The conversion of sedigraph size distributions to equivalent sub-sieve screen size distributions

TEglI ELSEV1ER

Powder Techtmlogy 95 ( 19981 1119-117

The conversion of sedigraph size distributions to equivalent sub-sieve screen size distributions H. Cho, K. Yiidirim, L.G. Austin * Mim ral Proces,ving Section, Department of Mineral Engineering, The Peml.D'h',mia State Umversity. 115 Hnsler Buihling, Unive~:~ityPt,'l~, PA 16A'O2-50fIO. lISA Received I January 1996: revised I March 1097

Abstrac¢ In many industrial applications a major part of the data on size distribtaions uses screening to define size. whereas an important part of tl~e size distribution inlbrmation is determined using sub-sieve size analysis. It is necessary to have an accurate me,hod of c,mverting one lype of da~a to the other. A technique is presenled for conversion of Sedigraph size distribution data to equivalen! screen size. The technique consist~ of first characterizing the material I~y determining the Sedigraph size distribution on a sample of the powder carefully wet-screened to lie betwee,i 270 and 4011 mesh ( 53-38 I.tm I :rod fitting the data by :t log-normal or log-logistic distribution fullctJon to give p. and ~r or p. and Z values characteristic of the ntaterial. The Sedigraph size distribution of a sample of less than 400 mesh powder screened from the total sample call then be converted to an equivalent screen size distrihtttion by a back-calculation technique, ushlg the ch,'tracteristic parameters nf the malerial in :~ consn'ai,led search program. The technique is ilhtstrated by application Io data taken :m}und an indtr~triai grinding mill. ¢'~ 1098 Elsevier Science S.A, KIW~VOI',IW SJ~t~ distril'qltions: ,~oligraph ~il~' dislrihutitm: Screen size dinlrihlltiol|" Data ¢oip,,¢i'sit~n

I. h l t r o d u e l l o n So,reline ill standard woven wire sieves is by he" tile most wklcly used rnetl'md of particle size analysis ill nliuly indus° tries, However, this method is only routinely tipplical~le for particles Im'ger than 400 mesh (38 pan nominal size). whereas often important inlbrmation is contained ill the size distribution of material less than 400 mesh. it is common. thel'elbre, to use screening down to 400 mesh ( or 325 nlesh. 45 p.m nominal size ) and a sub-sieve sizing method lbr that fractk~n below 4IX) mesh. For example, sedimentation melhads work well i,1 tile particle size range 3-75 gin and have tile advantage (unlike some other metllods which do not detect the smalt,:st particles l that they give a correct mass fraction below ~he smallest acorn"ate test size. However, Ihe delinitiou of size in sedimentation is the equivalent Stokes diameter, which is not the same definition us sieve size. Thus. it is necessary to have a methodology lbl' joining the two experimental deterlninatiol~S together with a consistent delinition of size ( usually sieve size ). For example, the consistent .joining of sub-sieve size analysis m sieve analysis is a well* Corresponding author. Tel.: + I 814 863 0373" fax: + I 814 865 3248. 0032-591(}/98/$19.00 LD 1998 Elsevier Science S,A. All rights reserved P!1S0032-5910(97 |03326-3

known problem ill the an:tlysis of the I'Pehaviol'ofline gl'iltding circuit.s I I i. It has I~een e o n l t l l O l l practice [ 2°°51 t,, ¢OllV¢l'[ ;,Ill IIISIrLI ° inent size to an equivalent sieve size ( or ,,'ice Vel'sa ) with the USe of LL 111einl 'shape lhclor', e,g. 8()c~; ( by 111ass) less than 20 ~lll instrullrlellt size heeonles 80c~ less that120/r ~lli sieve size. r being the 'shape factor'. However. this requires a knowledge of r, which must be obtained by an analysis of shapes or by overlapping Ihe size measurements by the two measurement techniques. Since a sample of irregular particles fl'om an industrial operation usually covers a wide range of shapes, the determination of r by shape analysis is very tedious. The overlapping inethod would work well if m:llel'kd of s e t ' t e n size x to x + cLx" gilve sedilnentalion size~ of" ~ p:t) It} trv) + d(r~'), since then the size distributions plotted on loglog scales could be made to match by sliding one in the x direction m join and lie on tile other, ltowever, becauseof the rauge of shapes, irregular particles in a narrow screen range give a wide i'a~lge of Stokes diameters ( some much bigger and sonle much smaller), and vice versa. Tilis very often makes it impossible to use tile sliding technique bee;rose the curves will never match for any value of r. Austin and Shah [ 61 attempted to overcome this problenl by measuring the instrument size distribution ( laser diffrac-

