tatistic A REPRODUCIBILITY STUDY ROLAND F. MUHNDORFF The Texas Company, Port Arthur, Tex.
A few of the tools of modern statistics have been applied to the data obtained €rom pilot unit operation during a reproducibility study. Control charts indicated statistical control was maintained during operation. A n L1 test showed no nonhomogeneity of variances. Analysis of variance indicated excessive variation of the average yield data; the assignable cause operating here was believed to be a time element. N RECENT years the application of modern statistical procedures to the problems of industry has been increasing steadily. The purpose of this paper is to apply only a few of the tools of statistics to pilot unit operation. Specifically, these tools will be used to study the results of runs made on a pilot unit for the purpose of determining the reproducibility of thc pilot unit operation. OPERATION OF PILOT UNIT
This study was made on a fluidized fixed-bed catalytic cracking pilot unit. The unit is automatic in that all temperatures are automatically controlled and a time cycle controller maintains the cycle described below. Figure 1 shows a simplified flow diagram of this unit. The reactor is a 4.25-inch inside diameter stainless steel tube 107.5 inches in length. The catalyst charge is 20 pounds maximum. Referring to Figure 1, the fresh charge is pumped through a heater into the bottom of the lead immersed reactor; the oil. vapors rise through the catalyst bed. The product vapors leave the catalyst bed and pass out of the reactor through a filter tube which retains entrained catalyst. From this point the vapors are direcJed by means of a three-way valve through a product cooler into a gas separator. Here samples of gas and total liquid product are withdrawn. At the end of the process period, the time cycle controller stops the charge pump and opens a solenoid valve admitting nitrogen purge gas. The purge displaces the hydrocarbons out of the heater and reactor into the product recovery system. After the purge has been. completed, the time cycle controller closes the purge gas valve, opens a regencration air valve, and switches the three-way valve to direct the combustion gases through a cooler, separator drum, sampler, and meter. At the end of the regeneration period, the time cycle controller closes the air valve and opens a second purge gas valve. The purge gas displaccs the air and combustion gascs out of the
heater and reactor into the regeneration tail gas system. Xitrogen is used for this purge, also. On completion of the purge, the time cycle controller closes tho purge gas valve, switches the three-way valve to direct the effluent, from the reactor to the product recovery system, and starts the charge pump. This cycle then is repeated. Generally, operations were not interrupted betmpeen test periods. The charge was a regular gas oil cut, having an A.S.T.M. 50% boiling point of 648 O F. A used silica-alumina synthetic catalyst was employed t o prevent activity decline. The catalyst was changed after each series of test periods. The operating conditions specified for the unit were: Reactor temperature, O F . Space velocity wt. oil/hr./wt. cat. Catalyst to oii’ratio, wt. of c?t./wt. of oil Pressure on reactor, Ib./sq. in. Process period, minutes Total cycle, minutes
895 1.3 7.3 6
6.3 60
The total of 27 test runs was made under these conditions. CONTROL CHARTS
Reproducible data cannot be obtained from the pilot unit unless the unit operates under statistical control during the time the data are gathered. One method of detecting this type of control is the use of control charts such as the X , 8,and R types. Theoretically, every measurement made or point required to control. the operation of the unit should be maintained in statistical control. Obviously, this would be impractical when one considers the large number of controls used during the operation of the pilot unit. Therefore, it was decided that only the primary variables would be maintained in a state of statistical control during the operation and gathering of data. These variables were: reactor
T-~BLW I. CONTROL CHARTLIMITS
-X *18,0 X-_X *A72..55
Reactor bed temp., O F . Charge pump, ml./cyclc Product gaa, CU. ft./cycle Liquid hydrocarbon, mi./cycle
1300
I
X *5.0 -
-
X
r2.O
I
X *60
-
- *1.0 X 130
3.0 10 1.0
0-7.0 0-24
40
0-90
0-3.0
1301
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1949
sora per
4
Three Way Valve
Water
I
I
; F l
Filter
Cooler
I
I
I__,
To Product Gas Meter & Sampler
Controller
.on
A lr
Draw Off L i q u i d Sample
Draw Off
Lead Immersed P r e h e a t e r Figure 1.
