appli~ation do not, at the present' time, exist. ll ι · comp1ιcated process of fracture and the lack of lιιΙΙι10gcneity in the structure ofthe majority ofmaterials Ι ιι I11 · giΌund, particul~rly n.ιinerals, makes the developιιι · ι ι ι of a genera1 relatωnship between size distribution Ι ι,ι · ι · ·duction and energy input extremely difficu1t. Thu~ Ι l ιt .ρ~·ocess of deducing energy relationships is largely ι ι ιφιπcal and r elated to particular conditions and types ιιl ι ·ι · luction machinery. ' I'11 pres en~ po~ition regarding the investigation of the I Ι\V.H Γ commιnutωn car;ι probably be fairly summed up ι lιι ll ws. The problem ιnvolves extreme difficulty in the I Ιιιιιl ;ιmental measurements. In addition, apart from any

138

Ο Ν ΤΗΕ COMMINUTION OF SOLID BODIES


questions of the accuracy of the actual measurem ηι , much of the existing work must be treated with res ι· v ι since it is generally impossible to differentiate betwιΨιι energy put in to the grinding system and that usefιιll · expended in comminution. Many of the published resιιlι are further confused by the crushing of deep beds ι ιl particles; th,ι::re . b~ing po ~theρreticall sound reason w Ι ι the crushing ofthe particles in a bed should obey the sηιι11 rufι;,~ a,s ,tJi~~ <;rusfϊi[fOFasΊz:~<;..,Bar IC e. ln gener:i, questωns ansrng rrom imperfect natun· ,,, individual crystals and the inhomogeneous nature ofrnί111~' industrially important materials have not received ι Ιιι attention which the subject merits. The truth ο[ Ιlιι assumption of geometrical similarity between a partiι lι and its progeny has not been adequately investigated. ΊΊ11 laws of Rittinger and of Kick are possibly extremes, Ιιι · tween which most ,practical ideal crushing processcs Ι ίι· It is scarcely possible at present, however, to statc Ιlιι conditions for one or other of these laws to be applical>lι ·, nor is it possible to state, in any particular case, the law 111 which the actual grinding operation will approxinι:ιlι most closely. It appears fairly safe to say, however, Ιlι.ιl coarse jrinding ~JL.<:I?.EΣ.~?i i,R~,.EJ.ψe closel to } iι · k ' law an fine fr~ rn~-more near ~.19.. !{\~~i~~r) l_<~;w. Tli.e extent to whrc the desrgn of a mifl, and. ιlιι physical dimensions of the crushing elements influen · Ιlιι crushing characteristics of a given material is unccrt:ιiιι , It ap_eears that1 in general, the efficiency of transfι:ι· 11l energy to tlϊe articrerans wrt ecreasrn ar rc c Νι:ι. ι an t rs actor a one ma es· t e overa e avrour ο ιιιι ι ι tl'.l?es οΓηnΊΤ aμr'ox'imafe to '1he"..~itlin~er" tylξ οΠοιw even tliought e fιind aήi.eαι'at......c'ΠisΠing process m rg more nearly in accordance with a law of the "Kick" ty)ιι . Finally, it may be said that the subject of cοmmiηιιιιιιιι offers a wide field for useful, interesting and fundamι• ιιt ιl investigation.

139

REFERENCES ' 'ι ι, ιι

ι ι οττιc, G . F. Dechema-Monogr., 1952, 21, 245 and 96. ι •'ΙΙΙ .ΙΧ, and GEIGER, τ. Sulzer tech. R ev., 1956, 38, 14 . Ητ ττ, F . D . Proc. phys. Soc. Lond. , 1949, 62, 418. ιιιι.ΕΥ, D. Ρ. Proc. Conf. Ultra-Fine Structure of Coal, 1943.

w.,

ι ι ιJMPHREYS-0WEN, s. F. Ρ. Proc. Phys. Soc. Lond., 1955, 68, 325. ι (οsΕ, Η . Ε . Unpub1ish~d research, but see J . app. Chem., 1957,7, 244. VoN RιττιΝGΕR, Ρ. ν. Lehrb. d. Aujbereitungskunde (Ber1in, 1867). Κ ι CJK, F. Das Gesetz d. proportionalen Widerstiίnde u. seine Anwendung

(Leipzig, 1876), p. 14. 11 ΗΜ ι ~Κ ΑL, Α. Ζ. Ver. dtsch. lng., 1937, Beiheft Verfahrenstechnik, Nr. 5. Sce a1so: Ost. lngArch., 1953, 7, 49 /70; Glastech. Ber., 1950, 23, "7/67; A cta phys. austr. , 1953, 7, 110 / 122. 111 ι ll.INIG, F. Forschungslι. Ver. dtsch. lng ., Heft 378, 1936. ι I ΛΧΙΙLSΟΝ, J. w ., and ΡιRΕΤ, Ε. L . lndustr. Engng Chem., 1950,42, 665. I 1 ι ΙοΝD, F . C. Chem Eng., 1952 , 59, 242. Ι Ι WΛLΚER, W. Η., LEwιs, W. Κ . , McADAMS, W. Η., and Gιι.ι.ιι.ΑΝD, 1\. Κ. Principles ofChemical Engineering (New York). 1937. 11 ( : ouι.soN , j. Μ., and RιcHARDSON,j. F. Chemical Engineering (London), Vol. 22, p. 665. ι 'ι ι >οωΕ, W. Β. Recent Developments in Mineral Dressing, 1952. Ιιι ( 1Λ ιιεΥ, W. F ., and SτAIRMAND, C . .J. Recent Developments in Mineral Dressing, 1952. 11 ΜΛR"ΓΙΝ, G. J. $oc . chem. Ind. , Lond., 1926, 45, Part 4, 160Τ. ΙΙι ( :ιιοss, J., and ZIMMERLEY, S. R. Bull. U.S. Bur. Mines, 1938, 402. 1'1 ι ι /\ ι. ι., R. G . Trans. Amer. Inst. min. (metall. ) Engrs, 1935, 112, 20. 11 Λx,LSON, J . w., ADAMS, J. τ., jOHNSON, J. F., KwONG, J. Ν. s., and ι > ιιιΕΤ, Ε. L. Trans. Amer. Inst. min. (metall.) Engrs, 1951, 190, 1063. ι Ι ι •'Λ ι Rs, G. Μ. Trans. Amer. lnst . min. (metall. ) Engrs, 1953, 63, 211. ι ι ι ΙοwDΕΝ , F. Β . , and TABOR, D. Friction and Lubrication of Solids, Camb. l] niv. Press. Ι ΜΑ 'ΚΕΝΖΙΕ, R. C., and Μιι.ΝΕ, Α. Α. Miner. Mag., 1953, 30, 178. ιΙ N Ι JREAS,!!Nl Α . Η. Μ . Trans . lnstn chem. Engrs, Lond., 1957 35, 104. Ι ι ΜΜι'ι'ιΝ,
PROCESS OF GRINDING WITHIN Α MILL

141

Ι ιο ιι l d be analysed and an effΌrt made to establish the t ι· nt to which the performance of a mill may be pre' I ι ι ι · d from theoretical considerations. The experimental ι ι ι ι Iy of the performance of mills has, in general, been ιl ι ι ιι ~ the following lines: ι

CHAPTER 5

ΤΗΕ

PROCESS OF GRINDING WITHI Α MILL

Ν

IN the previous chapter, the various "laws" of cοmmίιιιι tion which have been propounded have been studi d ίιι rela.tion to the physical process of size reduction and to ι Ι11 · avaιlable results of experiments into the fracture of hοιιι11 geneous and ~eterogeneous materials by impact ancl Ιι slow compressωn. ln this study it is seen that no sal ί factory "law" of comminution has yet been deduced Ιι 11 t tha~ a number of m~re or less empirical expressions : ι.ιι · avaιlable for the gUidance of the designer of crus hiιι ~ machinery. ι The presen~ c~apter. wμl be devoted to a study of ιlιι • process of gnndιng wlthιn a real mill. Since within 1 "tumbling" type of mill the size reduction can result fΓοι 11 dire~t cr~shing, from shearing or abrasion between ι 111 partιc!es ιη ~he feed, between the particles in the feed nιιcl the mιll bo~ιes or between the particles and the milllini 11 1-\, t~e process ιs much more complicated than in the casι: οΙ sιmple ."[ree" cr~shing c~ndi~ions considered previously. In addιtωn to ~hιs c.omplιcatωn, however, the moti ιι ιιf the ball charge ιtself ιs extremely complex. Thus the pr Ψ i Ν ι · mode of.transfer ofthe energy to the crushing surfaces :Ιιι not be sιmply represented. For these. reasons, . a rigid theoretical analysis ο Γ ι Iι ι • para~eters ιnvolved ιη the milling process becomes ίιιι pract~cable, a?d recourse is frequently made to pilot sc: ιlι expe~ι~ents ιη order to determine optimum milliιι l-\ condιtι?ns. Ν evertheless, considerable experimental ωιι Ι theoretιcal work has been done on the crushing proc sιι iιι the "tumbling" mill, and it is desirable that these c\ιιtιι 140

( I) T he eneΓgy relationships and theoretical efficiency of the ball mill from surface energy and thermodynamic considerations. (' ) The size distribution of the product in relation to the mill parameters. (3 ) The rate of crushing in relation to the mill parameters. Lual.ly these three divisions are not independent since, IΙ ' i ll 'ly, ιf there is a definite relationship between the ι ιΙΙ · ι · y input to a particle and the change of surface area, ιι ι • ι · · luction ratio, it follows that a given mill, to which Ι Iι ι · <~nergy input is constant, willlead to a definite rate of ι ι 11 ~ lιing ; the rate being a function of the relevant mill Ι ' ι ι ·: meters. ln fact the inαease in the surface area of the Ι ιΙ , ι ι c~i~l, prod_uced by the in pu t of a uni t of energy to the ιιιι ll , ιs ιncreasιngly accepted as a criterion of performance IΙΙΙ ' Γιηe milling operations. Without doubt this is due to the ιι Ψ iι nological importance of specific surface in the subM iι · ν · range, and to the establishment of reasonably satisl ι ιι · t ry techniques for specific surface measurements. It is Iι ιι• ι l1is reason that Rittinger's law, even though lacking a ι ι ι ιn d theoretical basis, is widely accepted as a basis for ι l ιι · alculation of the performance of such machines. Ι . ι~ t us aE l these ideas to the derivation of an eguation 1

I Ιι Ι ' Ll1e rate of p;rinding in a mill. lt follows that if,

ι ψ Ι Ι ~ars

as

reasonaole, tne cnange of SJ)eCific surface S 1 ιι,ι 2f m 3 , within a mill d epends uEon tl1e dιameter oftne ιι ιιΙΓ, upon the ail diameter and uEon the mean diameter ιι f' ι Ι ι ' h arti cle, D:aandb ~spectivel>J. upon the densιty of ι IΙ ι ' bal s and the ulE, ρ and σ respectively on gravity, g, " " ιbc speedofthe mill, Ν; upon u tfie diameter ofthe exit Ι ι ο ι the mill, upon Ε, the energy necessary to change the

142


Ι ' I{OCESS OF G RINDING WITHIN Α MILL

surface ofthe material by unit amount and upon the pa ι·: t • meters J, U, V and Ζ, where J is the fraction of the n1ill volume occupied by the balls (or rods) and the spac Ι ιι· · tween them, V the volume occupied by the powder (ο ι pulp) charge, including νoids expressed as a fraction ο[ ll11 volume between the balls (or rods), Uis the ratio ofsoliιl volume to total volume of the pulp, and Ζ is the ratin ι ιl circulating load to the total load when closed cirι· ιι ίι operation is considered, and clearly, S will d epend on ι Iι ι · time of milling, t. Then we may write symbolically : S = Φ (D, d, b, ρ, σ,

g,

Ν, Ε,

t, J , u, U, V,

Ζ)

143

1 \ι Ιιιι· · proceeding wi'th such an analysis of experimenta1 ι lιι '• Ιι οw ver, a digression will be made to intro_duce Ι ιιιι 11 y ιι theoretical treatment of the internal dynamιcs of ιl11 Ι ι : ι 11 mill propounded by one of the present authors, Ι ιιι , . (I) This, as far as the writers know, is the on1y ιιιι ιιφ to deduce the general behaviour, including such ι ι ι Ι ι ι ι·ι-; :ιs ball wear, fro m purely theoretica1 considerations. Ι ι ι ι ιl~ι·ι· more this theory can best be studied in conjunc-

llιιιι

wi ιh th; analy:;is of the experimenta1 resu1ts which is

.... (r.. l )

where, as before, φ d enotes "a function of". The application of the principles of dimensio rι : ιl analysis, already illustrated in Chapter 3, leads to l/11 following equation.

(S .D) =

Φ{ (Diρ), (i), (i), α) , (;) , (n~2)'

' } (Dgt2) '(J), (U), (V), (Ζ) Fιc.

which, as before, may be written

Φι(Diρ) ·Φ2(i) ·Φs (i) ·Φ4(n:Vz) ·Φ5(~ ) 2

(S.D) =

χ Φ6α). Φ1 (J). Φs( U). Φ9 ( V). Φιο(Ζ). Φη(~) .... (r:.•) This equation (5.2 ), is the dimensipnless equation gi v iιι the rate of production of specific surface in a tumbliιι H mill. Purely theoretical considerations are insufficient to t;ιkι · the ana1ysis further, but, in pr.actice, the various dim<• ιι sionless relationships may be established by referen c tιι experimental data.

ιΙΙ ·ι · ι·Ηsary ι ψ ι :ι ιίοη.

5.1

for the evaluation of the above dimensionless This theoretical treatment is of necessity eχ Ιι ι · nΙ ·ly tentative since the dynamics of a mill are of ~uc~ ιιΙΙη ρl xity as to defy rigid analysis. Nevertheless, ιt ιs lιι · liι ·vcd to form the basis of a more precise treatment. Ί'h basic idea underlying this treatment is that the ~ lι o l behaviour of a ball mill depends on the probability ιι l . ι particle arriving into a "zone of operation" between 1~ ι ι /)alls, and on the probability of occurrence of some I'' ιι<· ss when it arrives there. Outside this zone, no οpera Ιιιιι ι , such as crushing or metal wear, which demands the Ι ι .φ ρίηg of a partic1e between two balls can take p1ac.e. Ι •' ι·οm Fig. 5.1 it follows that the zone of operatωn ιs a ι i ι τ ular area of radius Υ, where Υ is given by

144

ι • ROCESS


Υ2 + α)

2

= (

d; b)

2

=

ιιιιΙitί ιη.

Υ= /(dbf2J (5 .3) provided b is small compared with d; d and b being t]H· diameter of the ball and of the particle respectively. Now the maximum number ofparticle s, Nmax> which caιι be in the operating zone are those which just form a rin ~ around the contact point. *

MILL

145

per per

-

< :ι'Ι' t : ιίη plausible probabilit y functions are derived for ιlιι ι φι antities and so an expression for the equilibriu m ΙΙιιιιιiΗ · ι· [ particles embedded in the surface of a ball is ~ι l ιΙ , ι ι ι Η ·d ; this expression incl uding terms to cover the Iι ιιι\ιιι·s of the ball relative to the hardness of the particle. I Ι ιΙΙ <ιy then be stated that a relatively very soft particle ιll I ιι· rushed when trapped between

Thus, from equation (5.3 ),

' Γwο

bare metal surfaces, ne bare surface and one surface "armoure d" with embedded particles, Two "armoure d" surfaces,

2πν'(dbί2) = dN max Nmax

Α

l11 ι ι · Ν; is the number pressed in to the metal su.{[ace ι ι ι ι Ι Ι i ι η • and Νο is the number of particles torn out

~ +~ +~

or

or

OF GRINDIN G WITHIN

= 4·5(djb) 112

.... (5.1-)

ln fact the probab e number ΝΡ of particles in the zonc· of operation is less than this; the maximum number occuι· ring when there is an infinite number of particles in tl1c· system. '" t ιs suggested that a likely form of the probabilit y fun<:tion is given by Φ(Ν) = 1- e-kv .... (5.5) and if it is assumed that about 40% of the maximuιιι number of particles are present when the void space bι· tween the balls is just filled, that is when V = 1, then ΝΡ = 4·5 (dfb )l/2(l - e-0·5V) .... (5.()) Thus, this equation gives the probable number οΙ particles in the "zone of operation ", in terms of the b:ιll diameter and the quantity of powder present in the rnill , Considera tion is then given to the equilibriu m numbc· ι ofparticle s,N,, embedded in the surface of a ball and tlιi H is clearly given by dNfdt =Ν-Ν .... (5.7) I

* The number would scarcely be expected

Ο

to be proportional to tl1c ΗΙ 'r of the circle of radius Υ, since a particle inside the circle automali rι Ιιl y relieves some of the others of load, but all of the particles in a ring coulcl Ι ι loaded.

Ιιι ι·ι·as a relatively very hard particle will be crushed ιιιι l y I tween two "armoure d" surfaces (it will be pressed Ι ιιι ιι : ιη "unarmou red" one) and probabilit y functions to 1 11 ι · ι · ιhese possibilities are introduce d. On the basis of ιlιι •Η ι ' Γunctions, expressions for the rates of milling, in Ι ι ι ιιι Η of the various mill parameter s, are deduced. , ΙΙΙΙi l ~ιrly it may be deduced that metal wear will occur 1 Ι ιι · ιι , during relative sliding motion between a pair of lι , ιii N, tl1e "armoure d" portion of the surface of one ball ι ι ι ι · r·ses the "unarmou red" portion of the surface of the ι ιι Ι l-\ lιbouring ball. From a knowledg e of the equilibriu m ι ι ιιιη I r of particles embedded in the surface of a ball the ΙΙ ο I>:ιbili ty of such "armoure d" to "unarmou red" contact

Ι

c·omputed and expressions tor metal wear deduced. ' I'11 analysis leads to extremely complex expressions for ι l11· ι·rι ιe of crushing; expressions which do not lend themι · l v ι·s to a general form ofsolution . For the two boundary 1 ιι ιι litions, in which the particles are either very hard or ι • ι γ ·oft compared with the balls, however, the equations , ιιιιιl<· r certain assumptio ns, become tractable and the ιι lιιtίοη for these two conditions can be written in the I ι ΙΙ ' ΙΙ ι

S• = Kt

.... (5.8)

146


ι>ROCE SS

where S is the sp_ecific surface of the particles,~ K is a ι:οιι stant, aiϊd α is -! for very soft materials and 4·5 for νι· ι ' hard materials; all other cases falling b etween thes ι wc 1 conditions . It is interesting to note that the case in which α =-! ι· οι responds to the " Bond Law of Comminution" dis cιι ssι • ι I previously. Furthermore, for some intermediate hardιH 'H there is an αpproximαtely (but not strictly) linear relaιioι1 aor---------~-.------------.------------,

~δΟΓ-----~~---r--~.-------r-----------~ ;:.

.

·= ~

~40r------------+-----------+-~~~----~ .. u

~

11'1

ιn/+ ~... Ι

u20l_____

-ι~τ==f==~~----~fi~====~~:1

"'

ο

1·0

2 ·0

Ball Diomot
5.2

lnι. )

b

3·0

ship between the increase in specific surface and thc llΙil · ling time. The analysis also reveals that the rate of grincliιΙ increases w1th decreasωg 6all s1ze; the Improvemι·Ιιl 15rouglit about --ya:-gΊven reductωn of size being gr <ιtι · ι with soft material. This result is clearly confirmcd Ιι Fig. 5.2, in which results of'tests by Coghill and Devan<'y(' Ι on chert and dolomite, these materials having hardncss ι · in the ratio of about 3 to 1, are plotted. Analysis of the fineness of grind in relation to the pοwι\ι·ι charge reveals that within limits, the fineness of grindiιι~ can be increased by reduction ofthe amount ofpowder in ιlιc·

OF GRINDING WITHIN

Α

MILL

147

111 11 ;

:ι ll other variables being held constant; a maximum l1 ι 11 • ι· ached when the powder about fills one-tenth of Ιl1ι ιφ;ι between the balls. When a soft material is being ι ΙΙΙΙΙΙΙ Ι , however, this effect is much less marked and also ιlι1 Ιl1 <ίximum is reached when the powder occupies about ιι1ιι lι ;ι lf of the space between the balls. The analysis also lιι1~VΗ ιhat the wear on the surfaces of the balls and mill is 1 /Ιιιιι· ιiοη ofthe quantity ofpowder present; this function ιι ι ΙΙ l1ing a maximum when about one-tenth of the space l1ι Ι w ιτη the balls is filled and thereafter decreasing rapidly. ( )ιιι' Γurther matter of interest arises in connection with

ΙIΙΙ ι ·once:et of a " zone of operation" in the b aH mill. It Iι Ι ll ιιwι; from ~ uatωn (i.3), that'""'iJie"""radius of the zone 1ι Ιι·ι; as the square root of the particle diameter. Other ιlι ι ΙΙ f.Ι'Η being equal;tFie'refore, there is a defΪnitely reduced ΙΙΙιΙiι:ιbίlity of the smailer partiCle entering the zone of ι ψι ι· : ιιίοη and, consequentiy, a preferent1a gniiaωg of the I ιι κ ιτ particles would 15e ~pected; rovided, of cours~ ιlι.ιΙ Ιl1 e partic1es are neitlί er s'Oi)Ig 1n relatωn to the ball ι , ι • Ιl1 a.!J~ are ex~ed from the we ge-sha ed s ace l1 ι Iw · η tlie bails nor tlίat the balls contain jnsufficient 1ιιι · ι 'ΑΎ for tlie rac ure ο t e RartiC es. η fact, the ι~ι ·c ific gnn Ing charactenstics' ' deduced by Theimer ιιιιl Moser <3> show a marked tendency in agreement with ιlιι·Η : conclusions. This preferential grinding would also " Kf.\'CSt that the size distribution of the material after ι ι ιιcling would not be geometrically similar to that ιιιili :ιlly, but that a "tail" initially on the right-hand side ι 11 Ι lι curve would be reduced; this geometrical similarity ltι · Ι wcen mill feed and product would not be expected. l•'r· m this theoretical treatment the following tentative ι 11 11 \Usions emerge. ( Ι ) The processes occurring in a ball mill are controlled Iιy ιlιe probability of particles entering a "zone of οpera ΙιιΗι" between two balls and by the probability ofoccurΙt ' ιι ι·' of some process after they have entered the zone. (2) The rate of grinding of the powder is a function of ΙIΙι; quantity ofpowder in the space between the balls and

148


this function reaches a maximum value when this spac · ί approximately full. (3) The fineness of grinding attained by a lιιιι·ι l material in a given time of grinding increases rapiclly :1 the quantity of powder between the balls is decreascd · : ι maximum being r each ed when the powder fills appro i mately one-tenth of the space b etween the balls. Wh n .ι soft material is being ground, however, this effect is mιιι· Ιι less marked and also the maximum is reached when 1Ιιι · powder occupies about one-half of the space between tlιι · balls. (4) The law giving the change of surface with tim ιιl grinding is of very complicated form and includes tcrιιι Μ depending upon the h ardness of the material undergoiιι κ grinding. (5) The reduced probability of the smaller parti I1·M entering the grinding area would suggest that there ίΝ .ι preferential grinding of the larger particles. Thus it woιιlc I follow that the size distribution curve of the material a Γιι· ι grinding would not be geometrically simi1ar to tlι:ιι initially, but that any "tail" initially to the right-h a ιιΙI end of the curve would be reduced; the n egative slop · 1ιl the curve thus being increased. (6) The wear of the b all surfaces is a function of ιlιι · equilibrium number of particles pressed into the surfac't' and this number depends upon the relative hardnesιι 1ιl the balls and of the particle. Thus it follows from ι lι1 · treatment, that there exists the possibility that sojt b:ιll will wear less than hard ones; since soft balls mί ι.~ Ιιι acquire an artificial "hard" surface by reason of the Ιι : ιι 1I particles embedded in the surface of the metal. This is iιι accordance with the practical observations of Engcls, l Ι) who states that the bodies ofmills for grinding soft m a Lι'l'i.ιl should be hard whereas the bodies of mills .for grindiιι~ h ard materials should be soft. (7) The wear on the surfaces of the balls and mill i~ .ι function of the quantity of powder present, this fιιη ιiΙιιι reaching a maximum when approximately one-tenιl1 ιιl

I'ROC ESS OF GRINDING WITHIN Α MILL

149

ΙI ι ι Ι ι : ι · between the balls is filled and thereafter de' Ι ιι , ι i ιι ~ rapidly. will be disconclusions thesc of validity IΊ ι ι · Α' ncral ι • 1 Ι Ι nιer in this chapter in connection with the study 1 11 ιι/ ι l ιc ι·ι·su} ts of grinding tests. I ι ·vι·~· ting ?ow to the study of the grinding tests by use 1 '' ι Iι ι '. lιmensιonless eqιιation (5.2), it is found that Gross<5J Iι ι I' ι νι~ η figures f?r the values of "Rittinger's Constant" f, ΙΙ 1-\ ιlc:na, sphalente, pyrite and calcite, as determined by

FIG. 5.3 11

ι

or a

laboratory ball rnill, rιιη under specified cοηdi

IΙΙΙΙΙ~, ~nd also figures for the Rittinger's constant as

ιlι ιι ; ι· ιηιηed .by the use ofa drop-weight apparatus.

( ι l ι: rly, sιnce, as follows from the treatment of Chapter '', ιlι ' po~er. to drive the miiΓis sensϊ6'~ constant, tfϊe rιlιιι· ο[ Rιttιnger's constant for tlϊe miiJ.s proportional !Ι ι t Iι ' ι~crease of specific sιιrfaces, S. Furthermore, the 1 11 ι ψ-we~ght. figures for energy p er unit increase in specific ιs dιrectly proportional to Ε, so when the Ritιιιl <ι IIΙΙ Ι-Ι'c•r's constan_t for the mill is plotted against the results ι ι/ Ιlι drop weιght test a curve of S plotted against l jE i ιΙΙ ' (/1C mill is obtained. Ί'lιis is done in Fig. 5.3 and a sensibly straight line is

150

PR OCESS OF GRINDING WITHIN Α MILL


obtained; thus it follows that the functiona1

(S.D) =

relationshiρ

Φl (D 2 gρfE )

is, in fact, a direct proportiona lity or

(S. D ) =

ο

K (D 2 gρfE )

•

•

•

(5.9)

This relationship is for a batch mill grinding a singlc· material. This expression can, however, be written as

(S. D ) =

K ( D4gρ. 1 fD 2E )

151

Ι ι ι l ο ιt d in Fig. 5.5, and it is seen that the influence of ΙIΙΙ ι.~ι· up is not large and it appears probable that the

ι lιllc'l'c · n e arises from the reduced efficiency of crushing ιι lιί~ Ιι nergy densities, which has been discussed in the I"' vio us chapter. These relationships are still further cοη Ιιι ιtιι· I, lιowever, by the results of comparativ e tests, carι ι ι l ou t by Coghill and Devaney, in which chert and ι lιιlιιιηίι are ground by use of balls and pebbles (Table ' I ) . Ί'Jι se resul ts are summarized in Ta ble 5.1, and i t is ι

I

ιι [ Jιat

(5.10)

the agreement

ΪS

good.

