Statistical Properties of Dry Blends Frank P. Vance
Most investigators have mistaken the true properties of the statistical terms they employ in predicting the performance of dry blenders.The author has developed a simple statistical procedure to provide a reliable indication of good dry mixing. The method includes the effect of sample size and the number of samples in promoting errors which indicate good blending. This aspect of dry-solids blending is not described elsewhere
revious investigators in this field have used statistical methodology in some form, and usually with small, horizontal cylindricial or twin-shell blenders. The present study evolved from a program originally designed to evaluate the performance of an in-tank blender of about 50,000-lb. capacity for polyolefin pellets. Subsequently, in an effort to establish a basis for comparison, tests were performed on double-cone blenders of 1500lb. capacity, ribbon blenders of 1500-lb. and 50-lb. capacity, and a twin-shell blender of about 10-lb. capacity. A simple statistical procedure was developed which provides a reliable indication of degree of approach to statistical ideality. Further, the relevance of individual sample size and number of samples to Type I1 decision errors is shown-i.e., control of the likelihood of an unmixed condition even though sample analysis indicates a mixed condition. This aspect of dry-solids blending is not described in the existing literature.
P
Statistical Analysis
The criterion for judging the quality of a blend or the performance of a blender is to measure the approach to statistical ideality. For particulate solids, the expected variation of weight percentages in samples can be determined beforehand. This variation in the inspection of liquid solutions cannot be determined in advance, since it will depend upon the uncertainties of sample assay or laboratory test error. The errors attending assay of samples of mixtures of particulate solids can be made zero in conditions which involve merely counting colored pellets or weighing a portion of the pellets, readily separable because of density difference. In situations where the variations within the blend are scrutinized by measuring some property such as melt viscosity of a polymer blend, test error affects the results. I n most of the cases discussed in this paper, samples were assayed by counting pellets or VOL. 5 8 NO. 6 J U N E 1 9 6 6 37
by weighing the more dense fraction. Finally, when development work on the blender design appeared complete, uniformity of blends was judged on the basis of melt viscosity. Decisions us. hypotheses can be represented graphically by the following diagram:
I
Var(x)=npq
I
I
I
I
l
1
I
I
UPPER 6 PERCENTILE FOR T H E RATIO S/u', GIVEN T H A T E(*) = u'
I
n = number of particles in the sample = proportion of the component represented by the sample assay q = l - p
If we observe numerical proportions in the samples rather than numerical counts, this variance is Var @) = p q / n
x'/h
- 7)
2.37 1.88
1.59 1.24
Frank P . Vancc is Manager,EnginCning Siati&s, PIanning and Cwdalion Branch, PhiUips Pelrolnun Go. He acknowIedges thc uaIuablc assistance of R. J . Emneit, R. R. Goim, 0 . R. W e a w , and V. Jones in thc cdIection of data, &sign and testing of thc in-tank blenders, and in thc pIanning and e x c m h h of tk tests.
.. ..
=
W@P- m W
m - @P) . . . . .
(2)
(3)
where P = weight percentage of the specified component W = weight of sample taLen @ = mean particle weight overall = mean particle weight for those of the specified component Note that, when mean particle weights are equal, Equation 3 reduces to Equation 2. Thus, we have an extremely convenient too1 for estjmation of the variance of a dichotomous system which, when @ e d y mixed, is denoted by uo'. Now the sample variance, 9,can be calculated from
AUTHOR
INDUSTRIAL A N D ENGINEERING CHEMISTRY
.
Budii (2) has shown that when the densities of the two components are not greatly different, the variance of weight percentages is given by,
(1)
TABLE 1.
38
l
p
Var )'2(
.....
