A proposed universal homogeneity and mixing index

111

Powder Technology 7 (1973) 111-I 16 Elsevier Sequoia SA, Lausanne - Printed in The Netherlands

A Proposed Universal Homogeneity and Mixing Index D. BUSLIK SusanProducts.

1500 Mahoning Acenue,

(First received February

P.O. Box- 494, Youngsfown, Ohio

The word “homogeneous“ is generally used qualitatively. A simple numerical homogeneity index is proposed suitabIe for expressing varying degrees of homogeneity quantitatively, namely, the negative log of the sample weight (g) required to obtain a standard deviation of I%_ The scale is universal in tlzat a homogeneity indes can be computedfor heterogeneities observable only or1 the atomic scale (e.g. sodium chloride crystals and air) andfor homogeneities observable only on the stellar scale. Values of the index, Hi, are computed for certain defined and certain theoretical mixture arrangements as follow: (a) pure hydrogen (maximum and limiting valrte of Hi), 23.5; (b) sodium chloride, 22.0 -C Hi < 22.2; (c) air, 19. I; (d) IO% oil in water emulsion, 9.4 < Hi < 12.4;

(e)

a I-g

medical

B homogenized

pill composed so that 99.99%

pills are within the limits (I5 + I)%A,

of 15%

A

of the I-g

1.2; (f)

some

arbitrarily defined, partially mixed industrial materials, considered as equivalent to random segregations of 1 lb., - 6.1; I ton, - 9.4; and (g) an approximate, order of magnitude, figure for the unioerse according to available scanty astronomical data, - 46.3. The index (and even more directly, its antiIog) sltould be a useful measurement of the degree of mixing, since

homogeneity

indirectly

as have

(U.S.A.)

28, 1972; in revised form June 12, 1972)

Summary

and 85%

44501

is nleasured previotzsiJ>

directly used

indices

rather of

than degree

of i71iGlg.

“I often say that when you can measure what you are talking about and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.” (Lord Kelvin)_ The homogeneity of their mixtures is a quality of considerable interest to chemists and chemical engineers. Nevertheless, heretofore it does not seem to have been measured in any direct way_ The chem-

ical engineers who have studied the unit operation of mixing have probably made the most intensive studies so far of homogeneity_ In their literature one reads the phrase “homogeneous within the scale of scrutiny”‘, which is very descriptive and will be useful in ihe following discussion. However, this phrase only implies a degree of quantitativeness. Furthermore, the need for numerical differentiation is indicated by the variation in context in which the homogeneity concept is used, from the composition of the universe2 to the composition of the interior of a proton3. A dictionary’ definition of the word “homogeneous” is: “composed of parts all of the same kind, not heterogeneous_” If it were a requirement of the homogeneity concept that all the “parts” of an aggregate or material be of the same kind, no matter how small the parts be taken, there would be no ambiguity in the term homogeneous and there would be no reason for the following discussion and proposal. However, in many fields of science the term homogeneous is used more broadly, so that when some material or aggregate is considered to be homogeneous by an author, the “parts” which must be of the same kind, or similar, are tacitly assumed by the author to be of, at least, a certain (minimum) magnitude. Thus t:ie chemist usually has in mind “parts” composed of molecules or an aggregate of a small number of molecules ; the chemical engineer considers the “parts” to be regions within the aggregate or mixture, slightly less than the scale of scrutiny; the astronomer’s “parts” are volumes in space encompassing clusters of galaxies. Basically, the following discuss:on and proposal is a method of specifying the type of homogeneity envisaged by specifying the size of “part” (or sample*, which is *To be explicit, by a “part” or sample we mean a group of contiguous particles such as would be lifted out by a scoop which might be of sub-microscopic or super-stellar size.

