I
FRED R. SHELDON
I
Inorganic Chemicals Research and Development Department, Food Machinery and Chemical Corp., Princeton, N. J.
Statistical Techniques Applied to Production Situations These statistical techniques can be used successfully for studying variations in the chemical research laboratory and are equally valuable for chemical industry production situations which may be even more variable
T,E
chemical industry, outstanding in many respects, can claim no superiority when it comes to freedom from the vagaries of fate. Chance and variation in general are certainly as prevalent in the chemical plant as in the factories of other industries. Therefore, statistical methods and the mathematical approach are becoming more widely used in the chemical process industries. Over the last ten years there has been a steady growth of statistical techniques applied to chemical problems. At first these techniques were used on problems related to chemical research and development. More recently, the early laboratory successes have led to statistical experimentation on production units. If statistical methods have shown themselves to be of value in the laboratory when variations are generally a t a minimum, there can be no doubt of their potential value in the plant where chance effects and random fluctuations are usually of a far higher order. T h e purpose of this discussion is to give typical examples of a few valuable methods in sufficient detail that they may be used as guides for solving similar production problems occurring in the chemical process industries.
Table 1. Incomplete Factorial Experiment unbleached Bleached Operator Operator B C D A B C 59.8 60.7 61.0 69.2 68.8 69.3 60.2 60.7 68.9 70.1 69.5 60.8 60.4 60.5 60.6 68.6 69.4 68.9 59.9 60.9 60.5 69.3 69.3 68.6 60.0 60.3 60.5 68.0 69.2 69.8 60.1 60.6 60.7 68.7 69.4 69.4
-
A 59.8 60.0 60.8 60.8 59.8 Av. 60.2
Buchner funnel, would be air-dried and once a day measured for reflectance on the standard instrument by the brightness meter operator. Discounting pulp differences, many variations can occur in this procedure; hence, these variations were explored by variance analysis of a designed experiment. A nested experimental design or, as Brownlee ( 3 ) terms it, an “incomplete factorial analysis” was conducted as follows : 1. All four shift operators made five handsheets each on two pulp samples (unbleached and bleached) which were especially prepared for the purpose. 2. The pulp handsheets were then read by the brightness tester, with the results given in Table I.
D 69.7 69.6 69.5 69.5 69.9 69.6
The experiment fulfills Brownlee’s definition of an incomplete factorial in that there is no relationship between, for example, sheet four of operator A and sheet four of operator C-Le., no one-to-one correspondence exists for handsheets. As information was desired on the uncertainty common to the brightness of each pulp separately, the analysis was split into two parts. The resulting variance analyses appear in Table IIA, and, from the mean squares values, separate variances were calculated, Table IIB. The within operator variance (.,”) is equal to the within operator (residual) mean s uare. The between operator variance ( u 9b ) is found from the equation : u;
+ 5 u.,”= between operator means square
Analysis of Pulp Handsheet Brightness Analysis of Variance (Data X 10) Sums of Squares Mean Squares d.f. Unbl. B1. Unbl. B1.
Table II.
A.
Sampling and Reliability of Routine Test Data With routine testing for control these questions frequently arise, as: How many samples should be taken? What does a change in average test value mean? How reliable is the test? Such queries can be answered only after a study has been made of the variations inherent in the test and in the way the test is applied in the situation being considered. For example, in a pulp mill producing both bleached and unbleached pulp bleach plant performance was based on the difference between bleached and unbleached pulp brightnesses as measured on a reflectance meter, as the standard within the industry. Each shift operator would make a single handsheet-Le., pulp padfrom samples of bleached and unbleached pulp. The handsheets, formed on a
Source Between operators 3 134.0 262.15 Within operators 16 170.0 222.80 19 304.0 484.95 Total Required for significance at 5 % error level. B.
Between operators (u:) = Within operators (u:) = u$ = ob” u: (.;)*‘* -
+
C.
Sheets 1 2 5 W
44.667 10.625
...
87.382 13.925
...
F Ratio B1. Unbl. 4.20 6.28 (3.24) a -
Variances Unbleached 0.06808 0.10625 0.17433 0.41752
Error Limits at 5y0 Level Unbleached k0.8 0.7 0.6 0.5
VOL. 52, NO. 6
Bleached 0.14691 0.13925 0.28616 0.53493
Bleached rtl.0 0.9 0.8 0.8
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The sum of the two variances is the total variance (ui), and its square root is the standard deviation for brightness on single handsheets from a single operator. The calculated standard deviations give rise, at the 570 error level, to uncertainties of 1 0 . 8 and 1 1 . 0 unit for unbleached and bleached pulps, respectively. In actual practice, if each operator made two handsheets and the brightness reported as the average of those two sheets, a small reduction in uncertainty would result. The within operator variance (u:) could be halved and the new total variance would then be the sum of this halved value and the former operator varianceu:. Similarly, five i.e., us = u2/2 handsheets per operator would reduce uw2 to one fifth its original value, etc. The 5% error limits for several numbers of handsheets are shown in Table IIC. With the same four operators, even an infinite number of handsheets would not reduce the uncertainty of brightness reports below 0.5 and 0.8 unit for the two pulps. This analysis indicated that for further improvement in the uncertainty associated with brightness readings an examination of operator technique is in order. Unfortunately, the experiment tells nothing about the effect of more than one reading on each brightness handsheet. Original instructions for conducting this test specified four brightness readings per sheetLe., the addition of brightness readings as another (incomplete) factor. With this additional information it is possible to divide the within operator variance into two parts-namely, that due to handsheets and that due to readings. The net result, as far as the operators are concerned, probably remained unchanged. Nevertheless, without an estimate of the effect of more readings per sheet, the entire study loses much of its effectiveness. Additional data might have been obtained with very little extra work, and their lack illustrates the need for careful planning and follow-through on the part of the investigator.