tt. Choet .I, / Powder Teclmology 95 (1998) 109-117

I tO

tometer) of ~ sieve fractions of powder, and defining the mean shape factor by the ratio of the 50% ( by mas,~) passing size to the geometric mean sieve size. They showed that this technique gave the ~ m e mean shape factor for different sieve fractions. However, there is no proof that this definition of shape factor is the correct one to give the correct value of r. ~ v also fitted the distribution of instrument size from a sieve size interval by a dimznsionless two-parameter function and showed that given a complete sieve size distribution the instrument size distribution could he predicted using this function, The reverse problem 'Given the instrument size distribution what is the equivalent sieve size distribution'?', which is the required methodology, was solved only for the special case where the sieve size distribution was of the form P(x) = ( x / k ) ' , x being the sieve size and P(x) the cumulative mass traction less than size x. In this paper we will present a tbrmal analysis of this method and show how data from two different size analysis techniques (one suitable for small sizes, the other for large sizes) can be interconvened and joined to ff)rm a single consistent size distribution, using relatively simple experimental and computational techniques. The approach does nnt use the concept of a rnean shape latter but is all explicit solution to the problem and is applicable to any shape el'size distribution. To our knowledgt~our group is the only one using sub-sieve size analysis on I/2 screened fntctions to establish techniques for conversion of sub-sieve analyses to equivalent sieve size distributions. In the work reported here the instrumem chosen for sttb-siev¢ sil0 analysis wlts the X-ray Sodigraph, model 5100, Solutions were it|ad0 with about 3 vol.'/; of solid and (I, I WI,G of st~Jiuln hexantetltphosf~hate a,, dispersant.

where p(x) is the density function p ( x ) = dP(x)/dx and X is constant in the integration. For example, consider a sieve size distribution which is a power function over some size range xt to x2: O 0

P(x) =

if a sieve interval is screened from this region its sieve size distribution will be

(:',is-t'" (:'g" Pix)=/t.V .

t ..... " xt'
(4)

-(;:)

where a, is the top size el' the interval and ,h, tile bottom size, and x,/xh = R. R will usually be ~ or ~vr2"2. Eq. ( 4 ) then becomes

and

f

l

x/xh > R

(x/.h,)"

-

I < x/x. _< R

(5)

xtx,, < I

(t

,'.

re.h, i

I

,

I ~,v/.h, -< R

t)

2. 'rhenry

(3)

(0)

~/.h, <~ I

Sul~stitttling into I.:q, ( 2 ) give~

Assume that particles of sieve size x to x ¢da will distribute into instrument size X with a function that does not vary with relative size, This it~tpliesthat particles are of the santo d~'nsity and distribution of shape lbr all sizes. A suitable fimction 16 ] might be the simple tog-logistic function I

F(X, x) ~ I + (X//,Lll)

a,

II

Q(,'¢)~

f

I

'" ('f

,,I\.'q,ll

I"(X,.~)~,,,I

lh

i.e. /¢

m J( I "' ( X ,. v ) v" i dy CBX~ ~A"'--'--[

,~,>11, -~ >ll, /~t>(t, A > 0

(7)

I

(I) where I,'( X, .x) is the cunluhttive nlass fraction less than partide instntment size X, and p. and A are dim~'nsitmless con. stunts, The value ol'a indi~:atestile spread of the X sizes, with lower values of A gi~ing a v¢ider spreud, If tile tested sieve silo distribution is denoted by t)~,v) and the cumulative instrument size distribution is QI X), the relation between (~(X) and/'iX) ~conles

Q(X) = i F(X, x ) / , ( x ) @ % :|)

Two functions were investigated for F( X, x). The first was tile function of Eq. ( I ), which has the property that I.'( X, .v) is 0,5 when X ~-/~, i,e,

P = .¥,,,/.~

(8)

where X~, is tile particle size detenuined by instrument at which F(X, x) =0.5, i.e. the 5()~ passing size. The value of /,t can be considered to be an effective shape factor. The second was the log-normal function

'I

hll

(2) FI X, x) = ~

,~'I/£~

)/re

exp -t

(-":) T

du,

tr > (l

(9)

H, Cho et al. I Pou'th,r Technology 95 ( 199,~'1109 . I 17

Table I Sedigraph analysis of 270 x400 mesh 153-38 bum nominal sieve sizes) quartz Sedigraph diameler (bum)

Mass fraction in size interval: Expt.

Log-h)gistic

Log-nt)nnal

> 80 80x70 70 x 68 68 x 66

0.003 0.019 0.007 0.009

0.011 0.017 0.006 0.008

0.003 0.011 0.006 0.008

66 X 64

0.012

0.010

0.010

64X62 62 x 6 0 60 X 58 58 x 56 56 × 54 54 x 52 52 x 511 50 x 48 48 X 46 46 x 44 44 x 42 42 x 411 411x 38 38 x 36 36 x 34 34 x 32 32 x 30 311x 28 28 x 20 2(') x 24 24 x 22 22 x 211 < 20

0.015 0,019 0,024 0,030 0,037 0.045 0.1153

0.013 0.OI7 0.021 0.027 0.034 11.042 0.1152

0.014 0.018 0.023 0.030 0.037 0.(140 11.055

(1.062

0.062

11.(104

0.071 0.1178 I).1183 0.086 0.083 0,076 11.11o5 0.051 I).1135 11.1121 0,11111 0.0113 11.(I 0.001

11.072 0.080 0,086 0,(186 0.U81 0.U72 0.0o0 0.1146 0.034 0,024 0.016 0.1110 11.006 (1.003 0.003

0.072 (I.080 11.1184 (1.085 0.082 0.075 0.064 (I.1150 11.0311 0.023 0,013 0.0117 0.002 11.001 neg.