112

D. BUSLIK

perhaps a more appropriate term in this discussion)_ It migh? be noted that the term homogeneous is also commonly used when there is a statistical (if very slight) probability that some of the parts might not be of the same kind (see section on the Homogeneity of Air). The usual methods of studying degree of mixing have been to use statistical measures of heterovariance. geneity5*6, e.g. the standard deviation, chi-square test and complicated functions of these statistical measures. The very names of some of these functions indicate that it is the heterogeneity that is being measured_ Homogeneity is then implied as there is a complicated inverse relation between the heterogeneity and the homogeneity_ Excepting for comparatively minor operations such as determining sampling requirements, it is the homogeneity of a mixture, rather than the heterogeneity, which is of direct concern and more meaningful. It would seem, therefore, that a method of measuring homogeneity more directly would be more useful in studying the mixing operation and in clarifying the homogeneity concept which at times becomes somewhat hazy. The usual procedure in studying homogeneity involves taking samples of some given weight (or volume) and obtaining a statistical measure of the average deviation in composition (color, weight, density, etc.) from sample to sample. The method here proposed is to take as a standard (instead of a given weight or volume of sample) a definite degree of variation and to determine the weight of sample required to obtain this degree of variation. The weight (W) of sample (g) necessary to satisfy this requirement will serve as an inverse measure of the homogeneity of the given material or aggregate : the smaller the sample, the greater the homogeneity_ Initially it is proposed for simplicity that the basic standard of deviation be taken as a standard deviation of 1 o/Oin sample composition or weight. Then if H= homogeneity, and W, =weight of sample (“part”) required for a standard deviation of 1%. (A

H=

l/W,

(1)

Now in discussions of the chemistry of acids, the strength of acids is measured by the hydrogen ion concentration, which in practice ranges over a wide magnitude (from about 1 to lo-r4). To cover this enormous range a convenient scale is generally used, i.e. the negative log of the hydrogen ion concentration (pH). Analogously, to cover the even more

enormous range of magnitudes in which the term homogeneous is involved, it is suggested that an index of homogeneity, Hi, be used as a measure, where

Hi=logH=log

l/W, = -log

giving the homogeneity TABLE

W,

(2)

spectrum shown in Table

1.

1 index (Hi)

Homogeneity

arrangements

for

certain

defined and theoretical

and materials H; > 23.5

Pure hydrogen*. maximum limiting value of Hi Pure sodium chloride Air* 10 y0 oil in water emulsion* One-gram medical pill (85 “/, A. 15 >; B). 99.99 :J; within + 1 “/_ composition limits Partially mixed industrial materials*. with random segregations of 1 lb. 1 ton Universe’ * Values based on assumptions in the Appendix.

22.0 < Hi < 22.2 19.1 9.4tHi<11.4

and approximations

1.2

-6.1 - 9.4 - 46.3 as defined

In random mixtures of solid particles (of uniform size) and in certain simplified partially segregated mixtures (discussed below) the homogeneity ratios calculated by eqn. (I) for a given pair of mixtures will be in obvious accord with our usual understanding of homogeneity_ Under certain conditions some modifications of this proposal for a homogeneity index might be advisable, namely : (1) Use of another standard of variation than a standard deviation of I %_ (2) Use of volumes rather than weights as a measure of homogeneity, e.g. where gases or stellar conditions are being considered. (3) Use of gram-molecular weights rather than actual weights, if it is considered that the influence ofmolecular and atomic weights on the homogeneity index should be eliminated. (4) Tn the case of random mixtures or arrmgements (which is a very important case in mixing or homogeneity studies) the concentration would be an important factor in determining the proposed homogeneity index, as described below: There is a well-known equation for the standard deviation of the binomial distribution a” = @P&

(3)

PROPOSED

UNIVERSAL

HOMOGENEITY

AND

MIXING

This equation is applicable to a random sample of n balls from a population of black and white balls, where c,= standard deviation of the mdwr of black or white balls in the random sample of II balls, from a population when p=proportion of black balls, and q = proportion of white balls. For the problem of interest in this discussion, it is more appropriate to consider a mixture of two ingredients, G% of the ingredient g and (100-G) “/, of the other ingredient, where lv=particle weight (assumed constant) and W = sample weight. Making the rather obvious substitutions, p = G/100, q = (lOO-G)/lOO, op.= CJll x 100,

II = W/IV

and

eqn. (3) can be written in the more useful form Go = [G(lOO-G)w/w]*

(3a)

where o, =standard deviation of percent of the g ing-Aient in random samples of weight W ii-om the mixture. If cp = 1 y0 as required by our proposed homogeneity (or index) definition, and W, =sample weight required to attain such a standard deviation, then W, =G(lOOG)\v, and substituting from (1) and (2) H=l/W,=l/G(lOO-G)\v

(3b)

and since Hi = log H, Hi = -log

WI = -log[G(lOO-G)\v]

(3c)

In comparing two similar cases, as similar as possible excepting for different values of G (concentration), the homogeneity indices might differ largely due to concentration effects. We do not ordinarily consider that the homogeneity of a mixture should be affected by concentration, although it should be recognized that the homogeneity concept has heretofore been a rather vague one_ not susceptible to measurement. If it were considered worthwhile to eliminate concentration effects one might use a modified homogeneity index, namely Ri = -log

W,/G(lOO-G)

(4)