ι·ο ,r--ι==::::::=τ----,

~ t----/ι-------1J?jf------l ι (~σ'}

ο

3

2

FIG. 5.5 2

6

4

Sp~ιcilic Croνity

Ot

8

TABLE 5.1

Crindinc,ι M~diα.-p,

FIG. 5.4 1\ Ι

where D 4gρ is the potential energy in the rods and D ~ /•.' is the energy associated with the powder, and a furιl1ι · ι check made on this relationship since, in a given mill, tlιc · increase in surface, SJ er unit time will be 12ro ortioιι ιl to the density: of the grinding bodies, ρ. _ Coghill and Devaney, (6) have given the results (τabl<· 1) for the grinding of chert in a rod mill, in which the densi t of the rods is varied and these results are plotted in Fig. 5.'1. It is at once seen that the two quantities are almost exa tl proportiona l, as required by equation (5.10). IC tlιι deviation from linearity is assumed to arise by reason ιιl the existence of the group Φ 6 ( ρ fσ), then this relation s lιiιι may be computed. The relationship Φ 6 (ρfσ) so determinc·cl

ιιι· ι· ίal

Density (g/cm3)

Increase in surface per hour

Balls

Pebbles

Balls

Pebbles

7·8 7·8

2·7 2·7

15·4 46·0

5·5 16·9

Ratio of surface ιn crease

houι·

Ratio for ρ =

7·8

ρ=

2·7

and

From Fig. 5.4

I il ιr • ι · ι ll ι ι l ιι ιηίιc

2·8 2·7

2·58 2·58

ΊΊιc function given in Fig. 5.5 is probably not highly

ιι c · ιιra te, since it has been derived from a combinatio n of ι r 'Ν ιιlιs for tests on drop-weigh t crushing, on a ball mill

Ι ι Ι I η a rod mill, but it is ofvalue in that it indicates the

11

Ι'Ιl

Js which may be expected to exist.

ΊΊιis treatment leads to the interesting idea that, for a

?

153


) ROCESS OF GRINDING WITHIN Α MILL

given material to be ground there pos~ibly exis~s :ιιι optimum density of rriaterial for the gnndiΠg matenal. Thus, from Chapter 3, the power Ρ to rotate a ball ωill ο<: is given by

ι h e same figure is also shown the correspondi ng ι ι ι ι νι· [ι r :;ι rod mill; and it is seen that the same general Ι ιι ι ιιι or relationship holdso Ί' Ι ι ι•s curves are not of great practical importance since ιlιι Ι yρ s of materials suitable for grinding bodies are very lιιιι iΙ ι ο< l , and so the range of densities available is re-

152

Ρ= ρ(l +0ο4σf ρ)

7

~ ρ+0-4σ =J<:..("+0·4-<:r

(5011 )

Also, from Figo 504

Iι ι

I. curves do show, however, that the use of lum s of ο 1es 1s a practice which, at least on ι ι ι ι ;ιs grinding ιι ιιι ι 1ds of miΠϊng effiCiency, cannot be recommende do Ί' Ιι · next p air of groups which may well oe stu 1e are 1 ι Ι ι I co

Ί' I Η:

ο ο ο ο

Sf t oc 50ρ- ρ 2

(5o I

1 )

ι οο

-

/........//

ο

1/

.."' c

ο

ο

ο

I

3

8 δ 4 Dιrnsity Of Powdt:r oσ'- GmjCm3

ο

05

<'!ιο> FιGo

οοιο

ο ιs

507

FIGo 5ο6

•/• (rl / D) and Φ 3 (bfD) and these are most convenientl y cοη

_§_

?

~

Pt-

50 ρ-

ρ2

ο ο ο ο

ρ+Οο4σ

On differentiati ng with respect to simplifying,

ρ,

(5o I 'I)

equating to zero

:ιιιcl

ρ2+0°8σρ-20σ =Ο

or

ρ= -Ο ο4σ ±y(Οο016σ2+20σ)

ιι l ι·ιο d together. From Table 32 of the work of Gross the ι l ι ι · of Rittinger's constant for various diameters of ball, Ιιι .ι ιnill of specified size, is given. Since the diameters of ι11i ll and balls are known a curve of Rittinger's constant

ψ,. ι inst (d/D ), Fig. 5.7, may be plotted; this curve being ι ψιi valent to a curve of Φ 2 ( d fD) plotted against (dfD). Ι'lιι ο variation shown, however, arises in part from the

of (b /d) which occurs with the change ofball size; arising from the use of the same feed material Iι ιι ' c~ach ball sizeo (: ghill and Devaney, however, give sufficient data, in Ί ', ιl ι l s 2 7 and 28 of their work, for these effects to be ι ι ι;ιι·a ted, and the basic data from these tables are plotted \ι ι ι · i :ι ιiοη

ο ο ο ο

(5o l•l)

From this equation the optimum values of densiιy οΙ balls for grinding materials ofvarious densities, are a lc · ιι lated and these figures are plotted in Fig. 5ο6.

Ι I ι iHvariation

154

PROCESS OF GRINDING WITHIN


Α

MILL

155

ι J •' i ~ . 5.8; a base of dfD being used since, again, the ball ι lι , ιιη tcr and the mill diameter are known. Since the time ι ι l •ι·inding is comparative1y short, it is assumed that the ι llι · ι · ι ive particle diameter is the average of the mean 111 11 ι i I size of the fe ed and of the product; the mean ( Ι Ι ΙΙ ' ( i lc size of the feed and product being assumed to be ι l11• Hi ve size which passes (or retains) 50 % of the sample. ι l11 tllis basis the data of Table 5.2 may be calculated. Ι ι · now the curves of Fig. 5.8 are studied, it will be 11 ιι Ι ίι· ·d that the curves are similar in form and that they ιιι . Ι bc brought into coincidence by :

( I ) Multiplying the abscissae of each curve by a suitable constant, such that the maxima are brought on to the same verticalline ; this constant being related to the mean particle size of the material to which the curve relates. (' ) Multiplying the ordinates by a suitable factor.

e-

-65Muh .

( ) - -35M
- -20

M
8 - -14 M
ο

0·05 b.

FIG. 5.8

Ι •:xamination will show that the curves are brought into lιιιι ί:ι. ntal coincidence by multiplying the value of (d fD) lι y (D/b)l/ 2 and this product, after squaring, becomes (ιl ''j f)b) which may be treated as the new independent . ι ι ί;ιble. This may be regarded as a correction to the (d fD) Ι ι ι ιη to allow for the effect of the ratio (bfD ) upon the ι llι · · ι of this term . W lιcn the data are plotted for the dolomite and chert ιι Ι ltis way, as in Fig. 5.9, good horizontal coincidence is ιιlιi : Ιincd. (The root scale ofthis diagram is used merely to ι ιιl , ιι·g the scale for low values.) It will also be noticed Ι l1.1 t ιhe maxima for these materials occur at, as closely as ι , ιιι I> expected, d2jDb = 2·00 ( = l-42 2) for the dolomite ιιιιl 2·56 ( = 1·602) for the chert.

Si11 e the mill is 19 in. diameter, these values give the ' ΙJΙΙ : ιtions : d2 = 38·0b for dolomite .111 ι I . . .. (5.15) d 2 = 48·5b for chert

}

156


e-

PRO CESS OF GRIND1NG WITHIN Α MILL

-δ5 Μ<ιh .

( ) - -35 Mcsh .

@- -20M .. h. • - -14 Mcsh .

w

50~--~~~--,_~r-τr-----+-

& - -10 M••h. Upρ c: r S«Γit'ι- Oo!omitc . Lowcr

Sc:riιs

- Chcrt

u σ

L ~

"'

~u 40~--~~~--~~--~~~~~---i----~~----

157

ι ιιΨi lι d value of (d2fDb), are plotted agaiι:st the corι ι ι ιιo nding values of (bfD ), the curves of F1g. 5.10 are ιιlιι.ιίη d . μ;~ιίη it will be seen that the curves may be brought ι ι ι Ι ο almost perfect coincidence by multiplying the ιιιι l ί ιι a ι c for chert by 2·8 and, as a confirmation of the ιι ι ιιι · a y of the treatment, this is almost exa~tly the ratio ι ι l ι lι c rates of milling obtained from the Independent Ιι IN, ιnaking use of balls and p ebb1es, summarized in 11

w

ο.

~

"' : 30~~~4-----~-----t-----t----_,~~~r----

ο

~

• L

u

c

ο

) ο

d_'2

- = 1 ·4 2

Db

.

1

)2

}· 42\

2

ι-~

-r: 3

FιG .

ί

""

.......

~ ~ΟΜΙτΕ . /CHERT.

v

4

5

ο

6

0·001

ο/ο Fω.

0·002

0003

5.10

5.9

Ί'.ιl> le 5. 1. It will, however, b e remembered that Ε for ι Ι ιι·r ι is about 2·8 to 3·0 times that for dolomite, and it

d2 = 35b d2 = 55b

}

(5 . llι )

given by Coghill and Devaney,(6) and it is seen tha ι ι lι ι · agreement is satisfactory. These formu1ae are also sirnil:ιι to those developed by Starke. (7) · If now the ordinates of the curves of Fig,: 5.9, for a ιι

Ιν ill . ι ~~ ι ·ι~ ΙIΙ :ιι IH' l ι

also be noticed that 3 1 /3 χ 38=55 which is in fair ment with equation (5.15 ). This will at once suggest the curves of Fig. 5.10 will be brought into even r agreement if the quanti~y d 2 / Db is multi:plied by (/':) Ι /3 . However, since in the present work chert IS taken .ι~ f unit h ardness, this result is best expressed as (ιl 2fDb)(ξ /E ) ' I \ where Ec is the value of for chert. Ί ' Ιιί quantity is then used as the complete llldependent

!i

ι ιri able .

158


PR OCESS OF GRINDIN G WITHIN

TABLE 5.2 'Ί.

Feed Ball mesh size

Mean feed size (in.)

d

pa~ticle

b

d

lJ

D

( ίη. )

( ίη. )

0·0098 0·0098 0·0098 0·0098 0·0098 0·0098

0·0089 0·0085 0·0079 0·0072 0·0071 0·0070

0·0094 0·0092 0·0089 0·0085 0·0085 0·0084

0·0034 0·0046 0·0065 0·0085 0·0113 0·0138

0·145 0·105 0·072 0·052 0·039 0·032

0·00050 0·00048 0·00047 0·00045 0·00045 0·00045

2·75 2·0 1·38 1·0 0·75

0·0210 0·0210 0·0210 0·02\0 0·0210 0·02 \0

0·0140 0·0138 0·0138 0·0134 0·0126 0·0146

0·0175 0·0174 0·0174 0·01 72 0·0168 0·0178

0·0063 0·0087 0·0127 0·0172 0·0225 0·0285

0·145 0·105 0·072 0·052 0·039 0·032

0·00093 0·00092 0·00092 0·00092 0·00089 0·00094

2·75 2·0 1·38

0·0400 0·0400 0·0400 0·0400 0·0400 0·0400

0·0156 0·0119 0·0126 0·0109 0·0145 0·0301

0·0278 0·0258 0·0263 0·0255 0·0272 0·0150

0·0101 0·0129 0·0191 0·0255 0·0362 0·0565

0·145 0·00147 0·105 0·00135 0·072 0·00139 0·052 0·00134 0·039 0·00144 0·032 0·00185

....:-6- f-(j7s

0·0480 0·0480 0·0480 0·0480 0·0480

0·0152 0·0151 0·0186 0·0166 0·0329

0·0316 0·0315 0·0333 0·0326 0·0405

0·0115 0·0158 0·0240 0·0326 0·0540

0· 145 0·105 0·072 0·052 1·039

0·00167 0·00167 0·00175 0·00171 0·00210

2·75 2·0

0·0690 0·0690

1·0

0·0690

0·0226 0·0232 0·0230 0·0286

0·0406 0·0460 0·0460 O·Ms8

0·0167 0·0230 0·0330 0·0488

0·145 0·105 0·072 0·052

0·00240 0·00240 0·00240 0·00258

2·ί5

65

2·0 1·38 1·0 0·75 I 0·62

_,. 35

·. s __.

tm"

20

·~- ~ 0·75 0·62 14

Ι

--· ~

0 ·04-8

I

I?

b

(in.)

1

CHERT

Mean

- -- -

20 ·8

10

2·75 2·0 1·38 1·0

sιze

Ιnnι·ι ι Η ι in ~ι ιι Ι ι ιι ι

I

t'

-

5.

'· 75

Ι
4·2·!1 51HI 57 · 1 ι

pa~ticle

sιze

Mean product size

( ίη .)

( ίη.)

(ίη . )

0·0098 0·0098 0·0098 0·0098 0·0098 0·0098

0·0075 0·0075 0·0071 0·0070 0·0070 0·0074

0·0184 0·0184 0·0184 0·0184 0·0184 0·0184

Mean

b

I

Increase surface per minute

d

b

d

JJ

J5

0·0086 0·0086 0·0084 0·0084 0·0084 0·0086

0·0031 0·0043 0·0061 0·0084 0·0 113 0·0139

0· 145 0·105 0·072 0·052 0·039 0·032

0·000455 0·000455 0·000440 0·000440 0·000440 0·000455

0·0 122 0·0116 0·0108 0·0109 0·0107 0·0132

0·0 153 0·0150 0·0146 0·0146 0·0145 0·0156

0·0056 0·0075 0·0106 0·0146 0·0193 0·0252

0·145 0·105 0·072 0·052 0·039 0·032

0·00081 0·00078 0·00077 0·00077 0·00076 0·00082

15· 1 18·7 20·9 23·0 -τrϊ 18·3

0·0374 0·0374 0·0374 0·0374 0·0374 0·0374

0·0142 0·0138 0·0138 0·0 142 0·0174 0·0285

0·0258 0·0256 0·0256 0·0258 0·0274 0·0329

0·0094 0·0128 0·0185 0·0258 0·0365 0·0530

0·145 0·105 0·072 0·052 0·039 0·032

0·00134 0·00135 0·00135 0·00136 0·00145 0·00171

14·4 16·4 19·3 18·4 17·4 12·1

0·0700 0·0700 0·0700 0·0700 0·0700

0·0328 0·0390 0·0512 0·0530 0·0622

0·0514 0·0545 0·0606 0·0615 0·066 1

0·0187 0·0272 0·0441 0·0615 0·0885

0·145 0·00272 0·105 0·00286 0·072 0·00320 0·052 0·00325 0·029 0·00350

13·4 15·3

Γeed

s ιz e

ίη

11·1 15·0 19·5 21-4 23·3 10·6

6(Ηι

:.ι · 75 '·Ο

3S·O 4:\ο/Ι

I <18

57·0

1·0 0·75

67 ·rΙ

m 3!ΗΙ

1Ηί2

'11

54 ·!ι

57 ·!1

-;,.,, 3(ΗΙ

!ΗΙ

47 ·11 4 7·'• 5().11

";ffi.

:.ι · 75

:.!·0 1·38 1·0 0·75 0·62

47 ·11

111

2·75 2·0 1·38

1·0 0·75

.12:.7~

15·7 11-4

3:1·11 37 · Ιι 4!ΗΙ

:Ι ···

Thus, the_func~iona1 relationsh ips (d /D ) and (b/D ) m:ι summanz ed 1n~o the two curves of Fig. 5.11 aιιι l F1g. 5.12; where F1g. 5.11 consists of the curves ο[ Ι •ϊ/-\ . 5.10 brought into coinciden ce by multiplyin g the ordinzιtι ·H ofthe curve for chert by 2·8 and then plotting the resu1tiιι~ curve on a scale such that the vertex gives 100 %, anιl

~ ~ ~Q.f1o\-e ~) d...:::. · Ιl ' ..D~ ' ' I (

( i ιι . )

I :ιι - u

2rι·Ιι 3ΙΗΙ

I

b~

M ean

\ι

mίιιιιlι

I

1\ ι ι ΙΙ " ' '1.('

ι ι ιl

1111

pt"Ι"

I

~ ~r 0 ·0690

159

MILL

DOLOMΠE

Mean product size

d.

Α

J ι' ί κ . . ιι ·ι ·

5.12 is the resu1ting curve when the data of Fig. 5.9 brought into coinciden ce by the use of the curve of Ι •'ίκ. 5.1 1, the resulting curve being p1otted on a per~ ι ι · ntage scale. : nsider now the form of the curve Fig. 5.12 . FQr a ~· i νι η size mill and ~artic1e, but with an extremely: smali l ι: ΙII the rate of rin ing -:Ι.1IlJ e 1ow; since tfie a w1ll not ι 11ι1tain sufficient energy to proauce much new surface. VViιh increasing ball size, however, the rate of grinding

160


ι>ROCESS

('~

ο

I

ο

•ο

20

~

I I

ο

0001

<%)

1------

0002

Sj t W lι.cn

I~

1/

0·6

~ 0 ·4

0·2

~

~

~ J'-..._ ..........

I I

~

!/

ο

5

dΧ ~(~ Υε ) ~ FIG. 5.12

MILL

161

α:.

d3 ( l jd3) = constant

.... (5.17)

the balls are excessivel lar e, however the rate a ter 4, the 111 oclu ctωn of WILa.\e-εer unit ο erier χ · ecreases w1t ι • ι y I1igli ener~ ,9-epsities and also the efficieήcy of the I ιlow is impaired When th e balls are unduly large in relaIΙι ιιι to the mill size. Thus, the falling branch of the curve Ι ol)lained. (In fact this _reduction is over-emphasized by ι Ι ι ι · use of the root scale; ση a natural scale the decrease is ιιΙΙι h more gradual.) Furthermore, as the ball diameter ι ι ΙΙ ' Ι ' ·ases, the number of particles which may be trapped, ιι Ι :ι point of contact, is increased and since an impact \ν lιί h contains enough energy to crush a single particle ι l ιιι·Ν not necessarily contain enough to crush two, and so 11 11 ; ιhis cause in itself can result in a reduced rate of li ' iι1ding. Thus several causes operate to reduce the rate of grindΙΙι ι.( with excessive ball size and this is exactly wha_ t has Ιιι τn fo und by experiment. It is interesting to note that ιlιι : results of Gross, Fig. 5.7, are in close agreement with Ιlιοs d educed from the work of Coghill and Devaney, in a suitable ch ange of the scale ratios will bring th e

ι ι l •ι · ιn ΙΠg ecreases Since, as ex a1ne 1n

1· 0

0·8

Α

Ί ' Ιιιι s

FIG. 5. 11

~

OF GRINDING WITHIN

1\ ill ίtt Γ ease since the increased energy will bring about ιιι ι ·ι · :ιs d surface production per blow. This gain is in part ιι ll ι · Ι by the decreasing number of balls, and so blows, 1 • i ιlι iπαeasing ball size but, nevertheless, the result is an ιιι ι · ι ·: ιsc in rate of grinding. When the balls have reached ι ι ~ ι · such that there is suffiαeήt energy to crush p raclιι , ι ll y ~artic e w~en, theoreticaiJΎ:"" Ι lιι · ι·;ιιe ο gnn ing wou rema1n cons ant Since t e ι ιιι • ι '/.\Υ producea 15y aπ impact ο a al 1s proporfωnalϊO Ι Iι ι • ι · ιιb e of the ball diameter, d 3 , whilst the number of Ι ιιψ : ι c ls, per unit time, varies as the number of balls, that , : ιr· ί s inversely as the cube ofthe ball diameter, that is, ι lιι · ι · ι· tly as l fd3 .

100

162


PR OCESS OF GRINDING WITHIN

curve of Fig. 5. 7 into extreme1y c1ose agreement with that of Fig. 5.12. Turning now to Fig. 5.11; ifthe ratio of (d/D) is constant then in fact (bjD ) is direct1y proportiona1 to (bjd) · the ratio of partic1e diameter to ball diameter. It follows from equation (5.3 ) and the exp1anation given ear1ier however, that the number of partic1es which wou1d bι: crushed between two balls 1s g1ven by t e expressωn Ν

cc

b

.... (5. 18)

when b is small compared with d. Thus, for the initia1 part of the curve, the rate of grinding wou1d be expected to vary as the square root of th · partic1e diameter and, in fact, this part of the curve /' Fig. 5.11 coincides c1ose1y with such a re1ationship. As thr partic1e size increases, however, the ang1e α, Fig. 5. J, increases unti1, at a certain critica1 partic1e size thι· partic1e is expelled from between the balls; because th · frictional forces are insufficient to ho1d it in p1ace. Clear1y, when this condition is reached the partic1es are expelled, instead of crushed, and the rate of grinding should , ideally, fall to zero. Since in practice, however, the paτ· tic1es have a range of sizes and the ball surfaces are η ι p erfect1y smooth, the curve shows a steady d ecrease in grinding rate, instead of a sudden drop to zero. This view can, to some extent, be confirmed as follows. For the maximum, at which proj ection commenc s, (bfd) =0·00075. Assuming this condition a1so coincid ~ with the maximum of the curve of Fig. 5.12, (d2jDb ) = 2, or (d/D )2 = 2 χ 0·00075. Thus djD = 0·04 and so (bjd) = 0·00075 /0·04~ 0·02. Thus, cos α= d j l·02d = 0·98 or α= ll ο 30' approx. Also a study of the forces acting on the partic1e shows that th1· coefficient offriction, μ, is given by μ=tan α or μ~Ο·2 ; this value being quite as close to the va1ue which wou1d bι· expected as can reasonab1y b e hoped . The infiuence of the ratio of rod diameter to mill dia meter on the rate of grinding for a rod mill may now w 11

Α

MILL

163

Ι ιι · <' nsidered, and in Fig. 5.13 are given curves ofthe rate ιι f ιωrΓace productiαa re1ated to (d/D ) ratio, based upon f' ιlιf 11 of the work of Coghill and Devaney. A1though ιlιι · l. ata on the point are 1imited, it apP.ears that the rate ι ι f f.\ Ι'inding increases steadi1y with the diameter of the rod , ιιιι l Γσr 1arge rods becomes sensib1y constant. This resu1t Ι ιtσι surprising, however, since the rods act as crushing ιιι ii Η and all the partic1es have to ιιι • surfaces. Thus, the probabi1ity ιι ο ι such an important variab1e as

pass between the crushof trapping a partic1e is with balls as the grind-

.

100

-~

;;;::.

v

50

,./

//

ο

v

Rod ΜίΙΙ- Ορεn Cιrcuιt • _Ν/. •0 ·7

@ -Νiι - Q-4

c 0 ·02

0 ·0 4

%

I

Ο · Οδ

INc

Q-08

0 ·10

FIG. 5.13

ι " Α'

media. Clear1y, thus, the rate of grinding will increase

ί 11 i ιially, after which the increased production per impact ι offset by the reduced number of impacts per unit time;

ι lιi s

1eading to a constant rate of grinding, as suggested for ball mi11. In all cases the production of specific surface wou1d be ι · x. pccted to be direct1y proportiona1 to the time of grinding . ι 11 ι so it follows that ι Ιι

.... (5.19) Finally, there is the question ofhow the rate of grinding with the group (gjDN2) ; that is, from Chapter 3 with (NfNc)· '

ν: ιries

164

In Fig. 5.14 is given the values of Rittinger's constant, at various speeds, as determined by Gross for batch milling a dry powder. From this curve it is seen that the value of Rittinger's constant, which is proportional to the rate of increase of surface of the powder, varies almost directly with the mill speed. Theoretically, the charge would centrifuge at N f Nc = 1·0 and, since there can then be no grinding, the curve should theoretically drop to zero along a line coincident with the line Ν f Nc = 1·0. Ι η 40

...

30

<:

~<:

/

ο

u 20

...,.

-~ <:

·oc

10

ο

/

/

/

/

/

MILL

165

Ι ι ι Ι ι ι~ιll occurs. This decrease iη the rate of griηdiηg at lιi ~· lι sp eds ofrotatioη, wheη wet-griηdiηg, would be eχ ιιι • ι ι(' ι . Thus, for a VClΎ wet pulp or for a dry powder, Ιlιι'Ι'ι · would be ηο appreciable cohesion of the powder ι lι , ι ι '.'{ aηd so ceηtrifugiηg of the griηdiηg media would

l ιι · ιι o i nflu eηced by the preseηce of the pulp aηd would ιιι ι ι ι r

at, practically, the critica1 speed.

ιnod erately dry pulp would, however, exhibit conN Ιι l ι • ι ·;ιbl e cohesive streηgth aηd so the balls would teηd to 4 0r------r-----.~----,------τ----~

~ c

::ι

~2 0Γ-----Γ----?&L----~----~~~

..

v

u

Ε.

"

ιn

u

..

10

t-----~~---+------+------+--___j\---1 Dotσ OΙCoghill and ~νqιι_
u

Data ΟΙ Cross. Ball Mill -Dry Crinding .

I

a.

ιn

•

I 0·8

0·2

Α

P H. OCESS OF GRINDING WITHIN


1·0

0·2

11

11

8-Ο'Dιο

0 •8

1·0

FIG. 5.14

practice, sin ce there is s1ip between the mill shell and charge, th e charge does not centrifuge at N f Nc = 1·0, buL the operation takes p1ace progressively over a range ο[ speed. Thus, the curve would be expected to gradually fall to zero for speeds of rotation greater than the critical. In Fig. 5.15 are giveη the results oftests, by Coghill aηd Devaηey, ση a rod mill wet -griηdiηg chert ση opeη circuit. Ιη this case the rate of griηdiηg falls rapidly for speeds iη excess of about 65 % of the critical ; this result beiηg iη coη trast with that for dry milliηg, just quoted, iη which ηο

ιι ll1 re to the mill shell. For ι ι ι ll1 sioη might b e sufficient to

very cohesive pulps, the allow a coηsiderable por1i ο ιι of the charge to b e carried arouηd wi th the shell, even w lι η the actual speed of rotatioη is coηsiderably below Ι lι · ritical. This explaηation is sufficieηt to accouηt comι ι l ι: Lcly for the observed results. Ιη fact this phenomeηoη is ! • τquently ?bs:rved i~ bal! mills used to mix the pigmeηt ιιι t the lιquιd vehιcle ιη the maηufacture of paiηt. ,c. ; i ιni~ar. results for tests ο η ball mills are showη iη Fig. 5.16, . ιι1 d rt ιs seeη that curves of the type showη in Fig. 5.15 rI ι Lhe exp erimeηtal poiηts well.