+
T h e incomplete factorial experiment (or nested design) and its subsequent analysis of variance are recommended as powerful tools for resolving complex sampling and testing problems and for help in answering the sometimes embarrassing questions posed above.
ment of the active bleaching agent concentration. Unfortunately, average values were useless in comparing the two methods because many other variables were acting to affect the final pulp brightness. In other words, as so often happens on production equipment, things did not remain constant from July to November. Unbleached pulp brightness varied, daily tonnage requirements changed, higher brightness pulps were needed for some orders, lower colors for others, etc. Although the simpler bleaching method could give equal performance at no increase in cost, there was little direct evidence to support this or to contradict the comparison which showed it at a cost disadvantage. Bleach plant data were subjected to regression analysis ( 7 ) and of the many variables examined-unbleached pulp brightness, amount of bleaching agent, and daily pulp flow rate (to a minor extent)were mainly responsible for variations in final bleached pulp brightness. Regression equations were derived as follows :
+
Std. dev. B1. Brightness Complex
48.39
(1)
0.56
+ 0.332X1
+ 1.09Xz - 0.008X3
where XI = unbleached brightness
Xz
(2)
57,O to 61.0
70 HzOz, 50% on pulp
X I = throughput in tons/day
1 . 4 5 to 2.20
40 to 80
These equations plotted (Figure 1) clearly showed that the simpler bleaching method is at least equivalent to the more complex system. When the pulp mill finally adopted the simpler bleaching treatment, a three dimensional model was made of Equation 1 setting X S a t a constant 70 tons per day. This model had a t its base the two remaining independent variables, unbleached brightness and per cent bleaching agent, while the third dimension
74
t
Routine plant records serve to ensure proper plant operation and, of course, become the historical record of unit performance. Like other historical documents these plant records often do no more after their filing than gather dust. Frequently they are put to better use as sources of vital information on process behavior and on the relative importance of process variables.
508
=
=
Std. dev. = 0.63
Use of Plant Records-Multiple Regression
For example, in a similar groundwood bleach plant, two slightly different bleaching treatments were compared over 5 months by running the two treatments on alternate two-week periods. One of the treatments was slightly more expensive than the other, but supposedly more efficient and more simply operated. Thus, the comparison was made by running this simpler treatment at cost equivalence to the more complex method through adjust-
+
B1. Brightness = 16.37 0.818X1 Simple 3.29x2 - 0.007X3
BRIGHTNESS
SIMPLE
PROCESS
67
t
TONSIDAY.
1.4
60
18
22
%HI 0, 5 0 %
Figure 1. Final brightnesses at various peroxide levels show that the simpler bleaching method is equivalent to the more complex system
INDUSTRIALA N D ENGINEERINGCHEMISTRY
represented final pulp brightness. Thus, it was possible to predict bleach plant results within reasonably narrow limits, or, more importantly, to prescribe correct amount of bleaching agent needed to achieve a given bleached brightness from a specified original pulp color. Although considerable effort is needed to boil plant operating records down to useful information, regression analysis offers a scientific means for so doing. With some prior knowledge of system behavior, a quantitative measure of the importance of variables frequently pays for itself in better control and more economical operation.
Experiments leading to Optimization-Response Surfaces I n production situations not only is there a great amount of variation, but also there is a great reluctance on the part of production supervisors toward excessive experimentation on a producing unit. Such units are generally behind schedule and the merest suggestion for use in experiments is abhorrent to those in charge, as they immediately see visions of high rejects, off specification material, etc. Thus, when a productionscale experimenter finally gets a chance on a full scale unit, he must get as much information in the shortest possible time, not only because of the high cost per test but also because if he interferes too much with production, he will not be invited back. Modified response surface methodology (2) offers one means of getting much information in a few tests and can be used advantageously in production. As an example, let us take the "superbleaching" of bleached kraft pulp. Kraft is bleached at the rate of 200 to 250 tons per day in multistage bleach plants using chlorine, caustic, and more recently chlorine dioxide. After the fifth and often final bleaching stage, the pulp, at between 80 and 85 brightness, is washed and sheeted for storage and/or shipment. The sheeting operation uses drying ovens or cylinders to reduce moisture content to 5 or 10% and the pulp is generally baled at 120' to 140 F. and about 83 brightness. During the course of storage and shipment the pulp temperature within the bale falls and generally so does the pulp brightness. Superbleaching is to prevent this bale from brightness loss during aging and consists of spraying an HxOz-polyphosphate solution onto the wet pulp web ahead of the dryer. It is sometimes advantageous to include a sequestering agent of EDTA (ethylenediaminetetraacetic acid) type in the superbleaching liquor. In general, the optimum amounts of the three chemicals, HzOz, polyphosphate, and EDTA, vary with each pulp and with prior bleaching treatments so that best results are not achieved in the pulp mill until a variety of combinations have been explored. The ideal approach is to examine the three variables in a 33 factorial experiment -Le., each variable at each of three levels. This design's 27 tests would present various production problems, the least of which
STATISTICS IN CHEMICAL PROCESSES A.
