x ) ! :intersection of the operating line with q-line k,: intersection of the operating line with the equilibrium curve
L + qF V - (1 - q ) F C = 1 + (a - l ) k s XA = X O - k, X; = ~2 - k,
i\
M=
for the stripping section
(1
x : intersection of the operating line with q-line k : intersection of the operating line with the equilibrium curve 1 2
Q2
D n =K,U,
where U , = Kk
.
=
( L +D)( C p
tz -
0 ,0’ = reflection factor in Complex method $(. . ) = constraint function y(. . ) = penalty function
-
mean temperature difference
where Q3
GREEKLETTERS
SUPERSCRIPT
Af=QfIUL(%)f (i=1,2,3,4j where ( ~ t , , ) denotes log a t i t h heat exchanger
e, = constant to be reset inside an explicit constraint = objective function a t kth vertex on j t h iteration K = number of vertices of a simplex n = exponent of objective function value R = gas constant T = temperature, OK t = holding time, sec x,, = independent variable x,.,,~ = ith independent variable of kth vertex on j t h iteration xh,, = centroid of xt,,.i; excluding the worst point y. = dependent variable f.i
Cp, TI),Q4 = L (Cp t 5 - Cp t i )
and Q, and Q. are given above.
= centroid
SUBSCRIPTS i = independent variable number j = iteration number k = vertex number n = new vertex literature Cited
Utilities consumption
WI = Q i I C p * At,, W , = Q4 / C p Ws = Qi/hs HP1 = k,, ( L + D ) * N HP? = kp.L . Z
-
In the above equations, a, C,, C,I, C,, C,,, C, C,, C,,, C,, F , , h l , H,, k,, K : , k, , k;, k,, k:, L,,, m , Po, P,,, P,,Pi, g,,,, qrn,,, R,,,, U , , B , h,, and Atcu are constants. Detailed explanation and nomenclature have been given by Umeda (1969).
Box, M. J., Computer J., 8 (l), 42 (1965). Nelder, J. A,, Mead, R., ibid., 3 (4), 308 (1965). Robbins, T., Francis, N. W., AIChE, 64th National Meeting, New Orleans, Preprint No. 3413, 1969. Rosenbrock, H. H., Computer J . , 3 ( 2 ) , 175 (1960). Rosenbrock, H. H., Storey, C., “Computational Techniques for Chemical Engineers,” Pergamon Press, London, 1966, pp 102-5. Spendley, W., Hext, G. R., Himsworth, F. R., Technometrics, 4, 441 (1962). Umeda, T., Ind. Eng. Chem. Process Des. Deuelop., 8 (3), 308 (1969). Wilde, D. J., Beightler, C. S., “Foundations of Optimization,” Prentice-Hall, Engelwood Cliffs, N . J., 1967, pp 230-45.
Nomenclature
A = frequency factor of ith reaction rate constant E = activation energy of ith reaction rate constant
RECEIVED for review December 16, 1969 ACCEPTED September 25, 1970
A Method for Plant Data Analysis and Parameters Estimation Tomio Umeda’, Masatoshi Nishio, and Shoei Komatsu Chiyoda Chemical Engineering and Construction Co., Yokohama, Japan P r o c e s s analysis is necessary to improve the operation of a plant. Especially important is the quantitative knowledge of steady states. However, because of the great complexity of chemical processes, some systematic way of process analysis must be taken on the basis of the knowledge of process models and its structure. Since the operating data are generally obtained under nonideal To whom correspondence should be addressed.
236
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
conditions due mainly to inaccuracies in the measurements of process variables and to the fluctuations of the conditions, it is first necessary to adjust these raw data to satisfy the interrelationships existing between process variables of the plant. Use of a mathematical model of a physical or chemical processing system is the common procedure for expressing the behavior of the system. In addition, the linear relationships indicating the interconnection of these subsystems are also used to express the
~
~~
~
To improve the performance of a plant, it is necessary to have a clear understanding of its operating status. However, often insufficient accurate data make adjustment of the data necessary before estimating plant parameters, Since a plant i s a total system consisting of various subsystems or process units, useful information may be extracted by studying the interrelationships of the system. Adjustment of data and estimation of parameters may be made by solving them as an optimization problem under various constraints which are the mathematical expressions of the interrelationships between the process variables in the steady-state conditions. The Complex method of Box is applied to solve the optimization problem.
