SIZE DISTRIBUTIOiV OF PARTICULATE MATERIAL6
241
steady state 1 4 will approach a lower limiting value I,* > 0. If 1: < < 1: I:, equation 12 may be used, or equation 10 if R, is a fictitious resistance defined as #/IT. This value mill in general be much greater than the ohmic resistance of the capillary. In general, 1: will be larger if n large surfacc area is involved and if rip is greater than the decomposition voltage of the electrochemical reaction resulting from 1;. However, in many cases where the rate of depolarization is slow or the resistance of the capillary wall is high, a satisfactory measurement of flow potential should be possible, using a conducting capillary.
+
REFEREXCES (1) EVERSOLE,W. G . , A N D DPARDORFF, D. L.: Proc. Iowa .\cad. Sci. d,177 (1536); 44, 109 (1937). (2) KOEUIG,F. 0.: “Dic Elektrokinetischcii Erscheinungen”, in Wen-Harms Handbuch der Ezpen‘mentalphysik, Vol. XII, Part 2 (1933). (3) KRCYT,11. R., A N D OOSTPRMAN, J.: Proc. Acad. Sci. Srnsterdarn 40,4004 (1937), 41, 370 (1938); Kolloid-Beihefte 48, 377 (1938). ( 4 ) S W T H , G. w., . 4 S D REYER5ON, 12. H.: J . Phys. Cheln. 38, 133 (1934).
ST.lTIST1CAI. .%NALYSIS OF SIZE DISTRIBUTION OF YARTICULATE IVIATERIALS, WITH SPECIAL REFERENCE TO BIJIODAL AKD FREQUENCY DISTRIBI‘TIONS. C0RRE;IA’I’IOS O F CJUAIRTII,EWITH STATISTICAL VALUES’ PAUL S. ROLLER2 Bit, enu o/ ill?nes, Eastern Ezpertment Statzon, College Park, Maryland
Recezied March $8) 1.9@
Size data foi a particulate material are properly evaluated solely by a statistical m e t h t d a i this alone take5 into account the contribution of all sizes in the ensem1)lr. Statistical analysis of the data yields certain conbtants, such as euriiice aica per gram, coefficient of uniformity, coefficient of regression, and iiiimber of particles per gram, wliich properly summarize the distributioii and define the material as t u size. Knowledge of these constants is important for the correct physical valuation of a product rtnd of the mode and method of its preparation. The fruitfulness of the result easily justifies the effort involved in calculating the constants from dze data. Published by permission of the Acting Directoi, Bureau of Mines, Department
of the Interior. ?
Physical chemist, Bureau of Mines, Eastern Experiment Station.
242
PAUL 8. ROLLER
The basis of any statistical analysis is a distribution function, which must satisfy theoretical boundary conditions and of course the experimental data as well. A proper distribution function was previously described (16) and forms the basis of the present work. Originally only relatively fine materials, the data for which wcre cxpressed as cumulative weight per cent against size, were considered. We now consider in addition coarse materials for which the data are presented in the same way but which exhibit a bimodal distribution (the latter apparently escaped notice previously (E)in certain cases of incomplete or insufficiently accurate data) and also fine materials for which the data are presented as number frequency against size. The characteristics of these new elasses of materials have required a more extended and more exact procedure than before. TRANSFORM.4TIONS OF DISTRIBUTION FUNCTION
The distribution function3 is (16) =
az112e-b1=
(1)
in which y is the cumulative weight per cent of all sizes less than z, and a and b are the parameters of thc distribution. The particles are assumed to be smooth cubes, regardless of their actual shape; the question of a correcting shape factor is discussed later. To transform equation 1 into a more general form and a more useful one for certain purposes, we shall replace z by its fraction, 9,of the statistical greatest size xg (or size obtained on substituting 100 for y in equation 1). We have 3 The origin of this function was purely inductive, paying due regard t o boundary conditions (16), and quite independent. A comparison rrith Pearson’s theoretically deduced curves ( E a ) may be of interest. The frequency form of the present funct,ion, given by equation 8 for number frequency and by equation 2 in reference 16 fur weight frequency, is limited in range in one direction (at zero size) and skew, and is therefore of Type I11 in Pearson’s classification. Pearson’s equation for the Type 111 curve can be put in thc form (cf. Elderton @a))
ahcrt fox romparison
(:- :)or
has been writtcn for Pearson’s y.
This latter cqua-
tion, in whish p m d y apparently arc positive, somewhat resembles ours, which is more cc nipliczt,ed; specifically the difference between them lies in the algebraic part, in whicir R ? h&vetwo terms instead of one, each of which, moreover, is determined in its poacw. ail4 the second term in its coefficient given a and b . The surprising feature, one n ighL say, is not the difference b u t the resemblance, as Pearson (15, page 381) and hiderton (8a, page 128 e l al.) admit the possibility of other premises and various other corresponding theoretical curves.