where x = random variable whereby the samples are characterized-eg., pellet counts
5 10 20 100
I
Fact
These errors can never be completely eliminated, but they can be made to approach zero by increasing sample size-up to some economical limit. In the literature to date, confidence intervals and tolerable limits have been described which are based usually on assurance that Type I ermr rate is no greater than 5 or 10%. Type I1 error has not been considered. I t is apparent that both types of error must be taken into account. Ideally, these should be balanced economically, but the necessary information for such a balance is usually lacking. A formal procedure will be developed for specification of sample size and number of samples necessary to limit both types of error as desired. In a system of particulate solids in which the components are distinguishable only by a property that does not atTect their random motion relative to each other, such as color, the variation in counts of colored pellets will be according to the binomial frequency distribution when the contents are completely mixed. A quantitatative expression of variations from the population mean is provided mathematically by the variance, defined as the mean square deviation from the mean. I n dichotomies, such as those described above, that is,
m
103
(4) where
P, = individual weight samples
percentages obmved in
i = 1, 2, .....m m = number of such samples
The random variables f., are then assembled into the ratio of sp2/uo2which has the approximate frequency distribution of x2/(m - l), with ( m - 1) degrees of freedom. The expected value of this ratio is unity when the system is perfectly mixed. Upper limits for this ratio a t 5y0 probability (95% confidence; Type I error rate = 5y0)for different number of samples, m (each containing n particles) are shown in Table I. Stange (5) tested approach to ideality of blends by observing ratios of s/u-Le., the square root of the values listed for x”(m - 1). This provides assurance that the frequency of Type I errors will be relatively low (say, Pca or ? < Po, given E@) = PI 5 0.02 probability{P > Po, or f. < Pcz, given E@) = PIor P 2 ) 5 0.02
a = probability{p
@ =
Since the hypothesis test is two-sided with respect to Type I error, we can write the equation for a as, a
- = probabilityip < Pc,lE(p) 2
= P)
5
0.01
and
-a = probability{$ > Pca\E($) = P ) 5 0.01 2 Then,
p (Wt. %)
A
From tabulated values of the fractiles of the cumulative normal we have ta/2 = 2.33; tp = 2.06. We have then, by rearranging Equation 5a and substituting, uo
=
10 - 11 = 0.23; 4.39 VOL. 5 8
~ 0 = %
0.053
NO. 6 J U N E 1 9 6 6 39
Referring to Table 11, we have for P = IO, uo2
=
29 W
Accordingly, W = 29/0.053 = 550 grams. Further, the critical values, $cL and Fc2are found as follows: A
Pc, = P - (tcr/2) FC,
=
P
u =
10 - 2.33 X 0.23 = 9.46 wt. yo
+ (tcr/2) u = 10 + 2.33 X 0.23 = 10.54 wt. %
If the system is completely mixed so that the true system variance equals uo2, then one 550-gram sample can be taken and assayed. If 9.46 < P < 10.54, we conclude that P = 10 (CY 5 0.02). I f ? < 9.46 OY $ > 10.54, we conclude that P < 9 or P > 11 ( p 5 0.02). This procedure is reliable, of course, only if it can be shown that, in fact, u2 = u$. I n the course of the experiments, the results of which are discussed in following sections, no sample variances were observed as low as ao2. As will be shown, however, many blenders can be expected reliably to produce blends having s2 = 2 uo2, and this ratio could be used as a criterion for determining sample size. Thus, it would be a simple niatter to double the sample size obtained in the calculation just described in order to assure that u p would be of the order of 0.23 as required for the conditions of the sample test, so long as assay of the sample does not represent a prodigious amount of effort. Thus, in the illustrative example, a 1200-gram sample could be taken, and the inspection based on a noncritical region roughly would be equal to 9.5 to 10.5 wt. yo, Test Program
To find a more economical method of blending large batches (-50,000 lb.) of polyolefin pellets than that of using a double-cone blender or equivalent, an in-tank blender was designed in which it was intended that the pellets would be circulated from bottom to top. Internal baffling would promote lateral and longitudinal flow during circulation. Thus, although it was anticipated that several “turnovers” might be required, and that this would take many times the blender cycle in a double-cone blender, time is not critical for that part of the process, and the investment is much less. A test program was set up to evaluate the performance of different design concepts. The statistical techniques that have been described were used to effect quantitative comparison of the stepwise modifications in these blenders. Test 1A. No specific program had been designed for sampling and evaluation of results before the test was commenced. However, after the circulation was terminated, the blender contents were dumped into a series of 1000-lb. boxes, and approximately a 500-gram sample was taken from each box. Weight percentages were obtained by separating colored pellets from neutral. The hypothesized variance was calculated from Buslik’s equation, using data on pellet properties obtained during later tests and shown in Table 111. Test 1B. This was conducted in the same blender as 40
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
TABLE 1 1 1 . SUMMARY OF RESULTS FROM IN-TANK BLENDER TESTS Blender I Sample iVo. -_____ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Average Hypothesized variance Observed varianci Observed/hypothesized .-
a
-~
B
A
14.7 11.3 13.0
9.4 9.1 9.3
~
1
‘1
5.0 4.5 8.4 12.5
’1
9.8
i
11.0 9.8 11.2 13.7 11.2 11.3 11.5 11.4 9.9 10.6 11.2 10.2 11.1 8.7 10.5 12.2 12.7 12.2 12.5 15.9
10.69 0.068 5.34 78
1
I
9.01 10.25 3.45
3.65 5.20 11.11 13.34 14.78 12.95 13.86 12.61 11.05 11.78 11.05 10.80 10.01 10.82 10.18 10.49 10.44 9.93 9.51 9.72 9.82 9.66 12.62 12.98 12.96 12.36 10.06 11.09 11.20 11.60 12.05 10.99 16.83 4.32 10.26 0.052 11.15 211
~~
’
~
I
1
c 11.41 11.25 8.35 11.21 11.00 12.00 12.49 12.72 11.50 11.31 11.31 11.45 10.90 12.15 11.10 11.79 11.81 10 36 12.70 10.86 11.40 8.74 8.87 8.06 9.74 9.20 10.52 11.27 11.35 11.67 10.63 10.69 11.92 10.60 8.85
10.89
~
1
0.053 1.46 28
Blender
II
... ...
