y = liquid-phase activity coefficient, defined by v =
61 = w
=
$, =
Equation 4 pure liquid fugacity coefficient, defined by Equation 3 parameter in Chueh-Prausnitz equation acentric factor vapor fugacity coefficient, defined by Equation 5
SUPERSCRIPTS
-
0 = pure component parameter = molar property * -- ideal gas property
SUBSCRIPTS
a , b , i.
J,
k = specific component designators
L = liquid phase m = mixture property r = reduced state Literature Cited
Barner, H . E., Pigford. R. L., Schreiner, W. C., 31st Midyear Meeting, Amer. Petrol. Inst., Division of Refining, Houston, Tex., May 1966. Chang, H. L., Hurt, L. J., Kobayashi, R., AIChE J . , 12, 1212 (1966). Chang, H . L., Kobayashi, R., J . Chem. Eng. Data, 12, 517 (1967). Chao, K. C., Greenkorn, R. A., Olabisi, O., Hensel, B. H., AIChE J . , in press (1971). Chao, K. C., Seader, J. D., ibid., 7 , 598 (1961). Cheung, H., Zander, E. H., Chem. Eng. Progr. Symp. Ser., 64 (88),34 (1968). Chueh, P. L., Prausnitz, J. M., Ind. Eng. Chem. Fundam., 6, 492 (1967).
Curl, R. F., Jr., Pitzer, K . S.,Ind. Eng. Chem., 50, 265 (1958). Donnelly, H. G., Katz, D. L., Ind. Eng. Chem., 46, 511 (1954). Hildebrand, J. H., Scott, R . L., “The Solubility of NonElectrolytes,” Reinhold, New York, N. Y., 1950. Kay, W. B., Albert, R. E., Ind. Eng. Chem., 48, 422 (1956). Kohn, J. P., Kurata, F., AIChE J., 4 , 211 (1958). Natural Gas Processor’s Assoc., “New Constants for ChaoSeader Correlation for Nitrogen, Hydrogen Sulfide, and Carbon Dioxide,” Tulsa, Okla., 1965. Olds, R. H., Sage, B. H., Reamer, H. H., Lacey, W. N., I d . Eng. Chem., 41, 475 (1949). Preston, G. T., Prausnitz. J. M., Ind. Eng. Chem. Process Des. Develop., 9, 264 (1970). Price, A. R., Kobayashi, R., J . Chem. Eng. Data, 4, 40 (1959). Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem., 45, 1803 (1953). Redlich, O., Kwong, J. N. S., Chem. Rev., 44, 233 (1949). Robinson, R. L., Jr., Jacoby, R. H., Hydrocarbon Process. Petrol. Refiner, 44 (4), 141 (1965). Vaughn, W. E., Collins, F. C., Ind. Eng. Chem., 34, 885 (1942). Wilson, G. M., Aduan. Cg’og. Eng., 9, 168 (1964). Wilson, G. M., Barton, S. T., Tullos, H. J., 65th National AIChE Meeting, Cleveland, Ohio, 1969. Yarborough, L., Pan American Petroleum Corp., Tulsa, Okla., personal communication, 1969. Yudovich, A., Robinson, R. L., Jr., Chao, K . C., AIChE J . , in press (1971). RECEIVED for review June 3, 1970 ACCEPTED December 29, 1970
A Modified Complex Method for Optimization Tomio Umeda’
Chiyoda Chemical Engineering and Construction Co., Yokohama, Japan
Atsunobu lchi kawa Tokyo Institute 3 i n c e the Simplex method was introduced by Spendley et al. (19621, its extensions and applications have been made. Nelder and Mead (1965) proposed the method for making the Simplex adapt itself to the local landscape, determining reflection, expansion and contraction factors. Box (1965) extended the Simplex method to solve implicitly and/or explicitly constrained optimization problems. The method is known as the Complex method, which stands for the constrained Simplex method. The Complex method has been applied to an optimal control problem by Robbins and Francis (1969) as well as to an optimal design problem by one of the authors (Umeda,
I
To whom correspondence should be addressed
of
Technology, Tokyo, Japan
1969). This method has the advantage of easy handling capability for implicit inequality constraints, and of not requiring computation of any derivatives such as in the gradient method. I t has, however, the disadvantage of relatively slow convergence to an optimum point and requires many iterative computations. Therefore, improvement of the convergence rate of the method was attempted by modifying it without introducing many complexities. As modification, the nth power of function values a t the vertices forming a simplex has been weighted on generating a new trial point by replacing the vertex with the worst function value. This is considered as an attempt to take the function values into account. Further modification in handling inequality constraints has been attempted. Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
229
~~~
A new optimization method and three examples of its application are presented. The method has been developed by the modification of the constrained Simplex method. The modification has been made by weighting the nth power of objective function values at the vertices of a simplex on replacing the vertex with the worst function value by a new trial point. Compared with the original Complex method, this modified method i s shown to be more effective by solving three kinds of problems in which the optimal design problem of an absorber-stripper system was included. The results of the case studies for the examples show the recommended values for the expansion factor, a, and the exponent of the function values, n, to be a = 1.62 and n = 1.O, respectively.