The complexity of the problems involved in the study of possible mixing arrangements has been described by students of the mixing operations-6. It might be useful at this point to consider the various possible arrangements arrived at, after equilibrium is established (because mixing has

INDEX

113

been carried on for a prolonged

period). Uniformly sized particles mixed in the usual mechanical mixer will ultimately attain a random arrangement, of which the mathematical properties, at least, are simple. If very small particles are mixed with very large

particles (as an extreme example_ consider mixing fluids with solid granules) the surfaces of the large particles will be coated with the small particles at equilibrium. This could hardly be considered a random arrangement. If there is a gradation of particle sizes in the material being mixed, the arrangement that would result at equilibrium has been discussed in ref. 7 and in treatises on the structure of concrete. It is shown’ that the homogeneity of the smalier particles in the mix is less than would be obtained if particles of this size alone were arranged at random, due to the nature of the packing of multisized particles, after thorough mixing. If elementary sodium and chlorine are mixed in stoichiometric proportions under conditions in which chemical reactions result, there will be obtained 2 checkerboard alternating type of arrangement of sodium and chloride (ions) and a degree of homogeneity unattainable in the usual mechanical mixer. If one considers mixtures of uniformly sized particles of some given weight which are randomly arranged, eqn. (3b) shows that the homogeneity ofthe mixture will be inversely proportional to the given weight of particle composing the mixture. This conclusion is compatible with our usual concept of homogeneity. Partial segregation is the usual result when mixing has not been carried on long enough to reach equilibrium and may result at equilibrium with certain kinds of mixture (e.g. multi-sized particles). In some common cases of interest, the advantages of measuring homogeneity directly as here proposed are very obvious. Consider two common cases of partial segregation with segregations very much larger than the samples used to measure the homoor heterogeneity (Figs 1 and 2). In Fig. 1 segregations are each of nz particles (black or white balls) and in Fig. 2 of 2nr particles_ To simplify calculations of the homogeneity (or its index) assume that these segregations are arranged at random throughout the material. Samples are of n particles in either case, and m %- II. By the usual methods of assaying heterogeneity, these two cases would differ very little. The great bulk of the samples from either population would be either pure black or pure white balls. There would be a slightly greater frequency of samples from the borderline regions from the Fig. 1

113 type of population so that the standard deviation, variance, etc., from the Fig. 1 population would be only slightly less than the corresponding statistic from Fig. 2. The greater the difference between KI and II, the less the difference in the variance statistic

D. BUSLIK

standard deviations of sample composition Versus the sample size. Where segregations are very large compared to sample size it would be necessary to map the sample composition as a function of location and then compute the size of sample required to attain the standard deviation of 1%. The quality of homogeneity is not only of interest i.7 chemical and physical studies, but also in the biological and social sciences, being generahy Wrmed “integration” in the latter fields. The most appropriate application of the concepts in this paper to the social sciences is a matter for additional study.

CONCLUSIONS Fig. 1. Arrow indicates an aggregate with segregations of m particles in each. Segregations. in turn, randomly arranged. Sample of )I particles with m % PI.

Fig_ 2. Arrow indicates aggregate with segregations of 2m particles in each. Segregations, in turn, are randomly arranged. Sample of n particles with m %>n.

between the two populations. But our proposed homogeneity index would clearly show that Fig. 1 was twice as homogeneous as Fig. 2 and the homogeneity indices would differ by log 2 regardless of the ratio ~n/lr,since the sample size to get a 1% standard deviation would only be half as large for Fig 1. This result is more in accordance with our usual understanding of the homogeneity concept than the result we would obtain in inferring homogeneity from heterogeneity data. For an actual mixture or material aggregate the homogeneity, or its index could, at ieast theoretically, be determined experimentally. If the segregations are not large compared to the practical sample size, the weight of sample, W,, required to obtain the standard deviation of 1 oA could be determined by interpolation or extrapolation after plotting the