166



These results can be summarized in a curve of the form of Fig. 5.17; this curve giving Φ 4 ( Nf Ν,) in terms of (Ν/ Ν,) and the pulp composition. Unfortunate1y there are ~ο

ι

Brokt:n

I

,....._ \,.,..,.-2~Lb. Ch«t .

Lin~s Αι

Fϊq . S

-15.

I

11

30

I/

20

1/

10

ο

\

I

40

~

-·.,-so ιb. Ch crt .

//

--- ·-/

........-

ΙOOLb . Cht:r ι .

------'

0 ·2

Fισ.

:- ,

ι

~200Lb . C

'~J

Ι· Ο

0 ·8

5.16

1·0 Pulp Eithcr Dry

or Vcry Wcι .

y /

I

0·8

/___ /

/ /

v

', \

\

Pulp About 70% Solids. \ . By Volumc . '-\

\\ 0 ·4

0 ·6 NfNc

Fιο.

5.17

0·8

Ι ·Ο

Α

MILL

167

111 ιιfficient data for the effect of pulp Ιιι ΙΙ y va1uated but 1inear interpo1ation

composition to be between the curves ttl J•' ig. 5.17 is probably sufficient for most purposes; ι ψι• i.ally as but few mills run at much above about 75% ttl ι lι critica1 speed. Ι ι will be noticed that in Fig. 5.15 the increase ofspecific 111 1ηι e per unit contact time is plotted for the continuous 1 11ωit operation and that when p1otted on this basis the 1 ιιι·ν s correspond with those for batch mills. Thus it ltt ll ws that if the surface produced per unit contact time 1 ιι scd, a direct comparison between batch milling and ι o ι Hίnuous milling is obtained. Doubt1ess this is the basis Ι ιι 1 ιh e statement, often made without proof, that the perl ιι ι · mance of continuous rnills may be predicted from tests 1111 ba-tch mills provided that, in the case ofthe continuous ιψιτatiοη, the actua1 contact time is used. onsidering now the rnill filling parameter Φ 7 ( J), results Ι ι ΙΗ ·d upon tests by Gross (Table 32) and Coghill and l >ι·vaney (τable 19) are given in Fig. 5.18, the former for ltitl h milling and the latter for open-circuit milling. It is . ι ι nce seen that a simp1e change of ordinate will bring Ι IH:s independent curves into coincidence; and, since the I11Uιl surface, S 1, is given by S χ J, these results may be ι · η ralized into the curve of Fig. 5.19, for φ 7 ( J). These ι • nι·ves are of the form which would be expected since, wi ιlι no balls present (J =Ο) there would be no appreci.ι l > l grinding. Similarly, when the mill is packed full with lι : ι lls (J = 100%) there can be no ball motion and so no Ι' iιιding . In between these values the rate of grinding ι ιιust attain a maximum. Furthermore, for small ball l i ll ίngs the rate of grinding increases almost directly with 111 · ball charge and this would be expected since the Ι Η' bability ofimpact increases with the number ofballs. * T he next variable which may well be studied is the ι· ft·ct of the variation of the fraction of the void space • ι ιι ι Ι ιιι ll

Α

ball can only collide with its immediate neighbours, and so, p er unit , the probability of collision is not the probability of collision of a given with every other ball in the mill.

168

P ROCESS OF GRINDING WITHIN

J:3ALL , TUBE AND R O D M I L L S

roor----,----.------,-------τ------,

ω

801-----~------~~----,_----~~----~

::Σ

ι-=

...

lιι · Ι w

Α

MILL

169

η

the balls which is occupied by so1id particles, function Φ 9 ( V). In Fig. 5.20 are plotted the 1 t'. 11l ιs of a number of tests by Coghill and D evaney (Ί': 1IJI s 9, 13 and 14) for ball and rod mills grinding '' · Ι ι · io u s materials at various speeds; the results being cor' ι · Ι <ιt ·d by the elimination of the effects of critical speed I ι y t h · use of a multiplicative factor, and again the agree" tι ' ΙΙ t is satisfactory . Ι I Ι,Ι I

js

tlιe

:360~---~f--.~~~~-;-~~-i~~-_, 0:

ω

α.

,,, ,,, 11

I\ ....ι

~ ~

20

ι---nr--,~+---:---:ι----:---:-±:-:---=---:--1-:--:--~Η Continuous Milling-Boll Lood Vori~d Chorg~

Constont .

ο

20

6

40

Boll Lood

Ο

8

2 0 ~-~-+---~r-~~+---~---~---~

ι

)

Cf)

Curv~s 2&3 Botch MiU i nq· Chorg~ Proportionol Το

1 1

I•

Ι• Ι Ο ι-~--4~~-~---+--~~--~~~~-~

u

Ο

ΒΑιι CHARcε·{J) - PERCENT FIG. 5.18

I 00

"' IΙ

(J)

ο

0 ·2

0·4

0 •6

0 ·8

1·0

1·2

R Ατ ιο - V. Fισ.

Wlιen w iιh the

3 0 ·5 ι---1----11------1--""........::-t----1

-er:-

ο

20

40

Fισ .

( J)

5.19

δΟ

80

5.20

these results are plotted on a basis comparable theoretical curves as deduced from the theory of l{ose, explained earlier, Fig. 5.21a, it is seen that the ι • ρ rimental curves have the same form as the theoretical 0 11 s, shown in Fig. 5.21 b . The theoretical curves are caln Ιi a ted for materials of zero and infinite hardness, and 11 1 curves for real materials show intermediate characteriNti s, as would be expected. Furthermore, when curves ο /' lotal surface change per unit time, Fig. 5.22a, are plotted

170

P ROCESS OF GRINDING WITHIN Α MILL

BAL L , TUBE AND R O D M ILL S 1

Γ"

VChcrt (Hard)

v

Ιlι ι

I

ο

0 ·(1

<Η

0-8 Rat io

or tical treatment proposed by R ose; this treatment

lιι • ίιι g, as far as is known, the only one so far proposed by 1 Ι ι ί ·h th e effect of the mill parameters has been deduced

~ ~~lomi tc (sογ .)

0 ·2

mpared with those calculated theoretically, Fig.

ι , ' ' IJ, the agreement in form is again seen to be reasonably I\ ι ιι Η I. . Ί' Ιω s these results support the general validity of the

I I

ι ι I( I

171

10

12

1-4

-ν

α .

0 ·2

0·4

0·6

0 ·8

Rat io · V

1·0

1· 2

[\-ι... "+n· "'~Z t ro Hordnc:ιι . /

~ ο

f--

0·2

I

l 0 ·-4

0 ·6 08 Rotio- V

10

ι

2

Ι· -4

b.

Ro tlo- V

b.

FIG. 5.22

Iιηm theoretical considerations. U ntil a theory is suffi-

I

\ \

ι

\

ι

I ι

,,

ι

/

ι

ι

,

Ι lι ί s function from the basic physical data, the curve of

~Hord Orc: .

~' ..........._

I /-.. . Sott O rc.

ι/

ι iι · ntly developed, however, to allow the calculation of

I

I

1----

-

Ο ·β

Ι •' ί g. 5.2lc probably gives the value of Φ 9 (V) with an ιι · ι ; ιιracy

0-8

Ratio - V

10

12

Ι

4

sufficient for design purposes. The question of the effect of the consistency of the pulp ι/ιΗ( U) would appear to be, at least as a first approxima1iοιι, one which is fairly easily treated. H owever, the ιωblished data relating to this variable are in fact so ι ·r nflicting as to render impossible the drawing of firm ι · onclusions.

FIG. 5.21

172


When grinding a pulp, a mill ma~, i?~ally, be operat~d in accordance with either of two lιmltlng cases; one ιη which the pulp just fills the space between the balls and the other in which the solid material in the pulp just fills that spaceo In addition there are an unlimited numb~ro of other possibilities, in which neither of these condιtωns oare observed, and it is probably the grcat numbe~ of possι?lc conditions of operation which lead to the ι~conc~usιvc nature of the experim ental evidence on thιs vanableo From other evidence however, it is possible to draw tentative conclusions 'which are possibly of value until sound experimental evidence is available o In the first place, if the mill is operated in such a way that the space between the balls is just filled ~ith pulp, then clearly a reduction in the solid content wιll reducc the quantity of powder present and, at lea~t ~ο a first approximation, have the same effect as the ovanatι~n ofthe parameter V discussed earliero Thus, for thιs case, ιt would be expected that an approximate allowanc~ could be ~a~e by the use of Figo 5o2lco Clearly, since this calculatiOn ιs based ο η a knowledge of the volume of solid material present it can be extended to cases in which the quantity of pulp i~ either less than or greater than that required to fill t~e void space between the ballso In the second case, that ιη which the solid material is just sufficient to fill the space between the balls, it would be expected that the consistency of the pulp would have but little effect and, as a first approximation, may .be neglectedo In both cases the fluidity of the pulp will have s?me effect in that with a more fluid pulp the finer partιcles will be washed from between the balls and the efficiency of the crushing process improvedo This action will ~lso lead to an amount of selective grinding, but this subjeCt will be treated in a later chaptero Thus, an approximation to the value of Φs ( U) may be madeo ο The diameter u of the exit aperture from a mιll would be expected to h~v~ but little effect upω;ι theorate ?f~illing since, other things being equal, vanatωns ιη thιs dιmen-

PROCESS OF GRINDING WITHIN Α MILL

}73

Ιο ιιο ~ill m erely vary the flow rate through the mill. In

ι ι l c lιtωn

the effects of variation of this quantity are in-

ι lιι t l d when the calculations are based on the actual

ι ιιιΙt <~:ct time of the material with the mill. There might ι ι ο ιιΗηη, howe~er, some small effects arising from variation Ι ι ι ο Ι lιc operfect10n of the mixing in the mill, with varying ι . ιt dιameters, and similar minor influences; but such Ι . ιι · ι rs would be e_xpected too be ~nimportanto The experiιιι ιo ntal daota relatιong oto thιs vanable are sparse, but the ι ι ιιν of Fιgo 5023 ιndιcates the trends as far as they can be ι Ι ι • luced from the published evidenceo It is at once seen

ι Ιι : ι t the effect of this variable is small. It is clear that the lι · ιι g th of the mill is also a significant variable in this

~

~ O ·Sι------t--+--+--+-----1

-e.::

Μ111

ο

02

24" Long.

ΙοΟ

FIGo 5023

ιι ·:~ ρ ct; since this affects the angle of slope of the surface ι ι / the charge, but there are insufficient data for the relaι iorιships

to be evaluatedo The question of the influence of the ratio of new feed to ι ί ι: ulating load does not appear to have r eceived system. ι ι ι · ο treatment, but general conclusions based on the I"' 'ν1ous treatment may be drawno l "ro~ the previ~us treatment it is clear that, provided ι l1 c ratω of b all s1ze to average particle size is not too ~ι1 all,o th~ ratoe of grinding is not greatly influenced by the • ι zο dιstnbutωns of the granular material, and the foreo ιng work may be used for the estimation of the increase οΓ surface in the mill. Thus, the combined feed may b e ι l'('ated as a normal feed and the product size calculated


174


Α

175

MILL

'Ι .!I

Authority

Coghill and Devaney * (τab1e 25)

Materia1

Chert

" Do1omite "

Ε

19 χ 36 2 4 χ 24 68χ48 19 χ 36

1·0 1·0 0·36 0·36

0·60 0·60 0·80 0·80

I

u

Ν

Mill size Diam. (in.) χ Length

Ν,

0·60 0·60 0·77 0·77

Gow, Guggenheim, Coghillt (τab1 e 16)

Dolomite

78χ. 144

0·36

0·715 0·78

Maxson, Cadena and Bond:

Quartz 6-28 6-48 6-100 6-200

12 χ 12

1·08 1·0 1·0 1·0

0·89 0·89 0·89 0·89

1·00 1·00 1·00 1·00

Coghill and Devaney, Rod Mill (τable 9)

Chert

19 χ36 19 χ 36

1·0 1·0

0·40 0·70

0·60 0·60

36χ48

0·5

0·75

1·00

" Cement Clinker

Starke§

(~))

(b)

0·65 0·65

()o(J5 ()o(J5 0·05 0•05

0·00 10 0·0008 0·0003 0·0010

0·70

0·03

0·98 0·98 0·98 0·96

!HJ94· IHJ94 ( )ο( ) 4 11·094

0·40 0·70 0·87

Φ4(~) -- --

Φ2(~) Φa(b) 1·0 1·0 0·75 0·97

(V)

Φg ( V )

(J)

Φ? ( J)

(~)

φ6ωι

- -- - - - - -- -- - - -

κ

-

1·00 1·00 1·00 1·00

1·00 1·00 1·00 1·00

0·50 0·50 0·50 0·50

0·70 0·70 0·70 0·70

3·0 3·0 3·0 3·0

0·6 0·6 0·6 0·6

0·036 0·030 0·006 0·030

0·00013 0·83

0·50

1·00

1·0

0·47

0·72

3·0

0·6

0·0082

0·85 0·85 0·85 0·85

0·60 0·60 0·60 0·60

0·36 0·36 0·36 0·36

4·0 4·0 4·0 4·0

0·2 13 0·213 0·213 0·213

0·96 0·96 0·96 0·96

3·0 3·0 3·0 3·0

0·6 0·6 0·6 0·6

0·0079 0·0104 0·0121 0·0144

IHJ 6 0·0007 !1·0 6 0·0007

0·90 0·90

1·0 1·0

1·0 1·0

1·0 1·0

0·45 0·45

0·75 0·75

3·0 3·0

0·6 0·6

0·78 0·74

IHJ28 0·001

1·00

0·60

0·3

5·0

0·48

0·7

2·0

0·8

0·012

0 ·6 ιJ

0·005 0·005 0·005 0·005

I

* Ball Mill Grinding. U.S. Bur. of Mines Tech. Paper,

Ν ο. 581 .

t Trans . Amer. Inst. min. (metall .) Engrs, 1934, 112, 24. : Ibid. , 112, 130. § Rock Prod., 1935, 38, 40.

w l1 re

from it by the use of the various functional relatiσJnships already discussed. . . On the basis of the present analysι~ equatιon (5.,_ 2 ) for the computation of the change of specιfic surface, πnay be simplified. It has been suggested that Φι(D 2gρfΕ) = (DZgρfl.ιη and φ 5 (gt2D) = (gt2JD )ll2, so it follows that

s =

κ(D~ρt) -Φ2(;) - Φ3(i) -Φ4(~) .φ 6 α) -Φ; 7 (J) χ Φ 9 ( V) -Φη(Ϊ5)

(5.20)

-

0·95 1·00 0·73 0·95

0·68

. ιιιd

Φ 2 (d/D)

is read from Fig. 5.12, Φ 3 (b /D ) is read from Fig. 5.11, Φ 4 (ΝJ Ν, ) is read from Fig. 5.17, Φ 6 (σf ρ) is read from Fig. 5.5, Φ 7 (J) is read from Fig. 5.19, Φ 9 ( V) is read from Fig. 5.21c, Φ 11 (ufD ) is read from Fig. 5.23,

Ιlι graphs Φs( U) and Φ 10 (Ν) being omitted since it l>ι ~ η shown that their effects are largely included by ι

has the

rrect computation of the values of other variables. he value of the constant Κ must be determined by ι ·ι · Γι rence to the results of experiment and there is an . ι ιnount of data available for this purpose. ln Table 5.3 , ι· given a number ofvalues; based ση C.G.S. units, but

176


BALL, TUBE AN D ROD MILLS

with Ε taken as unity for ch ert. All these figures are seen to be of the same order, there being about a 3 to 1 ratio between the extreme values. This scatter probably ariscs from difference in the bases of calculation of the chang in specific suι-face; since the various workers have diffeι-en ι methods. The agreement is such, however, as to suggest that the genera1 method of approach is correct and tha ι an accurate series of experiments for the eva1uation of thc constant is the primary requirement. On the basis of thesc 1imited data, however, it appears safe to draw the provisiona1 conc1usion that the value of the constant Κ, in equation (5.20), may be taken as Κ

Κ

= 0·02 for a ball mill l. = 0·75 for a rod millf

(5.21)

with a deviation of, say, ±50%, but these figures will probably be modified as more data become availablc. Thus, the rate of change of specific surface may be calculated from the ι:nill dimensions ; provided the grindability of the material, r eJative to chert, is known. There now remains the question of the grinding of a mixed material; this case being practically important since, for example, any ore contains at least two components. The problem h as been studied by Holmes and Patching, (8) and some interesting tentative conclusions have been reached. The rate of grinding is written in the form:

kW Μξ

(5.22)

where Μ is the weight of product per revolution, ξ is thc bulk density of the material, W the weight of material in the mill and k is the specific rate of reduction, and it is shown that the ratios (k 1' /k2 ') and (k 1 /k 2 ) are approximately equal and sensibly independent of the proportions of the two components in the mill; k 1 ' and k 2 ' being th specific rates of reduction for the two materials millecl separately and k 1 and k 2 for them milled together.

Α

177

MILL

Thus, it appears reasonable to conclude that when two ι:ιaterials are g~ound in combination, the rate of producι ιοη of surface ιη each of the components is in the same

ratio as it would be ifthe materials were milled separately. Ι• ιιrthermore, the rate of change of specific surface in each omponent is the same as :when the component is milled scl?arate1y. This view is also confirmed by Gow, Guggenlι cιm, Campbell and Coghill, who show that for tests on ·hert-dolomite ~ixtures, the rate of size reduction is, very r ughly, proportιonal to the grindabilities ofthe chert and ιlolomite when milled separately. REFERENCES

I . RosE, Η. Ε.

Trans. Instn . chem. Engrs, Lond., 1957, 35, 87. 2. CoGHΙLL, W. Η ., and DEVANEY, F. D. Trans. Amer. Inst . min. (metall.) Engrs, 1930, 87, 51. 3. ΤΗΕΙΜΕR, 0., and MosER, F. Kolloidzschr., 1952, 128, 2, 68-74. 4·. ENGELS, Κ. Metall, 1954, 3- 4, 102. Crushing and Grinding, U.S. Bur. of Mines T ech . R ept 402 r.. GRoss, ] .

(1938).

.

6. Cocι:rιLL, W. Η . , and DEVANEY, F. D . Ball Mill Grinding, U.S. Bur. of . Mιn es Tech. Paper, Νο . 581, 1937. 7. SτΑRΚΕ, Η . R. Rock Prod., 1935, 38, 40. 11. HoLMEs,J. Α. , and PATCHΙNG, S. W . F. Trans . Instn chem. Engrs, Lond.,

1957,35, 111.

179

Α MIL L CHARAC T E RI S τiC S OF

CHA PTE R 6

OT HE R CO NS ID ER AT IO NS RE LA TIN G ΤΟ ΤΗΕ LL CH AR AC TE RIS TIC S OF Α MI

;.. --~~~--~+------4 ~δΟ ~~--+-----~--~~ 'Σ

.

α.

.

of the various IN the previous cha pter the influ ence char ge has mill and rnill the ning phy sica l qua ntiti es defi

orm ance of a rnill bee n stud ied in con nect ion with the perf in the pow der. ace as a device for the crea tion ofn ew surf desi rabl e that the For some purposes, however, it is also size dist ribu tion of form d erre pref a prod uct shou ld have it is usua lly necescurve. For exam ple, in min eral dressing mesh shou ld be a 200 sary that the mat eria l belo w abo ut ifications in the sizc min imu m. Το some exte nt such mod brou ght abo ut by be can cs risti distήbution char acte mus t be emp hasized, cha nges of the rnill dimensions. lt ted scope but, even lirni however, that such chan ges are of so, are often wor th-w hile . ed is the effec t of The first rela tion ship to be consider diam eter , upo n rod the vari atio n of the ball diam eter , or Fig. 6. 1 are shown ln . ined obta tion ribu dist size the by ball milling; typi cal size freq uen cy curves obta ined of Coghill and data these curves bein g ded uced from the (l>. Devaney of a coarse feed As wou ld be expe cted , the grin ding ctive grin ding effe the to rise give with sma ll balls does not does prod uce a conof the coarse part icle s in the feed but larg est ball sizes, the side rabl e amo unt offi ne prod uct. For tion is atta ined . frac se coar however, rapi d grin ding of the ever, it will be how , feed r fine the Tur ning atte ntio n to s a coar ser prod uct obse rved that the larg est ball size give s it app ears that Thu than does the nex t size sma ller balls. 178

+------4

..u 40 ι--~.-+-----~-τ+-~------~~~

c;

~

α.

ο

0·5

2·0

1·5

PortΊ clll

Dlam εt~r - Millinεtrli's .

400 r-- --- .--- --- r-- --- .--- ---

r-- --- .--- --- r-- , F~~d

(-35 Μ~ιh

Ch~rι .)

;.

.!Ξ

i 200

..

~~r---~~--~~~

t-~~i---~~--f7~~

α.

c:

--~--~~~~_μ~

~

~ιοο r-----+------Γ~--·-f----

-3~--~o~~----~o~~~--~~aJ

o~--~o~-~~--~0-~2----~o~ Part iciC'

Diamιιrt~r

FIG.

-

6.1

Millimιtrιιrs .

BALL, TUBE AND ROD MILLS 180 the coarser fractions may be ground, without the production of excessive fine material, by the use of balls of larg diameter; however, the particle size must not be ex siv or this ben efit is 1ost. From these rather limited data it would appear that th best selective grinding is obtained wheη the ball diameter is about 100-150 times that of the particle to be crushecl. This figure is very roughly in accordance with that whic]ι wou1d be d educed from Fig. 5.12, but close agreemenι

CHARACTERISτiCS OF Α MILL

ιιι ii Μ

1 aded to give a constant force. Thus, the lar~er

111 , ι· t i ·l s would be the first to be bro~en,. whatever th~ sι.ze

rods. Ι t is possible that the grιn~ιng chara~terιstιcs WΙΙ ΙΙI ι be different when the charge lS cataractιng, but . . . ι ,, iι l ι · nc e on this point is not available. Ε nmination of Fig. 6.3 shows that varιatιons ιη the ιlι • ιΙ Ν i ιy of the grinding media does not have great effect ι ι ι Ιι•lective grinding; although the use of the .more. de~se ιι ιι IH does very slightly increase the. preferentιal g~ι~dιng ιι l t 11 coarser fractions in the feed. Sιnce, however, ιt ιs not

11 1 t Ι 1 •

ιιΗ I

••

I

I

I

··~

---

ι---- .........,_

, ~

1

~

'

Ι ιι'\ι I

••

"

~

I""-

11 11

PARTΙCLE

Dι .Μ ε τεR - ΜιL L ΙΜετeεs . Fιc.

ιι

Q-2

ο •

Ο' δ

"" ·~

DεN stτvOF

~

Roos .Ρ= 2·89- • - 5 ·0 -@

- 7 ·3 -

-......

f",'

~+

Ι· ο

Q-8

ΡλRΤΙCΙ.Ε

6.2

cannot be expected since the two cases are not identical. Thus one is for the crushing of a powder having a singlc size of particle and the other is for the most rapid production of surface when a materia1 having a wide size-rangc is comminuted . For the rod mill, Fig. 6.2, however, marked selectiv grinding is not evident; in fact the size distribution curves of the products are very similar for the three sizes of rod tested. This is to be expected since the results are for NJNc= 0·4, which would suggest that the charge is cascading and, in such a case, the rods would act as crushing

181

""'

LFEEo .

"'< '-.....

~ --::.:--

I 2

1•4

Η).

DιAMETER.- Μιι.ι.ΙΜΕτRΕS .

FΙG.

χ

6.3

""

Ι·Β

2 ·ο

to select gri~~ing .bodies ~ade from materia!s of thιs should IH· noted that these tests have been run for times such that ιψι 1 work has been expe?ded in driving the mill. Thus, ι t (' Jlows that the exp endιture of equal amounts of work JΙΙ'Ο l.uce practically identical pro?uc.ts from ~he same feed, ι ι · ~ardless of the d ensity of the grιndιng m~dιa. ) Wh:n the Ιι : ιll clιarge is fixed, variation of the quant~ty <;>f ore ιη the ιιιίll brings about no significant selective grιndιng between t IΙ · different sizes of material in the feed.

ι ~ι ·: ι ιicable

ιι l ~ι wide range of densιtιes, ιnteres t ιη th: effects ν: ι ι· i able are theoretical rather than practιcal. (It

182

In Fig. 6.4 are plotted curves, again according to Coghill and Devaney, which show the effect of pulp density upon selective grinding and it is at once evident that ;ι more rapid reduction of the larger fractions is obtainccl with a pulp of moderate thickness than with a very thi k one. The probable explanation of this phenomenon jg that with a fluid pulp the fine material is washed from between the crushing surfaces, and so is ..not subject t further grinding. In addition, cleaning the grinding zon c of such materialleads to a higher efficiency of grinding ο/'

183

CHARACTERISTIC S OF Α MILL


ι~ fficiency being defined by the equation Εχ

η =

Increase of surface funit time Energy inputfunit time

. . . . (6.1)

(~ nsideration of the treatment of Chapter 3 will s~o7 t Ιιat, for practica1 purposes, the power ιnput to a mιll s κ i ven by

p

=

D2.5ρL( 1+ Ο·:σ) ·ΡΦι'(;) · PΦs(J) ·PΦn(V) (6.2)

~80~~~--+---+-~q---,_---r---r---r---+--~ u ~ ~60 ~--~~~--+---4---~-~-r---r---r---+--~ w

ι ι1

effect of variations in the other dimensionless groups h h Ι ι ·ing small. imilarly, consideration of Chapter ~ s~ows t at t e 1 ·:ι t· of production of specific surface, S jt, ιs gιven by

Ξ 4ο ~--+---~~d-~~--,_--~~~---r---+--~ ο

~

=

D~Sρ ·sΦ4(:.) ·sΦ7(J) ·sΦ9(V)

6-3)

···· (

ιlι • vanatιons of the other dimensionless groups being, ιιιι dcr normal conditions, negligible. ο

1·0 PAFH ΙCL E DιAMEfER

IN

ΜιιιιΜΕτRΕS

FIG. 6.4

the large particles remaining. Thus, both from the point of view of rate of production of surface and of improved grinding of the larger materials, it is desirable that the pulp should be of moderate consistency. Although there is not much evidence on the point, it appears that the size distribution curves of the products from mills with exit passages of laΓge and small diameter are very similar. Thus selective grinding is but slightly controlled by variation of this dimension. At this point it is of interest to consider, briefly, the conditions necessary for maximum efficiency in a mill; the

. J-!ere the pre-suffix Ρ or S attached to φ, de_notes ~hat Ιlι . power or the surface product~on, r~spectιvely, ιs a Ιιι η tion ofthe dimensionless quantιty whιch follows .. The total surface, S 1, produced is, however, proportιonal Ι ιι ιlιe product of the specific surface produce_d and the vo lume of the mill. Thus the rate of productιon of total tι r Γace is given by

~~

=

nz;ρL·sΦ4(:) -sΦ1(J) ·sΦ9(V)

· · · · (6 -4 )

Wlι η equations (6.2) and (6.4) are substitute_d into _(6.1)

Ιlιι ~ quantities D , ρ, L and Ε cancel and the mιll ,efficιency

i

f.\' ίvcn

'1

by

Κ {sΦ4 (NJN,)} fsΦ7(J)} {sΦ9(V)}

= ·

ΡΦι'(Νf Ν,) · ι.;,φs(J) · PΦn(V)

....