... 11.94
... 11.35
... 10.59 10.57
... 11.34 11.25 10.89 10.45
...
10.10
... 10.81
... 10.26 I
.
.
9.05
... 9.78
... 10.23
... 10.73
... 9.25
... 7.61 6.27 1.29
10.54” 0.052 0.59“ 11
Excluding last three values.
l A , but with a different pellet circulating device. The sample size was chosen so as to apprehend a deviation from the made-up proportion of 1.O percentage point. The noncritical region for acceptance was 9.45 _< P 5 10.55 wt. yo. The blender contents were sampled every half hour. After four hours, four sample percentages were observed sequentially within the noncritical region and blending was terminated. Upon dumping the samples into 1000-lb. boxes and sampling as before, we observed the values as listed. Clearly the conclusion that the contents were adequately mixed was erroneous. The variance observed for IA was 5.34; for IB, 11.15. The degrees of freedom in the former were 34; for the latter, 37. The ratio, 11.15 to 5.34 = 2.08, has the approximate frequency distribution of “Snedecor’s-F,” whereby the hypothesis that these two sample variances are estimates of the same population
TABLE IV.
Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Sample mean Sample variance Ideal variance Ratio
9rain __ ’est 7, 0 Lb. Red: 3750 Lb. ‘eutral __ 6 2 4 3 17 9 5 22 28 13 38 25 11 9 18 12 15 10 11 17 16 23 13 15 16 20 18 20 9 4 10 8 9 17 12 21 21 73 16
PELLET COUNTS FROM SAMPLES
- - Circulation Test 2,
70 Lb. Test 4,
Test 5,
Test 6,
Continue Test 3
Continue Test 4
Continue Test 5
10 15 18 27 26 15 26 41 42 25 13 16
15 16 17 12 22 22 26 46 37 21 12 24 15 15 21 21 27 15 35 27 15 26
30 30 14 17 15 18 10 12 24 17 26 21 21 11 18 35 15 25 19 18 13 18
20 29 27 36 21 21 27 13 18 19 38 25 22 27 17 32 28 29 15 27 20 23
Test 3: Continue Lb. nrieutral Test 2 - ___ 0 30 0 27 25 0 0 36 0 56 0 44 10 15 21 21 Black: 3760
8
6 1 3 14 19 11 5 2 8 13 19 24
10 10
15.8
..
24.9
22.1
19.4
24.3
43.2
..
156.6
74.5
43.4
41.6
19.5 7.3
.. ..
22.0 7.1
22.0 3.4
22.0 2.0
22.0 1.9
-
variance can be tested. By reference to tables, it is found that by chance alone, the ratio of two such sample variances will be as large as 1.77, 570 of the time. Accordingly the hypothesis E(sIA2) = E ( S ~ Bwould ~ ) be rejected (a 5 0.05) where E equals expected value. Test IC. During Test IB, the equipment had not been operated according to design-that is, circulation rate was not high enough to prevent accumulation in certain areas resulting in restriction of flow of pellets. Accordingly, the necessary steps were taken to correct this, and Test I C was conducted. As with IB, a sequence of values was observed within the noncritical region, circulation was terminated, and the contents were dumped and sampled. The observed variance, 1.46, is significantly lower than any observed theretofore, but is still 28 times as large as that hypothesized, namely 0.052.