Fluidized Fixed-Bed Catalytic Cracking Pilot Unit
bed temperature, charge pump rate, product gas rate, and liquid hydrocarbon product rate. From previous data taken during shakedown runs, standard deviations for the control points were calculated by the usual formula. The control chart limits as calculated by A.S.T.M. methods (1) are shown in Table I. Throughout the 27 test periods made during this study the unit operated reasonably well between the foregoing control limits., A control chart for a typical test period is shown in Figure 2. It represents the operation over a 32-cycle test period, the variable being reactor bed temperature. The chart shows that the central line together with the control limits for the X and R chart should have been about 1.4 F. lower. The values of the central line, during these test periods were calculated from data obtained by operating the unit for 8 cycles before starting the test period proper-for example, the average temperature of the &cycle prerun used in Figure 1 was 890" F. A slight shift between the average obtained during an %cycle prerun and the test run proper did not warrant it being considered out of control. Therefore the test period shown in Figure 2 was considered in statistical control during the time data were taken for the reproducibility study. Any such change, however, during the run proper would not have been accepted as satisfactory. The fact that there was a shift in averages between these two periods indicated an assignable cause was a t work. This was not investigated further. O
nus the gas oil. Yields of carbon, dry gas, butanes, pentanes, CSfree naphtha, and gas oil, as well as data on conversion, were obtained from: six test periods consisting of 8 cycles per test period; six consisting of 12 cycles; seven consisting of 24 cycles; and eight consisting of 32 cycles. This is a total of 27 runs. A cycle, it will be recalled, constitutes a process period followed by a nitrogen purge, then regeneration, and another nitrogen purge. The length of the total cycle was approximately 1hour.
TABLE 11. STANDARD DEVIATIONS AT VARIOUSCYCLESPER
TESTPERIOD
x,
REPRODUCIBILITY OF YIELD DATA
Test Periods Made. Carbon yields were determined by splitting off a fraction of the main regeneration gas, passing this fraction over hot copper oxide, and absorbing the dried carbon dioxide over Ascarite. Other yield data were determined from precision distillations and mass snectrometer analvses. The conversion is an uncorrected 430 conversion which is defined as 100 mi-
Sample Standard Deviation, Wt. % Yield (Single Test Period) --Cycles per test period-Carbon Dry gas Butane Pentane Cs-free naphtha Gas oil Conversion, vol. % Number of tests (No. of samples)
8 0.32 0.47 0.25 0.28 0.44 0.89 1.38 6
12 0.10 0.69 0.24
0.21 0.56 1.04 1.14 6
24 0.27 0.48 0.35 0.28 0.40 0.74 0.71 7
32 0.19 1.08 0.49 0.37 0.22 0.73 0.80 8
Standard Deviations of Yields. The standard deviations for a single test period as obtained from this study, at various numbers of cycles per test period, are shown in Table 11. These standard deviations are calculated in terms of weight per cent yields of carbon, dry gas, butane, pentane, C6-freenaphtha, and gas oil, and volume per cent uncorrected 430 conversion for 8, 12, 24, and 32 cycles per test period. The detailed data from which these were calculated are shown in Table 111. Normality of Distribution of Data. A rather simple test, the a test (2). was used to determine whether the s a m d e data came , from a normal distribution. This test is not sensitive for small
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1302
-
Vol. 41, No. 6
TABLE 111. DETAILED YIELDDATA Cycles per Test Period 8 12 24 32 8 12 24 32 8 12 24 32 8 12 24 32 8 12 24 32 8 12 24 32 8 12 24 32
Yield Carbon
Dry gas, C3 and lighter
Butanes
Pentanes
Naphtha, Cs-free
Gas oil
Conversion, uncorrected 430, vol. %
1
4.28 4.34 4.84 4.15 7.69 8.53 7.62 8.29 11.30 10.97 10.80 9.69 6.23 6.57 6.88 6.35 24.35 24.18 25.64 24.59 46.15 45.41 44.22 46.93 55,27 55.54 56.79 54.10
2 4.37 4.35 5.18 4.21 8.92 8.27 7.78 7.01 11.24 10.59 11.09 9.98 6.27 6.10 6.38 6.47 23.39 25.03 24.96 24.79 45.81 45.66 44.61 47.54 53.25 55.36 56.47 53.30
samples but may be used in practical applications as an indication of normality. The ratios found in this study are shown in Table IV; other tests that are available are not so simple to apply.