(6.5)

184

CHARACTERISTICS OF Α MILL


From this equation it follows that, for given values Ι (N j Nc), (J), and ( V), the efficiency of the mill is ind pendent of size . In Fig. 6.5 the curves of power input to a dry mill , ~omputed from. the results of Chapter 3, and the rate I' ιncrease of specιfic surface, from Fig. 5.1 7, are plotted on

..

. ό!:

ο

~

ο

0 ·2

σδ

Ο' Β

1·2

( t;YNc) Fιο.

6.5

a ?~se of (N/Nc) · From this figure it is mιllιng of a dry powder, or a very fluid

seen that, for the pulp, there is fair agreement between the curve of power input and that of the rate ~f.production of SJ?e~ific surface, for all speeds up to the. cntιcal. . For the mιll'ιng of moderately dry pulps there ιs ~~so faιr agreement for speeds up to about 70 % of the cntιcal. Thus, for these two cases the mill efficiency

185

ιιι . ιy b considered to be sensibly independent ofthe speed ιι l t Ι ι

mill. lt is also probable that for speeds above 70 % of the ι ι i t iι:al the power demand would fall away when modeι . ιtι· l y dry pulps are undergoing milling and so, in this ι . ι Η <' ιιlsο, the mill efficiency would remain sensibly indeιιι · ι ιιl nt of the mill speed. Experimental evidence ο η this ι ιοiηι is lacking, however, so a final conclusion cannot be . ιl ( ·ly drawn, for this case. Ι ι appears to be safe to state, however, that, for all ι ι : ι ιical milling conditions, the mill efficiency is sensibly ιιιιlι• pcndent ofthe speed ofthe machine. . . 'Ι'Ιι relationship between mill efficiency and mιll fillιng 1ν i ll now be studied. Ι η Fig. 6.6a is shown the curve relating the rate of l"'oduction of specific surface to the mill filling (see Fig. ' ι . Ι <) and in Fig. 6.6b is the curve of total surface producΙίιω , per unit time, deduced fr?m it. '!h~ curve Fig. 6.6c ι ιlιat ofpower input to the mιll and ιs, ιη fact, the curve ιιl l•' ig. 3.5. . . . . · ι 'lιe mill efficiency ιs then deduced by the dινιsιοη of Ιlιι: rdinate of curve Fig. 6.6b by that of Fig: 6.6c and ι Iι ι • urve so obtained is plotted in Fig. 6.6d. From this Ι 'il ρh it is seen that the mill efficiency is sensibly cons!ant IΙιι· flllings ofup to about 50% and that for greater fillιngs ι IΙ ι ' ·fficiency increases rapidly.Thus, it would appear that, lιι1· ιlιe greatest efficiency, a large filling, say 75%~ should Ιιι· used. This is only partially true, however, for wιth such . ι lilling the rate ofproduction of~pecific surface is re~u~ed . ιιιι l so to obtain a product of gιven fineness the mιllιng ι tΙΙΙ must be increased. If the energy to cause comminuι 1011 were all that is involved, this would be immaterial, Ιιιιt ιhc en ergy lost in friction in gearing, bearings, etc., is . ι! ι η . t independent of the charge in the mill and this loss ι Ιninimized by the reduction of the milling time to a ι1tiι1imum. Furthermore, the reduction ofthe milling time lιl H'rates the plant for other duties and so is economically Ιlι·Ηi ι·able.

186


CHARACTERISTIC S OF

Thus, on balance it appears that lower fillings, witl1 higher rates of specific surface production, are desirabl , even though the energy consumption is proportionately larger, and the statement by Gow, Guggenheim, Campbc11 1· 0

v !'>

""'-..

Ο 2

Mi tt

"""\

Ο 8

Ο 6

0· 4 F ιlli ng

-

J

I

~0 · 5

""' "" ο

v

10

I

/

\ ι

1/ I Ο

ο

Ο 2

~

Ο 4 Mιll

J,

-

b,

ι ·ο

1·0

/ \ ο

Ι· Ο

Ο· θ

0·6 Fιlling

α,

5

\

I v

I

0-2

~

\

ο

Mίll

ο 6 4 Fill ing - J .

-~

0 ·5

ω

~

ο Β

ι ο

ο

1

'

~

02

04

1

v

--v Ο·6

Mi ll Filling -

c.

σ8

I •ο

J.

d. FιG.

6.6

and Coghill (2> that: "Ι η commercial operation large charges of 55-60% of the mill volume have been found to give lower capacities and efficiencies than loads of 45-50 %", is probably well founded . Finally, the relationship between mill efficiency and the

MILL

187

ψι : ιntity of solid material present in the pulp, the paraιιιι · ι r ( V), can be examined. Ι η this case Fig. 6. 7a shows the curve for the rate of ιιι ·oduction of speci.fic s.u rface, ~η terms of this parameter (Νι' • l<'ig. 5.21) and ιη Fιg. 6. 7b ιs the curve of total surface ι l ι·ι luced from it. Ί'he curve of Fig.

~I\

Α

6.7c gives the power demand of the 111il\ in terms ofthe variable V, so deduced from the treatιιιι· ~t of Chapter 3. Although this curve is com~ute~ f~r ~ \Η: ific values of mill filling and powder densιty, ιt ιs ι ι ι·obable that the form of the curve is appli~able to all ι ι( \ι r practical cases, with an accuracy sufficιent for the . ιιι τsc ηt purpose. 'fhe mill efficiency is then deduced ι~ the same. way ~s ι ι ι · viously and it is seen that the maxιmum efficιency ιs ι ι· ; ι hed when the value of V is about 30%. ft must be pointed out, howev~r, that the red.uction of ι \ι · quantity of powder in the mιll leads to an ιncreased Ι 'ΙΙL ofmetal wear. Thus, the quantity ofpowder cannot be ι ι · cJuced unduly and, on balance, it ~s probable that the 111 timum value ofthe parameter (V) ιs between about 0·6 ιιιd 1·0. 'fests by Coghill and Devaney<3) show that the. ha~dn~ss of' ιhe balls has some effect upon the rate of gnndιng ιη ιιιί\1. These results are plotted in Fig. 6.~a an~ it is se:n ι Ιι : ιι, both for dolomite and chert, there ιs an ιncrease ιη Ιlι · rate of grinding with increasing hardness of ball. ΊΊι ·se data require correlating, however, and as a first ψρroximation this can be carried out as foll?ws: In ~he Ιιι ~ ι place it is clear that the greater .rate ~f.gnndιng. wιth ι \11• dolomite is due to the greater gnndabιlιty, that lS the ιι1 all er value of Ε, of this material. There is a.mple ι · vidence however that Ε, is 2·75 to 3·0 times Ed so ιf the ι ι ι· linates ofthe curve for dolomite are divided by, say, 2·75 ι ιιιτelation with respect to this vaήabl_e is introduced. It Ι Η : ιlso clear however, that Η, the relatιve hardness of the Ιι: ι\l to the :Πaterial undergoing grinding, is an important v. ιriable.

•

"'ό ω

ω

ο

ο

l:r

... .:- ~~

ο

"' ό

I

~

> -....

.ci

Ί

~."-..

I

ο

(Q

I ω

.;