In this paper a modified Complex method is presented and suitable parameter values, the exponent of function values, and the expansion factor, are given. These have been determined by solving three examples. The Complex Method
The original Complex method is briefly described here. I t has mainly two functions: one for unconstrained optimization, which corresponds to the Simplex method, and the other for making vertices satisfy inequality constraints. The Simplex method starts with the evaluation of objective function values a t vertices forming a simplex in the space of independent variables. After comparing the function values t o find the worst value and its position in the space, a new simplex is formed by replacing the worst vertex by a point a times as far from the centroid of the remaining points as the reflection of the worst point in the centroid. If this trial point is also worst, it is moved halfway toward the centroid to have a new trial point. The above procedure is repeated until a specified stopping criterion is satisfied. The factor, a , of reflection, expansion, and contraction may be varied according to the information on the space obtained by iterative computations (Nelder and Mead, 1965). For general optimization problems, however, the spatial structure of independent variables is unknown because of the high dimensionality of the variables, and, their modification does not always seem to be effective. If the factor a is not varied, the above procedure of the Simplex method needs some modification, for the factor a ( > 1 ) tends to cause a continual enlargement of the simplex. Also, the procedure of moving halfway toward the centroid tends to contract a simplex, especially in the neighborhood of an optimum point. The procedure of making vertices satisfy inequality constraints is given: If a trial point does not satisfy constraints, it is moved halfway toward the centroid of the remaining points. This does not require the introduction of any penalty functions, as in the gradient method, to have feasible solutions within a constrained region. This procedure of making vertices satisfy implicit inequality constraints is particularly useful for complicated optimization problems.
The first modification is concerned with the procedure for unconstrained optimization problems and it is considered a generalization of the original Simplex method. The further modification for handling inequality constraints will be given later. On determination of a new trial point by the Simplex method, the following equations are used for each independent variable: For the ith independent variable, -
z,
As described above, the procedure of the Complex method is simple and easy to use. The convergence rate, however, seems not t o be very great and many iterative computations are required, especially for high-dimensional problems. Thus a modification has been attempted. 230 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
(1 + a ) XL,, - ax,,w
(1)
where denotes the centroid of the points excluding the worst one on the j t h iteration. I t is calculated by K
1
j=1,2,
. . . ,J
I n Equation 2, K expresses the number of vertices of a simplex. I t is evident from this equation that the function values at each vertex of a simplex are not taken into account on generating a new simplex. To make use of the information of the function values, one of the simplest modifications is to replace Equation 2 by
-
k f u
(3)
XI.,= f;fk k = l ' k # w'
i
= 1,2,. . . , I
j=1,2,
. . . ,J
This modification, however, requires the function to have a positive value and is only applicable to minimization problems. Therefore, further modification has been made to generalize this approach. Since it is not necessary to weight x,,,~with the absolute values of functions, a normalized difference, A f A , may be used. Af, is defined as: Aflk =
fmax,]
- f, k
for minimization problems
fmax,j - fmm,]
or Afih=
Modifications of the Complex Method
XI, + 1,n =
- fj,mm f j , m x - f1,mm fi,k
for maximization problems
where f l h is the function value of the hth vertex on the j t h iteration, and f,, , and f,,,,, are, respectively, maximum and minimum of the function value, on the j t h iteration. By defining a weighted centroid, i , , with A h h by
x2
k f u
new trial points are determined by K
,
xi,
- I n = (1 + a )
I
k # U
-
K (Afi,k)n k = l sk f UI
CuXL,,,U
(5)
i = 1 , 2 , . . . ,1 j = 1 , 2 , . . . ,J
Figure 2. Modified complex method
By this modification, both minimization and maximization problems are uniformly treated without any complexity. Based on experience gained through its application to various kinds of problems, the following computational algorithm has been developed: In the first stage, where 1 5 0.05 is not satisfied, the criterion 1 (fl,,,, - f l m )/fl,,, the modified Complex method is applied and in the second stage, which is considered t o be the final step to an optimal point, the original Complex method ( n = 0) is applied. For this modified method, further modification has been attempted to make vertices satisfy inequality constraints. According to the Complex method, a new trial point is moved halfway toward the centroid of the points, excluding the one to be replaced, when it does not satisfy inequality constraints. Instead of moving halfway toward the centroid of remaining points, a more general way of moving is defined by Xii-ln
= a’x2,
+ (1 - LY’)XI,,.