11

(I0)

"rlds equality gives exact agreelnent at I,'(X, x ) = 0.25, 0.5 and 0.75, i.e. the three vahles ol' Xlx at these three points are the same for the two functions. The value of cr is given by = InlX84/X.~n)

( 13 )

i

sallle value of # and Ibr

cr = I n ( X . J X , , )

t 12

where q, is the fraction in the size interval indexed by i, and the sum is over all intervals. If the top size is indexed by I. the second by 2, etc., then q, = Q( XA - Q( X,,, ). However. this absolute least squares objective function i~ strictly valid only for a population where "errors" defined by q,(calc) - q~(expt) are randomly distributed about a mean of zero. In practice, tl:c error in a particular size interval can have several components, including model error, random experimental error and systematic enor. Model error arises because the use of the two-parameter equations with Atand A or At and tr is arbitrary and the chosen function may not lit the data. Random experimental error investigated by replicate analyses of the same material gives a standard deviation for each size interval i, but these standard deviations will generally be different for each interval. To reduce the errors in each size interval to a general population. Eq. (12) is rewritten as minimize SSQ = )2 14/,I q~(calc) - qi( expt ) I -'

which is another two-parameter function with X~, =/a,r. The log-logistic and the h)g-nornlal distributions give ahnost identical results over the region F( X, .r) = 1).25=0.75 Ibr the cr~ 1.6171A

minimize SSQ = y" [ q,(calc) - q,( expt ) i 2

I 1I

( II)

Table 1 shows a typical data set Ibr a carefully wetscreened ~ size fraction. It is clear that there is negligible material below 20 Ixm Sedigraph size. It is also cleat" that it is difticult to decide which is the best litting I'unction without an analysis of goodness of lit, as Ibllows.

3. Discussion of goodness of lit A search routine can be used to determine the "best-lit' values of/z and ,t or #,'rod tr for a given set of data. Packaged litting programs usually operate with an objective rune(loll delined by

where the W~ values are appropriate weighting factors. For example, Austin et al. 17 ] lbund weighting factors of W, = l / q , for reproducibility errors in size analysis by sieving. with a typical variance tor the population of 4 × 10 -4. Weighting factors of I give absolute least squares and those of I/q, " give relative least squares, so W, =.- I/q, is in between. Table 2 shows typical replicate analyses on a sample of ground quartz screened to a top size of 400 inesh. ]'he tbllowing conclusions were drawn. Setting a large maxinlun~ s i z e in I h e i n s l r u n l e n l g a v e i ) | s l r u m e D I reports which contained 1nosy than 10(F~ of material. This is presulnahly an m'lifact of the calcuhlliOn performed by the machine. In a distilled ware1" settling medium a choice of lop size _< I I0 Ixtn gave no discernible trend with the choice of lop size, e.g. at 38 I~m the percentage reported which was less than the top size varied froln 9{).6 to 91.4, and at 2.3 Ixm it varied fmln 15.3 to 15.7. The top size (Sedigraph) interval conlaining significant material, the 50 x 75 ixm size interval, showed a relatively large variance of I x 1 0 4 while the size intervals down to 3.3 Ixm gave variances of about 5 x 10- ". The two smallest intervals, 2.3 x 3.3 Ixm and 0 x 2.3 p.m. also gave relatively higher variances of about 5 x l0 s. as would be expected for such slowly settling material. If the upper range of sieve size which can bc tested were increased it might be possible to use the overlapping technique to measure a mean shape l actor. To this end an investigation was made of the sedimentation method suggested by Smith and Stanley 181 which uses water+dispersantglycerol solutions to extend the upper size ( with appropriate densities and viscosities). However, the use of glycerolwater media showed a clear trend of change in the size distribution ( see Table 2), with the more concentrated glycerol solutions indicating less line material. Consequently, further

H, (VIo el al, / Powth,rTecltm)lo~y95 f 19981 109-117

I 12

Table 2 Sedi:graph size dislributions o1' the same .,atmple o1" less than 400 mesh (38 ~n) nominal sieve size) quartz measured in various distilled water-glycerol concentrations ( D! ---di.,ailled water + disper.sant, GL = glycerol S.D. = starting diameter set in the Sedigraph ) Sit,: AI,tm)

1491) 1{15.0 74.9

51),0 37,t) 2(,J) I 8,5

13.0 9.2 6.5 5.0

4.6 3,3 2,3

Mass percentage less than size: 100~7, DI

100'~ DI

100q D[

IO0% DI

100%D[

( 150 txm

( I IO txnn

( 90 f.tm

( 80 p.m

( 70 I.tm

S.D.)