TABLE IV. u TESTRATIOS Cycles per test period Sample size Carbon wt. % Dry pa; wt. % Butane,’wt. % Pentane wt. yo Cs-free ;aphtha, wt. % Gas oil, wt. % Conversion, vol. % a Outside 5% limit but aitliin TSBLE
Sample Bize 6 7 8
v.
8 6 0.749 0.878 0.927a 0.746 O.699a 0.773 0.736 1% limit.
12 6 0,733 0.93Za 0,844 0.926 0.821 0.807 0.830
LIXITFOR
Upper Limits 1% 5% 0.9581 0,9262 0.9537 0,9224 0.9492 0.9186
U
24 7 0.870 0.825 0.837 0.67la 0.885 0,774 0,770
32 8 0,909 0,787 0.818 0.824 0.879 0,847 0.823
TEST
Mean of Distribution 0.82482 0.82347 0.82211
Lower Limits 1% 5% 0.7070 0.6521 0.7087 0.6552 0.7183 0.6573
A comparison of the values in Table IV with the 5% and 1% limits given in Table V indicates that the normality of only 4 of the 27 runs is doubtful. These are the runs outside the 5% limit. However, since all were within the 1%limit, i t was assumed that the indication that the data came from a normal distribution would hold for practical purposes. The sample sizes used for this test are too small to show much more. The next approach in a statistical analysis of this kind would be to combine the standard deviations shown in Table 11-that is, to pool each particular yield. Determination of Optimum Cycles per Test Period. Here the primary concern is the effect, if any, of the number of cycles per test period on the variation as measured by the standard deviation, It is possible to use suitable methods to determine if the standard deviations vary significantly from one another. Two such methods for homogeneity of variances can be found in the L1 test ( 2 ) and the Chi square test, Bartlett’s test ( 3 ) . The L, values calculated from the data are shown in Table VI. Since the sample size should be the same for each sample a weighted mean of 7 was used. For four groups and seven samples per group, the 1% and 5% levels of significance are 0.5634 minimum and 0.6697 minimum, respectively. Table VI shows that
Yields (Wt. % Basis Feed) Test No. 3 4 3 6 4.25 4.40 4.54 5.19 4.35 4.11 4.38 4.38 4.43 5.15 4.85 5.24 4.39 4.59 4.35 4.60 8.89 9.00 8.67 8.17 7.78 7.15 6.76 6.82 7.30 7.15 6.93 7.23 7.43 8.43 9.67 10.50 11.31 11.36 10.73 10.82 10.77 10.32 10.68 10.29 10.73 10.10 10.59 10.83 9.74 9.62 9.64 9.23 6.46 7.05 6.29 6.33 6.20 6.12 6.06 6.55 6.37 6.25 6.45 6.46 6.04 8.17 5.56 5.34 24.28 24.82 24.43 24.50 24.47 23.79 25.53 24.65 25.81 24.67 25.43 24.78 24.43 24.94 24.29 24.62 44.81 43.37 46.34 44.99 46.43 48.51 46.59 47.30 45.36 46.68 45.75 45.46 47.97 46.25 46.49 45.71 58.24 57.89 55.57 56.05 54.73 52.18 54.48 53.60 55,62 54.43 55.35 55.66 52.84 54.63 54.53 55.36
7
7
8
,, , , ,
.