(6)
i=1,2,.,.,1 j = 1,2,. . . , J
solving problem
inequali t
I
XI
Find new p i n t s ( 1 ) Use Eqn (6)for implicit constraints (2) Reset by e( inside constraints for exolicit ones t
H
I
function
Figure 1. Flow diagram of modified method
I
where x,, is computed by Equation 2 . In one of the tests, the golden section search has been applied to replace the process of moving the remaining points halfway toward the centroid. The reason for this test is given below. The golden section search is considered nearly as effective as the Fibonacci search method which is a sequential search plan for explicit object functions, that is, for the case being computed directly from independent variables. On the other hand, the procedure of moving halfway toward the centroid of the remaining points is, in principle, equivalent to the Bolzano method, an appropriate method for implicit objective functions. Wilde and Beightler (1967) have made a detailed description of these methods. When a vertex violates an inequality constraint which has the form 3(x1, x2, . . . , x,, yl, y 2 , . . . , y m ) 2 0, it is necessary to make the vertex satisfy the constraint. Hypothetically, if an explicit penalty function y(x1, x2, . . . , x,, y l , y2, . . . , y ? ) is introduced, an objective function becomes 8(x1, x?, . . . , x,, jl, y 2 , , y m ) + ?(XI, x ~ ,. . . , x n , y l , y ? , . . . , y m ) and this shows an explicit function of independent variables. Thus application of the golden section search is considered to be more effective than using the Bolzano-type procedure. Actually, it is not necessary to find a boundary itself between feasible and nonfeasible regions, but it is sufficient that a new trial point is in the feasible region. In Figure 1, the computational algorithm of this modified method is shown in a flow diagram. Figure 2 shows schematically the procedure for a two-dimensional case. Examples
T o examine the effectiveness of the first part of the modification, three optimization problems with or without constraints have been solved by applying the modified Complex method. As an unconstrained optimization problem, Rosenbrock’s parabolic valley problem (1960) has been chosen, and as a second problem, finding the optimal temperature and holding time in a continuous stirredtank reactor (CSTR) has been solved. This problem has explicit inequality constraints. Finally, as a complicated problem with equality and inequality constraints, the optimal design of an absorber-stripper system has been taken up. For each problem, case studies have been made by varying the exponent of objective function values, n, and the expansion factor, CY. I n addition, the further modification on handling implicit inequality constraints has been applied to the third problem. Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2 , 1971
231
Table I. Computational Results for Problem 1 a
n
0
1.o
J
200 0.3273 0.1092 0.4530 200, 120" 1.000 1.000 0.9 x 200, 150" 1.000 1.000 1.2 x 86b -0.2 2 40 0.0279 1.5480 200 -0.3013 0.0584 1.7980
XI
Xi fJ
J
1.62
XI X1 fJ
2 .o
J XI X1 fJ
J
3.0
XI X1 fJ
J
5.0
XI X2 fJ
Number of trials when I (f,,
1 .o
0.5
- f,
200 0.1737 0.0281 0.6832 200, 170n 1.000 1.000 2.0 x 200 0.9994 0.9988 4.4 x lo-' 89 0.1812 0.0165 0.6971 17b -0.2584 0.0409 1.6508
I5
) / f,,
3.0
2.0
200 0.2055 0.0270 0.6544 200, 150" 1.000 1.000 1.1 x 200, 190" 0.9993 0.9986 5.1 x lo-' 113b 0.1793 0.0083 0.7302 85b -0.1444 0.0019 1.3455
200 -0.0868 0.0737 1.1811 200, 160" 1.000 1.000 1.5 x lo-'" 200, 170" 1.000 1.ooo 1.6 x 10-lo 200 0.8385 0.6977 0.0289 96b 0.0961 -0.0003 0.8263
lo-' was satisfied. Number of trials when j (fmm - fmi, ) / f,,
A- KI
5.0
200 0.0073 0.0094 0.9936 200, 140" 1.000 1.000 3.6 x lo-" 200, 170" 1.000 1.000 2.6 x lo-' 88b -0.2667 0.0394 1.7052 75b -0.2738 0.0476 1.6975
B
I
5 10-l' was satisfied.
