(13) Ibid.,p. 213. (14) Ibid.,p. 248. (15) Egorov, A. I., J . Appl. Math. Mech. 27, 1045 (1963). (16) Gray, R. D., Jr., Ph.D. thesis, University of Delaware, 1965. (17) Gray, R. D., Jr., Ferron, J. R., “Optimal Heat-Removal Rates for Tubular Reactors,” Symposium on Optimum Process Designs, 149th Meeting, ACS, Detroit, Mich., April 8, 1965. (18) Horn, F., Chem. Eng. Sci. 14, 77 (1961). (19) Horn, F., Jackson, R., IND. ENG. CHEM.FUNDAMENTALS
simplified boundary conditions at r = 0 forcing function (Appendix) mu1tiplier ratio of length to radius multiplier dependent variable (Appendix) step size boundary conditions a t t = 0, 1 boundary conditions a t r = 0, 1
4. 110 - - - (1065’1. \ - . --, -7
literature Cited
(1) Aris, R.!“Optimal Design of Chemical Reactors,” Academic Press, New York: 1961. (2) Bellman, R., Dreyfus, S., “Applied Dynamic Programming,” Princeton Universitv Press, Princeton, N. J.. 1962. (3) Bellman, R., Osbdrne, H., J . Math. ’Mech. 7 (1958). (4) Butkovskii, A. G., Automation Remote Control 22, 13 (1961). (5) Ibid., p. 1156. (6) Ibid.,24, 292 (1963). (7j jbid.. p. 1106. (8) Butkovskii, A. G., Lerner, A. Ya., Ibid., 21, 472 (1960). (9) Denham, TV. F., Bryson, A. E., Jr., A.I.A.A. J . 2, 25 (1964). (10) Denn, M. M.. Aris, R., A.Z.Ch.E. J . 11, 367 (1965). (11) Denn. M. M.. Aris. R.. Chem. Ene. Sci. 20. 373 (1965). (12) Denn; M. M.,Ar’is, R., IND. ENG. C H ~ MF.~ N D L M E N T A L S 4, 7 (1965).
(20) Horn, F., Troltenier, U., Chem. Ing. Tech. 32, 382 (1960). (21) Zbid.,35, 11 (1963). (22) Jackson, R., Chem. Eng. Sci. 19, 253 (1964). (23) Katz, S., J . Electronics Control 16,189 (1964). (24) Kelley, H. J., “Optimization Techniques with Applications to .4ero-space Systems,” G. Leitmann, ed., Academic Press, New York, 1962. (25) Lure, K. A., J . Appl. .Math. Phys. 27, 1284 (1963). (26) .Pontryagin, L. S., Boltyanskii, V. A,, Gamkrelidze, R. V., hhhchenko. E. F.. “Mathematical Theorv of ODtimal Processes,” LYiley, New York, 1962. (27) Sakawa, Y., I.E.E.E. Trans. Automatic Control A-C 9, 420 (1964). (28) Wang, P. K. C., in “Advances in Control Systems,” Vol. 1, C. T. Leonedes, ed., Academic Press, New York, 1964. (29) \%‘zing, P. K. C., I.E.E.E. Trans. Automatic Control A-C 9, 13 (1964) . (30) LVang, P. K. C., Tung, F., J . Basic Eng. 86, 67 (1964). RECEIVED for review April 16, 1965 ACCEPTED August 6, 1965
OPTIMIZATION OF RECYCLE PROBLEM HENRY
HU NG
-Y EH
CH I EN
,
Monsanto Co., St. Louis, Mo.
A numerical example of a product recycle cross-current extraction process is given, which uses the technique of the discrete maximum principle in combination with the Fibonaccian search.