Note that two separate hypothesis tests can be constructed based on the variance. I n one case, we test whether the two sample variances are estimates of the same population variance by forming the ratio s12/s22. Probabilistic fractiles of this ratio are given by the cumulative frequency distribution of Snedecor’s-F, I n the other case, it may be determined (probabilistically) whether the sample variance is an estimate of some specific population variance, which in this case is listed in the table as “hypothesized,” and was calculated from Buslik’s equation. Thus, the ratio, s2/uO2is formed. Probabilistic fractiles of this ratio are given by the cumulative frequency distribution of x 2 / ( m - 1) where m equals the number of independent samples on which s2 is based. On these criteria it is clear that the variances of weight percentages from Tests IA-IC were not estimates of the same population variance (a < 0.05) ; hence the blender performance was different in each case. Further, the ultimate blend was far from statistically ideal, as judged by X 2 / ( m - 1). By chance alone, the ratio s2/u2will exceed 1.4 (approximately 35 degrees of freedom) only 5y0 of the time. As shown, the smallest ratio observed was for Test IC, namely 28. To obtain some guidance as to the commercial acceptability of such a blend, calculations were made of the blended melt viscosity, using two hypothetical values, unnaturally widespread. The variance of these predicted melt viscosities was several orders of magnitude less than the variance of replicate laboratory measurements. Meanwhile, however, a new blender design had been developed and tested. The results of this test are listed under Blender I1 and in Table 111. I t was not possible to operate this one exactly as designed. Upon dumping the contents it was found that a static region had existed so that a poorly mixed segment emerged as shown by the samples from the last three boxes. This flaw in the construction was readily corrected, and accordingly these three values have been excluded from the data analysis in order to obtain a comparison between this blender and the other. This results in a value for the ratio s2/uO2 of about 11, the lowest observed theretofore. With these observations as a basis, it appeared likely that a reasonable criterion for commercial acceptability could be in the region 5 uo2to 10 uo2. The blenders just described were stationary tanks through which the pellets were circulated by a pneumatic transfer system. They are capable of blending of the order of 50,000 lb., and, while it appears that it will require several hours to achieve blends of the quality just described, investment and operating expenses for these blenders are comparatively low. T o extend the comparisons of performance, data were collected on the properties of the blends obtained from a laboratory twin-shell blender (10 lb.), a double-cone blender (1500 lb.), and two ribbon blenders (1500 lb. and 15 lb.). The twin-shell blender was sampled periodically 38 times, with the sample being returned to the blender. The samples were comparatively small and were anaVOL. 5 8
NO. 6
J U N E 1 9 6 6 41
lyzed by separating and counting the colored pellets. The others were sampled at the end of 15 minutes, as the material was drained out of the blender. Ten samples were collected and the weight percentage of colored pellets was determined. As shown by Table VI1 these sample variances are significantly lower than for the in-tank blenders. I n fact, for the number of degrees of freedom, the ratios s2/uo2 do not exceed critical (a = 0.05) values of x z / ( m - 1). A prototype of Blender 11, modified to conform to most recent design changes, was tested using neutral and colored pellets. This was a special purpose test; no opportunity existed to set the sample size large enough to apprehend departure of any important magnitude from the specified made-up proportion. Weight proportions of the colored pellets were 10 lb. of each color to 3750 lb. of neutral. Since the samples were to be assayed by counting the colored pellets in a onepint sample, it was imperative that mean pellet weights be determined. Thus, although the weight proportion of each was 0.0027, taking account of mean pellet weights changed this to red: total = 0.00122: 1 and black: total = 0.00138:l. Further, the expected number of pellets in a pint sample was 19.5 red, 22.0 black. Note, however, the black pellets were not added until after the first run with red. This test was performed by loading the blender with successive increments of 750 