K4
200 0.3093 0.0907 0.4795 200, 130" 1.000 1.000 2.3 x lo-'' 200, 170" 1.000 1.000 2.1 x lo-" 200 -0.2642 0.0377 1.7014 89' 0.0446 -0.0808 0.9230
C
L D
Figure 4. Reaction scheme
,
-I 0 -I 5
-10
-05
05
00
IO
15
XI
Figure 3. Convergence behavior for Problem 1 Initial simplex --.CY = 1.0 n = 0.0
+
0 10th trial H 100th trial A 200th trial
-a
-.-
1.62
n = 1.0
a = 3.0
n = 1.0
=
+ +
Rosenbrock's Parabolic Valley Problem (Problem 1). The function to be minimized is given by
Minimize F = 10O(xi (Xi,
~ 2 ) '
+ (1- X I ) '
(7)
d x l / d t = -(ki kp k3)Xi dxpidt = klxl - k4Xp dxaidt = k4X2 - kbX3
(84
(8b) (8c)
with the initial conditions x l ( 0 ) = 1, x 2 ( 0 ) = 1, x 3 ( 0 ) = 0 where the h, is given in the following form:
x,)
This problem was solved for comparison purposes by Rosenbrock and Storey (1966), using the method of steepest descent with or without Booth's modification, the Newton-Raphson method, and Rosenbrock's method (1960). 232
Starting from the initial points ( x l , xz) = (-1.200, 1.000), (-1.200,0.700), and (-0.900, 0.700), the problem has been solved by varying n and CY. The results are shown in Table I. The comparison of the present results with those results obtained by the different methods tells us that the modified Simplex method will show a fairly good convergence rate. For this particular problem, the trial with CY = 1.62 and 2.0 for all powers of n studied gives a good result, however, the trials with CY = 1.0, 3.0, and 5.0 give unsatisfactory results. Figure 3 shows some of the convergent behavior on the x1 - x p plane. Optimization of an Isothermal Reactor (Problem 2). For a reaction scheme shown in Figure 4, a problem of finding the best isothermal yield of a product C in a CSTR was chosen. The kinetic equations for the reaction scheme are given:
Ind. Eng. Chern. Process Des. Develop., Vol. 10, No. 2, 1971
The isothermal yield of C can be obtained analytically by integrating the above equations. Thus the optimization problem is represented by
~~~~~
Table N
n
0
1.0
J T t
300 969.9 0.08273 0.42305 300, 90n 982.9 0.07601 0.42308 300, 110" 982.9 0.07595 0.42308 300 868.2 0.1780 0.41961 300 865.8 0.1820 0.41943
fi
J T t
1.62
f J
2.0
J T t
3.0
J T
fi
t f J
J T t
5.0
f J
Number of trials at
fn
= 0.42308.