It has been the practice heretofore to measure a degree of mixing by taking samples of a given size and determining the degree of variation in a sample property. The degree of variation is actually a measure of the heterogeneity of the samples (and consequently of the mixed material). Since the effectiveness of the mixing operation is determined rather by the resulting homogeneity, this must then be inferred from the complicated inverse relation between the homogeneity and the heterogeneity. A direct measure of homogeneity is here proposed based on the reciprocal of the weight of sample (g) required to obtain a specified degree of variation in the samples (a standard deviation of lo/J_ To avoid the unwieldy very large or very small numbers that would result, an index is proposed also, based on a logarithmic scale (analogous to that used in pH measurements). The measure proposed for measuring homogeneity is shown to be in accord with our usual (vague) concepts of homogeneity in at least one important case (the random mixture). Being a direct measure of the quantity of inter&t it should be more usef& where it can be determined, in measuring mixing effectiveness. In one definite rather common type of mixture (where the segregations are very much larger than the samples) the method initially described can be shown to be misleading in judging homogeneity. The proposed method is of universal applicability and a spectrum of index values for homogeneity has been computed for certain mixtures over a very wide range. It was found possible to compute approximate homogeneity indices for materials generally considered to be absolutely homogeneous (air and salt) as there are elements of heterogeneity

PROPOSED UNIVERSAL HOMOGENEITY AND MIXING INDEX in t h e i r c o m p o s i t i o n . H o m o g e n e i t y i n d i c e s were also c o m p u t e d for m a t e r i a l s g e n e r a l l y c o n s i d e r e d to be heterogeneous (emulsions and partially segregated i n d u s t r i a l m a t e r i a l s ) and an i n d e x w a s c o m p u t e d w h i c h m a y be a,r, plicable to the c o m p o s i t i o n of the universe. APPENDIX COMPUTATION OF Hi VALUES IN TABLE 1 B e l o w are c o n s i d e r e d s o m e c a s e s of a c t u a l mat e r i a l s and of t h e o r e t i c a l a r r a n g e m e n t s for w h i c h H i v a l u e s were c o m p u t e d ( T a b l e 1). (a) F i r s t , a l i m i t i n g and m a x i m u m v a l u e of Hi (for p a r t i c l e s o f a t o m i c , m o l e c u l a r or l a r g e r size) was c o m p u t e d by c o n s i d e r i n g the v a l u e t h a t Hi w o u l d a t t a i n when s a m p l i n g p u r e h y d r o g e n . If the s m a l l e s t p o s s i b l e s a m p l e to o b t a i n is c o n s i d e r e d to be a single molecule, then all such m o l e c u l e s from p u r e hyd r o g e n w o u l d be identical as they w o u l d all be p u r e h y d r o g e n and the s t a n d a r d d e v i a t i o n of s a m p l e c o m p o s i t i o n w o u l d be z e r o . T h e v a l u e of Hi, -- log w e i g h t of a single h y d r o g e n m o l e c u l e (23.5), is then the l a r g e s t a n d l i m i t i n g v a l u e w h i c h Hi can a t t a i n . (b) N o w c o n s i d e r a m i x t u r e t h a t is s o m e w h a t h e t e r o g e n e o u s a l t h o u g h not o r d i n a r i l y so c o n s i d e r e d , viz. the case of p u r e s a l t . We are all f a m i l i a r from e l e m e n t a r y c h e m i s t r y t e x t b o o k s with the rep r e s e n t a t i o n of the s o d i u m c h l o r i d e c r y s t a l as an a l t e r n a t i n g , c h e c k e r b o a r d a r r a n g e m e n t of s o d i u m and c h l o r i d e i o n s . A l t h o u g h p u r e s o d i u m c h l o r i d e w o u l d not o r d i n a r i l y be c o n s i d e r e d n o n - h o m o g e n e o u s , this is d u e to the fact t h a t o u r s c a l e of s c r u t i n y is c o a r s e . If we were M a x w e l l d e m o n s traveling t h r o u g h and a r o u n d sodium chloride c r y s t a l s they w o u l d l o o k as h o m o g e n e o u s as a c h e c k e r b o a r d does to us, so t h a t s o d i u m c h l o r i d e is not c o m p l e t e l y h o m o g e n e o u s , at l e a s t on the s c a l e of s c r u t i n y of a M a x w e l l d e m o n . T h e a l t e r n a t i n g , c h e c k e r b o a r d a r r a n g e m e n t of the ions in t h e s o d i u m c h l o r i d e c r y s t a l d o e s , h o w e v e r , p r o v i d e the u l t i m a t e in h o m o g e n e i t y for a m i x t u r e of t w o d i f f e r e n t ingredients. A s c o o p of size l a r g e e n o u g h to pick up t w o n e i g h b o r i n g ions will always y i e l d i d e n t i c a l s a m p l e s of s o d i u m c h l o r i d e . A s c o o p just l a r g e e n o u g h t o pick up only one ion at a time will half the time pick up a s o d i u m ion a n d the o t h e r h a l f the time a c h l o r i d e ion. This r e s u l t s in a d i s c o n t i n u i t y in o u r c a l c u l a t i o n of Hi for s o d i u m c h l o r i d e and we can only say t h a t Hi is then b e t w e e n the n e g a t i v e log w e i g h t o f a c h l o r i d e ion and t h a t of a s o d i u m chloride molecule, or