lb. of neutral pellets and 2 lb. of red pellets. The blender was then drained and sampled at approximately 100-lb. intervals. The number of pellets in each of the 39 samples is listed in Table 1V. The pellet counts are random with the Poisson frequency distribution, or closely approximated thereby. Thus, the hypothesized variance, uo2, of pellet counts equals 19.5. M’e note that the variance of these sample counts is actually 143.2, or 7.3 times the ideal variance. This is the lowest variance ratio reported thus far for our in-tank blender, and there had been no circulation a t all-the contents were simply drained. The blender was recharged with the same pellets, and the 10 lb. of black pellets were then added to the top of the charge (3750 lb. of neutral plus 10 lb. of red). Circulation was commenced a t a rate of about 160 lb. per minute, and the flowing stream was sampled at oneminute intervals. Circulation was continued until the contents “turned over” five complete cycles. These data are listed under Tests 2 to 6 in Table IV. Since no black pellets appeared before the seventh sample, data from the first cycle were not used to estimate sample variance. For the remaining cycles, however, it is clear that the contents are steadily approaching statistical ideality-i.e., thermodynamic equilibrium. The critical value for the higher 5 percentile of the ratio s2/uo2 with 21 degrees of freedom is 1.56. Thus, while these sample variances are significantly greater than the ideal, we are unable to assess the influence of the simplifying assumptions necessary in estimation of goz. However, after completion of the circulation tests, the blender was drained and 24 fairly evenly spaced samples were taken. Both red and black pellets were 42
INDUSTRIAL A N D ENGINEERING CHEMISTRY
TABLE V. PELLET COUNTS IN 300-GM. BLENDER- D RA I N SAM PLES“ Sample A‘o. 1 2 3 4 5 6 7
I 1 ~
1
8 9 10 11 12 13 14
15 16 17 18 19 20 21 22 23 24 Sample mean Sample variance Ideal variance Ratio
Pellet Count Black
15 28 17 30 28 31 26 28 20 26 14 16 30 25 24 23 25 26 26 17 20 16 25 10 22.9 35.4 22.0 1.61
Red
13 22 14 18 16 15 23 15 26 20 17 19 21 20 10 11 17 17.7 22.3 19.5 1.09
Sample taken at approximately 150-lb. interaals.
counted, and these data are listed in Table V. When we recall that in some earlier tests, the indications of ideality were not borne out when drain samples were assayed, it is particularly interesting to observe that in this case the ratio s2/uo2 for black is at the borderline of the upper 5 percentile and that for the red is well within the noncritical region for the hypothesis :
E(?)
= uo2
The tabulations in Table VI list results of a test in which the two components were alike in every respect to the extent possible, except for melt viscosity. There was available for this test a machine for continuous measurement of melt viscosity that has demonstrated a significantly higher precision than the ASTM laboratory method. The blender was charged with equal portions of two polyolefins; one having a melt index of 0.325, and the other 1.04. The melt index of the blend as predicted from the logarithmic mean of these two values is 0.58. The standard deviation of the continuous melt indexer, setup to setup, when looking at the same material had been measured at 0.062 melt index unit. Therefore, this served as a criterion for judging observed variations in melt index of samples from the blend. As shown by the data of Table VI, the indicated melt index of the blend quickly approached the predicted log mean, and the sample standard deviation was actually lower than 0.062. There was no way to compare sample and population variances on the basis of pellet counts, but it is clear that this in-tank blender quickly produced a commercially acceptable blend when judged on the basis of melt index.
TABLE VI. CONTINUOUS MELT INDEXER DATA ON BLENDER SAMPLES Sample Description Ref. Blend component 1B Blend component 1C Blend component 2 1-min. circulation 3 min. 5 min. Ref.” 7 min. 10 min. 15 min. Ref. 25 min. Ref. 35 min. 50 min. 65 min. 87 min. Mean, blender, 10 samples Std. deviation a
Readinj 0.925 0.33 0.32 1.04 0.34 0.51 0.51 0.925 0.53 0.56 0.62 0.94 0.57 0.925 0.62 0.59 0.58 0.58
Sample Description
Reading
After 500-lb. drain Ref.“ 1000 lb. 1500 lb. 2000 lb. 2500 lb. 3000 lb. 3250 lb. 3500 lb. 3600 lb.
0.62 0.94 0.59 0.57 0.58 0.52 0.59 0.53 0.44 0.43
Mean (7 samples) Std. deviation
0.571 0.035
Mean (9 samples) Std. deviation
0.541 0.068
0.567 0.04
For instrument calibration.
N.B.: instrument std. dev. was measured
(19 observations) at 0.062.