behavior of the total processing system. The parameters estimation should be made on the basis of the adjusted data which have satisfied the necessary relationships of the process. This paper presents a rational way for adjusting the raw data and estimating the plant parameters in the steady state on the basis of the information provided by the measurements and mathematical relationships mentioned above. The application of statistical techniques t o the analysis of data has been widely used in science and engineering. In chemistry and chemical engineering, extensive studies have been made on the estimation of reaction rate constants from a single response variable using nonlinear leastsquares methods (Hunter, 1960; Kittrell and Mezaki, 1965; Lapidus and Peterson, 1965; Mezaki and Watson, 1966). The estimation of parameters from multiresponse data has also been presented where variance and covariance of data are unknown. (Box and Draper, 1965; Hunter, 1967; Mezaki and Butt, 1968). Most of those analyses have been devoted to the development of a mathematical model for a system. On the other hand, the problem of adjusting flow and enthalpy measurements has been solved to satisfy heat and mass balances (Kuehn and Davidson, 1961). This has been dealt with under fixed operating conditions and plant parameters in related equations. Recently, Box and coworkers investigated problems of data analysis under the existence of linear constraints (Box et al., 1970). I t has been pointed out that careful analysis on linear dependencies among responses should be carried out t o delete dependent responses before analyzing data. An empirical eigen value-eigen vector analysis has been shown to be useful if the existence of linear relationships was unknown. I n practice, the raw operating data collected from industrial plants from incomplete records are due t o the unsatisfactory situation of experiments. This stems from the lack of sensing instruments or difficulty in obtaining careful control of some variables. The present method is an attempt to adjust observed operating data of direct and indirect measurements and to estimate parameters under linear and nonlinear constraints of a total processing system. On formulating the problem under study as one of optimizing problems, careful analysis of the system is necessary so that independent and dependent variables of the optimizing problem are determined. Mathematical Expression of Processing System at Steady State
A process consists of various physical and chemical processing systems which are considered to be subsystems.
Mathematical expressions of the process consist of the mathematical models of subsystems and the interconnected relationships of these subsystems. I n the steadystate conditions, the general form of a subsystem model is given by
The following set of relations describes how the subsystems of the process are interconnected: !.I,
Xnh=
c n z n m
h = 1,2, . . . K
(3)
rn=l
The steady-state behavior of the total processing system or the process is obtained by using Equations 1, 2 , and 3. These equations are useful when the transient behavior of the system is not important. I n practice, most commercial plants may be analyzed by-passing the studies of their dynamic behavior because of the complex structures and multivariable systems. Criterion for Adjusting Data and Estimating Parameters
Each set of raw operating data obtained by direct and indirect measurements is subject to errors caused by some or all of the following conditions: Faulty calibration of instruments, different observers in using instruments, fluctuation of environmental conditions, possible systematic errors, etc. The data obtained by well-designed experiments may provide useful information because of the careful control of the above conditions. However, the raw data collected from industrial plants will be incomplete because of several reasons-e.g., some of the records are missing because sensing instrumentation is not sufficiently installed or