,.. . . I
.,.
5.19 4.57
4.70
...
...
8:48 7.92
7:95
... 1o:ii 10.49
... .
, ,
5.87 6.25 ,
..
25:24 24.33
..
...
,.. 10:85
... ... .
, ,
6.38
... ...
.,.
2i:i4 ...
45:il 46.44
45:88
.,
.
5i:89 54.48
...
...
... 5i:i2
Average 4.51 4.32 4.98 4.41 8.p!
I
.a5
7.80 8.40 11.13 10.60 10.61 9.90 6.44 6.27 6.38 6.06 24.30 24.61 25.22 24.54 45.08 46,65 -15.31 46.65 55.71 54.32 55.74 54.30
no discrepancies exist. (Bartlett’s test can be used easily with unequal sample sizes. This test also when applied to these data showed no discrepancies.) These standard deviations, therefore, could be pooled into one for each particular yield. This was interpreted as meaning that the range of cycles per t m t period investigated does not have any practical effect on the variation. It was expected to have an effect, however, since the yields are cumulative and hence tend to average themselves as the lime on stream increases. I n other words, it was expected to find the standard deviations decreasing inversely as the square root of the number of cycles. Since this was not found, it was assumed that the unit was operating on the flat part of the curve for standard deviation plotted against the number of cycles per test period-that is, the time effect might be significant around 2, 3, or 4 cycles per test period. The pooled, weighted, estimated population standard deviations of a single test period as determined from consecutive test periods are shown in Table VII. Variation from One Period of Time to Another. Any significant differences between the average yield obtained under each group of 8, 12, 24, and 32 cycles-that is, how the avcragc yield of, say carbon, from the 8-cycle test periods rompares with the 12, or 24, or 32-cycle test periods-can be shown by the method of analysis of variance (2-4) since it has been determined that the variances are homogeneous. Consider, for example, the
TABLE VI. L1 TESTVALUES Carbon, wt. Yo Dry gas wt. Yo Butane,’wt. % Ca-free daphtha, Pentane wt. Yc wt. % Gas oil, wt. Yo Conversion, vol. % TaBLE
1’11.
POOLED
Carbon Dry gas Butane Pentane Ca-free naphtha Gas oil Conversion, vol. %
Li Values 0.7265 0.7340 0.7846 0.8845 0.8733 0.9878 0.9079
STANDARD DEVIATIONS Estd. Population Standard Deviation of a Single Test Period, Yield, Wt. 70 0.25 0.81 0.39 0.32 0.45 0.92 1.10
INDUSTRIAL AND ENGIN.EERING CHEMISTRY
June 1949
I
.. . . .. . .... 0
1
5
10
15
1.
30
25
20
35
Lower Limit = 885
X-Chart (Individuals)
L; Upper Limit
892.5
Central Line
890
Lower Limit
887.5
11. -%-Chart (Averages of Four)
5
10
15 Ill.
Figure 2.
20
7.0
Central Line
3.0
Lower Limit
0.0
35
30
25
Upper Limit
R-Chart (Ranges of Four)
Sample Control C h a r t for 32-Cycle Test Period
Data indicate central lines (I a n d 11) should have been 888Aa F.
data in Table VI11 presenting the carbon yields on the basis of weight per cent fresh feed obtained in this study. The problem is to determine if the column means or averages vary significantly among themselves-that is, are the variations in the values 4.51, 4.32, 4.98, and 4.44 normal or is some assignable cause responsible for their variation?
TABLE VIII. c
Test period 1 2 3 4 5 6 7 8 Average
CARBON YIELDS
Yields (Fresh Feed Basis) % Cycles per test period
8 4.28 4.37 4.25 4.40 4.54 5.19
12 4.34 4.35 4.35 4.11 4.38 4.38
4.51
4.32
.. ..