Proposal for Characterization in Future
The random variable represented by the ratio s2/uo2 has the frequency distribution of x 2 / ( m - 1). This enables us to judge approach to statistical ideality by comparing observed values with fractiles of the cumulative X 2 / ( m - 1) distribution. Thus, it may be possible to establish that, so long as the overall proportion is correct, a blend may be commercially acceptable, even though apparently far from statistically ideal, by the usual criteria. For instance, to apprehend that the blend variance is some specified order of magnitude larger than the ideal, sample size depends upon the ventured rates of error. If a = p = 0.05, the sample size for orders of magnitude ul2/uZ2 = 2, 3, 4, 5 can be read from the intersections of the curves of Figure 1. The curves plotted represent the upper 5 percentile of the random variable SS/uo2 where SS = Zx,2 - ( Z X , ) ~ = / ~ (m - 1) s2,and lower 5 percentile of S S / q 2 ,for u12, for u12/uo2 = X2 = 2, 3 , 4 , 5 . Now, while the conditions illustrated by Figure 1 provide systematic decision rules, it is difficult to use because it is rarely possible to specify the alternative variance that is representative of a commercially acceptable blend. Further, it may not be of greatest importance to determine precisely what the system variance is, and, in fact, for most actual systems, it may well be impossible to determine what the ideal, base variance should be. Thus, in blending pelleted polyolefins, the base variance is estimated from Equation 3. However, certain physical characteristics of the pellets make it extremely doubtful that the motion of individual pellets is truly random. Accordingly, the properties of a n ideal binomial distribution could rarely be achieved. Wald (6) developed sequential procedures for testing hypotheses that can Be used to construct a n explicit decision rule in the face of this uncertainty. As shown by Figure 1, a minimum of about 10 samples would be needed to discriminate (at frequencies of Type I and
10-
CONTINUE SAMPLING
5
7
9 11 13 15 17 19 21 NUMBER OF INDEPENDENT SAMPLES IN SEQUENCE, n
23
25
Figure 2. Conditions f o r testing null hypothesis
Type I1 errors of 0.05 each) between a base variance uo2 and one that equals exactly 5 no2. Now suppose that a variance of 5 uo2 is commercially acceptable-that is, when the material is used for its ultimate purpose, variations of this equivalent magnitude do not cause perceptible variations in the product. I t is not important to know precisely how much less than 5 go2 the true variance may be. However, it is important that no material be produced that has a variance greater than, say 10 uo2. Accordingly, let the null hypothesis be that u2 5 5 uo2; and the alternative, u2 2 10 go2. The conditions for testing these hypotheses sequentially are shown by Figure 2. Thus, for the polymer pellets we have been discussing, 10 wt. yo of a given polymer, and sample weight approximately 600 grams (typical result from Equation S), we have uo2 0.05. The null hypothesis is, then, u2 5 0.25, and the alternative is u2 1 0.50. An observation is made by taking a sequence of, say, 1000-gram samples during blending and finding the weight percentage of the specified component therein. When we denote values as 8,,the statistic computed is the c c corrected sum of squares,’’ SS = 28: - ( Z p $ / m . If the sum enters the upper region, the null hypothesis is rejected; that is, conclude the material is not satisfactorily blended (a 5 0.05). I n that case, start over with the calculations while continuing to blend. It may not be necessary to discard all of the preceding observations. Certainly, the mixture has been improving continuously in uniformity. Therefore, if the last few values appear to be leveling off somewhat, start the new calculations with an observation a t or following the point where the slope levels somewhat. This may lead again into the upper region of the chart, but nothing will have been lost. Eventually, if the equipment is capable of achieving a blend such that u2 5 5 uo2, the cumulative sum will enter the lower region (/3 5 0.05). For any level of go2,a! and @, and defined null and alternaand XA2u02),the conditions for tive hypotheses (X,+02 VOL. 5 8
NO. 6 J U N E 1 9 6 6 43
constructing charts such as Figure 2 are described in detail in Chapter 8 of reference (6). While practical difficulties exist in application of this method in evaluating every blend, it is a systematic procedure for evaluating blenders whereby it may be determined precisely what operating conditions are required to achieve a blend of some specified quality using the variance ratio as criterion. I t must be emphasized that the random variable being scrutinized is weight percentage, the frequency distribution of which may be approximated only very roughly by the binomial. Based on the close agreement between s2 and g o 2 obtained in the series of tests illustrated by Table VII, however, one might conjecture that it is possible to evaluate an empirical relationship between the variance as determined from the assumption of existence of an ideal binomial-both from the standpoints of physical properties of the individual particles, and of random sampling-whence a numerator could be found for Buslik’s equation. For instance, in a rough test of the Buslik relationship, the sample size was reduced for the 50-lb. ribbon-blender test (Table VII) whence the hypothesized variance g o 2 was increased to 0.032. The observed variance was higher, and the ratio s2/uo2observed was 1.38, well within the expected random variation. For practical purposes, it may be found that the variance as estimated from Buslik’s equation should be multiplied by something on the order of 1.5 (average of the five values shown in Table VI1 is 1.52). (This would be an interesting project for future study, say in a graduate student research program.)
