INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1948
paper represent a laboratory development only. materials described are commercially available.
Sone of the
ACKNOWLEDGMENT
The authors gratefully acknowledge the assistance of staff members of the Organic Chemicals Department of this company in testing some of tho vulcanizates doscribed in this paper. LITERATURE CITED (1)
Carothers, W. H., and Collins, A. M., U. S. Patent 1,950,431
(2)
Carothers, W. H., Kirby, J. E., andcallins, A . M . , J.Am. Chem.
(1934). \----,
SOC., 5 5 , 7 8 9 - 9 5 (1933). (3) Carothers, W. H., Williams, Ira, Collins, A. M., and Kirby, J. E., Ibid., 5 3 , 4 2 0 3 - 2 5 (1931).
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(4) Collins, A. M., U. S. Patent 1,967,861 (1934). (5) I b i d . , 2,264,173 (1941). (6) Fornian, D. B., and Radcliff, R. R., IND.ENG.CHEX..38, 1048-52 (1946). (7) Mochel, W. E., U. S.Patent 2,426,560 (1947). (8) I b i d . , 2,429,838 (1947). (9) Salisbury, L. F., U. S. Patent 2,416,456 (1947) (10) I b i d . , 2 , 4 2 6 , 7 9 2 (1947) ; other pending patent applications. (11) Sebrell, L. B., and Dinsmore, R. P., I n d i a Rubber W o r l d , 103, 37 (1941). (12) Starkweather, H. W., and Collins, A . &I., U. S. Patent 2,227,517 (1941). (13) Yerzley, F. L.. and Fraser, D. F. IND.ENG. C H E M ,34, 3 3 2 (1942).
RECEIYED March 6, 1948.
Contribution No. 226 from t h e Chemical Department, Experimental Station, E. I. du Pont de Nemours & Company, Wilmington. Del.
Logarithmico-Normal Distribution in Breakage of Solids BENJAMIN EPSTEIN Coal Research Laboratory, Carnegie Institute of Technology, Pittsburgh, P a .
In this paper a statistical model. is constructed for breakF,(x); a second step of the age mechanisms and a breakage process is conceived of as process operating on Fl(x) under investigation in depending on two basic functions: P,(y), the probability will lead to F&) and so on this laboratory is the characof breakage of a piece of size y in the nth step of the breakterization of the strength of with the nth step of the breakage process; and F ( x , y ) , the distribution by weight of coke, or more precisely, the age process operating on particles of size x less than or equal to y arising from the F " - ~ ( z )and leading t o F n ( z ) , factors affecting the resistbreakage of a unit mass of size y. Under certain hyance of coke to size degradathe size distribution after n potheses about P,(y) and F(x,y) it can be proved that the tion. The study of these steps. Basically the equadistribution function F,(x) after n steps in the breakage questions has led in a natural tions describing the effect of process is asymptotically logarithmico-normal, a form of any given step of the breakway t o the consideration of distribution frequently observed. the mechanism of the breakage process express the fact t h a t t h e change in the cumuage of solids in general. It lativeweight finer than a given is the purpose of this paper size, r, is given by the amount of material of size less than or to report some of the results found thus far. I n the course of this work it has become increasingly evident equal t o x which arises from the breakage of material of size that certain types of crushing and grinding operations lead t o greater than or equal to x in the given step under consideration. I n addition t o the concept that a breakage process can be the logarithmico-normal distribution (6)-that is, a distribution which plots as a straight line on probability paper. This has considered as a succession of discrete events, two basic functions which are essentially statistical in nature are introduced. These been recognized empirically in the literature on the grinding and crushing of solids-for example, Austin (1) who gives an extensive functions will determine the progress of a breakage process. literature, and Hatch and Choate (9)and Hatch (8). However, no They are: (A) P"(y), the probability of breakage of a particle of attempt has been made in these articles t o give physical reasons size y in the nth step of the breakage process; and (B) F(z,y), which would make the occurrence of logarithmico-normal size disthe cumulative distribution by weight of particles of size x 5 y tributions plausible. Indeed, the only published attempt t o conarising from the breakage of a unit mass of size y. struct theoretical breakage mechanisms basedon thetheory of probThe introduction of these basic functions, coupled with the ability which will lead t o logarithmico-normal distributions seems underlying assumption that a breakage process can be broken to be due t o the Russian mathematician Kolmogoroff ( I d ) . up into steps, gives a framework within which the changes in the This paper, however, is relatively inaccessible and sketchy and is particle size distribution can be studied as a function of the based on the introduction of a number of complicated functions. number of steps in the process. It is recognized that consideration of the size (in the linear DESCRIPTION O F A THEORETICAL BREAKAGE MECHANISM sense) or dimension of the piece is not strictly justified except when dealing with essentially one-dimensional solids such as Any breakage process may be conceived of as composed of discrete steps and therefore it makes sense to talk of a breakage thin rods. In actual practice the pieces are three-dimensional event which will consist merely of a single step in the degradaand of irregular shape. I n this case the logical procedure would tion process. Viewed in this way, it is clear that the breakage be' to classify pieces according to volume because the principle process can be studied logically after any finite number of steps: of mass conservation then could be applied t o each piece broken. n = 1,2,. , N , . More precisely, if the original cumulative Unfortunately there are practical difficulties involved in measursize distribution by weight-that is, the per cent by weight less ing volumes accurately and quickly, and therefore it is common than or equal t o size x-is F,(z), then the operation of one step practice 60 divide a sample into linear size compartments by of the breakage process will give rise to a new distribution sieving through a series of wire-mesh screens of decreasing square
0
NE of the problems
'
.. .