.. , .
7
24 4.84 5.18 4.43 5.15 4.85 5.24 5.19
..
I ~
4.98
32 4.15 4.21 4.39 4.59 4.35 4.60 4.57 4.70 4.44
If a simple analysis of variance is applied t o the data in Table VIII, tests of significance can be made through the F test. The results of such an analysis are given in Table IX which lists the F ratios found. An F table shows the 1% and 5% levels of significance as 4.76 and 3.03, respectively. Comparing these values with the ratios in Table IX it is found that, with one exception, all are outside the 5% limit and all but three are outside the 1 % Limit also. OF VARIANCE ( F TEST) TABLE IX. ANALYSIS
F-Ratio
Gas oil, ,wt. % Conversion, vol. %
8.96 3.09 11.69 1.96 5.19 5.66 3.75
1303
standard deviations calculated from test periods made from one period of time to another, with shutdowns or operating conditions changed between test runs, will be expected to be greater than when calculated from consecutive test periods as in the case of this study. The excessive variation in the column means is judged by the varihion within the column means. This variation within the column means is none other than the pooled standard deviations shown in the previous section. The pooled standard deviations were calculated from consecutive test periods and do not take into consideration as much variation due to longer time intervals as the column means do. Therefore, had the within variation been based on runs from one period of time to another instead of consecutive runs, it is doubtful if any significant variation would have been found in the column means. A study of the variation from one period of time to another is being conducted currently. I n view of the foregoing discussion i t would be wise t o use the total variation, which includes the column mean variation, for measuring the standard deviations of the different yields; the total variation would be a better measure of the expected variation as i t does include more of the time element than the within or pooled variation shown previously. This total variation will suffice until the study of the variation from one period of time to another is completed. Standard deviations based on total variation are shown in Table X. CONCLUSIONS
From the results obtained in this study, the following conclusions may be drawn: 1. Data obtained in test periods of 8, 12, 24, or 32 cycles are equally satisfactory and, therefore, there is no reason for extending the run past 8 cycles unless additional product is required. 2. The estimated standard deviations of data which are a measure of the variation obtained in this study are shown in Table X.
TABLE X. STANDARD DEVIATIONS-TOTAL VARIATION Estimated Population Standard Deviation of a Single Test Period, Yield, Wt. yo Carbon Dry gas Butane Pentane Cc-free naphtha Gas oil Conversion, vol. %
0.35 0.90 0.58 0.34 0.54 1.14 1.27
3. The use of control charts showed that operations generally conformed to normal distributions. This was further substantiated by the a test which indicated that the yield data from each series of runs came from a normal distribution. ACKNOWLEDGMENT
The author wishes to express his thanks to H. L. Madinger for his help in obtaining the data from the unit and to J. J. Kent who assisted in making the calculations. LITERATURE CITED
(1) Am. SOC.Testing Materials, “Manual on Presentation of Data,”
Committee E-1 (1945).
It is believed that the assignable cause responsible for these nonhomogeneous means is the changes in equipment or operating conditions which take place during a long time interval. This may be explained as follows: It is desirable to introduce the concept of variation from one period of time to another. The variances determined over a long period of time will, in general, be larger than if measured over short intervals of time. Therefore,
(2) Freeman, H. A., “Industrial Statistics,” New York, John Wiley & Sons, 1946.
(3) Rider, P. R., “Introduction to Modern Statistical Methods,” New York, John Wiley & Sons, 1939. (4) Snedecor, G . W., “Statistical Methods,” Ames, Iowa, Iowa State College Press, 1946. RECEIVED January 17,1949. Presented before the Fourth Annual Southwest Regional Meeting of the ArmRxcAN CHEMICAL SOCIETY,Shreveport, La., December 9 to 11, 1948.