I I . Computational Results for Problem 2 0.5 1 .o 2.0
300 954.3 0.09163 0.42290 300, 190" 983.3 0.07581 0.42308 67' 868.7 0.1770 0.41964 300 868.7 0.1771 0.41964 300 869.2 0.1766 0.41967
Number of trials when 1 (f,,,
300 929.3 0.1100 0.42241 150', 80" 982.8 0.07600 0.42308 58b 868.6 0.1776 0.41963 65 868.3 0.1779 0.41961 210' 870.2 0.1750 0.41974
- ,f
300 879.8 0.1617 0.42034 1236,100" 983.4 0.07570 0.42308 9gb 983.3 0.07572 0.42308 300 870.3 0.1749 0.41974 300 873.9 0.1696 0.41998
I
)/f,,,
3.0
5.0
300 933.6 0.1057 0.42252 300, 80" 983.0 0.07592 0.42308 300, loon 983.0 0.07589 0.42308 300 872.1 0.1721 0.41987 6ab 878.0 0.1641 0.42023
300 932.5 0.1076 0.42249 150b,100" 982.5 0.7617 0.42308 300, 90" 983.0 0.07592 0.42308 300 871.8 0.1725 0.41985
120b 871.7 0.1728 0.41984
2 lo-'' was satisfied.
Table I l l . Computational Results for Problem 3 (nth Power Weighted Method) a
J
1.0
f j
J
1.62
fj
f2mb
J
2.0
f j
f?mb
Figure 5. Schematic flow absorber-stripper system
O
sheet
/I-
ofan
0.5
1 .o
2.0
3.0
250" 67,224 67,315 211" 66,840 66,840 437" 68,743 68,820
380" 66,776 67,184 347" 66,486 67,313 334" 66,484 66,678
232" 66,757 66,800 260" 66,491 66,521 242" 66,959 66,972
500 76,448 83,041 253" 66,490 66,551 332" 66,530 66,987
500 79,536 82,667 421" 66,568 66,723 500 72,911 74,950
"Number of trials when I cf,,, - fmin ) if,,, 1 5 0.0001 is satisfied. bf2,, is the objective function value at J = 200.
:ir
I
1750
1500
200
F?
-z Q
6z
0
n
1250
2
w
E
1000
0 -7 W
m
0
i!
i?
750
1
400
~
0
1
I
100
50
I50
200
250
Number of trials
Figure 6. Convergence behavior of objective function
-a
= 1.62, n = 1.0
--- a
=
1.00, n
=
0.0
Figure 7. Convergence behavior of flow rate of adsorbent
-a
---
= 1.62, n = 1.0 a = 1.00, n = 0.0
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
233
56 m a W
52-
k 3
4
X W
40-
lb
LL
0
c"a
2 t a a x
44-
a
i LL W
a
40-
36-
32l 0
80
40
50
,
100 I50 Number of trials
200
I
00
250
Number of trials
Figure 8. Convergence behavior of flow rate of exhaust gas
-N --- N
=
1.62, n = 1.0
=
1.00, n
=
Figure
9. Convergence behavior of reflux ratio -N
=
---
=
= 1.62, n a = 1.00, n
1 .O 0.0
0.0
subject to 1 < t 5 10 (sec) and 200 5 T 2000 (OK) where h has been written for ( h l + k r + hi). This problem has been solved by Rosenbrock and Storey (1966). Starting from the initial points ( T , t ) = (1073.0, 0.500), (1000.0, 0.400) and (950.0, 0.300) to form a simplex, this problem has been solved by varying N and n as before. The computational results are shown in Table I1 where the number of trials, J , required to reach the ' optimal point, x i = 0.42308, are given together with the temperature, T , and the holding time, t. The optimal result obtained by using the hill-climbing method (Rosenbrock and Storey, 1966) is x 3 = 0.42308, T = 978.96, and t = 0.0781 a t the 146th iteration for the initial point ( T , t ) = (1073.0, 0.500). They reported that it was not always possible to find the optimal point because of the existence of a ridge. A similar result has been given on Table I1 (in case of N = 3 and 5 for all n ) . In this example, N = 1.62 for all n , and N = 2.0 for n = 2, 3. and 5 are suitable. Optimal Design of an Absorber-Stripper System (Problem 3). To examine the effect of CY and n on constrained optimization problems, and t o test the effectiveness of the further modification in handling implicit inequality constraints, the optimal design problem of an absorberstripper system (Figure 5 ) has been studied. This problem has been solved (Umeda, 1969) by applying the Complex method. Included are five independent variables: flow rate of an absorbent, L ; exhaust gas rate from an absorber, G; composition of an absorbent fed to an absorber, x r ; reflux ratio in a stripper, R ; and g-value of a feed t o a stripper, q. These variables are to be determined so as to minimize the production cost of the system. For 234
120
3
3
2
I6 0
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
the purpose of showing the complexity of the problem, necessary equations with brief descriptions are given in the Appendix. Using the same design conditions and initial points as those of the previous work, the optimal design calculations have been carried out by varying N and n. I n Table 111, the object function values computed are given and, in Figures 6 to 9, the convergence behavior of the objective function and some of the independent variables are depicted for two cases with respect t o the number of trials. In this example, n = 1.0 and 2.0 with CY = 1.62 give better results than those with other values of CY and n.