.. .
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INDUSTRIAL AND ENGINEERING CHEMISTRY
or circular openings for pieces greater than 200 mesh or 74 microns, or to use a sedimentation process with pipet withdrawals for subsieve material which again divides the material into size compartments. The pieces in each compartment then usually are weighed (particularly if they are small) and there is obtained a size distribution where the fraction of material lying between the linear dimensions z = z i and 2 = z2 is given by the weight of material in this range. I n this paper particle size distributions are taken in this sense. This actually is not an important restriction because i t is generally accepted that pieces arising from the breakage of solids can be taken as homologous so that the volume of a piece having diameter 5 is related to z3by a shape factor-see Austin ( I ) . It turns out that under the hypotheqes stated explicitly in the following section, the size distribution will approach more closely a logarithmico-normal distribution the longer the process is continued no matter whether size distribution is considered by volume, linear dimension, or count. This is a consequence of the fact that if the characteristic is logarithmico-normally distributed then so is zn for any finite n. CORDITIOSS URDER WHICH BREAKAGE 1IECH4UISBI L E I D S TO LOGARITHMICQ-NORMAL DISTRIBUTPON O F PRODUCT
Clearly the shape of the di3tribution function after S steps is dependent on the functions P,(y) and F(z,y). Consider now a particular arbitrary set of assumptions about the functions P,(b) and F(z,y): (A) P7,(y),the probability of breakage of any piece during the nth step of the breakage process, is a constant T,, independent of the size of piece (but possibly dependent on t z ) and of the presence of other pieces; and (B) F ( s , y ) , the distribution of pieces of size x 5 y arising from the application of a single breakage event to a given piece of size y, is independent of thc dimension of the piece broken in the sense that the fraction by Teight of mateiial having dimension < ICy(0 5 IC 5 1) arising from the breakage of a unit mass of size y is indcpendent of y. It should be mentioned that in certain breakage mechanisms, P,(y) is certainly dependent on the size y. Consider, for example, the drop shatter test for coke where large pieces are almost certain to be broken in one drop whereas small pieces are far less likely to break. Under conditions (A) and (B) the folloning theorem can be proved:
THEOREM A. If the initial distribution of pieces is described by the cumulative distribution function G(y\, 0 5 y 5 yo, with some finite maximum yt, and if assumptions A and B are satisfied, then the cumulative distribution function, F N ( z ) , after S steps 1x1 the breakage process IS asymptotically logarithmico-normal. The pioof of this thorem (6)_dependson the fact that thc distribution function, F.v(z), after A steps of the breakage process 1s the distribution of a random variable zw which is the product of -Il ) ,independent random variables. Therefore, log z . ~ is the sum (S f l),, independent random variables. One can then apply the central limit theorem of mathematical statistics ( 5 ); i t follows that log X N is asymptotically normal. Therefore, T V is EL random variable which is asyinptotically logarithmico-normal. Theorem A explains in a compact way why so many size distributions arising from intensive grinding operations appear to be logarithmioo-normal. Por such processes condition (A) may bc a reasonable approximation to the true state of affairs Furthermore, assumption (B) seems plausible and in fact is tacitly a= sumed to be true by a nunibcr of authors ( 7 , I O ) . It should be mentioned that other authors ( 2 , 3, 4 , 7 ) have made the observation that a degradation process can be considered as a sequence of breakage events. But all of thebe Trriters have missed the essential point that under certain hypotheses the l a m of probability operate in such a way that size distributions ohtained from the continued repetition of the breakage process will belong to a definite class of distribution functions -namely, the logaritlimico-normal distribution. It has by no nieans been proved that conditions (A) and (B)