measurements are not easily made due to too much noise, and some of the data may be obtained with considerable error because there exists incomplete functioning of instruments or no control actions against fluctuation of some operating conditions. As the criterion, it has been common t o use a weighted sum of the squares of deviation to adjust the data. However, the general criterion for fitting of multiresponse data has recently been given by Box and Draper (1965). When the operating data of the total system are to be adjusted, it is necessary to deal with both direct and indirect measurements which are statistically independent and (or) dependent. This problem may be defined mathematically as follows: Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
237
section, Equation 4 or 5, has t o be minimized under equality and inequality constraints. Equations 1, 2, and 3 are the equality constraints. Inequality constraints would be necessary to have the adjusted data within some feasible regions. For example, the reflux ratio in a distillation column must be larger than the minimum reflux ratio of the system and vapor load must be less than that corresponding to the flooding point of the distillation column. Inequality constraints are generally expressed by
When statistically independent measurements exist, those corresponding terms become the conventional weighted sum of the squares of the deviations. In practice, however, all the variances are unknown and Equation 4 may not be useful. For these cases, the following equation would be used instead of Equation 4, with the assumption that the Bayes theorem is applicable (Box and Draper, 1965):
where V t ( Z ~ ) ,Vr(783
l
i -
'
/
PRESSURE DROP
Min. Liquid l o a d
Max.Liquid load LIQUID
Inequality constraints are given as follows:
n = 3, and 4
Li; 5 L, 5 L,L'
In addition, the stable region of the CSTR operation must be given as an inequality constraint. The proper operating range of a distillation column is also given as a function of its internal structures. A typical range is schematically shown in Figure 3. Other inequality constraints exist for the stable operation of the plant in a dynamical sense. The operating data given in Table I1 have been generated by using normally distributed random numbers around each specified data. These specified data are actually what we want to have after the adjustment of the data. These data have been treated so as to minimize the criterion, Equation 10, under the constraints. This optimization problem has 18 independent variables and the Complex method has been applied. The solution given in Table I11 was obtained under the assumption that the data were statistically independent. These results show
LOAD
Figure 3. Schematic diagram for a distillation column operation
that this method of adjusting data and the estimation of parameters is practically useful. Although, in this example, only one set of parameters was obtained, there is no difficulty in having additional sets of them on the basis of different sets of operating data a t t = 1,2, . . . , N,. Discussion
Although the general form of the criterion, Equation 10, may give more reliable results in principle, the iterative procedure of direct search cannot proceed to a final solution. This is considered due to high nonlinearity of the cross terms in Equation 10. At first, the example in the previous section was tried in an effort to minimize the general form of the criterion, including the cross terms. However, considerable computation time (about 30 minutes by IBM 360/75) was required and the results were unsatisfactory. On the other hand, the assumption
Table II. Generated Data Specified 1
2
3
4
5
6
7
48.92 40.63 81.05 77.44 0.996 0.996 163.37 162.91 242.79 0.986 0.988 1.000 0.0012 0.0000 0.5038 0.4831 0.0153 0.3893 0.5945 0.7734 0.2293 0.0000 0.0011 0.9978 0.002
49.16 40.97 80.93 77.75 1.023 1.ooo 163.36 163.47 241.59 0.989 1.001 0.995 0.009 0.0000 0.5023 0.4647 0.0149 0.3749 0.6129 0.7745 0.2286 0.0000 0.0010 0.9942 0.002