S.D.)

S,D.)

S.D.)

S,D.)

102.O 101,5 I(X).6 98.t) ~~,t)

90.9 82.7 71.5 59.2 45.5 35.1 32.0 21 .~ 13,7

99.2 99../. 98.5 96,5 91.4 83,2 71.9 59. I 45.5 35,1 31,9 22. I 15.3

100.4 98,7 96,2 90.8 82.8 72.0 59.0 45.3 35.3 322 22.3 15.9

99. I 97.5 96,0 90,6 82.t~ 72.2 59.2 45.9 35.4

32.2 21.8 15.5

98.2 96.0 90.9 82.6 71.9 59.t) 45,5 35,2 31,9 22.1) 15.8

experiments were performed only with a distilled water medium, Since the replicate analyses showed no signilicant trend with particle size (except as noted) Ihe weighting factors of Eq. (13) were taken as unity, that is, absolute least squares wa~ used. The end objective in the work here is to produce a sub=sieve size analysis fl'om tile Sedigraph analysis of le.~s than 40(1 mesh material which would dttplicat¢ the sieve size an:tlysis if ,~tandard sieves were available below 41111111esh ( 38 pJn ).

4, Determination ot'purumeters Analysis of the data of Table I using Eq. (7) and tile suggested lilting functions, with searches fiw appropriate values of#, ,L and rr. gave the lbllowing conclusions, The values of # and A in the simple log-logistic functkm were insensitive to the value o f m chosen in Eq, ( 7 ), for reasonable ',alues of m, This is demonstrated in Table 3, This means that Eq. 12 ) can be replaced by Table 3 F,fti.,ctof ~lop., m tm the ,,haW laclor #. and H~l~ad tactor A in 1he IogJogi,,tic IUIK'lion required to tit the Scdi~l~ll~lt alltll}sP, Oil tl ',,uuple ot 2711~ 41H1 tl~sh qtl{U'lt,tt',ittg l!q.I? m

kt

A

0.5 11.8 1.0

t).963 11.961 0.959

8.(~,20 8,~I I 0 81~, I I

1,2 2,0

0.957 t1,949

8,(d~ 8,579

100~ DI 50 I.tm S.D.)

lit1,4

96.8 91.3 82.8 71.8 58,9 45.4 35.2 31.9 21.8 15,9

90';~ DI -IO% GL (70 l~m S,D. }

97.4 95.5 90.6 82.3 71.5 58.3 43,8 33.4 30,4 20.8 15,0

80% DI -20% GL (80gin S.D, )

70c.,t DI -30% GL (90~m S,D. )

98,5 97.5 95,5 89.9 81.3 711. I 56.8 42.5 31.9

98.9 97,2 94.5 89.3 80.7 b9.3 55.9 41.3

29.0

28.2

20.2 14.3

19.7 13,9

3t.()

q(x)--~'(x, .~,)/,,

6097 DI --40% GL (ll0~m S.D. )

509~ DI -50% GL (150 tim S.D, )

99.9 99.1 97.4 94,3 88,8 79.8 67.9 54.5 4(1,3 3o.4 27.6 18.5 13.4

I01.3 100.3 9,',I.9 96.6 93.5 87.4 782 6(~, I 52.5 38.4 28.5 25.8 17.9 13.2

(]4)

]

with

/"( X, .~',) =

I

I + (XII.t(:,)

"~

(15)

the geonletl'iC m e a n o f tile ~ s i e v e s i z e i n t e r v a l indexed by./. # and A are effective overall values based on a size interval, and fl, i.., the mass fraction itl ,.ieve size interval j. Similarly, tile log-normal fanclit)ll of Eq. ( 9 ) can be used in Eq. (14) in the form w h e r e ,L is

hLI .~ , ~ 0 , I ' f P

116)

thus eli,alluring the double integration required in tile solulion of Eq. q2) or Eq. (7). For a ~ sieve san)pie, tile experimental values are directly tile new delinition of F( X, .f~), and # and A or bt and cr are found as best-lit values using lhe appropriate .~:,value. The data of Table I from 70 to 20 I,tm were lilted with an ab,,,olule least ~quai'es regression o0 q values using either Eq. ! 15 ) or Eq. (16), and the resulting parameters were/.t = 0.96, A=7.40, S S Q = 2 . 6 6 × IO a and/z=0.96, ¢r=0.22, SSQ ==0.56 × I 0 "~, indicating thai the log-normal distribulion gave the better lit. Table I shows the predicted data for the two fits, As expected, the value of the spread factor, A = 7 . 4 0 , is less than that of A = 8,6 shown in Table 3 where full integrations were performed.