w
"
2
500
~
b
50
lbo
I50
200
250
Number of triols
Figure 10. Comparison of the modified methods (a
=
1.62, n
- nth power modified method
=
0)
_ _ _ nth power modified method with golden section seorch
Table IV. Comparison of Two methods: nth Power Weighted and Its Further Modified Methods (Problem 3)
cc
1.62
n
0
J f J
f200
1.62
1.0
J f.J
fm
nth Power weighted method
Further modified method by golden section search
211 66,840 66,840 260 66,494 66,521
234 66,952 66,937 236 66,958 66,960
Appendix
The optimal design problem of an absorber-stripper system is briefly described as follows: For given F , yland xa,
subject to the following constraints, the design equations and utilities consumption: (Material balance)
Fy1 + L x = ~ GyZ
+ (L + D ) x ~
for the absorber The results of testing the modified method using the golden section search for handling implicit inequality constraints are shown in Figure 10 and Table IV. Comparing these results with those obtained in the case of CY’= 0.5, corresponding to the original procedure, there seems to be no significant difference between the two.
( L + D ) x i = L x +~ D x ~ for the stripper
F=G+D for the whole system (Heat balance)
( L + D)Cplt, + Qz = DCp,t3 + LCp,t, + Q i
Discussion
Through the case studies made for the three examples in the previous section, it is found that there exist best values of CY and n. The case studies carried out by Box (1965) were restricted to the more narrow range of CY between 1.0 and 1.5. As shown in Tables I to 111, the combination of CY = 1.0 and n = 0 has not given better results than those computed by using larger values of CY with n = 0. These larger values of CY would not give an easy contraction of a simplex. However, overexpansion will necessitate an increase in the number of trials to achieve the contraction necessary to satisfy a stopping criterion. In addition, overweighting by using a larger value of n causes too much deformation of a simplex and will result in difficult convergence. By the results for the examples in the previous section, cy = 1.62 and n = 1.0 seem the best combination of parameters included in the modified Complex method. With respect to the further modification in handling implicit inequality constraints, the parameter CY’in Equation 6 has been specified in such a way that the golden section search can be applied in solving the third example. Figure 10 shows that there is no considerable difference between the two ways of moving toward the centroid of the remaining points. Since it is considered that the effectiveness of a way of handling implicit inequality constraints might depend on many factors such as the relations of those constraints with the position of the optimal point, initial position of vertices forming a simplex and so on, further investigation would be necessary to confirm the effectiveness of this further modification. The Complex method has been modified without introducing many complications, and the exponent of function values and expansion factor in the modified method have been properly determined through the examples.
around the stripper where Q , = D h , ( R + 1)
( L + D)(CP,tl- CPlt2)= L E p & - C&, for the heat exchanger Inequality constraints x3 - Y1 x3 - mx2
F>G>F-
> R > R,,,in
R,,
=
x3 - yc yc - xc ’ where yc =
Qmax
>Q >
1+
ax, (a - l)Z
Qmii
Equipment design equations
G D2=
K,U,
where
U, = K : exp
where Acknowledgment
The authors wish to thank Akio Shindo for the support in solving the examples. They also express their gratitude to Chiyoda Chemical Engineering and Construction Company for the support and permission to publish this work.
M = R ( R + 1) X:
= + (a = x3 - k,
xk = X, - k,
/
for the rectifying section
i
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 2, 1971
235
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
(i
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 -
CY, 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.
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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