48.69 41.45 80.52 77.80 0.995 1.008 161.11 165.64 235.78 0.993 1.011 0.992 0.0011 0.0000 0.4819 0.4806 0.0144 0.3840 0.596 0.77515 0.2280 0.0000 0.0010 1.000 0.002
49.43 40.66 81.48 77.50 1.006 1.019 164.25 162.66 237.57 0.998 1.004 0.9915 0.00125 0.0000 0.5124 0.4880 0.0153 0.3733 0.614 0.7771 0.2267 0.0000 0.0012 0.990 0.002
49.60 40.71 81.40 77.61 0.994 1.000 163.21 165.01 237.93 0.999 1.003 1.000 0.0013 0.0000 0.5091 0.4818 0.0150 0.3847 0.5590 0.77475 0.2260 0.0000 0.0009 0.9895 0.002
49.27 41.42 81.27 78.19 1.013 0.997 162.26 165.90 241.78 1.000 1.002 0.997 0.0016 0.0000 0.5249 0.5012 0.0154 0.3867 0.6146 0.7683 0.2269 0.0000 0.0010 0.9894 0.002
49.69 41.35 80.70 78.18 0.997 1.006 163.39 164.86 240.71 0.992 0.998 1.000 0.0012 0.0000 0.5082 0.5029 0.0149 0.3746 0.5718 0.7763 0.2272 0.0000 0.0010 1.000 0.002
a 49.72 41.10 81.08 78.13 0.999 1.008 165.19 162.57 237.41 0.994 1.014 0.991 0.0012 0.0000 0.5145 0.4830 0.01445 0.3634 0.6119 0.7624 0.2298 0.0000 0.0009 1.000 0.002
value
49.1 41 .O 81.0 77.9 1.0 1.o 163.3 163.3 240.0 1.o 1.o
0.999 0.0012 0.0000 0.506 0.479 0.015 0.379 0.599 0.773 0.227 0.000 0.001 0.997 0.002
Ind. Eng. Chem. ProcessDes. Develop., Vol. 10, No. 2, 1971
241
Table 111. Optimal Solution Optimal result
Specified value
49.3 41.16 78.6 78.3 1.016 1.016 164.5 164.5 238.5 1.015 1.024 0.9988 0.0012 0.506 0.479 0.015 0.379 0.600 0.022 0.774 0.2255 0.0015 0.998 0.000 0.20 x 10' 0.25 x 10' 0.298 x 10" 0.292 x 10'0.679 0.661
49.1 41 .O 81.0 77.90 1.00 1.00 163.3 163.3 240.0 1.00 1.00 0.999 0.0012 0.506 0.479 0.015 0.379 0.599 0.025 0.773 0.227 0.001 0.997 0.002 0.20 x 10' 0.25 x 1@ 0.30 x 0.30 x 10'' 0.667 0.667
Summation of square error
0.97 0.95 48.97 2.30 0.199 x 10 0.150 x 10 ' 20.89 14.3 53.95 0.364 x 10 0.412 x 10 ' 0.187 x 10 ' 0.630 x 10 0.108 x 10 0.140 x 10 ' 0.106 x 10 0.929 x 10 ! 0.322 x 10 ' 0.946 x 10 ' 0.195 x 10 I 0.556 x 10 ' 0.234 x lo-' 0.263 x lo-' 0.311 x 10 '
... ...
... ... ...
...
a,,
0.250 0.250 0.563 0.563 0.100 x 10 0.100 x l o - ' 0.640 0.640 5.76 0.100 x l o - ' 0.100 x 10 j 0.400 x lo-'
I
I
i
I 40
80
l
120
160
200
I60
200
I60
200
160
200
N U M B E R OF T R I A L S
Figure 4. Variation of
Jd
... 0.100 x 1 0 - J 0.920 x lo-'
...
0.578 x 10 ' 0.144 x lo-'
...
8,
I
I
I 40
1
I 80
120
NUMBER OF T R I A L S
0.237 x l o - ' 0.203 x lo-'
...
Figure 5.Variation of P t s
0.390 x lo-'
...
...
...
...
... ... ...
of statistical independence results in the elimination of the high nonlinearity and gives much more satisfactory results (Table 111) within 15 minutes of the computation time. Since there is a recycle in this problem, it is difficult to adjust smaller amounts of quantities, such as x12,x3j, n53, xx,, and xRI, to satisfy the mass and heat balance relations imposed as equality constraints with use of a plate-to-plate calculation for the distillation columns. Nonlinearity in the reactor also causes it to be highly sensitive to the feed composition of the reactors. Therefore, the weighted summations of the squared errors for such small quantities were treated as unimportant terms in the criterion. If these unimportant terms were heavily weighted, other important terms in the criterion become relatively less important. This causes much difficulty in reaching the optimal solution. The plant parameters, A I , El, A ? ,E L ,q l , and 72, coverage to the specified values respectively as the results of computation. This shows that the adjustment of the data and parameters estimation' can be made simultaneously by the Complex method. Figures 4, 5 , 6, 7, and 8 show how some of the data are adjusted and parameters are estimated with respect to the number of iterations. I n Figure 9, a weighted summation of square errors is plotted against the number of iterations. Conclusions
The problem of plant data analysis and parameters estimation has been formulated as a constrained optimization problem. The present method gives meaningful results since the adjusted data satisfy all the constraints in the 242
I
9
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