ο

ο

~

r

~~~

ο

400

s·o

2·5

ο

ι!ΟΟ

7·S

Min
Hardn
ιn

Φ

-

~

I

Ι· Ο

/

ι2

I/

'

"' "'" 11 11

"

.....

>

~

I

~~

ο

----------4----~L-~ό

b

0...

- 6

500

"'

ό

Ο

'

....

ο

ο

\

Ν

()

"'

ο . J)MOd

ιw!l }!unjιJoμns =>!ιριdς

FIG.

6.7

ο

soo

Ηι ιοοο

ISOO

FIG. 6.8

.

...

ό

ο

1000

~IlM I! Ι ~ιrdn aaOr θα ΙI Χ Μ~~~;~~7~~;~ΙΗ::~:~ΜO~l~;~t .

()

-ι ~ ~-ό --'---J_ ό

θ r l n
ι S'

'#

f.....----------------1----ι/ ο~ '!'

"'

200

11

α

I I

,

----- - ·

ο

,!

1----------+--------1----ι ο

·'

~

Ί(JU» ! J! ~ J3 I/!~

1-----------1-------...ι-ι ~

..

δ ~

ο

σ

"'ό

ο

r---------τ-------,-, ~

r

1·0 ~-----..-------.

16

',, -::1_- "'

<:'

ο

ιw!1 Hun..ιιd ι:>ομns: ιο~ο1

-~ /ι.

189

MILL

ο

ο

k~~

Α

ι .:

~

ο

CHARACTERISTICS OF

Ν

ο

σ

'

<:'

~

~

~~

ιΙ

9

I I

I

'

'!

Now a curve by Fahrenwald, <4 > shown in Fig. 6.8b, ι ι l , ι t ί ng grinda bili ty to mineralogical hardness, suggests ιl ι , ιι ~ι reduction in the value of Ε from 1·0 to 1/2·75 corι ι ι ιo n ds to a reduction of mineralogical hardness from Ι • Ο ( ο 0·5 and so the relative hardness of the ball to the ι l ιι l oιni te is twice that of the ball to chert. Thus the 11l ι

ι · issac values of the curve ΙΙΙιιltίρli ed by 2·0. When this

relating to dolomite must be is done the curve ofFig. 6·8c

190

is obtained and it is seen that there is fair agreemc nt between the results for the two material s. From this it may be deduced that the rate of grinding is approxim atcly halved when changing from grinding the softest ore wit11 the hardest possible ball to the grinding of flint with ;ι very soft cast-iron ball. This result is reduced to the form of a multiplicativι· function in Fig. 6.8d. Unit value of Φ (Η') has been taken when Η'= 450, since this corresponds to chert undergo ing grinding by moderat ely hard cast-iron balls, which is, a~ far as can be judged, the condit~on of the tests which hav<~ been examine d in the present work. The rate of grinding is not the whole story, however for it has been shown, from theoretic al consider ations by Rose(5) and from tests on industria l mills by Engels, (6) thaι ball and liner wear is reduced by the use of sojt metal for the balls and mill shell when very hard material s are to b ground. Thus, although the rate of grinding is reduced, the replacem ent costs of the balls, and also contami nation of the product, is reduced by the use of soft metal for thr grinding surfaces. The reduced rate of grinding is doubι less due to the energy absorbed by the plastic and elasti distortions of the soft baJls reducing that available for crushing the particles . Whilst dealing with the question of the influence of thc grinding media upon the grinding characte ristics of a mill, it is of interest to consider the available data relating to tests upon grinding bodies of various shapes. These data are collected in Table 6.1, and it is seen that, ο η all counts, spherica l grinding bodies give the highest performa nc figures . Thus, although it might be economi cal to use larg lumps of ore, or "cylpebs " made from scrap drill-rods, when such are available at the plant, there appears to b no economi c justifica tion for the use of grinding bodies οΓ shape other than spherica l when the l&.dies have to bc ' purchase d at compara ble prices. about by thc brought demand power of increase The use of bodies of non-sphe rical form would be expected ;

Α

CHARAC TERISTI CS OF


...

-

"""

"'....~""'....

-

"""

. . ο.

-

~

~

~

"' "'

::g~

MILL

Μ

U")

ω

""

- - - -- - ----- -..ou

ti

~- ~

ο

Έ

ο"Ό

~

p..,§

.Ω

*....

"'....

"Ό

"8 ~ :::5

ο

Ε Ρ.., ο

c

·::g5

ω

U")

ω

Μ

"""

""

Μ

""

U")

Μ

u

~"Ό

"'

-~ "' :::5 .... u

-

~-g

c: ....

"""

U")

ω

""

σ>

ο U")

U")

,_

Μ

Μ

;::::>ο.. ι

~ c:ο c:.:::. ο .Ω .....

.;-ο

σ>

~6

.Μ

,_ I'-

ι--

<> Ρ..

u

Ξ~ CJJB

U")

ι:ό

ό

Μ

-

Ρ.,

,_

-ι-

-ι-

-

:Ε

U")

U")

"'""" "'"""

~

~~

;σ

u-

"' · "dY~~ ~

::g"'

'5 <:

.;-,

-c-ι

'<:>

""

<>:J · -

..c:

+

u:->'7'

-"'!"

ο

ο

ω

ω

"'~

~

"'

co

+ +

MU") U")

....

· ι::

ω

-ι- -ι-

- ....

~ ..C::!

-+

-ι-

-

~~

"Ό!:::.

c:ε ~

c:

-~

:-:::::> ..C:OJ ~~

ο

-..ο "' ""'

~8 "'

~ ....

..c:"Ρ..

CIJ

~u

~

~

z

·~

· ι::

....

ο

"'

:.§

:>-

ο

"'

"Ό"

~ "Ό

~

.Ω

:::5 ο

"'

θ

....

~lj

"ΌΡ..

OCIJ

~ ::g

Έ~

u:::..o

;σ=' οΟ

::g

192

CHA R ACTERISTICS OF

BALL , T U B E AND ROD MI L LS

since tl1e interlocking is tantamount to an increase in thι: coeffici ent of friction, which brings about an increasrcl angle of slope of the free surface of the charge, and so a n increased displacement of the centre of gravity, whi l1 leads to increased power demands. The size distribution of the balls in the mill chargι· appears to have no significant effect upon the size distribution of the product, nor upon the rate of grinding provided that the size distribution of the balls is not to unreasonably chosen. Thus, no significant difference iιι the grinding characteristics can be d etected in the published data on the subject between grinding with balls /' uniform size and with balls having a ratio of diameter οΓ largest ball in the charge to the diameter of the smalles ι b all of about 2 to 1·0 or even greater. This comparativc insensitivity of the grinding characteristics to ball-siz<· distribution is fortunate since it means tha t any convenien\ method of ball rationing may be adopted. · This at once gives rise to the question ofthe equilibrium size distribution of the balls in a mill; this distribution being in equilibrium under the conditions of wear whicl1 hold during grinding. The problem has been studied by Davis, Ο Ο ) and, on the assumption that the rate of wear i~ proportional to the weight of the ball, it has been shown that the size of distribution of the equilibrium charge is given by . . .. (6.6)

where W is the p ercentage by weight of the balls falling in the size interval d. to dh, d,n is the diameter of the balls fed to the mill and d0 is the dia meter at which the balls arc r ejected. The results of sieve analysis of the b all charges of largc mills, quoted by D avis, are in extremely close agreement with those calculated from equation (6.6) and so th general truth ofthe theory appears to b e established. From

Α

MILL

193

ιl 1iH ί ι would appear that a definite equilibrium of the size ιl ι ι ι ι · ί bution of the balls exists. So all that can be done in Ι " . ι ι· ι ί c is to feed replacement balls of diameter d," at the , '" ι ·ι · · ι rate and to remove the balls smaller than diameter , / 11 ι · ithcr continuously or at fairly frequent intervals. The ιlιι · ο ιΎ of Davis leads to an expression for the size distribulιo ιι

or the balls in a mill but does not give information as

rate of loss of m etal. This matter will be discussed , lnter chapter. ' Ι 'Ι Η: factors upon which the rate of flow of the pulp ι Ι ιι o 11 g l1 a mill depends appear not to have received 1 Ιι • ιι ~ίνe investigation. In an article by Anselm,OI) transI , Ιiι · c l by Pearson,f the ιιιίΙΙ , ofthe ball and the p artιcle, D, d and b respe ctιv_ely, ι ψοη the length of the mill, L , and the speed of rotatωn, Ν . Ι ι would also be exp ected that the throughput would Ιlηι 'nd upon the kinematic viscosity of the pulp ν, the Ι1ι Ι IΗ: ιιι

194

CHARACTERI STICS OF


density of the pulp σ, upon the exit diameter of the mill ιι, and upon the mill filling J. When the method of dimensional analysis is applied to these variables it is found that

....

195

MILL

"

ο

r

";;:- 0· 11----1-----b~=----+---f----1

..."

(n5ι2~gιι2) = Φ{ (n3ι~gιιz ), (Ζ} (~), (i), (i)·

(];), (J)}

Α

ο

Mill

0 ·5

0·3

0.2

(6.7)

(JJ

Filling . -

FIG. 6.9

The kinematic viscosity of the pulp would doubtlcss depend upon the consistency of the pulp, the size of thι~ particles in the pulp and on the cohesive properties of th · solid materials when immersed in the liquid phase. Unfortunately there is not nearly sufficient published data to allow a complete study of this equation, but the data ι· Coghill and Devaney, (13) plotted in Fig. 6.9, indicate tlυιt the throughput varies directly as the mill filling, J, at least for values of filling up to 50 % - Similarly, from Fig. 6.10, i ι would appear that the throughput varies directly with thι~ speed of rotation, for speeds up to about 70% of thι· critical. At some point above this speed the curve woulcl be expected to become horizontal, since at the critical speed the charge would be spread around the rnill ancl horizontal flow would then be unlikely. Finally, it woulcl be expected that the ratio of the average size of the passages between the grinding media to the particle size woulcl be a controlling factor in the flow of the pulp; since if th · particles are larger than the passage dimensions there js m echanical sieving, whilst, if the particles are relativcly small, the flow would be limited only by the effectivι· viscosity of the pulp. Now, from the geometry of a bed of granular materials, the mean pore size is directly proportional to the particl · size. Thus, the ratio (dj b) would be a controlling variablc. This group is not inconsistent with equation (6.6) sincc (djb) = (djD) 7 (b /D). In Fig. 6.11 are plotted the quantities of flow through a

Ο·SΟr---.---..-----,---τ---ϊ

-;; Q.

.c

Ο"

"e ,_

.c

1·0

Q-6

0 ·4

ο

( N/Nc)

FIG. 6.10

0·4 . . . - - - - - - - - ; - - - - . . . . . . , . , - - - - - ,

+-+.!1Nc"0 .7 χ-•---0·6

..:

" ΙΟ · 2 ':;;-

. - - - 0·5

ο

0·4

..." ο

Rod ο

Dιamrtι

Rotio

r.

lnch~s

50

FrG. 6.11

2

100

196

CHARACT ERISTICS OF


rod mill, in terms of the diameter of the rods. Α subsidiary scale also gives the correspon ding values of (d/b) and it is seen that the rate of flow is substantia lly ind pendent ofthe ratio of (dj b) when the ratio is greater tl1aι1 about 50. This result is in accord with a finding by one οΙ " the present authors, Rose and Barnacle, 04) working 0 11 the flow of suspensions of solid materials through pip s, that the hydraulic resistance is independe nt of the particlc size provided the ratio of (D jd), where D is 'the pipe di ameter and d, the particle diameter, is greater than aboιιl 30 to 40. It is improbab le, however, that, in practice, the ratio (dj b) would exceed about 40- 50, since this would givι· rods of excessively 1arge size in relation to tl1e particl . Thus, it is probable that the lower values of (djb) on thι~ curves of Fig. 6.11 would be relevant and, for this casc, Qocdj b. Thus, insofar as there are the data available for tl1ι· analysis, it appears that equation (6. 7) can be reduced ιο .. .. (6.8) Q = KD 2.5J. ( Ν/ Ν,) . (djb). From equation (6. 7) it is seen that, all other things b eing equal, the throughpu t should vary as D 2·5. Unfortuna tely data for the verificatio n of this equation are practically non-existe nt, but those given in Table 6.2 suggest that tl1ι~ observed throughpu t is roughly in accord with this expression. TABLE 6.2 Authority

Mill si ze

Ν

J

-

Di a m . χ

length (in. ) Coghill and

Devaneyιl3 )

Norris\9)

19 χ 36 68 χ 48

0·50 0·50

90 χ 68

0· 30

Q/Ql 9 ίη. χ 36 Ιη.

Q

Ν,

observed (tonsfhr)

0·8 0·8

0·79 28

0·7

15·20*

ι

* Estima ted figures.

Calculated

I

1·0 25 26

Obse ι·vrcl

1·0 35 19- 25

Α

MILL

197

cffect of liner design upon mill performan ce Ψ Ι Η': ιι ·s to l1ave received 1ittle attention. Clearly, the main IΙιιιι · ιί ο n of the liner is to form a removable surface to the ι ι ι Ι ! I I> dy, which may be replaced when seriously worn. Ι ι Ι Μ Hi so clear however, that the metal plates which serve ι Ι 11 ριι rpose may have a surface which ranges from smooth !ι ι ι ω ι: which carries an intricate pattern of raised bars or ι 11 ι Ι τ η depressions. The merits of the various types do not ι ι 1 ' Ι 11 ·: ι r, however, to have b een studied. Λ~ η first step in the analysis of the problem it would be ιι · ι· ι d that liners would fall into one or the other of two 1 ι Ι ιιι ι : ι ι l lasses: ΊΊιι ·

( I ) mooth liners, (' ) " Lifter" liners, 11 lιι · ι ·ι · " smooth" liners are those which have projection s 111 ι1 /Γι ient to give appreciab le keying between the liner 1ιιι I ι h ball charge, whilst lifter liners are those which are ι ι lιι ·nvil y ribbed as to give rise to appreciab le interlocki ng l ιι • Ι Ι.Υ<" η the balls and the liners. :ιι· i o us common types of liners are illustrated in Fig. Ι ι I' . Λ lthough these liners have various patterns of prο ι ι ι Ι ions, or depression s, to give an amount of interactio n Iιι Ι 1-νι· η the liner and the grinding medium, it would be ι ι >< · ι· ι c d that wear would round the edges. It is doubtful ΙΙ iι ι · ΙIΙ · r, after some time in service, the performan ce ofa 111 ill with these lincrs differs appreciab ly from that of a ΙΙΙill with a smooth surface. Liners furnished with heavy lιlιι · ι · bars are also sometimes used and in such a case the Iι 1ι king σf the ball charge to the sl1ell must be very effective. N ι • v ι · ι· ιheless, although a few vague general statement s to Ι lι ι · ι· Πcct that a lifter mill gives a product with different Ιι. ι · ι· Ι1 racteristics to that of a smooth mill have appeared, ι Iι ι · ρ int does not appear to h ave been widely investigat ed . I! tH ρωbable, howeνer, that, ΟΠ the grounds of differences 111 111 size characteristics of the products, ther,e exists η ο ΙΙ Ιιι1 cl reason for the use of lifters in preference to the ιΙ ιι Ι'Ιηal smooth liners.

(d ) (Hardinge G'o. Inc. )

Wave liners

( Usines Ernile

Henricoι

S .A. )

H enricot studded liners

(Hardinge Co. Inc. )

(c)

' Vedge-bar liners

( 1-lardin.~e

Cornpaι·t ιηtnt

Co. Inc.)

grate

(f)

(Edgar Allen Ltd.)

Chrome steel step FIG. 6.12

lineΓs

( Usines

W orn solid liners FrG. 6.1 2

Ειηί/e

1-/enricot S.A.)

BALL, TUBE AND ROD MILLS 200 It is possible that, when a materia l with a low coeffici nι offricti on is milled, the charge might slip on a smooth mill shell, with consequ ent loss of grindin g capacity , and ίιι such a case the use oflifter bars might well be the soluti ιι . It has also been suggested by one ofthe authors , Rοse,<Ι ΙΙ ) that the use oflifter bars might elimina te the "surgin g" I the charge sometim es encoun tered in mill operatio n. An entirely differen t concept ion ofthe duty ofthe mill liner underlie s the design of the studded liner develop ed by Usines Emile Henrico t of Count St. Etienne . These lincr~, illustra ted in Fig. 6.12 and Fig. 6.13, consist of compaπι tively thin plate liners with uniform ly spaced studs on tl1<· working surface; these studs being integra l with the plaι '. Provide d the spaces between the studs are not allowed ιο become choked with tramp-i ron, etc., the studs furnish ;ι good key betwee n the shell and the charge which, it iH claimed , leads to a greater power consum ption and lo improv ed grindin g. Further more it would appear that thι· studs impose a definite geomet rical arrange ment in tht• outer layer of balls which, in turn, brings about a closc ι· packing , through out the ball mass, than obtains witlι conven tional types ofliner. This effect would also lead to improv ed perform ance. Evidenc e of this effect of the studH upon the packing of the charge appears in Fig. 6.13b, fοι' the balls are clearly seen to lie in rows in the mill insteacl of in comple tely random array. An inciden tal merit claimed for these liners is that tlH: high bearing pressure between the balls and the studs ι · the liners leads to work hardeni ng of the studs; with ;ι consequ ential reducti on of the rate of metal wear. The Henrico t liners, which have been discussed in :ι paper by Belwinkel, (Ι6) appear to be the only attemp t so far made to influenc e the grindin g charact eristics of ;ι mill by means of correctl y designe d liners. It would ther fore appear that there is some room for develop ment in this directio n.

(a) liners ~l ι ll 2· 4 m . d iam. χ 12 m. lon g. Fitted with H enricot classifying

( Usinrs Emile Henricoι S.A.)

ι ΙΙI Ι

11 0 1"

(b) of M ill fitted wi th H enr·icot studded liners (underfill ed to show ordered a rray imp osed ση th e ba lls.

Fro. 6.1 3

202

BALL , TUBE AND ROD MILL S

REFE RENC ES EY, F. D. Ball Mill Grinding, U.S. Βuι·. DEVAN and I . CocHILL, W . 581 . ο. Ν Paper, of Mines, Tech. and CoGHILL, W.] Ι . 2. Gow, Α. Μ., GuGGENHEIM, Μ ., CAMPBELL, Α. Β., 5 1. Trans. Amer. Inst. min. (metall.) Engrs, 1934, 112, ref. I. 3. CoGHILL, W. Η., and DEVANEY, F . D. L oc. cit., ) Engrs, 1934, 112, 91. 4. FAHRENWALD, W . Trans. Amer. l nst. min. (metall. 5, 87. 1957,3 Lond., Engrs, . hem c . nstn l . Trans 5. RosE, Η. Ε. 6. ENGELS, Κ. Metall, 1954, 3-4, 102. ref. I. 7. CocHILL, W. Η., and DEVANEY, F . D . L oc. cit., 855. 118/1, 1924, ., J Min. Engng Η. D. ILD, 8. FAIRCH ) Engrs, IY54, 63, 197. 9. NoRRιs, G . C . Trans . Amer. l nst. min. (metall. Engrs, 19 19, 61, 250. .) (metall min. . Inst 10. DΑνιs, Ε. W. T rans. Amer. 11. ANsELM, W . ,Zement-Kalk-Gips, 1950, 3, 16. 12. PEARSO N, Β. Μ. Rock Prod., 1951, 54, 106. cit., ref. I . 13. CocHILL, W . Η. , and DEVANEY, F . D. L oc. r, Lοιιd., 1957, 203, 898. 14. R osE, Η. Ε., and BARNACLE, Η . Ε. Enginee 15. RosE, Η. Ε. Engineer, Lond., 1957, 203, 522. 16. BELWI NKEL, Α. ,Zement-Kalk-Gip.< , 1953, 6, 439. Η. ,

CHAP TER 7

,' URG ING AND VIB RAT ION MIL L

ΙΝ Α

a subje ct upon

is ι Ί ι ι r, ρiΊenomenon of "surg ing" in a mill mabl y becau se it presu en; writt een b has 1 • lιιΙ · Iι very little ted in mill opera tion Ι , ι ι· orιdition whic h cann ot be tolera tion of some or varia by ιιιι l wl1ich must be elimi nated cteris tics of the chara or s nsion dime ι ι ι , ι ι ι y of the physi cal impo rtanc e, such of is ct subje The e. charg ιιι 11 or mill a from book deallιt~ wι·v r, that it cann ot well be omitt ed An attem pt will mill. ling ιι ψ, wίιh the subje ct of the tumb a revie w of the ter, chap this in Ιlιι · ι ι-Γ re be made to give, ct. subje the of ι ι ι Ι ' ι: nt knowledge a phen omen on know n as "surg ing" consists of Ί ' Ιι mill, ofthe e charg e whol ιιι ιιιlιιlum-like oscill ation ofthe

al the mill shell. Thus for one part of the cyclic mill ofthe e centr the d aroun ng movi is e charg ιιιιιlί η the and durin g the ι ι ι Ιlι same direc tion as the mill shell, direc tion. As a site oppo the ι 1 ο ιιd part of the motio n, in charg e, the the of n motio atory oscill ι ιιιι H ι · guence of this in stead y shell mill the tain main to sary neces Ιιιιψι · mstan ces, circu in certa ι ιt1ΙΙ ίο η fluctu ates wide ly and, in r that the moto ng drivi the ' ι Ι ι ι ιnake such dema nds upon of the out ing throw This out. n throw are ι 11 ι · ιιί t brea kers er dang the evil since ι ι ι 1 ι ι ί ι break ers is, however, a lesser more Α died. reme be ι ι ίιι such a case, appa rent and must torqu e whic h are ι ι Iιt Ι • dang er is hidden in fluctu ations of devices into ctive prote t cϊrcui the bring ιιι ιιffi ient to existence of such fluctu ation s 1 ψι · ι·<ιtiοη . In this case the ce of fluctu ation s may ι ιι οι appa rent, and the main tenan ng geari ng throu gh drivi the of n Ιι .ιι l to the destr uctio

1 · ι ιlι ί 11

203

204

S U RGING AND VIBRATION ΙΝ Α MILL

BALL, T U BE AND ROD MILLS

fatigue failure. Although, for obvious reasons, bu t littlc ί ~ m ade pu blic concerning such failures, it is Γairly well know 1ι that failuι-e of the gearing of large mills is e ncounten• ιl with a frequ ency such as to sug·gest that factors σther th ίΙ Ι Ι mere wear and tear are operative. The oscillatory motion of the charge, which has b c ι' ΙΙ described above, is complicated by the collapse of t!Ι< · leading toe of the charge when the charge comes to r ~ ~ at the end of the forward motion. This collapse impos\·~ fluctuations of small amplitude but higher frequency up 11 the fluctuations arising from the m ain motion; this, i11 turn, gives rise to a complex motion which involves : ι great number of harmonics in the frequency spectrum. Consider now the motion of the charge in some d eLBil and, since, in Chapter 2, it has been shown that t iH· equilibrium surface of the charge approximates to : ι straight line, the charge may be considered to be a m ass ο Ι the shape shown in Fig. 7.1. The following m athemati ίιl treatment, which has been suggested by Hinsley( l), is th ι ι applicable. The charge in the mill is assumed to b e a coherent mass with the centre of gravity at G. The gravitational force 11Ι.f!, and the centripetal forc e mrω 2 have a resultant R whi !ι meets the shell at Ρ. This resultant can, in turn, b e resolv ι! into tangential and normal components at Ρ .

Rιs. Fιa .

,

+ g sin θ )

.. .. (7.1)

since ε is small. Also, from the equations of motion mk 2θ

=

mgι·

k2

FR ·· gr θ = k2 cos θ - mk2

gy'(r2+μ2R2 cos2 ε) . k2 (sin ξ cos θ- cos ξ sιn θ )

k2 gives

. . ~ . (7 .3)

1 lιι • Ι' '

or .. .. (7.2)

2

"'ιΙ I /(ι·μ cosε {j2 lι iι ·lι

cos θ- FR

R

θ = gr cos θ - μRg cos ε sin θ - μk rB 2 cos ε k2

~ mμR cos ε( rθ 2

7.1

( :oιηbining (7.1 ) and (7.2)

Then,

FR = mμR (rθ2 cos ε+ g sin ψ )

205

Ιι

μrR

cos

= __k_2_ _

ε

and

206

SU RGING AND VIBRATION


rι ' Ιιι· tίme,

Let Φ=ξ-θ, so 0= -Φ and Β= -Φ, so that (7.4) Φ -ΑιΦ 2 = -Α 2 sin φ This equation can be integrated once by use of thι: integrating factor 2e-2ΑιΦ, whence

ΙΝ Α

207

MILL

t, ofthe descent ofthe charge is given by

Γ'''· dΦ φ

J,,,,

dφ

Γφ·

2 ) φ, y [{2A 2/y( l +4Α 1 2 )} sin (η +Φ) +C 1 e ΑιΦ]

(7 .6)

Ιlιι ι :quation may be solved by graphical and numerical ιιlι • •t ϊllion.

φ.2

=

. ( 2Α2 y(l + 4Α 12 ) sιn η+Φ) +ce2Λ 1 Φ

(7.5)

where η and 2Α 1 are related as shown in Fig. (7.2).

FIG.

7.2

a

c

b

d

FIG. 7.3

If now the angle θ, at which the charge just slips, ί ιι observed by experiment and since, for this condition , Φ =Ο the value of C1 the constant of integration, c, may bι· obtained from equation (7.5). Since equation (7.5) is :ι quadratic . there are two solutions, Φ 1 and φ 2 , to thι · equation Φ 2 =Ο which, in practice, are one negative an(l the other positive. The expression (7.5) is the equation for the velocity ι · motion of the charge and the solution can be taken no further by formal methods. For practical purposes, however, it is desirable to know the time for a complctι· oscillation, since the frequency of the oscillation may thcιι be calculated thus.

W l1 η the function on the right-hand side of equation ( /, ()) is plotted, a curve of the form shown in Fig. (7.3) 1 ι ι i JLc ined. The portion of the curve between b and c may 111 • ί ι ι ιcgrated by Weddles' rule, or any similar method, Ι ι ι ι 1 owing to the infinite values, the strips above ab and cd ι , ιιιι1 t be so treated. ιι ίnverse sine function is therefore fitted to these end I ι ί 1JS so tha t _ [ . _1 / ]Ρ dx P . . . . (7. 7) - Slll Χ α α

J

α y(α2-χ2)

ι ι ι ι Ι by the correct choice of α, β, α and χ a very close fit

208

S URG ING AND VIBRATION ΙΝ Α MILL


may be obtained. Thus the time of descent of the load m:ι y be computed. After the load has gripped the shell, the ascent is mίΙ
= 1·035 sec

This gives a frequency of 1/1·035 χ 60 =58 cfmin, whirlι is in close agreement with the observed value. The above treatment is of great interest in that it givΙ 'N the frequency of surge with considerable accuracy, buι ίι is not of such a nature as to decid e, whilst the machinι~ iM still on the drawing board, the question ofwhether a mill of proposed design will surge or not. Ι η a theoretical study of the problem, one of the pres 11 ι writers, Rose, <2 > attempted to analyse the motion of tiH· mill charge by use of the ma thema tica1 method of topo1ogy ;

* Determined by use of an oscillograph.

209

Ι ι ι . ι ι iο ηs of the coefficient of friction with rubbing I Ι I( i ι Υ bcing included in this treatment. Ο η the basis of ι l11 ι ο ρ logical diagram relating to a mill of about 8 ft Ιli ι ι ΙΙΙ Ι ' l 'r, together with an assumed but probably reasonΙI Ι I Ι Ιτ l tι ιionship between the coefficient offriction and the ΙΙΙ I Ι i ιi ΙΙ g speed, the curve relating the torque to rotate the I

ΙΙΙΙ II Η lι

'll at constant speed, on a base of time, was

cοm-

Ι ΙΙΙι ·< l ;ιnd this curve is showr_ι in Fig. 7.4. From this curv:e

that violent surges In the torque occur; a ΙΙΙΙΙ Ιιιι of about 2·5 times the m ean torque being attained Ι ι

maΧΙ

ι · ' 11

11 , ι Ι 'ι · quency of about 30 cfmin. This ratio of maximum 2·5 2·0

OL----------------------J Timt

Fια.

7.4

I ΙΙ Ι ψι to mean torque is in fair accord with an observa1 11 1ι l>y Hinsley, working ση the mill mentioned previously, Ιl ι.ιl during surging the current to the motor varied beΙν ι · Ι · ιι about 300 and 700 amperes. ll ;ιv ing regard to the theoretical nature of the treatΙΙΙ Ι ιι ι f Rose, which involved a number of assumptions as 11 ι ι Ι ι · values of important parameters, the agreement l11 ι wι· η the observed and calculated magnitude of the 1 11 ι · ι ηιt surge is as close as can be expected. Ι t appears ι ι . ι Ho nable to conclude that peak values of the torque of 11 lι ·;ιs t two or three times the mean va1ue may be en1ιι ιιιιt red during surging. Ί '/1 topological method is, however, not of a nature

210

such as to be readily used iη the desigη office. Ιη aη effoι ι to obtaiη a simple criterioη for the existeηce, or η 11 existeηce, of surgiηg, aη experimeηtal iηvestigatioη w:ι Η carried out by οηe of the preseηt authors with a co-workι·ι-, Rose aηd Bluηt<3) · use beiηg made of the small-s ~tlι · models aηd the t~st apparatus already illustrated ίιι Fig. 3.1. . The basis ofthis work is that the aηgle 2α through whιι· Ιι the ceηtre of gravity of the mill charge oscψates is ?ep !Ι deηt οη the "fluidity" of the charge aηd this quaηtlty, 101' reasoηs already discussed, depeηds upoη the mill diamctιτ, D the ball diameter, d, upoη the coefficieηt of frictioη μ , of the mill charge (if μ= Ο the mill char~~ would alw~yH be iη stable equilibrium at the lowest ροsιtωη ofthe mιll) aηd upoη the chaηge of the value of the coefficieηt of frictioη with the slidiηgvelocity. Thus if μv=μο(1 +ΦΒv), where φ deηotes some fuηctioη, the amplitude woul depenιl upon the mill filling, J, since bot~ ~he restonηg fo~cc· actiηg on the mill charge aηd the fnctωηal forces, whι l1 depend οη the number of coηtact points, depend up 11 this quaηtity. Thus .. .. (7.8) α = Φ (D, d, μ, t, Ν, Β, J) where

SU RGING AND VIBRATION ΙΝ Α MILL

BALL, TUBE AND ROD 'MILLS