115

22.0 < Hi < 22.2 w h i c h is one of the l a r g e s t v a l u e s of Hi t h a t a h e t e r o g e n e o u s m i x t u r e can a t t a i n . ( c ) A i r is a n o t h e r m a t e r i a l not o r d i n a r i l y cons i d e r e d to be n o n - h o m o g e n e o u s , a n d a g a i n this failure to r e c o g n i z e h e t e r o g e n e i t y is d u e to the c o a r s e n e s s of o u r s c a l e of scrutiny. If the p r e s e n t t r e n d c o n t i n u e s t o w a r d s a n a l y s i s of g r e a t e r a n d g r e a t e r sensitivity and of s m a l l e r and s m a l l e r samp l e s , s o m e d a y a c h e m i s t will a n a l y z e a s a m p l e of air and m a y find to his s u r p r i s e t h a t his s a m p l e is p u r e n i t r o g e n or p u r e o x y g e n (due to s t a t i s t i c a l fluctuat i o n s w h i c h will o c c u r with g r e a t e r frequency in small samples). We h a v e c o m p u t e d a h o m o g e n e i t y i n d e x for air, w h i c h is a s s u m e d for simplicity to be 80~oo n i t r o g e n and 2 0 % oxygen. N o w the o x y g e n and n i t r o g e n a t o m s in air are not a r r a n g e d in the n i c e l y o r d e r e d , a l t e r n a t i n ~ a r r a n g e m e n t of o u r p r e v i o u s e x a m p l e , but in an a r r a n g e m e n t m o r e typical for a wellm i x e d a g g r e g a t e , and t h a t is a r a n d o m a r r a n g e m e n t . F o r this a r r a n g e m e n t o u r eqn. (3c) is a p p l i c a b l e if we m a k e the simplifying a s s u m p t i o n that the p a r t i c l e s (oxygen and n i t r o g e n m o l e c u l e s ) are the s a m e w e i g h t and size. A m o r e a c c u r a t e e q u a t i o n is a v a i l a b l e7, but this r e f i n e m e n t is not c o n s i d e r e d a d v i s a b l e at this p o i n t . In eqn. (3c) we s u b s t i t u t e for w the a v e r a g e w e i g h t of a m o l e c u l e in the air m i x t u r e and a r r i v e at the v a l u e Hi = 19.1. ( d ) T h e o i l - w a t e r e m u l s i o n r e p r e s e n t s an int e r e s t i n g b o u n d a r y in the h o m o g e n e i t y s p e c t r u m . E m u l s i o n s are m a d e to a t t a i n a high o r d e r of h o m o g e n e i t y of difficultly m i x e d m a t e r i a l s (e.g. oil and w a t e r ) . N e v e r t h e l e s s , h e t e r o g e n e i t y may be obs e r v a b l e with a p p r o p r i a t e i n s t r u m e n t s and s o m e effects of the h e t e r o g e n e i t y m a y be o b s e r v a b l e even to the n a k e d eye (cloudiness). We c o m p u t e , t h e r e fore, the h o m o g e n e i t y i n d e x of a typical oil (10~/o) in w a t e r (90 9"o) e m u l s i o n . A c c o r d i n g to G l a s s t o n e9, " ' T h e g l o b u l e s or d r o p l e t s of d i s p e r s e d l i q u i d (oil) are f r o m 0.1 to 1 m i c r o n (10 - 4 cm) in diameter.?" C o n s i d e r i n g t h e s e d r o p l e t s to be p e r f e c t s p h e r e s , of d e n s i t y 0.8, they w o u l d x~ ,ht from 4 x 1 0 - 1 6 g to 4 x 1 0 - l a g . We a s s u m e t h a t the oil d r o p l e t s are a r r a n g e d at r a n d o m in the e m u l s i o n . F r o m eqn. (3c) we c o m p u t e Hi to r a n g e in v a l u e f r o m 9.4 to 12.4. T h e e r r o r i n t r o d u c e d by c o n s i d e r i n g the oil a n d w a t e r p a r t i c l e s to be of the s a m e w e i g h t is f o u n d by the e q u a t i o n in ref. 7 to h a v e a relatively s m a l l effect ( a p p r o x i m a t e l y 10~o) a n d is, t h e r e f o r e , neg l e c t e d in this a p p r o x i m a t e c o m p u t a t i o n .