I
A. E. HOERL E. 1. du Pont de Nemours & Co.,Inc., Engineering Service Division, Wilmington, Del.
Statistical Analysis of an
Industrial Production Problem I In the operation of chemical processes a major problem exists in determining performance criteria which approximates the true optimum conditions dictated by all of the economic considerations, To statistically approach the optimization of these applications requires:
b derivation of a reliable mathematical representation of the process behavior as a function of operation variables b incorporation of the important economic factors into the basic model of the process numerical optimization of the model to determine the operating conditions which will result in the economic optimization of process operation I
A is discussed in which this procedure was carried out. FollowSPECIFIC PROCESS
ing the application of the recommendations a plant test has shown a savings amounting to several hundred thousands of dollars. One concept of universal interest in the operation of process equipment is the optimization of the process operation-how to operate the process in such a way that maximum earnings relative to sales potential and quality restrictions are attained. Obviously, it is standard practice through engineering design and analysis to approach this goal of optimization within the limitations of the available scientific knowledge. However, the question arises whether beyond this limit of scientific knowledge a closer realization of the potential optimum operation can be obtained through a supplementary approacha statistical analysis and interpretation of actual operating data. I n principle, the use of statistical methods in this type of application has the potential of complete performance characterization beyond the ultimate limits of theory. That is, the over-all assessment of actual performance (which includes those side effects and deviations from theoretical performance which are not estimatable from theory) in terms of the operating conditions. What is implied here is that while known factors and approximations cannot in themselves
usually be explicitly identified by statistics, their ultimate combined effect is directly accounted for. This is based on the premise of relating actual operation to actual process output, and therefore all inclusive in the sense that output is the factual result, and must therefore include inherently all facets (side effects and deviations from theoretical engineering basis such as partial mixing, nonideal solutions, others) of the process mechanism. One such process study involved the characterization of a fluidized bed reactor. I n this study the operating conditions of the reactor were statistically related to the various daily productions collected over a period of time. Results of this analysis indicated that decreasing the reactor temperature (within the limits of operating history) had a marked improvement on the production obtained. This was exactly contrary to known theory and therefore it was concluded that the statistical result must have been wrong. However, testing was carried out at the recommended lower temperature level. After several weeks, production started to reverse its downward trend and actually show the anticipated improvement. This delay was actually a third mechanism over and above the two described. I n this case, theory was correct, that higher temperatures caused a higher catalyst activity; but the theory did not reckon
with the severity of a second mechanism-that is, the simultaneous degradation of the catalyst caused by higher operating temperatures. Visually LIMITS OF OPERATING DATA
3
0
RESULTANT 1 I
CATALYST
TEMPERATURE
Thus a somewhat underestimated effect, based on theory, showed a major effect in the actual performance. This is an example of the types of side effects which may be omitted in a purely theoretical analysis but picked up by analysis of the actual production data. I n terms of the detailed application to be described five basic steps were followed : 1. The definition of objective, cost factors and process limitations (safe temperature limits, crew sizes, sales potential for increased production, etc.) 2. The collection of operating data over a period consistent with the reliability of the data 3. T h e statistical analysis of data as a basis for characterizing process performance 4. The determination of optimum VOL. 52, NO. 6
JUNE 1960
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operating conditions relative to sales value, operating costs, and quality restrictions 5. Application of the results to the process and subsequent evaluation of the recommended operation The process itself, which was to be studied, involved a multireactor production operation. I n this process the production volume for each of some twelve individually controlled reactors is measured daily. The unique features of this problem are concerned with the catalyst requirements of the reactors. I n this regard a new catalyst change is made in each reactor about once a year. I n addition, during this period the catalyst is regenerated every four or five weeks. Because of the many operating problems it was difficult to measure accurately the daily production rate of a reactor. For this reason no precise measure, in terms of production, of catalyst activity loss with age was available. As a result, when this study was undertaken, the best frequency for both a new catalyst change and regeneration was not known. Regeneration and catalyst changes were scheduled largely on the basis of the experience of supervision and general rules of thumb. However, with an accurate estimate of production fall-off with catalyst age, this frequency could be determined based on the economics of the process. That is, given the various incremental values: New catalyst Catalyst regeneration Production
$N/pound $R/pound
Cost Cost
$P/pound
Value
The dollar loss of production could be balanced against catalyst costs. This then defined the objective of the study. From a statistical standpoint the main problem was to obtain an accurate estimate of the catalyst decay rate with data having a high degree of variability. However, this could be overcome to a major extent by the large volume of data available. For this reason approximately 40,000 individual daily productions, collected over the catalyst lives of the reactors, were selected for the analysis. In a regression analysis in which plant data are used to estimate the relationship
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between variables and the output of the process it is necessary to assume some type of mathematical relationship. Unfortunately, a proper form for the decay rate relationship was not known. I t was estimated by various engineers that any of the following forms might be appropriate:
In this form, for any one data point, all X s are zero except the appropriate one and similarly for the Z’s. I n addition, other production data, such as operation temperatures, were inchded in standard regression form. In all 50 terms were defined by the equation. By the method of least squares analysis, values of the coefficients were computed based on the tabulated production data. This required approximately 12 hours of computation with a Univac computer. With the now computed values of the coefficients an accurate estimate of the decay rates could be made. Graphically this was arrived at by analysis of the individual B values:
1
CATALYST AGE
Also, two possible decay rate mechanisms might occur simultaneously-during the over-all life of the catalyst and also within a regeneration life. That is, a catalyst can not necessarily be regenerated to the activity level of a new catalyst. Because the over-all evaluation depended on an accurate estimate of the decay rates, a special analysis was used. This involved the use of individual time increment estimators. To describe this, consider that each individual production figure has associated with it two ages of catalyst-first, the total number of days age, and second, the number of days since the last regeneration. For the total age 24 increments, each 15 days long, were selected. Thus, each age was classified into one of the 24 increments depending on whether the age was 1 to 15, 16 to 30, and so on. Secondly, the age since last regeneration was classified into 12 increments, each of 3-day duration. Mathematically, this can be stated as
where P = production - (day) C = relative constant Bs = catalyst total age coefficient b t = regeneration age coefficients Xi = zero or one Zi = zero or one
INDUSTRIAL AND ENGINEERING CHEMISTRY
15 30 45 60 75 90
AGE
345 360
This completed the third phase of the analysis. With the estimates of production falloffs with catalyst age, and the associated dollar values, an economic balance was computed. These results indicated that a major increase in both total catalyst age and periods between regenerations could be made at significant savings. These results were reviewed with the plant and the appropriate recommendations made. The last phase involved plant tests with the recommended operations. As this study was concluded several years ago, a sufficiently long period has elapsed to assess the significance of the recommendations. Based on the records kept to date, results show that the full savings are being realized. These savings amount to $150,000 per year. RECEIVED for review September 15, 1959 ACCEPTEDApril 19, 1960 Division of Industrial and Engineering Chemistry, 136th Meeting, ACS, Atlantic City, N. J., September 1959.