H, Cho el al, /Powder

T('chmdogy 95 11998,l 1 0 9 - I 17

Table 4 Comparison of Sedigral)h size distributions predicted frorn a sieve size distribution of P(x) = x l 3 8 , 0 <_a ~ 38 Fro, using A = 7,40,/x= (I.96 and

100

o'=0.22, #,=0.96

Instrumem size

ran"

(pm)

80

I

78 76 74 72 70 68 66 04 62

I I I 0.999 0.999 0.999 0.999 0.999 0.998 0.998 0.997 (13)90 0.995 0.994 (I.992 0.989 0.985 0.980

60 58 56 54 52 50 48 40 44 42 41) 38 36 34 32

0.972 (1.902 0.947

0.927 0.899 (l.8(~5

3O

0,824

28 2(~ 24 22 23

0.777 0,728 0.O77 tl,h22 l),50h

I~

0511

I() 14 12 10 8 () 4

0,455 0,397 (I.341 0 284 0.227 0.171 0.114

Log-normal

S~ve Size Dist. ,

//

Seai|lraph Size Oist.

//

Log-Logistic Log-Normal

_m tL I'-Z W O er Ud D. I.g

Cumulative mass fraction less than size: Log-logistic

113

10

.

.

.... ....

//

.

.

.

10

100

SIZE, # m

0.999 0,999 0,998 0,998 0.996 0.994 0.990 0.985 0.977 0.966. 0.950 0.929 0.90 I 0.867 0,82(~ 0,779 0.728 0.075 1),619 O563 0,5(17 O.45 I 0.394 0.33N (l.282 I).225 0.109 o.I I'~

5. Comparison e~' use of log-logistic and log.normal functions

Fig. I. Computed Scdigraph size analysis from an assumed sieve size distribution of P(a ) = ( x / 3 8 ) " with m = I.

As expected, the resulting Sedigraph size distributions are also power functions below about 15 I~m, with a slope of 1. The resulting curves on log-log scales ( see Fig. I ) lie higher than the assumed sieve size distribution, that is, they appear shifted upwards. They can be made to fit in the linear region by using a dividing factor r to change instrument size to sieve size. The factor was 0.926 for both the log-logistic and lognormal functions. Eqs. (15) and (16) state that the 50% passing size of the instrument size distribution resulting from a yr2 screen interval is related to the geometric mean size of the screen interval by X.~.= g.f. so that X~o//z=.fr Since /.t = 0.96, it is seen that dividing Sedigraph size by X~,/i) to obtain sieve size Call be used to correct the instrument size t,) sieve size only its a lirst approximation. In addition, Austin 19 ] has shown analytically that the correct method is to slide tile curve in tile vertical dil'cClion, that is, Q(.~.) = KP(x) at a given size x which lies on the straight-line region. A simple geonletric constnlction gives K = ( I/r)'". Thus tile values of Q(x) in the linear region should be divided by I,(18 to give P(x). The value of K is a shift factor which depends on particle shape, but it involves no explicit definition of shape.

6. Predicting sieve size from sedigraph size Eq. (14) can be discretized with respect to size and written

The determined parameters were used to predict the instrument size distribution which would result IYom a sieve size distribution of

as

./

[x.~"

/'(x)=--~,~.], ()_
where j indexes the sieve size intervals and i the Sedigraph size intervals For the log-logistic function I

d°=l+(X/iJ~j)-^

i I + (Xi+ i / / ~ P

(18)

i.e. d e is the fraction of material in sieve size interval,/' which appears in Sedigraph size interval i. Similarly. the d o values

H, C/.,' e: tfl. / P o l r , h ' r 7i'chllolo,~y 135 / ll#jS) 109-117

i 14

can be calculated using the log-normal expression if desired. It should be noted that Eq. (15) applies for a geometric sieve size interval, whereas the smallest size interval (e.g. 0-2.3 Ixm) is not geometric. Thus, the number of sieve size intervals n must be chosen large enough to give negligible error, that is, the top size.~;, of the smallest sieve interval must be small enough to give p,, = small, it is not necessary lhat the Sedigraph size intervals be in geometric n:tio or of the same number. In matrix form. Eq. (17) becomes

q--de

(~9)

Thus the conversion of a vector of a screen size distribution to the equivalent vector of a Sedigraph size distribution is straighttbrward. However, the conversion ofa Sedigraph size distribution to the equivalent sieve size distribution is not so simple because although mathematically the set of p, values can be found by multiplying q by the inverse of d (i.e. p---d ~q), this calculation is usually unstable due to experimental error in q values and produces p values oscillating widely between positive and negative numbers. This problem can be avoided by a least squares error minimization method. Two computational procedures were used. The simpler method is to assume th~lt the sub-sieve size distribution is of the fornl

Table 5 Experimental and predicted Scdigraph size distributions lilr ground quarlz I less than 38 ~ln sieve fructiun of mill discharge) and the corre,,,ponding sieve size distribution ( p, = 0,90, A = 8.08, (D= 0.79, mt = O.89, s.i, = 2.61) I (?ulmllative illi,lSS pereenklge less Ihan

size:

Sedigraph

Sieve

Size I tim )

Q<',l~,

(../<~l,

Size ( i.tm )