243
SIZE DISTRIBUTION O F PARTICULATE MATERIALS
9 mag for convenieiice be called the fractioiial size. Tact us now defiiw a new parameter, p , as follows:
(3)
P = (&)b
In the case of a bimodal distribution, the parameters appropriate to equation 3 are a'. b'. TABLE 1 Coitiputataon of p/f ( p ) ngaznst p Plf(?])
P
____ 0.0 0.002506 0.01265 0,02562 0.03887 0.05242 0.06347 0.08037 0.1094 0.1395 0.2026 0.3041 0.3760 0.4897 0.6016 0.6932 0.8053 0.9123 1.004 1.145 1.341 1.648 2.192
11
00 0.002494 0 01233 0 02440 0 03616 11 0 04767 0.05667 1 0 06985 0 09114 0.1115 0.1492 1 0.2028 0.2350 0.2800 0.3182 0.3466 0.3781 0.4058 1 0.4271 1 0.4582 0.4908 0.5493 I I 0.6270 I
'
'
I 1
1
I ~
~
1 j
~
' 1
,
I
'
ll
1
2 773 3 384 4 024 5 376 6 811 8 318 9 886 11 511 13.19 14 91 16 67 18 45 20 31 22 18 24 08 26 01 27 97 29.96 40 24 51 02 62.22 73 80
I
1
I ~
' ' ~
'
' ,
I
I ~
1
P/f(P)
0 6931 0 7520 0 8047 0 8961 0 9730 1 040 1 099 1 151 1 200 1.242 1.280 1.319 1.352 1,388 1.41s 1.447 1.472 1.498 1.610 1.700 1.779 1.845
I I
1
I
-
97.80 148.6 202.5 230.3 345.9 436.6 593.3 855.6 1198 1554 2293 3454 4254 , 5902 7601 1 1 10670 14280 I 17990 26100 46050 56360 99040 1
1 ~
1.956 2.123 2 250 2 303 2.470 2 568 2 698 2 852 2 996 3.108 3.276 3 454 3.546 3.690 3 800 3.950 4.080 4 183 4.350 4.605 4 695 4 953
____-
Instead of the approximate evaluation of
2,
that was previouslv given
(16), we now have ail exact equation as follows, in whichf(p) is a fuiiction
of p subsequently to be determined,
100 xs =
f(P)
From equations 3 and 4 the following identity results:
4
'
244
PAUL 8. ROLLER
Eliminating x, between equations 2 and 4,substituting for the resultant value of x into equation 1, and making use of equation 3, a reduced general form of the distribution function is obtained, in which fractional size is the variable, and only one parameter p is present, as follows:
Substituting 9 = 1 and y = 100 into equation 6, the following equation for f(p) is obtained:
Unfortunately this equation cannot be solved for f ( p ) explicitly; a numerical solution for p / f ( p ) as a function of p has, however, been made and is given in table 1. For ready determination of the desired quantities p / f ( p ) and f(p), a graph of the values of table 1 is useful; owing to the wide range, several sheets are necessary. An expression was previously derived (16)) and is required, for number frequency as a function of size. If now, more correctly, we consider number of particles in a weight interval, instead of in a weight-per cent interval as before, the previous expression becomes
In the above equation, wois the weight of sample, z the size inmicrons, dz/dz the number frequency, and p the density of the material in grams per cubic centimeter. DETERMINATION OF PARAMETERS
Ascertainment of the parameters is a prerequisite to calculation of the statistical constants. Three cases are to be distinguished: cumulative weight-per cent unimodal distribution, cumulative weight-per cent bimodal distribution, and number-frequency (unimodal) distribution. The method of determining the parameters is graphical, although least squares' may be 4 The method is given in standard texts, and has the advantage of furnishing a definitive result. We have succinctly
c
1 " log a = log n
--+--E; 4 2 2.303n y
b l " 1
SIZE DISTRIBUTION OF PARTICULATE MATERIALS
245
where n is the number of valid experimental values, A log v / d x is the algebraic difference between a given value of log y / d / z and the arithmetic mean of all values, and Al/z is the algebraic difference between a given value of l/x and the arithmetic mean of all values. In discussing the precision of the numerical results ( I egardless of the procedure whereby obtained) we shall ,follow Bond (6a), whose assumption implies that size z is free of ci-ror. We have for the fractional error of pirnmeters a and b,
where 6 log y / d x is a residual, or the difference between the experimental and eetimated values of log g/