NUMBER OF T R I A L S
Figure 6. Variation of R s
* W'O
40
I I20
80
NUMBER OF T R I A L S
Figure 7. Variation of A2
01
%- 0
11
I 40
I20
EO
NUMBER
OF
TRIALS
Figure 8. Variation of
El
-
8
u n , = adjusted value for weight fraction of com- ponent _ - i in stream n Z n m i , and Zt Woht = X n k l , yn\2, Xnb = hth i n m t vector of unit n compdnent i of sth system output of unit n component i of Y,, lower constraint function for y,,.g upper constraint function for yn52 mth output vector of unit n component i of Zn,, lower constraint function for Fnm, upper constraint function for Tnmz
x,~
GREEKLETTERS Lagrange multiplier vector plate efficiency of column n latent heat of stream n latent heat of steam density of stream n density of component i in stream n density of steam covariance between wok,and &‘ob
NUMBER OF T R I A L S
Figure
9. Convergence of
criterion
system, and the parameters are estimated on the basis of these adjusted data. The applicability of the method has been demonstrated by carrying out a numerical experiment on data analysis and parameters estimation of a typical processing system consisting of reaction and distillation units. The Complex method has been successfully applied to solve the optimization problem. Acknowledgment
The authors thank Nakanishi, Niida, and Shindo for their support in solving practical problems and for their encouragement in the development of this method; Ichikawa of the Tokyo Institute of Technology for valuable advice; Chiyoda Chemical Engineering & Construction Co. for permission to publish this article.
PARENTHESIS ( t ) = tth series of measurements SUBSCRIFTS i = component i of a vector n = unit n nh = hth stream vector of unit n ns = sth stream vector of unit n nm = mth stream vector of unit n
SUPERSCRIFTS L = lower limit or lower constraint U = upper limit or upper constraint u = uth set of data
Nomenclature
OVERLINES
A , = frequency factor of nth reaction in Arrhenius C,
K
= = = = = =
hn
=
Dn &, E,, FE
LE =
L
=
L(X,i,) = L(y,,,) = L(Znm,)= M, = M, =
P,
=
P,,= PtE =
P,, R:
R, vt(iiah)t!
= = =
= =
S,
=
4,=
type of rate constant matrix expressing input-output relationship decision vector in unit n component iofil-, activation energy of nth reaction u the measured flow rate of stream n adjusted value for the flow rate of stream n enthalpy of stream n uth measured value of level height in unit n adjusted value for level height in unit n lower limit for &,, constant lower limit for gTL, constant lower limit for 2,,,,, constant molecular weight of component i molecular weight of stream n parameters in unit n, vector component i of P, uth measured pressure of column n adjusted value for pressure of column n uth measured reflux rate of column n adjusted value for - reflux rate column n covariance on Wohc and ish) a t tth series of measurements uth measured steam rate of column n cross-sectional area of reactor n uth measured weight fraction of component i in stream n
-
-
= adjusted value = average value over components
literature Cited
Box, G. E. P., Draper, N . R., Biometrica, 52, 355 (1965). Box, G. E. P., Hunter, W. G., Erjavec, J., McGrefor, J. F., “Conference of Model Building Techniques.” University of Wisconsin, Madison, Wis., June 1970. Box, M. J., Computer J., 8, 42 (1964). Deming, W. E., “Statistical Adjustment of Data,” Section 28, Wiley, New York, N. Y., 1943. Japanese edition: pp-39, translated by S. Moriguchi, Iwanami, Tokyo, 1950. Hunter, W. G., Chem. Eng. Progr. Symp. Ser., No. 31, 56, 10 (1960). Hunter, W. G., Ind. Eng. Chem. Fundam., 6, 461 (1967). Kittrell, J. R., Mezaki, R., Ind. Erg. Chem., 57, 18 (1965). Kuehn, D. R., Davidson, H., Chem. Eng. Progr., 57, 44 (1961). Lapidus, L., Peterson, T. I., A.1.Ch.E. J . , 11, 891 (1965). Mezaki, R., Butt, J., Ind. Eng. Chem. Fundam., 7, 120 (1968). Mezaki, R., Watson, C. C., Ind. Eng. Chem. Process Des Develop., 5, 62 (1966).
RECEIVED for review October 3,1969 ACCEPTED August 2. 1970
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
243