P,,,i,

74 53 37 26 18.5 13.1) ~J.2 6.5 4.(~ 3.3 2.3

IO0 9t).8 95.7 75.3 51).9 34.3 24.7 18,2 13,t) 10.3 7.3 5,3

IO0 t)tJ.8 96.0 74.2 51 .tj "~6.3 2h,0 18.8 13.7 I0. I 7.3 5.3

38.0 27.0 19.0 13.4 9.5 6.7 -1.8 3,4 2,4 1.6 1.2

100 66.6 46. I 32.7 23.6 17. I 12.5 %2 h,7 4.~) 3.6

1.6

100

|

tL

I

I~ l)<,l'~,

~h>0, m~ bit, m,>>l)

~o

( 201

where k is the ,~cl'een size used ft~r ~¢plinltillg Ihe sliniple for ~i¢,eiinal!csis ( usiiillly ,IN i&nl ). This fornl ~vas¢lltlS¢ll he¢Inlse ledilrllph diilii on slilliples lilken around n~illin 7 ch'cuils I),pi¢lllly show ii ,~iinple plower filn¢lion lbrnl Itlr lille sizes, lind I~q, ( -~01 I~duc'es to Ihis fornl ~lt small ,r when lit: ;~ m i . ~ince k is known, Eq. (201 is a Ihrec-pilralilelCr lilting lilnclion, A plil~kllged program can be used for the inliirix nluliiplicaiion of Eq, ! 19i, using Eq, 120) to calcuhite P(xi and hence i,, vlilues und Eq. ( 18 ) I or the Iog-norinal equivalent i to calculate iI~j vlllues fronl the known lit lind ~ values, DcIinhig an unweighied objective function by Eq, I 13 i, the values of di, siiu and m: for least squares nlhiilnization can be found by ;i search procl~du~, Table 5 shows the Sedigraph size distribution of ii sanirlle

of the less than 4IX) mesh sieve fraction of a mill discharge t~im the diT grinding of quartz, The values of A and/.t flu" this quartz were 8,1 and O,t)0, respectively. The Sedigraph region ti'oln 1,5 to I 3 t,tlll gave fill estiniatc of m~ = 1),89 and exirllpohition gave esiimlites o1' i/i = I),7bl and itl, = 2,5, U~iii l the log-logistic runt:lion, values ofd~ = 0,79 i,nld Ipl, :: 2.(i gave a reasonable lit with m held constant at 0,89 to give the correct shape at the liner sizes. The liner regions of the COlilputed S~igrtiph and sieve size Schuhma~m curves had slopes of 0,89 as expected and the shift factor h'=Q(.rilPIxi was 1,12 (see Fig, 2),

~

'l

I StavllIlze ilill. • ComputliltIIIvl Illl illll, for lhli ,$7 ~m frliotion . . . . . . . . . .

i

10 SIZE,/ma

n ,, ,

,

I I *l,

100

Fig. 2. Setligrlillh ~ize d i M r i b t l l i i l l l o l ii les,~ Ihlill 4lilt illu'sh q U a l ' l l sal|llllL' alld lilt' t'qui% alclll sil.'~,e silt' tli5ll'ibulion,

The .,,¢ct)nd n1¢thod usod for the eonverskin was a hackcalculation of the set ol'pi values directly by a seardl for the oplilnuni set ol'p values inhihllizing the least squares objective fuuciion I Eq. ( 13 ) } subje¢l IO ¢onslraints ~ , / l / = I and p, >_O. The Wolfe reduced gradienl ineihtld I I01 was chosen ti'oni Ibe various techniques for ,solving a nonlinear progranluliu 7 problem with Ihlear conslrai0ls. This technique was protranllncd in C + + and applied to the size distributions oblailled front another test with Ihe sanle quartz on tile closed. circuit industrial mill, using Sedigraph dlita for the less than 38 I~ln sieve size Iraclions of the mill discharge, the classifier prodnct and the classilior recycle. The r~sulis are shown in Fig. 3. This technique worked well, giving good Iits of predicted Sedigruph size analyses with the experhnental values, as shown in Table 6. As niighl he expected, the values of the

it. ('he el a L I P,lu'th,r "l'e~'hn,lo,i4v 95 t Iql.C';l 109-I 17

100

I 15

Table fl (7olnrJl.irJ,,Ull of CtllnpUIcd 'gcdigraph ~ize ~ilh expel'inlelilal '.alu¢~ i id,Inpuled ScdJgr:lph ~ ahics deri~,cd Irlull haek-¢alculalion of the ettui~ alelii sic~c silt: %'eL'ttlr ) I a ) Mill discharge (2unnilalivc illas.,, percentage less than ~.ize:

,.=,

'

/

Sedigraph

,,/Recycle

Size ( i.tnl I

{-~,'.e,

O<.,l<

Size ( t*m )

P,

74 53 37 26 18.5 13.{) ~J.2 (~.5 4,6 3,3 2.3 l.fi

I11(1 99.8 95,7 75.3 511.9 34.3 24.7 18.2 13.9 1(I.3 7.3 5.3

99.6 99.7 95,7 75.3 51).9 34.3 24.7 18.2 13.9 10.3 7.3 5.3

38 26 18.5 13 9.3 6.5 4.6 3.3 2.3 1.6 1.2

t 011 6fi.7 42.1 30.1 22.(t 16.3 12.9 ~.9 6.5 4.8 4.1

Sieve size dist.