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are the only conditions that would lead to a logarithmico-normal distribution. It was found, however, that changes in condition (B), while leaving condition (A) the same, lcd t o processes which generated size distributions which were not asymptotically logarithmico-normal. The statistical approach as outlined here seems to be preferable to a purely heuristic and unmotivated approach from several viewpoints. The most esseiitial point is that attention IIUW is focused not on fitting data by purely empirical procedures, but on trying to construct plausible mechanisins and attempting to see where t’hese assumptions lead. I n the light of the results sketched in this paper it would be far more reasonable, for csample, to attempt to fit the size distributions studied by Roller ( I S ) by plotting the data on logarithmic probability paper rather than by using the Roller distribution function F ( z ) = ax1/%- b/x where F ( z ) is the per cent by weight of material of size _< R: and where a and b are constants to be determined from the data. As a niatter of fact the data given by Roller (IC) wcre plottcd on logarithmic probability paper by this author a,nd there was a pronounced tendency for the data to fall along straight lines on this paper. Such lines give a useful mag of characterizing the data as the mean size and the standard deviation of the distribut’ion in logarithmic units have a definite statist,ical meaning. These values can be read directly from the graph or determined more precisely by conventional least squares procedures. The two parameters-mean size and standard deviation in logarithmic units--give a rational basis for comparing one logarithmiconornial distribution n-ith another. It is important to have clearly in mind what is meant by tho mean size and the standard deviation when dealing with 1o:;arithmico-normal distributions. By definition, a random variable x is said to be distributed according to the logarithmiconormal distribution if log 2 is normally distributed. This means that so long as one works in logarithmic units, the problem rcduces t80t,he c.ase of the normal distribut’ion. If m and u arc thc mean size and the standard deviation of the size distribution in logarithmic units, then the mean size Af (the size Jf for which 50% of the distribution 5 1%’ on a weight basis) in ordinary units is simply the ant,ilogarithm of m. u must be kept in logarithmic units since its antilogarithm has no statistical meaning in this problem. A numerical example for which the linear dimensions are given in microne map make clear nThat is meant. Suppose that using the base 10, a logarithmico-normal distribution has m = 2 and v = 0.30, then: 50% of the distribution will lie within m * 0.675, u = 2 * 0.20 (that is, between 1.80 and 2.20); and 95LT, of the distribution will lie within m 1.96, u = 2 * 0.59 (that is, between 1.41 and 2.59). Thcse sizes are in logarithmic units and it is necessary to translate them into linear dimensions by taking antilogarit,hms. In ordinary units the mean size is 100 microns with 507, of the material lying between 63 microns and 159 microns and 95yo of the material lying between 26 microns and 389 microns. The linear dimensions are quoted in niicrons in this illustration because the dimensions of the finer sieve openings are generally given in microns. THEORETICAL BREAKAGE MECHANISM SATISFYIUC, ASSUMPTIONS A AND B
EXAMPLE. Conditions (A) and (B;, as stated, ale relatively abstract and it may be of interest to describe a hypothetical breakage process for JT-hicli it is evident that (A) and (B)are satisfied and to give data showing that such a process does genei ate distribution- which are asymptotically logarithmiconormal. As a simple example assume that one starts with 30 pieces each of size 1. In terms of theorem A , G(y), the initial cumulative distribution function is: G(y) = 0,O 5 y < 1; G(y) = 1, y 1 1. CIearly there is no Ioss in generality in considering !IO the size
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1948 9
/-
I 01
I
2
IO
5
CUMULATIVE
20 3 0 4 0 5 0 6 0 7 0 80
90 95
9600
OW
P E R C E N T O F TOTAL NUMBER OF PISCCS
Figure 1. Cumulative Distribution of L e n g t h s on a C o u n t Basis
of the largest piece to be equal to 1 since this amounts to a normalization which cannot affect the end result. The breakage mechanism is defined in the following way: for each piece in the size distribution obtained after n steps a coin is thrown. If a head appears the piece is broken into two halves; if a tail appears the piece is not broken. If this is done for each piece in F,(z), the distribution after n steps, it is clear that there is generated F,+i(z), the distribution after (n I) steps. For this particular choice of assumptions the probability of breakage P,(y) = 0.5 independent of n and of the size y (provided, of course, that the coin is unbiased). F(z,y), the fraction by weight of particles of size z 2 y arising from the breakage of a unit mass of size y, is given by the distribution function:
+