TESTS
AND
CONDITIONS:
B. T E S T
RESULTS:
AB
(TREATED-CONTROL)
(CALCULATED C.
REGRESSION
A B = 2.35+
l.ZX,+ 0.23X:
Po + + PZZx2 +
PlXl
+ p2xz + pax3 + + $.
@33x,”
PlZXlx2
+
P23xZx3
+2.25X2+0.06X~-l.IIX3-OO.9X~
0.9X,X3+
0.2Xp3
This is a twelve-test design for examining three variables at three levels
would be the storage and periodic examination and testing of 27 bales (plus controls) at 400 pounds per bale. The 33 design would, however, permit the expression of bale aged brightness in an equation such as: Y =
While the validity of the first assumption can be tested after the data have been gathered, the second requires patience and a somewhat empirical approach to find suitable designs. One such design appears in Figure 2,A. Figure 2,B shows the 12 experimental brightness differences obtained after bale aging (and also the values calculated from the Equation 3). Figure 2,C shows the equation derived from the 12 experimental values. The equation of Figure 2,C leads to a graphical representation of the superbleaching system as in Figure 3. This graph clearly shows the interdependence of the variables-Le., how the brightness response varies with increase in H202 concentration depending on levels of phosphate and EDTA. Such a picture of the system materially assists optimum operation of the process in the plant.
PllX?
P13XlX8
+
8
(3)
where: the P’s are coefficients of the variables : = ( 7 0 Hz02 - 0.25)/0.15 XZ = (% Polyphosphate - 0.5)/0.2 X3 = (70 EDTA - 0.075)/0.075 E = error Y = bale aged brightness difference (super bleached-control) = AB Xi
There are, however, only 10 unknowns in the desired Equation 3, and thus only 10 pieces of data are needed to calculate these unknowns, plus a couple additional for estimating the error. In the 33 design there are 27 pieces of data. Thus there appears to be some 15 to 17 unnecessary tests-i.e., 27 - 10 = 17 or 27 - 12 = 15. This is actually the case, based on the following assumptions : 1. That Equation 3 is an adequate representation of the system being examined. 2. That the 10 (or 12) tests selected for performance allow calculation of coefficients for Equation 3.
Naturally, when time and money permit it is preferable to conduct the full 27 tests of a 33 factorial for a better estimate of Equation 3. T h e 15 other tests may contain valuable additional information and when carried out, provide the basis for adding terms to the equation. Thus, with all tests run, a more complex equation may be assumed as a model. I n any case, additional confirming tests should be run for comparing the
arvARIOU
s
-
PHOSPHATE
05%-
Figure 3. The interdependence of variables is shown by the effect of HzOa and EDTA on AB at various phosphate levels
LEVELS
0.4 0
0.25 o’z5
$ 3
3 \
0.10
0
’
0.07B
’
0.15 % E D T A
equation derived from the experimental design, and modifications in the equation made if necessary. Periodic comparison of results predicted by the equation should be made with actual plant conditions to make sure that the equation still applies and that the process is being run at its optimum. While the technique described provides a means of defining the important variables in a process and their relative effects, the equations so derived may not represent the actual interrelationships which pertain. Such equations, however, as per the basic assumptions, are often adequate models of complex systems and allow useful work to be done on or by the system. Adequacy of the “working equation)’ as a representation of actuality can be tested by comparing actual us. calculated valuesor by a regression analysis of the data. I n review, statistical techniques have been shown to be useful in improving plant operation by reducing the uncertainties associated with sampling and testing, dusty plant records, and complex multivariant processes. The statistical approach is designed to sort out real effects from chance effects. Nowhere is it more important to separate fact from fancy than in production situations.
01 %
0.4 0
f
VALUES)
EQUATION:
t0.107XI X2-
Figure 2.
AT 2 WEEKS
0
’
0.0’15
’
1
0.10
o.is
literature Cited (1) Bennett, C. A., Franklin, N. L., “Statistical Analysis in Chemistry and the Chemical Industry,” Wiley, New York, 1954. (2) Box, G. E. P., Biometrics 10, 16-60 (1954). (3) Brownlee, K. A., “Industrial Experimentation,” 3rd ed., Chemical Publishing Co., Brooklyn, N. Y . , 1949. REOEIVED for review September 15, 1959 ACCEPTEDApril 12, 1960 Division of Industrial and Engineering Chemistry, 136th Meeting, ACS, Atlantic City, N. J., September 1959. VOL. 52. NO. 6
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509