agaiη φ deηotes

"some function of".

ι/ IIOW the methods of dimeηsional aηalysis ι, ι ι I ι ι ·~ variables it is fouηd that

α

=

Φ{ (D fd) , (Nt), (μ), (J), (BDN)}

211

are applied .... (7.9)

ι ι 11 : ι rιalysis may be carried further thus; I ,c·l Ν be some fraction of the critical speed Ν, of the ιιιιΙΙ 110 tha t Ν =kN,. Now it is easily shown that, for a Ι ι • ι ι ιrιill filliηg, the periodic time of oscillation of the ball I lι ι ι 1'1-\(' varies directly as νD and also that the critic~l I" ι cl Ν, varies as y' ( l /D ) aηd so the value of (Nt ) ιs IΙΙit ι > • ndeηt ofthe size ofthe mill, thus it follows that, as Ι ι ι ι Η ιhis group is coηcerηed, the results of the model Ιι Ι Μ directly applicable to the full-size mill. Ί'ιι ι· rιίηg atteηtioη now to the group (BDN) it follows ιlι,Ι/ , :ιs before Nocy'(1/D) so this group becomes (By'D). Ιl ιι ι Ν, ίt follows that there is some "scale effect", the value ιιl ι Ι ιi ι; group iηcreasing as the root of the mill diameter. 1\ιι ι οrι the other hand, the value of Β is usually smιιll aηd ιι ι /ι · variations in the value of the group would be 1 1ιι ·c · tcd to be small. Furthermore, the effects risiηg from ι lιι νι· locity dependeηce ofthe coefficient offrictioη can be, ι ιι ιιιne exteηt at least, absorbed by the 1wι• •η the coefficients of static aηd

I ιι

I l ιιι Η, ίt would be expected that, in aηy

use of the average dynamic friction. practical mill, the

ι I I ΙΨt f this group will ηοt be large. Ί ' lιu s, the basic equation for surgiηg iη ι11 , ι ι 1 ast for a first approximation,

α

=

Φ{(D fd) , (μ), (J)}

a mill reduced .... (7.10)

ι ι ι accordance with this equation, tests were carried 111 1ι Ι n stablish how the existence, or otherwise, of surging 1 / ι 1H'nds οη the dimensionless quaηtities on the right-haηd 11 Ι ι < Γthe equation. Ι ι l ιas already been shown, iη Chapters 2 and 3, that ι' " (I uidity of the charge decreases with aη iηcreasiηg ιιιι ιι Ι \) r of coηtact points aηd with iηcreasiηg coefficieηt ιι l Ιι · ί tion . Thus, the mill filliηg, J, aηd the coefficient

of the ch arge iH of friction, μ, are, as far as the fluid ity ld be expe ct cl con cern ed inter chan geab le, and it wou f~rm of tl~ι· the in that these' quan titie s wou ld o~cur πmental rι: expe the of ysιs prod uct J μ . Then , from anal across tl1ι· n draw be d coul line a that sults it was foun d nate aιι
0 · 10 ::ι, >(

....,

1\\

0·05

Non -Surg inq Rι:qion .

""

~

Surg1ng Rι:gion .

I ο

20

213

ΙΝ Α MILL S URG I NG AND VIBR ATIO N

S BAL L, TUB E AND ROD MILL

212

oper ating poin t lιιιιιlιl be varie d in orde r to brin g the

"non -surging" zone . of the oing treat men t involves a know ledg e Γoreg 'IH· Ί ion of frict of ts icien coeff ιιιι '"' οΓ the stati c and dyna mic henc e and , essed proc g bein rial mate ι l11 lι:ι ll char ge and lly. idua indiv ts ι1l ι Ι ι · static and dyna mic coefficien e ofth es valu al reve not Ι ιιιιίrι a tiοη ofth e liter atur e does to le icab appl on fricti mic dyna and 1 111 111ι·. ί nts of stati c ofpo wde r. How ιιιι Ν ι·ιι of balls load ed with vario us types ofth ese auth ors have mad e roug h dete rmin ation s 1 1 ι , 111 icab le appl be to y likel are ιl ιιι ·~ unde r cond itions whic h in an ed plac were der pow and balls I ι Ι Iι.ι 11 mills; that is, the and ce surfa ιιιιι Ιι - 1 > ttom ed box on a smo oth steel ns mea by d sure mea Ι ι ιιι ι i v ' force necessary to mov e.the box give to ved belie is od meth ιιl , ι sprin g bala nce. This for the pres ent purp ose and a 11 ιι lι ι-ι of suffi cient accu racy purp oses ofill ustraιιιι111Ιι · ι- ofsu ch results are show n, for llιιιι, ίη Tab le 7.1. Ιllιί ιι ιhe

TABL E 7.1

I' --

4.0

ι---

150

Ν ιlι1re

80

Rat io • Dlι,

FIG. 7.5

necessary . tιι surg ing is likely to occu r or not it is only of the mιll, ng fillι osed prop the J, establish the prod uct of the bal~ charl-\< ' and μ, the effective coefficient offri ction of plot thιs νa lιι ι · to and material unde rgoin g milli ng, and If the opcr;ι I d. / D ratio agai nst the prop osed valu e of the ing". zο ιιι · -surg '_'non .the e insid well ing poin t so obta ined is ι[ tlιι • but e surg then it is impr obab le that the mιll wιll t.Ι ιι · near sly erou dang poin t is inside the "sur ging " zone, or ι :ι.ι · s the and r. occu to y likel is boun dary line, then surg ing g, or bot Ι ι , fillιn mill the of e nitud mag of the balls or the

of

11 ι ι Ι ι~ ι;ι system

10

11 I

Coeff icient of dynam ic frictio n

μs

μd

Mean value of coeffic ient of frictio n μ

0·18

0·18

0·18

0·25

0·18

0·2 1

0·37

ό· 30

0·33

0·42

0·4 1

0·4 1

0-44

0·42

0·43

ι / ιιιι .rteel: ιιlι ιl 1 ιt d with 111 ιι ι y Ιιdenum ι l l ~ ι ιl μ l1id e

Ιιιl1 ιiΙ • ιι ι cd

"'

with

I

ι ι, ιιοι ·ι ι

a nd

ιl ι ln l

I

Coeff icient of static frictio n

Ι Ι ι Γr<"c

silica

I" ιwι l rr Ι\ ΙΙ Ι ι /l·ce Ι" ιwιl τ

emery

SU RGING AND VIBRATION

BALL, TUBE AND ROD MILLS 214 It might be of interest to note in passihg that tlι1 · dependence of surging upon the coefficient of friction, :ι. revealed by model tests, is as shown in Table 7.2, and th;ιl

ΙΝ

Α

215

MILL

ο ο

I .;

-g

Observation

Conditions in mill Mill-Cieaned and dried Mill-Oiled Mill- Treated with molybdenum disulphide Miii-With about 1% of voidage filled with silica powder Balls and Mill- With about 1% of voidage filled with emery

Balls Balls Balls Balls

Surging Surging Surging

and and and and

.z=

:; .!!!

Q.

ο

Ε ... ...·-"' ·-...."' -g~ -- .z= "'

•

occurκ occuι·κ

ο

occuι·"

Ξ

Νο surging

I Νο surging 1

this observation is in accord with experience, sincc it is unusual, though not impossible, for surging to arisc iιι mills grinding cement clinker, limestone or such simil : ιι materials which give rise to a high coefficient of frictinιι . Confirmation of the general accuracy of this criterion i contained in Fig. 7.6, in which the operating points for ι large number of mills are plotted; these points being :ι\ culated from the data of Taggart(4) and relating to mill between about 4ft and 10-! ft diameter. Additional d:ιt ιι relating to mills between 2 ft and 8 ft diameter grindi ι ικ molybdenum disulphide and graphite are included; thι· 1 being supplied by Messrs. Acheson Colloids Ltd. The data have been reduced on the basis that the coι·flι cient of friction is, for the metalliferous ore, 0·4; a figιιιι· which is reasonable for such materials. In fact, the d:ιt ιι also refer to mills fitted with various types of linι·ι· , "Shiplap", "Wave", etc., but it is probable that aJI of these liners would approximate to "smoath" after s ιιιι· wear has taken place and so, for safety, these mills should Ιιι operated under "non-surging" conditions. Examinatioιι of Fig. 7.6 shows that, although some closely appr :ιι · Ιι the boundary, all the points, with one exception, t:ιll within the non-surging zone. Thus, there is fair, thouκlι

ώ

ο

... .,::;,

TABLE 7.2

e

Q. ο

ο φ

Δ

>-

"' ()... ~

ο ~

•

()

e

-

ο ιο

.

~ e

••

• •

., • •

••••

•

(

ο ~ ο

e

•

•• • • • 48

e

v / ()

-·-· .., . ...v . • .:ι

•••

~

~

•

•

v

-

11)

ο

ο ...,.

ο Ν

•

-ο

1/)

ο ο

ο

216

SU RGING AND VIBRAτiON ΙΝ Α MILL


not abso1ute1y conc1usive, evidence that the resu1ts of tl~ι· small-sca1e tests app1y direct1y to 1arge mills,_and S? _tlι:ιl the effect of variations of the group (BDN) ιs neg1ιg1blι • , Strict1y, the above criterion is app1ic~b1e <_>n~y for valιιι · of the ratio (Ν/ Ν,) greater than ~-15 sιnce ιt ιs found t!Ι:~Ι surging does not occur fo~ :un11:ιng_ speeds ~ess than ιlω va1ue. This further condιtωn ιs, ιη practιce how νι· ι , trivia1 since industria1 mills are rarely, if ever, operatι·ιl . . at such 1ow speeds. In the derivation of the curve of Fιg. 7.5, the exιst η ι-ι ·, or non-existence, of surging was determined by obscrv:ι tion of the sound of the mill and of the fluctuatωns of tlιι · dynamometer arm. It is improbab1e, therefo:e, that Ι~ιι · surging was completely ~uppressed; although ιt was so Iοι practical purposes. It ιs probable, however, that t Ιιι boundary line on Fig. 7.5 corresponds to a small fluctιι:ι tion oftorque; say 5% or 10% ofthe mean value. It woulιl , therefore, appear that further lines. could be. added to t I ιι· graph, as shown in Fig. 7.7, in whιch each lιne represcι.ιl a fluctuation in the driving torque, equal to some definιtι· percentage of the mean value. If this i~ a tru~ pic_turc c ιl the case, then it wou1d appear _that a gιven mιll mιght IH subject to considerable fluctuatωns oftorque, eve~ thoυ~lι the gross manifestations of the occurrence of surgιng, sιιι Ι ι as the throwing-out of the circuit breakers, are not at οι ι< ι · .. . apparent. From the tentative graph, Fιg. 7.7, ιt ιs clear that sm:ιll changes of operating conditio~s cou1d bring a~out ap] ι·c • ciable changes in the fluctuatωn of torque. It ιs probalιlc· that it is these minor differ~nces in, for examp1e, t Ιιι· coefficient of friction of the charge, which give rise to t Ιιι· often inexp1icab1e failures which occ_ur in the gearing οΙ some mills, whilst identica1 mills working under appar η tI identica1 conditions suffer no such breakdowns. As : ιιι example of this the attention of the writ.ers ~as bιτιι drawn Anon. (5) to the case ofa number ofιdentιcal tιι\ΙC· mills, ~orkini under app~rently ide~tica_1 conditions, ίιι one of which mills the maιn gear paιr faι1ed after aboιιl

217

111 ,!10(} hours' service; whereas the other mills have given ι ιι ιιψ l ·tcly satisfactory performance over a number of ι ι <ι . T he reduction gearing of these mills, which were ι l ι ι rιc· d and constructed by makers experienced in the • ιι ιι Ι ι · ιι tion of gear trains such as those used in rolling

ιιι 11 :ιnd in ship turbine installations, were generously 1" ''1ιο ι · ιίοηed and apparently adequate for the heaviest ι lιιl . ι r now it is assumed that the fundamental fluctua1 ιι ιι ο Γ torque is of the order of 30 cfmin, which the tests •ιl 11 i ιι ·ley and the calculations of Rose indicate as being I'' ι ι l ι : ι ωy reasonable, and that the reduction ratio of the

..,

FrG. 7.7

l ιιΙν speed pair is 5 to 1, then 10,000 hours of service ΙΙt tl ι l correspond to about 3,000,000 cycles of stress on 1111 1-{ι~ aring. It is significant that it is at this order ofnuml ιι ι of stress cycles that fatigue failure would be expected

ur. 11 , iη the absence of an exact knowledge of the com1"' i Ι ion of the gear material and the heat treatment em~ Ι Ι i ι ιyι d, it is assumed that the fatigue characteristics of the ι ι , ι ι · ιnaterial are approximately the same as those for 1 lι iι : l1 the diagram Fig. 7.8, quoted by Morley,<6> was ι l ι ι iv ·d, then this dia?;ram may be used for the further Ι ιι ιιc ·

SU RGIN G AND VIBR ATIO N


218

Fig. 7.I\ analysis of the prese nt probl em. The diagr am of befoι ·ι· sals rever of er numb inite is such that, for an indef shac!ι·ιl the n withi fall must es stress ing failure, the work area. in~ Now suppose that the mini mum stress, correspond , thrιl s follow It . in2 tonsf 20 is e, torqu to the mean drivi ng er οΙ if failure is not to occur withi n a limit ed numb not must stress appli catio ns of the stress, the maxi mum atioιι exceed abou t 25 tonsfin 2; that is, the allow able fiuctu νalιιι •, mean the of 25% t abou d excee of torqu e must not 7.7, tiΙHt Howe ver, it has alrea dy been suggested by Fig.

20r---------~~~~~-----~ ~ ... ~ -~ ~

c ~

c

ω

,_

ιOf-------;~~M~>L--::.__

____ ___

ι ~

ω

--

0 1--------t~~_.:....-----------

Vl ω ·

0..

Ε ο

U ι O'L---~--------------------Fισ. 7.8 qιιίtι fiuctu ation s of this magn itude may well aήse from J Ο 11 11 say, by, bly possi , mode rate chang es in mill conditions on of tlι varia tion in the value of the coefficient of fricti brou~lιt be well could es chang such that charg e, and surfiιι size, le abou t by differences in pulp density, partic ιο Ιι as small so are h whic finish ofthe balls and mill, etc., ι apρι· d woul it , Thus nt. porta unim norm ally regar ded as practiι in ed unter enco es failur gear ed plain that the unex tioω; οΙ migh t arise from differences in the plιysical condi iι·N, terist charac er powd and e the mill liner, ball charg buι ιlο, nt porta unim as ded regar be to as Thes e are so small atioιι N in fact, have a profo und effect on the torqu e fiuctu furtlι still is em trans mitte d to the geari ng. The probl

ΙΝ

Α

MILL

219

ated by the fact that the stresses invol ved will upon such factors as the rigid ity ofthe drive , the moto r, etc. Thus , for Ι ιι ιφιι·-SJ eed chara cteris tics of the are such that the mill tics cteris ι ιι φΙ , if the moto r chara 1 e increases, the load torqu the as speed lose ιl l ιιw d to the moto r main tains Ιιιι ΙIΙι : gcari ng will be lowe r than if const ant. For these less or d of rotat ion more ι ι Ιιι e fiuctu ation s are torqu the to as s 1 ι ι 1 Ιt ι s •xact calcu lation nable doub t reaso no ins rema there but e, Ι11ψ ι 1 · ιίcabl n the "ηοη withi ι/ι ι ι ι mill shoul d be opera ted as far ble. possi is ΙΙΙ fΙ ΙΙ Ι ~j" zone ofFig . 7.5 as to the 11 lι as alrea dy been point ed out that, owin g n withi e charg the of n motio ng surgi any of ιιι Ι Ι ιιι l ί ity the e, charg the of toe ι/ιι Ι ι ll and to the collapse of the d by the drivi ng Ιl ιιι Ι ι ι ; ιι ίοηs of the torqu e trans mitte er ofhar moni cs. numb large very a in conta ιιιι 1 /1 :1nism will r at , ιherefore, possible that reson ance migh t occu Ιι ratus appa y nearb and ι 111Ιt ' Γr·equency, betw een a mill sensitive to vibra /ι ι Ι ι , if the adjac ent equip ment were origi n of whic h the , ulties diffic to rise give ιlιιι ι , ould natur e have this of ιιιl μ, l ιΙ not be easily trace d. Probl ems whic h it is in ,<7> Anon. e, l ιι ι ιι liscussed in a recen t articl mitte d trans be may tion vibra e l ιι 1wιι that appre ciabl traηs being tion vibra the ; ation found piled ι l ι ι ιι ιι ~h a bedthe gh throu mill, Ιιι ι ΙΙι ·< l down the piles bene ath the the ath bene piles other gh 1ι ι ι /1 :ιnd then emer ging throu exbe not d woul t effec This . ment 111 11\ l ιbouring equip there since 1ι ι ι I ι ·< l to be so mark ed with friction piles, trans mit the vibra ιΙ ιιl ι / then be no unde rlyin g rock to duce consi derab le intro d woul ι ι ι ΙΙ Μ <.ι nd the clay itself that the corre ct ar appe d woul it s, heles ι I ιι 1ψing . Ν evert point of the trans ι / 1 Ι /-\ 11 of found ation s from the view Ι ι ι ι iοrι of vibra tion, is essential. that some vibra tion Ι1 ι ιhe same articl e, it is ment ioned was suggested that it and ng geari mill the from ι Ιιι . ιη ;ι ted use of fibre or the ι· nditio n could be allev iated by ι /ι Ι η view of train. gear the in ιι ι .ι / tooth ed drivi ng pinio ns howe ver, ter, chap this of part r earlie ι 11 ι liscussion in the

ι ιιιιψlί ιlι I" ι ιι l

Ι

220


n thι· ιιιιl\ it appe ars that atten tion to the cond itions withi cati o ιι H Ι ι ι i modif such than ive shell migh t be moΓe effect wlιlι lι tion vibΓa of type er furth Α the gear tΓain itself. ~ Ι ιι ll ι mill migh t well be ment ioned is the vibra tion of the lι J>ι·o e. charg g fallin the as a b eam, unde r the influe nce of ) lι cl b absor is tion vibra this ably the great er part of 11 Ι Iι ι etwe b oints j the shell, the of etal m damp ing in the Γe ap ι ι·ι ιι liner plates, etc., and, in a b all or tube mill the ίΙ ι· i Ι ures meas al speci to be no n eed to intro duce any ι·ocl idual indiv the ver, howe mill, suppr essio n. ln a rod wil l ι ct conta make they when ct impa nt viole give a more cοmιιιιιιι the liner or the toe ofthe charge. It appea rs to be :ι ιιι I liners the een betw wood pract ice to inser t billets of l Ι cι an< shock the uate atten to order in shell of such mills .intro duce some damp ing. fa lliιι The p articu larly d estru ctive actio n ofhea vy rods ι · ι ιl rtan impo the ver, howe s, on t~e milllin ~ rs emph asize cιιι g<' impin rods the that such speed a at runωng rod mιlls . the toe of the charge and not upon the liners

J.

ion to the Autho rs.

4. 5. 6. 7.

Instn mech . Engrs Lond. 170 1

ME TAL WE AR

ΙΝ

MIL LS from the work -

Ι 1111 <: st ofrep lacem ent ofthe meta l worn of a mill often repre sents 1ιιι ιω ι:Γaces of tl1e b::ιlls and liners of opera tion of the plant cost the of ωn fract ble ιdera N ι ι ιι ιι rtanc e. In spite of ι ιtι l ~ is ofsom e r eal econo mica l impo of infor matio n nt amou the ver, ιιη.portance, howe ιlι ι and, unfor tusmall ingly amaz is ct subje this ι ιi l :ιb l e on re ofmo than a very ιι ιlι · l y_, is insuffi~ient to form the basis far as is know n the As lem. Ιι 111: t ι ve analysιs of the prob etal wear is that of m for y theor a e ι ιιι l y <ιltempt to deriv the relev ant parts and ω Rose, rs, autho nt prese 11 1ιι • ο Γ the y 111 Ιlι ί· treat ment will b e outli ned briefl here. is that parti cles ι I'11 basic idea unde rlying this treat ment conta ct be upon will, g millin g rgoin ιι f I Iι ι : mate rial unde

nds ι Iι ιvι·η into the_ surfa ce of the b alls at a rate whic h dep~ the cle, parti and ball ι ιcι ιι ~he relatιve hard nesses of the

REFE RENC ES F . Privat e Comm unicat ι . HιNSLEY, 2. RosE, Η. Ε . Unpub lished R esearch. 3. R osE, Η . Ε., and BLUNT, G. D. Proc.

CHAP TER 8

'}Ι Ι ' Ι

ι ' ' 1957. ~) Υιιι (New Wiley n ng,Joh Dressi l TAGGART, Α . F. Handbook of Minera ι 1948. s. Privat e Comm unicat ion to th e Author ΑΝΟΝ . reen & Co. (Lο η ι l ιιιι ), MORLEY, Α . Strength of Materials, Longm ans, G 17. 68,4 1953, Lond., Age, ΑΝΟΝ. Chem .

ιι other such factors. ι ι ι ι , ιιιlιty of powd er prese nt and upon relati ve tangential is there ded provi that, ed Ι Ι Ι :ιl so assum or other of the emι ιι ι ι Ι ion betw een a pair of balls, one

Fig. ·d partic les will,_ upon maki ng conta ct, as in that usion concl the to leads then 11 Ι ι ι , b e torn out. Thιs l surfaces at a rate ι lι c ι >artic les will be torn from the meta ball and Iι i ι· Ιι is a funct ion of the relati ve hardn ess of the relati ve the , dded embe les ι ιι ι ι Ι i le, of the numb er of partic rs. facto ar simil I ι Iiιι g velocity, and val of partic les \ιV Iιen the rates of embe dding and remo er of partic les numb ibriu m 11 ι · c·q ual ther e will b e an equil word s a cerother in or, ces ι ιΙΙI ><' lded in the m etal surfa ' . "arm oured " be will ball each of ce surfa the ι ι 1 11 ρ ortωn of now, an embe dded partic le 1 ι ΙI ι •mbe dded partic les. If part ofthe surfa ce Ι ιι ι JΙ ι • ball trave rses the "una rmou red" l ιι c l ι l

22 1

222

BAL:L, TUBE AND ROD MILLS

of a neighbouring one, as in Fig. 8.1b, then metal wι ·, ιι will take place. ln the work cited the calculation of ΙΗΙΙ Iι wear has, to a first approximation, been carried out Ι Ιιιιl it is suggested that the relationship between rate of wι • ιι and relative hardness of ball and particle has th Ι i ιι ιι ι sl;ιown in Fig. 8.2, whilst the relationship between ηιιι · 1ι l

(α)

.ι ιιιιιι·

ΙΝ

223

MILLS

iably greater hardness and so ~he _reduction
ι ι1 lιy .increasing the hardness of the gnndιng surface ιs ιιιι Ι" ιHsible . ln such a case, however, the rate of w~ar can lι ι ιι •ιlιι d by the use ofΎery soft metal (by reducιng Η)

ιιιιl ο arrying the operating poin~ ~ell to the left
Ιι 1 ιι:-race of the balls by the particles. Conversely when 1 1ι y s ft material is being ground, the ratio Η will be ιι , ι (( • ι· than unity, even with soft metals, and so the ι ψι ι , ι ι ing point is ση the right-hand side of the peak. Ι η Ιιi Ι ' ιs reduced wear is obtained by increasing the hard'" ol' the metal.

(b) Fι c .

METAL WEAR

8.1

>

-θ.._Ο 5 1 + - - - - - " t - - + - - - - - - i

3

1·0~---+--~+-----i

"θ:" Ι·Ο

ο

ι Relatiνr

Fι c.

2

3

(VJ FIG. 8.3

Hardnrss -Η

8.2

wear and quantity of powder in the mill has the forι11 οΙ Fig. 8.3. Although the derivation of these curves is complι · tι · l theoretical there is some qualitative evidence in suριιιιι Ι oftheir validity. Thus for example, it is clear from fj g, 11.', that ifa material is being ground by balls ofapproxim : ιιι · l eq ual hardness (Η= 1), then the ball wear may be re Iι ιι ι ι I either by increasing or by decreasing the value of 11. 11, however, a hard mineral, say of Brinell Number ~lιolll 500 to 600, is being milled, it is not practicable to us Ιι.ι\1

't' l ι<~sc conclusions are confirmed by the practical ob' ι ;1ι ί ns of Engels(2) that for the grinding of soft materials 1 /ιιιιιl mill body andoalls should be used, but for grinding Ι ι 111 Ι ma terials a soft bu t tough metal should be used for ι l ι1 ι ι · inding surfaces. Similarly, the c~r~e of Fig. 8.3, is ιιιι lιι ·med by Smith, (3) who states that ιt ιs well known to ιιιΙ Ι Ι op rators that aπ impr0~ed fineness, in_a given t~m~, ι t t rι.ined when the quantιty of powder ιη the mιll ιs ιιι l ι ι . d, but that this improvement is obtained at the ι I " ' ΙlS of increased iron wear. I ιι rder to analyse the various published data, advanι ι ι · ιnay well be taken of the simplification offered by thc

META L WEAR


224

metho d of dimen sional analysis and this will be uscd Η ι basis of the subseq uent treatm ent. of n1ι • ι , ιl , Ι t would be expec ted that the total wear οΓ .ι ι ιll ss hardne e W 1 lb, would depen d upon the rel~tiv ι. pa~tιι and ball mιll, of size the and partic le, Η, upon I, m1ll! the. of length the upon tively, respec b D, d and IΙΙ ι ι y densιt upon the speed of the mill, Ν, and upon the ball mater ial, ρ. It would also be expec ted to depen d ιψοιι the energy necessary to increase the surface or ι I ι ι Ε rr{~tal by unit amoun t, upon the mill filling, J, upon t Ιιι fractio nal volum e of the powde r, V, upon the pulp αιιι sistency, υ, upon gravit y, g, and upon the time ofmillin ~o~, ι This may be writte n symbo lically as W = φ{D, d, b, L, Η, Em, ρ, Ν, g, t, J, V, U} .... (11. Ι )

225

MILLS

ΙΝ

the rate of ball wear, insofar as these on the speed of rotatio n of the mill . < 0·6, the ι I 1 ;ι l so found that for slow speeds, ( ΝI Nc) able reason with ed Ι ι ι ιιι Ι ίonal relatio nship can be replac

ι,, ι ίιι a mill and ψι , ιιιιίιίcs depen d

I

and, after applyi ng the usual metho ds of analysis, this leads to

Sρccific

dimen s ίoιι ι ιl δ

(~~ρ) = Φ{ (~), (i), (i), (~), (;;~g)' (D; (J), ( V), ( U)}

ο ·4 t----+---τr

Ball

Wιar

ο

( // ),

ιr.

fΊ·οιιι

. . . . (11. 'I) Φι (Nf Ν,), First exami ne the functi onal relatio nship 5 > relatin g to < Gross, and Data by Coghi ll and Devaney<3J metal wear, are plotte d in F~g. 8.4, and on the same gr:ψlι the curve relatin g the rate of produ ction of specific s ιιι face to the group (N fNc), taken from Fig. 5.17, is sl1owιι for compa rison. It is at once seen that there is clo ι agreem ent betwe en the rate of produ ction of specific sιιι

Cross

>

(11. ~ι)

Φι (Η) .φz(~·~g) ·Φ3(~) ·Φ4(~) ·Φ5(i)

Data Of

Coqhill And Dιvanιy

α:

)'

2 This can be simplified, however: since (D Ν jg) is, 2 equati on (3.7), equiva lent to (Nf Nc) .

=

Surfacc .

. 8 @ .. ~ 0 ·21-----i"-----t---t----τ------1 .

2

(~~ρ)

lncrcasι ΟΙ

Ratc Of

1

ο

0 ·2

0·4

σa

0 ·6

Ι ·Ο

(N/Nc)

FIG. 8.4

ιιι ιιracy by Φ 1 (Nf Nc) =K(N fNc)l .30 JΙΙ ' ι ' ds, ( Ν fN c) > 0·6 by

Φι(Ν/Νc)

and for the highe r

= K(NfN c)O·B , speeds

' Ι Ίι decrea se in the rate of ball wear at higher tangen tial οι ιld be expec ted since, whilst decrea sing the ncy ~nd freque the se increa ιιιιιtί η, the catara cting would ιng. armour te promo would which ts ιιιtι·ιιsity of impac

\

\ t l1 e centrif uging condit ion is appro ached the numb er , ι l l>alls in action decreases, which again would reduce

227 e has a shar ply defi ned ll .' ι :.ιnd it is seen that the curv exp lana tion of the form of this 111 ι Jrnu m. The prob able becomes less effective but ι ιιι ν<: is that the "arm ouri ng" easi ng part icle size unti l incr Ιlιι • :ιbra~ion in~reases with r this, even thou gh the ~ !'' ' max~m~~ ιs . re~ched . .Aftethe part icle s sepa rate the ι ι ι · ιnounng IS still ιneffectιve, bear ings , and so, ball like ιιι.ι · Ιη l. surfac~s, ac.ting rath er wea r of the sive abra the ce redu Ι ιι l ι ιncreasing sιze, righ t-ha nd the that , for ιιι ι· ι , 1. It shou ld be noti ced e, wea r rang l tica prac I ιι ι .η h of th~ curve, :ννhich is the dire ctly , thus or, /D) (b as y · ιιι ι:s approxιmately ιnversel ely, for vers Con . eter diam mill the ι (D/b), and so :νith alm ost es vari r wea the Ι l ιι • I ft-h and portωn of the curv e MET AL WEA R ΙΝ MIL LS

226

LS BAL L, TUB E AND ROD MIL

8.4 app ears to I Jι ' ιιl ball wea r. Thu s the curv e of Fig. reas ona ble form. poin ts and ι · Ι Ι ι The close agre eme nt betw een the met al wea r per ιιιΙΙI curv e of Fig. 8.4 suggests that the app roxi mat ely ίιιιlι incr ease in surface atta ined shou ld be y ofC hap ter : ιl 11 stud Α . pen den t ofth e spee d ofth e mill for speeds of ιψ Ι 11 and size n give of shows that , for a mill er inpu t to a mill '" abo ut 70% of the critical, the pow . Aga in insofar a~ Ι Ιιι prop orti ona l to the spee d of rota tion a stra ight linc ονι· ι by aced curv e of Fig. 8.4 may be repl y gen eral ly accψlι·ιl this rang e, it also leads to the fairl prop orti ona l to tlιι conclusion that the rate of wea r is ears that th Ιι. ιll app pow er abso rbed in a mill. Thu s it rate ofpro
~{Ζ] ?Ε1 0

ο · οοο ι

< Υο>

ο· οοο2

ο·οοο3

FIG. 8.5

)-1·0. The left- han d

ι . ι tly . as (b /D ); or thus as (D/b ding a fine pow der and Ι ιι ·. ιη~h ιs clea rly for larg e balls grin s with coarse mat eria l ι lι ι · ngh t-ha nd side is for sma ll ball

nt. hard ness on the ( : nsid erin g nex t the effect of rela tive has been preship tion rela al retic theo ι . ι Ι c of _ball .wea r; a king of this rela tion ι '. ' ι d ιη Fιg. 8.2. Dat a for the chec but a few values of ing, • lιψ are alm ost com plet ely lack es of ball , pub ness hard ell Brin Iι. ι 11 wea r, for diff eren t Fig. 8.6. This curv e lι lι ·d by McLeod<7J are plot ted in definite corr elat ion lι ows the gen eral form of Fig. 8.2, but eral s wer e mill ed min ed 1 ι ι ιnο t be mad e sinc e very mix is unk now n. ness hard eral min ι : ι ι l t~e corr espo ndin g plet e data it mus t be Ιlιιι s, ιn the abse nce of mor e com repr esen ts this rela tion . ι , ιιme d that the curv e of Fig. 8.2 ι ι ι ι ·R

lιφ .

228

METAL WEAR