CJ

* •

.

.

.

.

.

.

.

I

Sedigraph size dist. Back-oalculatea sub-sieve size dist.

,,

n

n

|

|

. . . .

10

100

SIZE,/~m Fig. 3. i-xperi menhil sieve size tlistl'Jl~tllions ( do,vii Io 314 i.t nl i and equjvillOll sie~e si/e dislribulion dei'ited li'~uli SedJgrliph analyses I,I~S hi - 3 laUlnI: nlJll dialneler 2,~ ill, nlill length 8.4 IlL litq'nl;.il closed-cJrctiil dl'}' grinding nf qua 'lz with telantic niedia,

>~

( b I Recycle froln air nepal'ahlr

1.0

('tllnlllalk'e nla,ss pl21CClil;ige It's,,, than silt,:

o.8

Sedigrapil

0,8

Size I fun I

Q,.,,,

Q, ,.~<

Size ( bun )

P. ,,,,

74 53 37 2f~ 18.5

I(lll ~19.2 t12,2 58,N 27.1

13,11

I .:t.~

92 1~.5 .1.~ 3.21 2.?, I.(1

*),5 7.3 5.8 4.7 .%.7 3.7

I00 919,3 92,2 5S,g 27.1 I ?,3,1 IJ.5 7.3 5.:,I 4,7 3.7 3.7

38 2t~ IS,5 13 ~.13 t~5 4.t, 3.3 2.3 I,O 1,2

Illll 425 1~2 II ,,~ s.2 ~'~.'~ 53 4,~ 3.4 3,3 3,11

0.7

i

0.6 0.5

t

0,4

I

0,3 0,2

u.

Sieve

0,1 0.0

.

.

.

.

.

.

.

I

10

I

I

I

I

I

I

Sie~,e

I

100

SIZE,~m Fig. 4. ('lassilier partition curve ftH" lilt+' air separator, predicted from sieve atli.llx,,is and tile tl;iti.l ul' Table 6- selectivity values f(q' ~ size inlcr~ails. phltted at tile tlllper ~.i/¢ of tile inter% als.

equivalent sieve size distribution show some irregularities, but it is easy to pass a smoothed curve through the data. The classilier partition (air separator) curve predicted from the smoothed data is shown in Fig. 4. It ,shows no urlusual feature. indicating a correct conversion of the Sedigruph sub-sieve analysis to equivalent sieve analysis over the complete si:,e range. The high selectivity values Ibr the very line material are often observed in air separators, typically due to entrainmen( of line material in the recirculated classifying air I I I I and agglomen'ation of fine particles due to van der Waals threes.

( c ) |:Jlle I'il'tidu¢l froln air ~,epal'ator Cti]llulalJ~.e nl;.iss i~en'centage le.,,~ than st/e: Sedigralph

Sieve

Si/e I I,LIn }

/,-)¢w,

Q, .,,,

SJIC I t.tln )

I', .,,.

74 53 37 26 18.5 I 3.11 9.2 (~.5 4,¢~ 3.3 2.3 1.6

It)O q¢,L2 9f~.7 gS.0 73,2 5(~.7 41.8 31,1) 22,5 16.4 I 1.8 8.3

I()l) t)tL¢, qfi.(~ 88,O 73,2 56.7 .-1-I .I,I 31.(1 22.5 16.4 11.8 8.3

38 2f~ 18.5 13 9.3 (~,5 4.6 3,3 2.3 1.6 1.2

H)¢) Xf~.3 ¢,7.7 52.4 3f~.5 2h,¢~ 19.5 14,5 IOA, 7.5 43)