TABLE VII.
44
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
I
I
Sample NO.
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Conclusions
I t has been shown that the statistical ideality of blends of dry, particulate solids can be judged by the properties of the x2 statistic, and that severe departure from statistical ideality can be tolerated in certain classes of blende.g., polyolefin pellets blended to a specified melt viscosity. An equation due to Buslik relating mean particle weights to the variance of the proportion in a wo-component system expressed as weight percentage vas applied in a variety of systems and found valid. rhis equation, accordingly, provides the denominator, g o 2 , of the x2 statistic; ( m - 1) s2/uo2. The numerator, ( m - 1) s2 can be calculated from the observed weight percentages P, from the expression, BP,2 - (BP,)2/m. Whatever the tolerable departures from ideality, the degree of approach to ideal blending can be judged by the x2 statistic, or by the ratio, sample variance to ideal variance-i.e., S ~ / U O ~ . Further, conventional statistical significance tests can be designed for the hypotheses : E (?) = P (the expected value of the sample proportion .?‘ equals the made-up proportion, P ) against E ( p ) = P + M P (the expected value of the sample proportion equals some other proportion, discrepant from the made-up proportion by It M P , due to holdup in the blender). Depending upon the magnitude of M to be apprehended, the magnitude of Type I
’
S U M M A R Y OF RESULTS FROM BLENDER TESTS
Average
~
~
Twin-
~
Shell.
;[ 40
24
2;
29 31 29 27 35 18 23 28
i
1
1
70%-
Lb.
70% 50 Lb.
.61 . 00 .95 .I9 .83
9.72 10.26 10.58 10.20 IO. 71 10.55 10.17 I O . 29 10.23 9.96
9.79 9.79 10.12 9.53 9.67 10.18 10.19 IO. 22 10.18 10.38
10.34 9.89 10.15 9.92 10.02 10.33 9 .75 10.02 10.63 10.89
2.86
10.27
10.01
10.19
2.88 2.64 78
.90 .80
I
I
7500
:;
I 23 41 ~
30 23 45 37 34 28 43 31 23 23 33 28 25 26 32 21 23 29.5
variance 29.7 Observed variance 4 2 . 8 Ratio 1.44 Pellet counts:
__.__
707,~
Hypothesized
a
~-
Rtbbon
Dou ble-Cone 3.22%
I
__--
0.019 0.029 1.53
0.032 0.087 1.67
’ ~
0.052 0.081 1.56
0.092 0.127 1.38
colored to neutral ratio = 7:79 ( 2 02. in 10 16.).
and Type I1 decision errors ventured on the stated hypotheses, sample size, W , weight of sample (or number of particles in the sample) can be determined, under the assumptions either of statistical ideality, or some tolerable ratio, s 2 / u g , Previous investigators have misinterpreted the properties of the statistical terms they employed to assess the properties of dry blends and have ignored the role of Type I1 error in constructing their hypothesis tests. REFERENCES (1) Bennett, C. A., Franklin, N.,“Statisrical Analysis in Chemistry and the Chemical Industry,” Wiley, New York, 1954. ( 2 ) Buslik, “Mixing and Sampling. with Special Reference to Multi-Sized Granular Material,” A.S.T.M. Bull. No. 165, 67, April 1950. (3) Dixon, W. F., Massey, F., “Introduction to Statistical Analysis,” McGrawHill, New York, 1951. (4) Hald, A., “Statistical Theory with Engineering Applications,” Wiley, New York. 1952. (5) Stange, K., Chem. Ing. Tech. 26, 150-55 (1954). (6) Wald, A., “Sequential Analysis,” Wiley, New York, 1947.
n.,