The influence of the ratio of mill size ο η the ratc Γ wι ''' Φ 4 (d/D ) can more conveniently be written in th Ιοιιιι Φ 4 (D fd ), and in Fig. 8.7 are plotted the data of :οi-\ Ιιι\1 and Devaney<4> and Gross, <5> relating to the wear or \ ) , ι\1 when the charge consists of balls of uniform size, ~ΊΙΙΙI ι •I McLeod <7>, relating to the wear of balls of size d in a Ι ι : ιι ι•1 consisting of balls of mixed sizes. Α simple chang cιl Ι I~ι vertical scale of these curves brings them into coin iclι · ιΙ Ι 1 and so they may be generalized into the curve of Fip;. 11 11. this curve representing the function Φ 4 (Dfd) insofaι· : ι 11 may be derived from these data. It is also seen frοιη Ιlιι figure that, over the practical range of lO < (D /d) ιιΙ , the function may be replaced by Φ 4 ( Djd) = 0·556 (D/ rf) ΙΙ 1 , with an error of less than about lO % The relationship between rate of wear of the meta l Η ΙΙΙ faces and the ball charge in the mill is shown by thc (I I ι ο Ι of Gross, (5) plotted in Fig. 8.9. This curve is seen to \11 , very roughly, ofthe same form as that ofFig. 5.18, r J :ι Ιiιι to the total surface prod uced . The agreemen t wi t l ι Ι Iιι general form of Fig. 3.5 r elating the power input to ι lιι mill, is not so good for the higher mill fillings; the ηι t ι · 1 ιl metal wear being much higher thaρ the power input. ΊΊιι observation is in accordance with the hypothesis r "o ll rhouring" since, with high mill fillings the balls an~ ι · ιιιι strained to roll, with considerable abrasion, but im 1 ι,ιι ι , which leads to effective armouring, is suppressed. The only data relating to the effect ofthe quantity ι11ιl consistency of the pulp on the ball wear appear to \ιι those of Gross.<5) Unfortunately, these are insufficiι:Ιιl ιι ι form the basis of a complete analysis. Furthermor ι l ιι ι data appear to be defective in that they are based on tΙΙΙΙ of 21 minutes' duration and it appears that this pcrioιl I insufficient to allow equilibrium to be reached; that iNΙιιι the armouring to have become established. Howeν<'l' , 111 the absence of more complete data these must be ac <Ί•ιι ιl tentatively. It appears that, as a first approximatio ι Ι , t Ιιι effects of both pulp quantity and pulp density cηιι l11 stated in terms of the volume of solid material pres ι)( Ι ι

ΙΝ

. - - - - , - --

0·10 r - --

229

MILLS ----,----,

i

~

/

...J

~

0 ·05

~-~-/-t----'"""'=---t-----i

;ι:

δ

ο

6 00

4 00

200

800

Br inr ll Hardn r ss Or βal ls .

Fω. θ Ο

;) v li

?.....-

60

"

~

·~

σ

"'40

ο w

"

α:

ο

20

8.6

---

Cross

Ι

χ

Cog hill

I

andD<νon
~ @-

@ Mc l
ψ

ο

....

j ο

10

40

30

20

<% >

Fιa .

8.7

Ι· Ο

/

-v-=

50

-

"'""'-.Υ • 0·556t%>0·ΙS

( ο

10

2οι%) Fιa.

8.8

3ο

40

50

_

-

- -

230

METAL WEAR


use of the function ( V ) where, as before, the quantity is defined by

V=

\ν l ι

o·z

ο

t.ΝΙ

ο· •

o·a

is read from Fig. 8.2, is read from Fig. 8.4, Φ4(Djd) is read from Fig. 8.8, Φ 5 (bfD) is read from Fig. 8.5, Φ 7 ( J) is read from Fig. 8.9, Φιο( V) is read from Fig. 8.10. Φ 3 (ΝJ Ν,)

ιιι

1

(f now Em and ρ are regarded as sensibly constant for steel used in the manufacture ofballs and liners and if

. ιιιy

•fι 1 (Djd) is replaced by (D fd)O·l5 and φ 5 (b jD) by' (Djb), w lιich may be done over the practical v: ιriables, then equation (8.6) becomes

ι ·ο

J.

Fω.

231

MILLS

Φι (Η)

re

Volume of particles + Volume of space between the particles Volume of space between the balls (or rods)

~::vrr ιJS

ΙΝ

-

range of these

W 1 = K 1DH5d-O·l5b-l ·DLt. φ 1 (Η) . φ 3 (ΝJΝ,)

8.9

χ Φ1(J). Φ10 ( V)

.. .. (8.7)

In. practical milling the ball diameter is usually ap1)I' XImately a constant multiple of the particle diameter ' ο this equation becomes

W1

= K 2D4·65d-1 · 15Lt.φ 1 (H).φ 3 (NjN,) χφ7(J).Φιο(V)

(8.8)

If the relative hardness, Η, the mill filling, J, and the of solids, V, are maintained constant then from the ι• ·lationships Φ 1 (ΝJΝ,) =K(NJN,)1 ·30 for iow speeds and ι/ι 1 (ΝJΝ, ) =K(NJN,) 0·8 for high speeds, this relationship ι) comes: For low speed, ι ·<ιLiο

1•0

ο

2·0

ν

Fιο.

8.10

and the curve of Φ 10 ( V) so d educed is given in Fig. 8- 10. This curve should, however, be taken as only the rοughι·ιι ι approximation to the actual relationship. Collecting these data, the equation for ball w ·ω· becomes

W1

= ΚD3·5Εm-ι.ορ 1 1 2 Lt.φι(Η) ·Φa(~,) .φ4(~) .φ5(i) χ Φ1(J) ·Φιο(V)

· · · · (8.(ί)

W 1 = K 3D4·65 (NJN,)1 ·3d-1·15Lt .ιnd

ρ

.... (8.9)

for high speeds,

W1 = K 4D4-65 (NjN,)D·Bd-1 ·15Lt .... (8 .9a) The number of balls in the mill oc D 2L jd3 so the wear r ball wb is given by For low speed,

Wb = K5D-2 ·65dl ·B5 (NJN,)I ·3t

....

( 8 .1Ο)

232

METAL WEAR

BALL, TUBE AND · ROD MILLS

and for high speed,

Wb = K 6D2·65dl ·B5(NfNc)O·Bt

. .. . (8. 10 ι )

These equations should be contrasted with Hukki
W1 = KD 2·5 (N fNc)d- l·OLt

thosι~ ι ιl

wt

11ιc l

Λη ~ quation

( 8.Ι Ι . ι )

which lead to: F or low speed

Wb

= KDH5(NfNc) d-l·OLt

=

KD0 ·5(NfNc) d2t

.... (8. 1' )

and for high speed,

Wb = KD0 ·5(N/Nc) d3t

... .

(8.13) (8.14)

.. .. (13. 11 )

....

233

ll aving regard for the general uncertainties in the data d, the equations can be simplified to, for all speeds,

Ιν ι ·; ιr ,

given by Mortsell <6J, for the rate of metal

IS

W/ = KD 2 · 5(NJNc)Lt

MILLS

Ι ιι c lί

and for high speed, Wι =

ΙΝ

(8 .1 2.ι )

For low speeds the present equation is in fair agreen~t · ιιl with that of Hukki insofar as ball diameter is conccrιJΙ • ιl , but at high speed, and in all other respects the agre mι · ιιι is not so good. The formulae of Hukki are, however, derived from fi1 Ι principles on the basis of clearly defined assumptio1ι . These assumptions are that, for slow speed, the raιι· ιιl wear is proportional to the rubbing velocity and, for Ι ιίι~ Ιι speed, the wear is proportional to the force of the blo'A' The first assumption appears to b e quite reasonabl Ιιιιl the second is not so free from objection. In the first plt~ι ι ·, the question of the comminution of a material by a ω ο~\' is involved, and this has already b een shown to be a 0111 plicated question. Even, however, if the damagι· ι assumed to be proportional to the intensity of a sjJΙ~Iι blow, the "armouring" of the ball surface, and als : ιιι , work hardening ofthe metal, would be expected to rccl ιιι ι the rate of ball wear at high speeds.

(8K:b)

... . (8.15)

is. the total weight ?f balls, W/ is the wear per d IS as before and Κ Ιs a constant which generally lιlls between 9 and 16 for Swedish ore but, in extreme ι • . ι κ ·s, between 5 and 23; W/ being in kilograms per hour . ι ιιd Mb in metric tons. 'ince Mb α:.D 2 L it follows that, from this equation the lotal. wear is proportional to D 2; not to D 4·65 as sugg~sted 1 Ι ι. ' r~ Mb 111 ι ι ι tιme,

1ι ι·ι:vωusly .

It is .interes~ing now to exa~ine equations (8. 13) and (11. 14) ιη the lιght of the publιshed r esults of the various

. ι ιιlhorities.

F'irstly, it has b een suggested by Davis, (9) that the rate

ιι (' wear of an individual ball is proportional to d3. But Ι •: ΙΙίs, <10 > Prentice, <10 Garms and Stevens,
b, <13) and Norquist and Moeller(l 4) state that the rate is proportional to d2. Furthermore Bond, (15) gives figures, such as 2·0, 2·21 and 2·29 whilst Tag: ι rt (16) gives 2·0 for cascading balls. Thus, on this point, 111 results ofthe present analysis are roughly in agreement wi tl1 the view of the majority of other workers but more w rk is required before this question is settled.' The question of the influence of the mill diameter does ιι ο t appear to have received more than passing study but 111 suggestion that the ball wear varies as D2 ·6 where D ix the mill diameter, appears to be reasonable.' Α serious ιτiticism of all the work cited is that no effort appears to lι ave been made to maintain similarity between the ο Ι ' ~ear : ιnous

METAL WEAR ΙΝ MILLS


234

various tests. The most satisfactory type of equ~tioιι wou1d appear to be equation (8.6) and, whenever possΙIJiι ·, the resu1ts shou1d be presented in this, or simi1ar, fοπη . Since however the balls are invaήab1y of iron or s ιι· ι · Ι the va1~e of Em ~nd ρ may be regarded as sensibly οιι stant, so the equation may be reduced to

W 1

KD3· 5 Lt.Φι(H) .φ3(;) .φ4(~) ·Φ5(~)

REFERENC ES

....

(8.1/ι)

An effort has been made to ca1cu1ate the va1ue of Κ iιι this equation, ο η the basis of the various pub1ished wοι k, augmente d with reasonab1~ ass~med va1ues where η '(' ι ·.Η sary, and these va1ues are gιven ιη Tab1e 8.1; all va1u s ιιι C.G.S. units. TABLE 8.1 Va1ue of Κ

Authority McLeodl71 test: ( 1) Chill-cast balls (2) Sand-cast balls (3) Annea1ed balls

1·6 χ 10 8 1 ·6χ 10- 8 2·3 χ 10 - 8

Coghill and Devaney<41

0·6 χ 10 - 8

Mortse!llβl

2·6 χ 10 - 8

(pot mill test)

Mortsetl (from eqn. (8 .15) for 6-ft

χ

6-ft mill)

235 iman I ι shou1d a1so be pointed out that corrosion has ιιoι· ιant effect on the rate of liner wear. Thus, if corrosive ιιl)stances are present, the products of corrosion will be ι ι · ιnoved from the wσrking surfaces by abrasion and new 111 · ιa1 will be constant1y exposed to chemica1 action. The r arising from this cause will be proportion a1 to the . ιι ·ι· of meta1 exposed, but there is insufficient data for an ωΛysis ofthis prob1em.

1·2 χ 10- 8

The va1ues ca1cu1ated from equation (8.16) show faiι·ly wide disagreem ent but this is not surprising since νalιιι· for many ofthe parameter s hz.ve been assumed. lt app :ιι· , however that Κ= 1·6 χ 10-Β with an error of about ±50 ι;{,. Thus.it ~ppears safe to conc1ude that the ~oregoing analyHi inc1udes the major factors in the mechaωca1 wear of b:ιll and 1iners and it is possib1e that it can form an adeqιι:ιι basis for an extended and improved treatment .

Ι . tιosE, Η. Ε. "Symposium ση Gi:ίnding", Trans . Instn chem. Engrs., Lond., 1957, 35, 87. ' , ENGELS, Κ. Metall, 1954, 3-4, 102 . :ι SΜΙτΗ, Ε. Α. Trans. lnstn chem. Engrs, Lond. , 1957, 35, 144. Ι . OGHILL, W. Η. , and DEVANEY, F. D. Engng Min. J., 1937, 138,337. ROss, G. Bull. U.S. Bur. qf Mines, Να. 402. 'ι , Ιι . MORΊ'SE LL, S. Engng Min. J ., 1948, 149, 91. Ί . McLEOD, Ν. Α. Trans. Amer. Inst . min. (metall. ) Engrs, 187, 33. 11. Ηuκκr, R. Τ. Trans. Amer. Inst. min. (metall. ) Engrs, 1954, 190, 642. '1 . DAvιs, Ε. W . Trans. Amer. Inst. (metall. ) Erι,grs, 1917, 61, 250. 10. ELLIS, Ο . W. Trans. Inst. Brit. Foundrym, 1937, 40, 137. 11 . PRENΠCE, τ. κ. Trans. Amer. Inst. min. (metall. ) Engrs, 1946, 169, 147. I ~ . GARMs, W. 1., and SτΕνΕΝs, J. L . Trans. Amer. Inst. min. (metall .) Engrs, 1946, 169, 147. I Ί . NORMAN, τ. Ε. , and LOEB, c. Μ.. Trans. Amer. Inst. min. (metall.) Engrs, 1949, 183, 330. Ι ιl . NORQ.Uιsτ, D. Ε., and MoELLER, J. Ε. Trans. Amer. Inst. min. (metall.) Engrs, 1950, 187, 712. Ι 'ι . BoND, F. C. Disc. Trans. Amer. Inst. min. (metall.) Engrs, 1943, 153, 373; ibid., 1946, 169, 138. lli. ΊΆααΑRτ, Α. F. Handbook qf Minιral DreSJing (New York).

ΤΗΕ ROLE OF ADDITIVES ΙΝ MILLING

CHAPTER 9

ΤΗΕ

ROLE OF ADDITIVES MILLING

ΙΝ

term "additive" is applied, in milling practic , ιιι various substances which when added to the content~ οΙ a mill bring about an improvement in the rate of grincliιιι\ and, therefore, a reduction in the time of grinding reqυ i ιΨι I to attain a specified degree of fineness. Additives could ;ι I ΜΙ • be introduced in order to attain some special chara Ιι· ι istic of the final product. However, such applications :ι ι 1 d ependent on the purpose of the final product ~nd s : ιι 'Ι outside the scope of the present c~apter. An ιnter s ιιιψ, feature of additives is that the quantity required is g η ι · ι ally very small; usually being between about 0·01 Χ1 Ιιt . . 0·1 % of the weight of the charge treated. That additives are of sufficient importance ιndustrι ίιll ' to warrant consideration is shown by recent work 1ιl Skaupy, ω in which it is shoν.rn that the rate of millin μ ι~ I iron powder in iso-amyl alcohol is no less t?an twι· ιιι } times the r ate in dry air. These tests were carned out οιι ι vibration mill, and so might not be directly comp at'ίΙIΙIΙ with a ball mill but nevertheless the result is signifi : ιιιι In view of the Ίarge energy requirements for g~incliι,ι ι.ι,. especially fine grinding, a saving ?f eve~ 20 {!ο In .)IJ.ιll ing time could make a substantιal savιng In mtllιιι costs. In view of the potential importance of these subst ιιι ι · it is perhaps surprising that their mode of action i~ ιιοΙ precisely known. It is impossible to. esta?lish ~rom . Ιlιι literature on the subj ect whether theιr actωn anscs fι ·οιιι improvement in the conditions of crushing by the prι:νι · ιι tion of a coating of powder on the ball surface, from Ιlιι

'ΥΗΕ

236

237

ι ι· ·vention of agglom:r~tion in dry milling, or flocculation ιιι ι~e case of wet mιllιng, or by some action within the ι Ht~tιcle, such as the promotion of the spread of the fissures ~ I Hch occur naturally in all solid materials. It is possible, ~ ιιd ed probab~e, that. all these modes of operation are of ιιnρortance, eιther singly or in various combinations ι Ι~Ί. ending on the circumstances and the controlling con~ ι Iι ιιοηs presen t . . For. dry milling, the additive may be present in solid loι· m, ιη the form of a solution added to the dry feed, or in . ι gaseou.s or vapour state. For wet milling, the additive is ι ι ι· sent lll the water; .the concentration generally being ι ·ss than 1.% of.the .solid n:ιaterial present. , ~efore dιscussιng In detai1 the use of additives in milling ι ι lS perhaps relevant to recall what has been said earlier. Λs shown 1n Chapter 4, for a subs~an.ce in which the yield Ν Ι ress and the stress for fracture cωncide, the strain energy ι·~entually r eappears as heat and the energy associated ~ ιth. the creatωn of new s.urface re.presents a very small I~ actωn. of the total .e nergy Involved ιη the crushing operalι n i th1~ energy ?eing only about 5 % ofthe total when a ιιarticle ιs commιnuted in a simple impact crusher. For v ry ~ne powders, howev~r, th~ surf~ce energy becomes ιηοre ιmportant and manιfests Itself In high adsorbtivity : ι?~ the tendency to agglomeration observed in finely ι l ιvided subs.tances. Even so, t~e energy associated with ι lιe surface ιs a small proportωn of the total involved Β cause ofthis it is difficult to see how the reduction ofth~ N.ιιrfac~ e~ergy can significantly affect the energy required I r gnnd1ng but, nevertheless, surface active dispersing :ιgents are often very effective grinding aids. The reduction ο Γ th e surface energy of the particle itself is not however lhe o~ly possible explanation of the action' of thes~ ιna t enals .

Α possible direct mechanism for the operation of these Ν LΙ bstances has been suggested by Rehbinder. ( 2) Assuming ι.hat the molecu~es of the additive are adsorbed ο η the surface of the partιcles, then the cohesive forces which bond

238


the surface ceηtres of the crystal lattices, at the surface or the particle, will, iη accordaηce with Gibbs' equatioη, b lowered by aη amouηt ηecessary for the adsorptioη to tak<: place. Thus, the microscopic cracks aηd fissures, which ar always preseηt iη brittle materials, will, after the adsorptioη process, be mσre easily wideηed. Frσm this hypothesi s twσ teηtative cσηclusiσηs, regardiηg the effect σf such ageηts ση the crushiηg process, may be drawη. Firstly, wheη the particles are very small the physical streηgth would be expected tσ be affected. Also, witiΙ larger particles, the cσηditiσηs existiηg at the crack tipιι are mσdified aηd this cσuld lead tσ higher stress cσηceηtra tioηs iη these regioηs: these higher stress coηceηtration s leadiηg to reduced physical streηgth σf the particle. Thus, the streηgth of particles σf all sizes wσuld be expected to be reduced but, ση the whole, the fiηer particles would bc~ expected to be mσst affected. Secoηdly, the dispersive effect of the additive would bc· most marked with the small sizes of particle aηd so agglomeratioη would be reduced. This would lower thc• so-called "limit of fiηe griηdiηg", observed by mi\1 operators, at which equilibriu m is established betwc 11 the rate of productioη of fiηe material aηd the rate of pr<>· ductioη of larger particles resultiηg from agglomera ti 11 aηd, possibly, from recrystallizatiση, of smaller oηes. Α11 iηcideηtal effect would be that cσhesioη betweeη the ball aηd layers σf particles surrouηdiηg it would be reduc cl aηd that this would, by reductioη of the thickηess of t!H· layers, reduce "cushioηiηg" of the blow, with coηsequ 111 loss of eηergy iη the form of heat. Stroηg evideηce supportiηg this hypothesis is giveη by the work of Baker aηd Prestoη, (3) ση the teηsile streηg t l1 σf glass. Experimeηts show that the teηsile streηgth of gl aιι~ is 20 % greater wheη the specimeη is dried iη aη ον('ΙΙ thaη wheη the surface is wet aηd is as much as 200 % greater wheη baked iη a vacuum thaη iη the wet conclitioη. The tests also show that moisture is the chief factor iη the loss of streηgth; althσugh the preseηce of carbon

ΤΗΕ ROLE OF ADDiτiVES ΙΝ MILLING

239

dioxide also has some effect. Thus the surface coηditioη σf ι lιe glass affects the average teηsile streηgth at failure eveη ~ r the case of ro?s ?f comparati vely large diamete;. The surface of glass ιs, ιη fact, always covered with microracks, aηd the hypothesis put forward was that the water molecules ~re active at t~e crack tips aηd, that there is a local cheffilcal attack, wιt~ the.formatioη of a gel, which weakeηs the structure at this pωηt. However, this may be, the tests d:moηs~rate that the surface coηditioηs of eveη large partiCles ιηflueηce their streηgth iη a decisive ma ηηer.

It is clear that, irrespectiv e of the questioη of additives ' s su~h, the ηature of the fluid medium iη which the rushιηg process take~ place would be expected to iηflu η~e the eηergy reqUired to produce a specified fiηeηess. lt ιs commonly .stated that, all other things beiηg equal the eηergy reqUiremeηts for wet griηdiηg are as much a~ 20% less thaη fo~ d~ griηdiη~. Practical experieηce also ~ uggests t.hat the lπJ~ιtoffiηeg~ιηd~η~ is lower iη wet griηd ιηg th~η ~η dry .. Thιs observatιoη ιs ιη geηeral accord with ιhe pnncιples dιscussed above. The ge~eral pictu.r: wh.ich teηds to emerge coηcerηiηg ~he opera.tιoη of addιtιves ιs therefore that this is primarily 1η r;duc~ηg the effective eηergy at the particle-fl uid medιum ιη~erface, due to the adsorptioη σf the additive ση the partιcle surface. This could result iη decreasiηg the s treηgth of the larger particles, but a much more proηouηced effect would be expected iη the very fiηe size raηges, whe~e the surface of the particle becomes very much more ι~portaηt iη relatioη to its volume. Also iη t~e very fiηe. sιze raηges, the reductioη iη the agglomera tloη teηdeηcιe~ amoηg .the particles would be expected tσ have. a beηefi.cιal effect ιη reduciηg the eηergy for griηdiηg aηd lJ?- reducιηg the griηdiηg time. It can also be deduced that, ιη accordaηce with the above geηeral priηciples the mo~t effective additives would be expected tσ be surface actιve substaηces.

Such coηsideratioηs do ηοt, however, explaiη the

240

ΤΗΕ ROLE OF ADDITIVES ΙΝ MILLING


operation of solid additivesin dry ~rinding; for exampl ιiΗ improved mill performance obtaωed when a.small qu:~ι1 tity of carbon black is adde~ to ~he. cement cl1nker unclι'l" going grinding. Here aga1n, 1t IS su~gested th~t ~ I ι ι • function of the additive is in the preventωn of cushωnιιιl\ since Schweitzer and Craig(4) have observed that adclitioιι of carbon black in proportions as low as 0·08% is hi lιl effective as a cleaning agent for the surfaces of the b~ιll . and mill. Α rather different explanation of the operation of aclι lι tives is put forward by Bond<5J; it again being suggcstι ·ιl that ball coating is an import~nt factor. Th~se t<'H IH showed, in the first place, that w1th a soft m~tenal, s ιιι· Ιι as limestone excessive ball coating occurs w1th pow clι · ι 'M of much larger particle size than for hard mat.eria~s, suι· lι as quartz. Furthermore the occurren:ce of c~at1~g IS pr:.ιι · tically independent ofthe character ofthe gnnd.Ing mccl~:ι ; cast, forged and ball-bearing steel balls show1ng no S Ι J.I nificant difference other than that, when new, the smoollι hard balls are not subject to coating. Once abrasion Ι1:ι removed the surface, however, these balls show the saιιιι• behaviour as the others. According to Bond, the ball coating consist~ of a.n innn most layer of particles, of about 0·5-50μ 1n d1am tn, wedged into the surface of t~e ball. The outer l~y~rs οΙ particles making up the coat1ng are keyed to th1s 1nιιc • ι layer and are of much larger size. The effect of these layn is to cushion the grinding surface and so to greatly ~ecr η Η ι ' the grinding rate. It is possible that. other mechaω.sms :ιι·c · operative; for example the benefiaal results obtan~ed . ι' the use of small quantitites of carbon black as a gηηdι.ιι aid with cement clinker suggests some analogy wιl Iι electrostatic precipitation. ln this application it is knowιι that the addition of a small amount of carbon, gencr<ΙII in the form of unburnt fuel, can bring about the precipi Ι Η tion of a flue dust which otherwise cannot be treated; t lιc· function of the carbon apparently being to form a
ι · l cc trical charges leak away to i ι not possible, therefore, that ι πes ent in this application?

241

the collector electrode. Is some such mechanism is

T he question of additives will now be considered in detail; commencing with dry grinding. As previously Ntated, in dry grinding the additive may be present, :ιι least in theory, in solid, liquid, vapour or gaseous Γ rm and some of the most interesting results of the use of these additives are reported from the cement industry. According to Berry, (6) advantages have been shown by ι he use of vinsol resin, cod oil, beef tallow, aluminium s ιe arate and glycerol in the grinding of cement clinker. ln ρ articular colloidal carbon and two dispersing agents, known as R.D.A. and T.D.A. have proved effective. Λccording to Kennedy, (7) R.D.A., which is arylalkylHulphonic acid, and T.D.A., which is a mixture of triι· th anol-amino salts and highly purified calcium salts of li gnin sulphonic acid, were introduced as dispersing agents :ιnd catalysts to improve the properties of cement. These :ιclditives were introduced in the form of a solution in water, to the raw clinker feed of the tube mill in the proρortion of about 0·06% by weight of the charge. Apart Γι ·οm the improvement in the setting properties of the rcment, the output of the mill, for the same specific surΓace, was reported to be increased by 30-40 %· The ι·cse arches of Schweitzer and Craig<4J into the grinding of ι: cment clinker, in a laboratory ball mill, with carbon ω ack as a grinding aid show a continuous increase in the ι·a te of grind, or fineness attained in a given time, for : ιdditions up to about 1 % by weight of the feed. The rate f grinding thereafter fell with increasing amounts of the ιnore

:ιdditive.

These results, which are summarized in the curve of 9.1, show that the time required to grind to 85 % through 325 mesh is reduced by 35 % when compared with that required when no additive is used. The curve <ιlso shows that the optimum quantity of additive is fairly lcfinite; small deviations from the optimum quantity Ι • ig.

ΤΗΕ


242

ars to bc havin g appre ciabl e effect. This concl usion appe true for most addit ives. low as As previously noted , quan titite s of addit ive as effect ed mark a had 0·08 % by weig ht of the solid prese nt cing produ ut witho es, surfac mill on the cleanliness of the of rate The ct. produ the of on iorati deter r notic eable colou 100

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this sm:ιll grind ing is not, howe ver, mark edly impr oved by quan tity of addit ive. a sm:ιll The work of Bond(5) shows that the addit ion of t Η abou ght brou t, weigh by % 0·88 r, wate of amou nt ., D.Λ ofT. ion solut Α reduc tion in the coati ng of the balls. lιt brouι.ς ver, howe ed, fe the of ht at abou t 0·13 % by weig thercf(ιι·ι· abou t comp lete inhib ition ofba ll coati ng. It was tiιι preven in concl uded that the actio n of the addit ive ·t eff<·ι than rtant impo coati ng ofthe ball charg e was more

ROLE OF ADD ITIVE S

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MILL ING

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partic les. nns1ng from adsor ption on the surface of the amou nts small the that fact the by orted 'Γhis view is supp mark ed a t [ addit ive whic h are sufficient to bring abou allow to ient suffic are i ΙΏprovement in the rate of grind ing, 20% than less on layer cular ιl1e forΙΏation of a mono mole that fact the by orted supp also is t Ι ce. surfa f the prod uct ce produ d woul ·ven large chang es in the surface energ y nded dema y energ the in but small propo rtion al chang es energ y by the whol e grind ing process (since the surface nd dema y energ total ofthe on fracti SΙΏall repre sents a very tives, addi solid [ the process). Ι t is also significan t tha t the coati ng; proba bly Ιtlentioned earlie r, all tend toinh ibit ball ing ΙΏedia since grind the to ce surfa y by iΙΏparting a greas ting prope rties. brica lu ed mark it exhib to n know all ι hey are wate r in preof s The effect of very small quan titite catio ns. For appli other in n venti ng ball coati ng is know n in the know well is it that states xamp le, Manson enam els, s paint indus try that, in the grind ing of dry-proces effective be can r wate ιhe addit ion of as little as 0·0625 % of ermo re, Furth ng. coati liner and !η the preve ntion of ball d balls, coate badly from ved remo be also Lhe coati ng can wet few a of unde r runn ing condi tions , by the addit ion tive selec that sts sugge (5) balls to the mill charg e. Bond the be t migh ce surfa ball the at les l1ydra tion of the partic of tities quan small by ng coati of ition inhib the ι-eason for tive. tenta water. Such a suggestion is, at the most, surThe reaso n for such prefe rentia l hydra tion at the ed mber reine be must it but , clear not Γace of the balls is le derab consi a is there ct, impa of ent mom the that, at this of some tless, moun t of energ y dissip ated and, doub physicoould be used to bring abou t vario us chem ical, or energ y. of input an nd dema h whic ions hemi cal, react n by vatio T his possibility is supp orted by an obser to a ged chan was ore Pryor, (9) who found that copp er the e wher s point at level y energ ompo und of a highe r * it. ing break ut witho ore the ed rubb balls of the mill