F (ql) = 0,o 5 z < Y 2; F ( z , y ) = 1,z 2 Y2. For convenience the pieces are.assumed t o be thin rods so that only one dimension need be considered. I n a three-dimensional case the analogous classification would be by volume. But, as mentioned, if the distribution of volumes (or weights) by count is logarithmico-normal, then so is the distribution of sizes by weight (provided that it is legitimate to use shape factors). Table I gives the distribution of lengths of pieces by count after N = 1 , 2 , . . , 10 steps and Figure 1is a graph of the cumulative distribution of lengths on a count basis after 7, 8, 9, and 10 steps. As predicted by theorem A, the percentage by count
TABLE I. DISTRIBUTION OF PIECES RESULTING FROM TENSUCCESSIVE BREAKAGE STEPS STARTING WITH THIRTY PIECES O F UNITSIZE Lennth of Piece- Given as Log of Length t o Base0.5
10
(Probability of breakage, 0.5)
1 17 26
2 9
28 28
Distribution of Pieces after Step S o . 3 4 5 6 7 8 1 0 0 0 0 0 1 32 13 8 0 2 38 63 49 15 11 37 73 85 28 48 84 59 153 154 109 60 28 28 132 214 63 30 80 176 24 94 20
9 0 0 5 49 112 286 285 229 104 20
10 0
0 4 25 98 285 397 414 285 108 28
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(or weight) plots approximately as a straight line on logarithmic probability paper. A x2 test (6) carried out on these data showed that the hypothesis that the size distribution is logarithmiconormal is not refuted by the data since the value of xz observed, or one higher, could have occurred by chance 50% of the time. It was found convenient and time-saving to carry out this statistical experiment by using random sampling tables (11, 16). Only single digits were used and the digits 0, 1, 2, 3, and 4 were interpreted as heads and 5, 6, 7, 8, and 9 as tails. The special example just considered demonstrates the essential characteristics of mechanisms satisfying conditions (A) and (B). The choice of P,(y), the probability of breakage, as 0.5 is arbitrary; it could just as well be equal t o any number between 0 and l. Experimentally this is easy to do with a table of random numbers-for example, using single digit numbers the probability of breakage 0.2 is obtained by breaking a piece if a digit is 0 or 1 and not breaking otherwise; greater flexibility is obtained by using numbers with more than one digit. The probability easily can be made to depend also on n, the number of steps in the breakage process. The condition that a piece breaks into two equal pieces on breakage is capable of generalization also provided only that the distribution obtained be independent of the size of the piece broken in the sense of assumption (B). Some such generalizations probably would lead even more quickly to the logarithmico-normal distribution. GENERAL REMARKS
The technique for generating synthetic size distributions as illustrated conceivably may be of value in indicating what sort of size distributions may be expected in the breakage of solids under various h,ypotheses. In this paper attention has been focused on special mechanisms for which conditions (A) and (B) are satisfied. It is an important question practically and theoretically to know what happens if these conditionq are not satisfied. I n the case of the drop shatter test for coke, for example, data analyzed in this laboratory indicate that the probability of breakage depends on the particle size and is proportional to second or third power of the particle size. Attempts are being made in this laboratory t o study both analytically and experimentally processes for which the probability of breakage is proportional to a power of the size and for which condition (B) is still satisfied. ACKNOWLEDGMENT
The author acknowledges the help of Lois Bylenok of this laboratory in carrying out the synthetic breakages used in the example of theoretical breakage mechanism. LITERATURE CITED
(1) Austin, J. B., IND. ENG.CHEM.,ANAL.ED.,11, 334 (1939). (2) Bennett, J. G., andBrown, R. L., J . Inst. FueJ, 14,135 (1941). (3) Bennett, J. G., Brown, R. L., and Crone, H. G.,Ibid., p. 111. (4) Brown, R. L., I b i d . , p. 129. (5) Cramer, H., “Mathematical Methods of Statistics,” pp. 213-21 Princeton, N. J., Princeton University Press, 1946. (6) Epstein, B.,J . FrankZinInst., 244,471-8 (1947). (7) Gaudin, A.M., and Hukki, R. T., Am. Inst. Mining Met. Engrs., Tech. Pub. 1779 (1944). ( 8 ) Hatch, T., J . FrankEin Inst., 215,27(1933). (9) Hatch, T.,and Choate, A.P., Ibid., 207,369(1929). (10) Hukki, R.T., 60. D. thesis, Mass. Inst. Technol., 1943. (11) Kendall, M. G., and Smith, B. Babington, “Tables of Random Sampling Numbers,” Tracts for Computers, No. 24,London, Cambridge University Press, 1940. (12) Kolmogoroff, A.N., Contpt. rend. acad. sci. U.R.S.S., 31, No.2, 99 (1941). (13) Roller, P.S.,J . Franklin Inst., 223,609 (1937). (14) Roller, P.S., J . Phys. Chem., 45,241 (1941). (15) Tippett, L. H. C., “Random Sampling Numbers,” Tracts for Computers No. 15 (1927),Cambridge University Press. RECEIVSD May 28, 1947.