IIfi

H, Choet al. / Powder Techmdogy 95 ~1998) 109-117

7. Discussion a n d conclusions

The way in which material in a ~ screen interval distributes into Sedigraph size can be described with sufficient accuracy by either the two-parameter log-normal distribution {/z, o') or the two-parameter log-logistic distribution (/.t, A). Values for ground material were/z = 0.96, o" = 0.22, A= 7.40 for one quartz and/z =0.90, tr =0.20, A = 8.08 for another. Assuming this distribution to be dimensionally normalized it is possible to back-calculate the equivalent sieve size dislribution required to give the Sedigraph data measured on a less than 400 mesh .sample, This can be converted and added to give the complete distribution of a sample down to ,,- 3 p.m (the Sedigraph data start to become less reproducible below 3 Ixm for quartz), it is not implied that sieve size is a more fundamental property than Sedigraph size but, historically, sieve size has been widely used tbr process decisions and has the advantage that powder within a desired (sieve) size range can be readily separated for testing {sedimentonteters, l a i r diffractometers, etc., measure but do not ,~parate). The technique given here is only applicable it" the distributton of particle shapes does not change with particle size. In fact, the values of the characteristic parameters/.t and A ( or/.¢ and tr) are measures of the shape distribution and can be used to show if shape is varying with size within the sieve range I 121. It is anticipated that other meth(~ls of sub-sieve size analysis can be treated in Ihe same way. Obviously, if Serigraph dala are ~onvertible to the equivalent screen size distribution, and the screen size distribution is convertible to the equivalent laser dilTraclometer size distribution, then the end result is a melhod of converting Sedigr,tph .size distributions to laser size distributions and vice vers:l, If a functional torm is chosen Ior the equivalent sub-sieve sen:ca size distribution, the best lit of the contl~uted Sedigraph size distribution It) Ihe actual Sedigraph size distribution gives the parameters of the function and a smooth curve of the equivalent sieve size distribution is obtained. However, the goodness of fit depends on how well the chosen functional tbrm represents the screen size data, in addition to random repnxlucibility errors, If the individual values of the equivalent sieve fractions are back-calculated with a constrained search, any .~hape of the size dislribution can be handled, but the points will not generally lie exactly on a smooth curve. The back-calculution pn~gram is available by request From the authors.

8, List of symbols

lq X, x)

cumulative mass fraction less than particle size X determined by instrument, resulting from particles of sieve size x integer index of instrument size interval integer index of sieve size interval

K m n pj q~ Q(X) r

R W, x -h, .r,

.fj X

Xs,, y

top size of size distribution P ( x ) = ( x / k ) " (p~m ormm) shift l'actor defined by Q { x ) / P ( x ) exponent in P ( x ) = ( x / k ) " number of sieve size intervals mass fraction in sieve size mterval indexed by j mass fraction in instrument size interval indexed by i cumulative mass fraction less than instrument size X mean shape factor defined by x = X / r , x being sieve size and X instrument size which give Q(x) = P ( x ) ratio of upper to lower sieve interval sizes, or 4~_9 weighting factor for material in instrument size interval indexed by i particle size as determined by sieving ( Ixm or ram) value o f x at bottom size of sieve interval ( p~m or ram) value o f x at top size of sieve interval ( p.m or mm ) geometric mean size of sieve interval indexed b y j ( Ixm or nun ) panicle size as detennined by instrument such as SedignLph or laser diffractometer ( ~ln or llnn ) value of X at which F( X, .v ) = 0.5 ( itl.mor toni ) d¢lined by y ~ x/,t,,

th'e,'k Ictt('rs A ,r bt

spread lacier hl Iogologistic ftu1¢tion dispersion coefficient in Iog-nonnal function constaut in log-logistic or log-normal functions

d~

defined by/~ = X~,,I t coeflicienl in lilting equation. Eq. (20)

References I I I L,G, Aus0,, R.R, Klimpel and P.T. l,uckie, l)n~.'e~,~Engineering of Site Reduction: Ball Milling, St~.'i,:t.,,oJ' Mining Engineer.,;, AIMIL New York, lq,~4, Ch. I?.. 121 K. L,:~d~onski. Powder Tcchnol., "4 ~ 1979) I15-I"4. 131 R, Clift, in PJ. Ll~)yd ( ,:d. ), Particle Si,'e Analyst,,,.Wiley, C'hichester, UK, I'988, pp. 3-17, I'I I PJ. Fcrreinl, M,G. Rasteiro and M.M. Figueiredo, Part. Sci. Technol.. II (1993) 199-206. 15 ] T. Allen, Particle Size Mcasureuzcnt,(.?llapmanalltl Hull, l,ondon, UK. 4th edn,, 1990, p. 169. l(q L,G. Austin and [, Shah, Powder T~chnol.. 35 (1983) 271-278. 171 L.G. Austin, R.R. Klimpel and P.T, Luckie, Process Engineering of Site Reduction: Ball Milling, Society of Mining Engineers, AIME,

New York, 1984.pp. 222-223.

H. Cho et al. / Powder Techm~togy 95 ~19r)8) 109-117

181 D.W. Smith and H.D. Stanley, Particle Size Analysis of High Density, L~rge Diameter Powders by the Sedimentation Method, Micromeritic~ htstrument Corporation, Norcross, GA, 1994. 191 L.G. Austin, Conversion factors to convert sub-sieve particle size distributions measured by one ,nethod to those tx~easured by another, submitted for publication.

117

{ 10l M.S. Bazaraa, H+D. Shersali and C+M.Slleuy+Nm~iinear Pr, gramming. Theory and Algorithms, Wiley, Chichester, UK. 1993, p. 458. [11] L.G. Austin and P.T. Luckie, Zem.-Kalk-(iyps, 29 119761 452457. 112] L,G, Austin, O. Trass, T.F+ Dum.n and V.R. Koka, Part. Part. Syst. Character.. 5 < 1988 ) 13-15.