has a piece of this ore, presen ted to him this transfo rmatio n at a numbe r of points shows clearly by Mr. Pryor, which the balls. with made been has where contac t

* One of the author s (H.E.R .)

244


There are other examples of the increase in the raι οΙ grinding brought about by the use of additives in Ιι milling. For example, WaeserOO) states that both R.D .Λ. and ammonium salts are effective in the grindin οΙ graphite and that oleic acid is an effective additive for ι l11· grinding of zinc blende. Gotte and Ziegler
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ΤΗΕ

ROLE OF ADDITIVES

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of this anomaly is forthcoming but it is possible that it might be connected with the strong adsorptive properties of finely divided glass powder. Whilst it. is realized that the tests of Gotte and Ziegler :ιre not strictly relevant, in so far that they were carried out in a vibration mill, it is considered that they are of interest in that they are, as far as is known to the authors, tl1e only tests carried out with a controlled atmosphere onsisting of a surface active agent in vapour form. The ιne thod may well have applications for certain specialized batch dry-milling operations . Turn now to the use of additives in wet milling. Accordi ng to Berry, <6 > silicates, phosphates and arylalkyls ulpho~ic acid are often used as additives in wet milling; the paιnt and colour industry, especially, offering a very wide field for the use of such agents (Anon. (1 2)). Probably the most thorough investigatio n into the ffects of additives in wet milling is that reported by von S zanth~( l 3 ) In this work, the effect of varying percentages of flotatωn additives and of oleonic acid on the wet milling f quartz and limestone, in a laboratory batch ball mill was investigated . In each case the characterist ic produc~ ρarticle size, d, resulting from grinding for 30 minutes was estimated fro.m the size frequency curves. The results, showing this characterist ic particle size plotted against the ρercentage concentrati on by weight of the additive are shown in Fig. 9.2 . It is seen that, in each case ' there is an . op tιmum percentage beyόnd which the grinding rate decreases with any increase in the quantity of additive. · Von Szantho considers that the effect of the additive is pri~arily d~~ to the lowering of the surface energy of the partιcles, ansιng from the adsorption of the additive with consequent widening of the elementary cracks and fi~sures ?n. the .s~r~ace ~f the part_icles. As discussed previously, it ιs ιmplι~ιt ιη thιs conceptωn of the operation of additives ~hat th~ιr effect should ~ecome progressivel y greater with ιncreasιng fineness of gnnd. The undersize curves shown ίη Fig. 9.3 do in fact show that the effect of the additive is

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concentration of additive exceeds a certain value is tentatively explained ση the basis of the lubricating effect of these substances at high concentrations. This explanation is supported by the data of Fig. 9.4 and Fig. 9·5, in which curves of surface produced and coefficient of friction are plotted against concentration of additive. These curves

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ve fσr which ιiΗ · shσw that fσr cσηceηtratiσns σf additi griηdiηg is hi A· Iι , σf rate the high is ση cσefficieηt σf fricti a t · ιιl but wheη the cσefficieηt σf frictiση decre ases the r frί<~., σf cieηt cσeffi σf s griηdiηg dimiηishes alsσ; the curve Η, τJΊιι r. simila y clσsel eiηg b iηg tiση aηd rate σf griηd ση i ~ οΙ fricti σf cieηt cσeffi the that udes cσηcl νση Szaηthσ aηd that, at hi A iι majσr impσrtaηce iη the griηdiηg rate, σf frictiση ιιt cieηt cσeffi cσηceηtratiσηs, the decre ase iη tiση iη surΓ<ιι · c · reduc the frσm d gaiηe weighs the advaηtages the additivι ·. σf ptiση adsσr e surfac the frσm g eηergy arisiη cσηteηts οΙ the σf This result is alsσ σf iηterest iη the light iη a nιίll ηg surgi that sted Chap ter 7, iη which it is sugge iη tfι, · tiση reduc small a by abσut ht migh t be brσug s ιιl curve the cσefficieηt σf frictiση σf the charg e. Thus, mίglιl tives σfaddi use Fig. 9.4 aηd Fig. 9.5 suggest that the tlιί reduc e the cσefficieηt σf frictiση by 40-50 % aηd mίll , a cause tσ ieηt suffic thaη mσre cσuld well prσve which σtherwise perfσrms satisfactσrily, tσ surge . ~ It is alsσ iηterestiηg tσ eηquire why the rate σf millin rl Clea . ση fricti σf cieηt cσeffi asiηg shσuld fall with decre t οΙ this must arise frσm a reductiση iη the effective heigh H tl1i aηd e charg fall σf the ceηtre σf gravi ty σf the mill · tlιι ση e charg the σf ηg wσuld prσbably arise frσm slippi ιl t<· expe be wσuld it theη sσ, fact, iη mill shell. If this is, ; ιl that the use σf adequ ate lifters wσuld, tσ sσme exteηt aiHo is there ly least, restσre the rate of griηdiηg. Pσssib ηg aηd the iηcreasι·ι l sσme relatiσηship betw eeη the slippi iηg as cσmpan ·ι l griηd wet with wear, ball rate σfliηer aηd with dry griηdϊηg, which is sσmetimes repσrted. οΙ It is σf iηterest that νση Szaηthσ cites the wσrk 4) iη suppσrt σf lιi N Rehbiηder aηd σf νση Eηgelhardt(l

cσrrelatiση betweeη tlιι • aηd their resistaηce Ιιι was also showη that th<:ι· ι • hiιι ι.: is a prσgressive decre ase iη the resistaηce tσ scratc bed ση tlιι · adsσr ve additi of es ti quaηti siηg with iηcrea cι· ιι S]Jrface σfthe bσdy, there beiηg a clσse correlatioη betw nι, r isσthe ption adsor the the resistaηce to scratchiηg aηd

cσηclusiσηs. R ehbiηder shσwed surfac e hardηess σf mater ials scratchiηg with a fiηe pσiηt . It

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ΤΗΕ

ROLE OF ADDI TIVES

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MILL ING

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ed with a minim um of resistaηce tσ abras ion beiηg reach % 70 as much As . ptiση adsor ete cσmpl of under coηditioηs the way, this in red measu was ess hardη e surfac iη ase decre maxim um d ecreas e correspoηdiηg to the greate st differ It e. surfac the aηd m mediu the cηce iη polar ity betweeη would is sugge sted by R ehbiηd er that ηση-pσlar additi ves aηd pσlar ials, mater philic hydro fσr ive effect most be the a dditiv es shσuld be used for hydro phobi c crystals. mσff, (15) who Eηgelhardt repσrts the work of Obrei the lamiηa tiηg separa in doηe work elastic the measu red to the equal y of mica and fouηd that this was nume ricall d. forme e surfac ηew the th eoreti cal surfac e energ y σf prethe iη out d carrie was a lamiη When splittiηg of the e exsence of surfac e active m edia, hσwever, the surfac iη ased iηcre , posed, for a giveη amou nt of strain eηergy by d cause y eηerg e surfac the proportiση tσ the d ecreas e iη the t he adsorptiση. Eηgelhardt furthe r coηducted tests ση iη disc, e carbid η silicσ a with z quart of iηg griηd abrasive polar that fouηd th e preseηce of variσus additi ves. It was iη reduciηg the aηd dipσlar additi ves were very effective quaηtities of small ular partic In ση. abrasi resistaηce tσ e iη octan ol or butyr ic acid in beηzol or sσdium silicat aηce. resist wear water were effective iη reduciηg outstaηdiηg Skaupyω has receηtly repor ted some r iη variσus pσwde η σfirσ g millin the frσm ηed results σbtai iη a fluid m edia. Altho ugh the tests were carrie d σut σf ds grσuη the ση here vibratiηg mill, they are iηcluded iηg griηd σf rate the that fσuηd was their impσrtaηce. It thaη of irσn was five times as great iη glycσl, aηd ησ less air. dry iη as l tweηty times as great iη isσ-amyl alcσhσ ially espec were ves additi The tests revea led that dipσlar sureffective as griηdiηg aids, as wσuld be expec ted frσm alσηe. ηs eratiσ cσηsid y eηerg face effect the Althσugh, iη view σf the mσre mark ed energ y as e surfac the σf evide nce pσiηts tσ a lσweriηg ves iη additi σf tiση σpera the iη factσr being aη impσrtant mσre is g millin dry σf case the in ηce evide the g, wet millin prσ with ials mater cσηfused . The effectiveηess σf variσus 16*

248


t/1c • show that for conce ntrati ons of additi ve for which A" Iι , hi is ing grind of rate the high is n coefficient of frictio ιι/ but when the coefficient of frictio n decreases the rat fri<· of cient coeffi of grind ing diminishes also; the curves Η, tion and rate of grind ing being closely similar. τJΊLΙ ιl ι i$ n frictio of cient coeffi the that udes concl von Szant ho hί~Ιι major impo rtanc e in the grind ing rate, and that, at ιιt n frictio of cient conce ntrati ons, the decre ase in coeffi ·ι· surΓ<ι< in tion reduc the from d gaine weighs the advan tages additivι· . the of ption adsor e surfac the from g energ y arisin nts οΙ This result is also of intere st in the light of the conte nιί/1 a in g surgin that sted Chap ter 7, in which it is sugge t/ιι · in tion reduc small a by about ht migh t be broug ιι/ s curv the Thus, e. charg the of n frictio of coefficient lιl g mί itives ofadd use Fig. 9.4 and Fig. 9.5 suggest that the tlιi reduc e the coefficient of frictio n by 40-50 % and mί/1 , a cause to ient suffic than more could well prove which otherw ise perfo rms satisfactorily, to surge. ~ It is also intere sting to enqui re why the rate of millin l Clear n. frictio of cient coeffi asing shoul d fall with decre ο/ this must arise from a reduc tion in the effective heighι H tlιi and e charg fall of the centre of gravi ty of the mill · t/ιι on e charg the of ng would proba bly arise from slippi l<·ιl expe be would it then so, fact, in mill shell. If this is, t ;ιl that the use of adequ ate lifters would , to some exten aiHo is there ly least, restor e the rate of grind ing. Possib Ψι/ some relati onshi p betwe en the slippi ng and the increa an ·ιl comp as ing grind wet with wear, ball rate ofline r and with dry grindϊng, which is somet imes repor ted. οΙ It is of intere st that von Szant ho cites the work Μ Ιιί of rt suppo in l4) hardt( Engel Rehbinder and of von t/ιι · en betwe lation corre a d showe inder conclusions. Rehb Ιο surfac e hardn ess of mater ials and their resistance ' thι'l'ι that shown also scratc hing with a fine point. It was iιιι-: scratch to ance resist the in ase is a progressive decre ιlιι· with increa sing quant ities of additi ve adsor bed on ·ΙΙ betwcι lation s~rface ofthe body, there being a close corre ιιι, isother ption adsor the the resist ance to scratc hing and

ΤΗΕ

ROLE OF ADDI TIVES

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249

ed with a minim um of resist ance to abras ion being reach 70% as much As ption. adsor lete comp of under condi tions the decre ase in surface hardn ess was measu red in this way, differ st greate the maxi mum decre ase corre spond ing to It e. surfac the and m mediu ence in polar ity betwe en the is sugge sted by Rehb inder that non-p olar additi ves would polar be the most effective for hydro philic mater ials, and ls. crysta c phobi additi ves shoul d be used for hydro Enge lhard t repor ts the work of Obrei moff, (15) who a measu red the elastic work done in separ ating the lamin the to equal y of mica and found that this was nume ricall d. theor etical surface energ y of the new surface forme prethe in out d carrie was a lamin When splitti ng of the e exsence of surfac e active media , howe ver, the surfac in sed increa y, posed, for a given amou nt of strain energ by d cause y energ e surfac the propo rtion to the decre ase in the the adsor ption . Enge lhard t furthe r condu cted tests on in disc, de carbi n silico a with z quart of ing grind abras ive polar that found the prese nce of variou s additi ves. It was the and dipol ar additi ves were very effective in reduc ing of ities quant small ular partic In ion. resist ance to abras e in silicat m sodiu or l benzo in acid ic butyr or ol octan water were effective in reduc ing wear resistance. outsta nding Skaupyω has recen tly repor ted some in variou s er powd n ofiro g millin the from results obtai ned in a d fiuid media . Altho ugh the tests were carrie out of ds groun the ση here vibra ting mill, they are inclu ded ing grind of rate the that found was their impo rtance . It than of iron was five times as great in glycol, and no less air. dry in as twent y times as great in iso-am yl alcoh ol ially espec were ves additi The tests revea led that dipol ar sureffective as grind ing aids, as would be expec ted from . alone ons derati consi y face energ the Altho ugh, in view of the more mark ed effect as y energ e surfac the of evide nce points to a lower ing in ves additi of tion opera the in factor being an impo rtant more wet millin g, the evide nce in the case of dry millin g is prowith ials mater confused. The effectiveness of variou s 16*

250


nounced lubricatin g properties such as carbon black, as an aid in the grinding of cement clinker, is much easier to explain on the basis of the inhibition of ball coating by means of the deposition of a layer of carbon on the balls. Furtherm ore, the small quantities of water which have been observed to remove the coating from a badly coated mill lends support for the idea put forward by Bond (5) that, in this case, the particles are removed from the surface by a process of selective hydration , a surface phenomen on . However, the small quantities of water which are effectiv · suggests that the adsorption occurs at the ball surfac !I only. It thus appears that the surface phenomen a involved are complex and that present knowledg e is insufficient t give more than a very imperfect picture of the mechanism which is operative. In conclusion, it is fair to say that any process of crushing must be influenced to a large extent by the nature of' the atmosphe re in which the operation is carried out. There is ample experimen tal evidence that in many cases, the energy or time of grind to a given fineness, and als the ultimate fineness of grind attainable , can be favourabl y ~odified by the introducti on of a suitable additive into the mill. In view of the very large energy requireme nts for grinding, and in particular fine grinding, and also the low theoretica l efficiency of the grinding operation as at prcsent carried out, it would appear that the use of milling additives could effect substantia l economies. It is als clear that the subject has not yet received the attention which its importanc e warrants, and it appears to be a field in which much interesting research remains to bι· done. REFERENC ES

I. SκAUPY , F . Kolloidzschr. , 1948, 111 , 123. 2. REHBINDER, Ρ. Ζ. Phys ., Ι93Ι, 72, Ι91. 3. ΒΑΚΕR , τ . c., and PRESTON, F. w. J . app. Phys., 1949, 180, 382. 4. SCHWEITZER, C . W. , and CRAIG, Α. Ε. J. industr. Engng Chem., Ι940, 32, 751. 5. BoND, F. C. Amer. Inst . min. (metall. ) Engrs, Tech . Paper, Ν ο. 1160 (1940).

ΤΗΕ ROLE OF ADDITIVE S ΙΝ MILLI N G

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Ι 3. Ι4. Ι 5.

251

BERRY, C. Ε . Perry's Chemical Engineer's H andbook (New York) . K ENNEDY, Η. L . J . I ndustr . E ngng Chem ., Ι936, 28, 963. MANso N, Μ. Ε. J. Amer. ceram. Soc., Ι 938, 21, 3Ι 6. P RYOR, Ε.] . V erbai contribution to the discussion of "Symposium ση Crushιng and Grinding" , Instn Chem. Engrs, Lond . ( Ι956) . W AEsER, Β . Dtsch. Farbenz tg., Ι952, 6, 434. GδττΕ, Α. , a nd ZIEGLER, Ε. Ζ. Ver. dtsch. Ing. Ι956 98 Ν 9 373 · ' ' r. ' ' An on ., Paint Oil chem. R ev., Ι940, 102, 70. VoN SZANTHO, Ε. Ζ. Erz berg b. M etallhiittenw., Ι949, 2, Ι2 . VoN ENGELHARDT , Ε. N aturwi,·senscha ften, Ι946, 33, 195. 0BREΙMOFF, Ο. Proc. rl!JI. soc., (Α), 1930, 127, 290.

APPENDIX

SUBJECT INDEX

Ι

i

_1__

b

MILL FILLING PARAMETERS

Δο

ο

15 30 45 60 75 90 105 120 135 150 165 180

J 0·000 0·003 0·028 0·090 0·194 0·335 0·500 0·665 0·804 0·910 0·972 0·997 1·000

afR

C/R

1·000 0·966 0·866 0·707 0·500 0·259 0·000 - 0·259 -0·500 -0·707 -0·866 -0·966 - 1·000

0·000 0· 259 0·500 0·707 0·866 0·966 1·000 0·966 0·866 0·707 0· 500 0·259 0·000

252

bfR 0·000 0·034 0·1 34 0·297 0·500 0·741 1·000 1·259 1· 500 1·707 1·866 1·966 1·000

xfR ι-ο οο

0·960 0·9"0 0·810 0·70() 0·572 0·4·30 0· 2811 0·17 3 0·091 0·02Η

0·004· 0·000

Abι·asi on , grinding by, 133, 140 Add iti ves, mi lling: Charge friction, effect on, 247 Cleaning of surfaces by, 242, 243 Common, 240-4 Function of, 236 . Surging, effect on, 248 Types ofGaseous, 237 Liquid, 237 Non-polar, 244 Polar, 244 So1id, 240 Vapour, 237 Aggregation, 114, 237 Aids, milling. See Additives Analysis, Dimensional : Metal wear, 224 Power, 71 Rate of milling, 142 Surging, 210 Armouring, 144, 189 Associated energy, 125

F1ying, 48, 60 Profile, 5 1, 64,65 Rolling, 49 Total, 48 Circle, Davis, 44, 58 Classification, 24, 25 Contamination ofproduct, 31, 179, 187, 223 Coefficient of friction, 54, 213 Comminution: Energy,distribution in, 115 Laws ofBond, 122 Carey and Stairmand, 125 Dobie, 124 Kick, 11 8 Rittinger, 117, 149 Modes ofAbrasion, 133, 140 Crushing, 137 Deep bed, 132, 133 lmpact, 130, 140 Sing1e particle, 133 of ca1cite, 130 Ball diameter, influence ση : of cement clinker, 30, 240 Efficiency, 183, 186 of coa1, 110 M eta1 wear, 228 of coke, 11 3 Power, 81 ,97 ofdo1omite, 146, 154 Product characteristics, 178 of drugs, 32 Rate of milling, 153 of Durham anhydrite, 134 Surging, 210 of feldspar, 136 Ba\1 mi\1 , 2 1 of galena, ,130 Ball rationing, formula for, 192 of glass, 130 Ball wear. See under Wear, meta1 of graphite, 109 Balls: · of gran ite, 11 Ο Cleaning, 242, 243 of moybdenum disulphid e, 109, Particles embedded in, 144, 189 2 14 B1ow, cushioning of, 11 5 of pigmen ts, I 09 Bond's law, 122 of quartz, 109, 130, 146, 154 Breakage. See under Comminution of rock salt, 130 of steel, 110 Charge: Con tact: Cascading, 38, 65, 75 Frequency of, 49, 161 Cataracting, 38, 65, 75 Time of, 163, 167 Coefficien t of friction of, 54, 2 13 Corrosion, 235 253

SUBJECT INDEX

SUBJECT INDEX

254

Pσwer,

81 Milliηg, rate σf (q.v.) Surgiηg, 210,213

Critical speed, 42 . . Crushiηg. See under Cσmmιηutιοη Crystallite, 112 Crystallizatiση, 134 Crystals : Imperfect, 110 Perfect, 109 Davis circle, 44, 58 Deηsity

σf griηdiηg

.

bσdιes;

Griηding:

iη-

Pσwer, 76 Deηsity σf particle; iηflueηce ση:

Mίlliηg , efficieηcy σf (q. v.)

σf, 150 Pσwer , 76 Diameter (ball, mill, particle, rσd);

- , rate

iηflueηce ση:

174

Wear, meta1 (q.v.)

Dim eηsiσηal aηa1ysis fσr: Milliηg , rate σf, 142 - , efficieηcy σf, 183 Pσwer,

71 Surgiηg, 210 W ear, metal, 224

Diffractiση patterη, 134 Drσp hammer, 128

Efficieηcy. See under Milliηg, efficieηcy σf Eηergy:

Assσciated, 125 Ba1aηce, 128 Elastic straiη, 128

Necessary, 116 Plastic straiη, 128 Sufficieηt, 116 Surface, 116, 245 Flight, time σf, 49, 62 F1yiηg charge, 48, 60 Fracture, geσmetry σf, 112, 125, 136 Cσefficieη t σf, 54, 213 Effects σf additives ση 247 Iηflueηce ση-

efrectσfadditivesση thestr Πf<Ιlι ' σf, 238

-

flu eηce ση:

Frictiση:

He terσgeηeσus, 109, Hσmσgeηeσus, 109

Geariηg, destructiση σf, 204, 2 1Ο Glass cσmminutiση σf, 130

Milliηg, efficieηcy σf (q. v.) - , rate σf, 150

P σwer, 81, 97 Rate σf milliηg , Surgiηg, 210

Materials:

Abrasive, 133, 140 Impact, 130, 140 .. Griηding aids. See under Αddιtι ν ~ . milliηg

-

bσdies :

Types σf, 106, 190 M erit σf, 190 Armσuriηg σf, 144, 18~ . - , selective. See under Μιllιηg Hadse1 mill, 129 Hammer, drσp, 128 Hardiηge mill, 26

.

Hardηess (griηdiηg medιa, pσwιlιΨ,

relative) ; influeηce Pσwer (q. v.) Milliηg, rate σf (q. v.) Wear, meta1 (q.v.)

ση:

Impact, cσmmiηutiση by, 128, Ι :10 Iηterfereηce ίη traJectσrιes, 50 - betweeη trajectσries, 48 Kick's 1aw, 118 Krupp mill, 28 Laws σf cσmmiηutiση. See ιπιι/Μ Cσmmiηutiση

Lifters, 22 Lifters; iηflueηce ση: Pσwer , 81, 85, 86 Rate σf milling, 197 Surging, 200,214 Liηers :

Armσuriηg σf, 144, Fuηctiση σf, 197

189

Types σfStep, 198 Studded, 199-201 Wave, 198 Wedge-bar, 198 Wear σf, 200 Wσrk-hardeηiηg σf,

200

121

Mixed, 110 Metal, wear σf. See under Wear Mill, types σf: Air-swept, 22, 24 Batch, 22, 29, 90 Ball, 21 Grate discharge, 22, 90 Hadse1, 129 Hardinge, 26 Krupp, 28 Pebble, 21 Rσd, 21, 30 Τruηηίση σverflσw, 22, 90 Tube, 21 Mill prσducts: Cσηtamiηatiση σf, 31, 187, 223 Diffractiση patte rη σf, 134 Mill surfaces, cleaηiηg by additives, 242, 243

255

- - particle diameter, 153 - - - deηsity, 150 - - - griηdability, 150, 157, 169, 189 - - pulp cσηsisteηcy, 165, 171 , 182 - - - charge, 169, 172 - - relative hardηess, 169, 189 - - rσd diameter, 163, 178 - - time σfmilliηg, 163, 167, 174 - - wet milliηg, 165 Reasσηs fσr, 17, 18 Recrystalliiatiση duriηg, 134 Selective; iηflueηce σfBall diameter, 178 Milliηg bσdy deηsity, 181 Particle diameter, 178 Rσd diameter, 180 W et, 25, .30 Mσdels, tests ση small-scale, 42, 69

Milliηg:

Chaηge σf

chemica1 state 243 Clσsed circuit, 24 Dry, 25, 30

duriηg,

Efficieηcy σfEquatiσηs fσr, 152, 183 Dimeηsiσηa1 aηalysis fσr, 183 Depeηdeηce upση mill speed,

183 - - - fillίηg, 183, 186 - - pulp charge, 183, 188 - - - cσηsisteηcy, 184 Mixed materials, 176 Opeη cίrcuit, 25, 165 Rate σfEquatiση fσr, 142, 174 Dimeηsiσηal aηalysis fσr, 142 Depeηdeηce upση ball dίam eter, 153 - - dry milliηg, 164 - - griηdίηg bσdy deηsity, 150 - - 1ifters, 197 -..,.- diameter, 174 - - - filliηg 167 - - ___.: speed: 164 - - - thrσughput, 163, 167, 173 - - milliηg aίds, 236, 245

Particle, size σf; iηflueηce ση: Rate σf milling, 153 Metal wear, 226 Pσwer, 87 Particles embedded ίη balls. See under Surfaces, armσuring σf Pebb1e mill, 21 Pelletiηg , 128, 133, 134 Pigmeηts, 33, 102 Piles: Eηd-beariηg, 219 Frictiση, 219 Pσwer, depeηdeηce upση:

Ball diameter, 81, 97 Charge cσeffi cieηt σffrictiση, 81 Lifters, 81, 85, 86 Mill diameter, 73, 96, 97 - filling, 81' 103 -leηgth, 72, 99 - speed , 73 Milling additives, 247 - bσdy deηsity, 75 Partic1e deηsity, 76 - diameter, 87 - griηdability, 82 Pu1p charge , 90 - cσηsisteηcy, 102 - viscσsity, 102 - , dimeηsiσηal aηalysis fσr, 71

256

SUBJEC T INDEX

Power, equation for, 71, 93, 98, 99, 100, 101, 104 Product: Contarnina tion of, 31 , 187, 223 Size Distributio n of, 179 Production , rate of. See under Throughpu t Quartz, 129 Recrystalli zation, 134 Reduction ratio, 117 Rittinger's law, 117, 149 Rod rnill, 21, 30 Duty οΓ, 30 Power to, 106 Vibration οΓ, 220 Rolling, tirne οΓ, 49, 62 - charge, 49 Sirni1arity, dynarnica1, 69 Slip : On rnill surface, 35, 67 Limiting condition, constructio n fοι·, 67 Speed: Critical, 42 InAuence onArrnouring , 144, 189 Efficiency, 183 Meta1 wear, 225 Power, 73 Rate of milling, 164 Surging, 75,210 SurΓace:

Arrnouring of, 144, 189 Profi1e of charge, 51, 64, 65 Rate οΓ production οΓ specific. See under Milling, rate of Surging: Criterion for, 212 Dependenc e onAdditives, 248 Ball diameter, 21 Ο Charge friction, 2 10,2 13 Lifters, 200, 214 Mill diarneter, 210 - filling, 210 Speed, 75, 210

Dirnensional ana1ysis for, 21 Ο Nature of, 36, 203 Tirne of, 208, 209 Throughpu t: Dependenc e onBall diarneter, 194 Mill diarneter, 196 - filling, 194 Particle diameter, 194 Pu1p consistency, 193 Speed, 194 Equation for rate of, 194, 196 Tirne of Aight, 49, 62 - ο[ rolling, 49, 62 Torque, fluctuation of, 209, 217 Trajectorie s: Constructio n for, 45 Equation of, 40, 50, 51 lnterferenc e between, 48 - in, 50 Velocity, irnpact, 137 Vibration: Frequency spectrurn οΓ, 2 19 Origin of, 209,217,2 19 Transmissi on of, 2 19 Water as grinding aid, 242, 243 Wet rnilling: Power dernand in, 101 Reasons for, 25, 30 Wear, rnetal: Dimension al analysis for, 224 Equations for, 224, 230, 231, 232 lnf!uence ofArrnouring , 221 , 232 Ball diarneter, 228 Corrosion, 235 Geornetry of trajectory, 35 Hardness of rneta1, 190, 222 - of particle, 190, 222 Mill diarneter, 224 - filling, 223 Particle diarneter, 226 Pulp consistency, 228 Relative hardness, 190, 222, 227 Speed, 225

ΝΑΜΕ

INDE X

Andreasen , Α. Η. Μ., 134 Anon., 216 Anon ., 219 Anon., 245 Anselm, W., 193 Axelson, ]. W., 122, 132 Baker, Τ. C., 238 Barnacle, Η. Ε., 196 Barth, W., 52 Belwinkel, Α., 200 Berry, C. Ε., 241 , 245 Blunt, G. D., 210 Bond, F. C., 84, 99, 122, Ι 74, 233, 240, 242, 243, 250 Bowden, F. Β. , 134 Cadena, F. , 174 Carnpbell, Α. Β. , 73, 97, 99, 103, ιο4, ι86 Carey, W. F., 95, ιο4, 125 Coghill, W. Η., 73, 77, 97, 99, 103, 104, ι46 , ι 5ο, !56, 156, 178, 186, 187, Ι9ι , 194, ι96, 228,234 Couιson, ]. Μ ., ι23 Craig, Α . Ε., 240, 241

102, 102, ι 74, 224,

Davis, Ε . W., 43, 98, ι92, 233 Devaney, F . D ., 77, ι46, 150, 156, 174, ι78, 187, ι9ι, ι94, 196,224, 228, 234 Dobie, W. Β., ι24 Ellis, Ο. W., 233 Engelhard t, Ε., 248 Engels, Κ. , ι48, 223 Evans, D. Ε . , 37, 42, 65, 66, 69, 76, 99 νοη

Fahrenwal d, Α . W. , 97, ι89 Fairchild, D. Η., ι9Ι Faiι-s , G. Μ., ι 33 Felix, W ., 110 Fobelets, Ρ, 52, 60

Garrns, W. Ι. , 233 Geiger, Τ., ι 10 Gilliιand, Ε. Κ., 123 Gotte, Α. , 244, 245 Gow, Α. Μ., 73, 97, 99, 102, 103, 104, 174, ι86 Gross, ]. , 127, 129, 133, 149, 224, 228 Grί.inder, W., 42 Guggenhei m, Μ. , 97, 99, 102, 103, 104, 174, 186 Hall, R . G., 129 Hancock, R. Τ. , 100 Hinsley, J. F., 52, 204 Holrnes, J. Α., ι 76 Honig, F. , 121 , 122 Hukki, R. τ., 232 Hurnphrey s-Owen, S. F. Ρ . , I ι3 Hί.ittig , G . F., 110, ι 3 0 Kennedy, Η. L ., 24ι Kick, F., Ι ι8 Kwong, J. Ν. S., 129, ι 3 ι Lee, Ε. Ε . , 97 Lewis, W . Κ . , ι23 Loeb, C. Μ., 233 Mackenzie , R. C., ι34 Manson,. Μ. Ε ., 243 Martin, G., 127, 137 Maxson, W. L. , ι 74 M cAdarns, W. Η., ι23 McLeod, Ν. Α. , 227, 228, 234 Mi1ne, Α. Α., ι34 Moeller, J. Ε., 233 Mor1ey, Α ., 2 17 Mortsell, S., 226, 233, 234 Moser, F., 147 Mί.iller, R., 137 Nordquist, D . Ε . , 233 Norrnan, Τ. Ε . , 233 Norris, G. C., ι9ι, ι 96

257

258

ΝΑΜΕ

Obreim off, 0., 249 Patchin g, S. W. F., 176 Pearson , Β. Μ., 193 Piret, Ε . L., 122 , 129, 131, 132 Prentice, Τ. Κ., 233 Preston , F. W., 238 Pryor, Ε. J., 243 Ramml er, Ε., 137 Rehbin der, Ρ., 237, 248, 249 Richard son, J. F., 123 Ri ι ey, D. Ρ ., ι 13 νοη Rittinge r, Ρ. V., 117 R ose, Η . Ε., 37, 42, 65, 66, 69, 76, 99, ι 13, 143, 196, 200, 2σs, 210, 22 1 R osin, R ., 137 Schweit zer, C. W., 240, 241

INDEX

Skaupy , F., 236, 249 Smekel, Α., 121 , 122, 130 Smith, Ε. Α., 223 Sperling, Κ., 137 Stairma nd, C. J., 95, 104, 125 Starke, Η . R. , 156, 174 von Steiger, R ., 50 Stevens , J. L., 233 Stott, F. D. , 112 von Szantho , Ε., 245, 248 Tabor, D., 134 Taggart , Α. F., 93, 102, 214, 233 Theime r, Ο ., 143 Waeser, Β., 244 Walker, W. Η. , 12 3 Ziegler, Ε., 244, 245 Zimmer ley, S. R ., 127, 129, 133

Ball, Tube and Rod Mills, Rose and Sullivan, Publication year 1958

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