ECENTLY the problem of cross-current extraction with R p r o d u c t recycle proposed by Rudd and Blum (6, 70) was solved by the method of the complex discrete maximum principle ( 4 ) , dynamic programming ( 9 ) , and the gradient method (9). This note shows a n alternative computational procedure which basically is the so-called “cut-state’’ approach proposed by Aris, Nemhauser, and Wilde (7). T h e serial part of the problem is solved with the discrete maximum which eliminates the necessity of large principle algorithm (8), computer storage ai‘ea for the return function tables as normally required by dynamic programming method ; then the Fibonaccian search method is used for maximization of the return function with respect to the recycle composition, thus reducing the dimensionality of the problem to 1 (9). The equation relating the initial and final adjoint variables derived by Fan and Wang (4j for the recycle problem may be shown t o be equivalent to the classical method of finding the extremum of a function by solving for the zeroes of its derivative. Therefore their equatiox gives only the stationary point solution rather than the maximum of the return function of the recycle problem. Figure 1 is a schematic diagram of the process. The material balance a t stage n gives
66
l&EC FUNDAMENTALS
where xn = concentration of solute in raffinate y n = concentration of solute in extract q = feed rate r = recycle rate W , = flow rate of solvent Equilibrium is assumed in each stage; therefore
Y , = yn*
yn*(xn)
The objective function is defined as
Figure 1 .
Schematic diagram of process
0 Read in Table of y*(x)
f
Smooth the y*(x) Table Calculate the
dy* Table dx
dY * Table Smooth the dx
t> Present Accuracy
c b
Fibonaccian Search Method; Fixed x4
4
I
Figure 2.
Flow diagram of computer program VOL. 5
NO. 1
FEBRUARY 1966
67
where Xn+io
= xno
f
Vn(3.n
- X)
Table I. Smoothed Equilibrium Data and Derivatives
(7)
defines the objective function. T h e adjoint variables satisfy the following equations:
The optimum control law requires that:
If V ,
0 0. .mooon . ... . . 0.020000 0.030000 0.040000 0.050000 0,060000 0.070000 0.080000 0.090000
$0 0,025000 0,050000 0.075171 0.099914 0.123343 0.142229 0 155086 0.163429 0,169743 0.173914 0.176257 0.177914 0.179000 0.178657 0.178686 0.179314 0.181514 0.184486 0.190743 0.199714
Table II.
2.492000 2.516000 2.500000 2.466000 2.326286 2.021429 1.587714 1.140000 0.780286 0.528286 0.354857 0,225143 0.122286 0.056000 0.024857 0.056857 0.144857 0.290735 0.497102 0.744694 1 ,040041
~
0.100000
or
dx
Y*
0.110000 0.120000 0.130000 0.140000 0.150000 0.160000 0.170000 0.180000 0.190000 0.200000
< V , < V*, we must have
dY * -
X
Optimal State and Control Variables = 0.12056. $8 = 0.69261
~4
du
Stage
For any given
may be calculated from Equation 4. it is necessary T o obtain the optimal return, f 3 [xq,x1(x4,xl)], then to search for the initial value of $-i.e,, $ ~ - s u c ~ that Equations 1, 2, 9, and 10 will bring the state variable from the given x4 to its corresponding XI.T h e parametric influence technique using linear interpolation method was found efficient in solving this simple one-dimensional two-point boundary value problem. After f 3 is obtained for a given x4, one must find the right x4 which maximizesf3 in Equation 5. I n the numerical example shown below, f 3 is assumed to be unimodal; the Fibonaccian search method was used for the maximization of f 3 . If one differentiates Equation 5 with respect to x4 and sets the derivative equal to zero, one has x4,
A'O ,
IX
Using the usual interpretation for the adjoint variable (3),
3 2 1
Yn
xll
0.16266 0.13465 0.10266
0.07894 0.05549 0.04110
+*-
dx
Vna
0 81300
0 25586
0. 62x67 ._..
1.78932 2.30112
0.77414 0,14015
0 51297 0,35590
1
= 0.2, X = 0.05, and 7 = q = 1. The equilibrium data were taken from their curve and smoothed before they were used in the computation. T h e derivatives of the equilibrium curve were calculated by the three-point formula of the smoothed data and were smoothed again after they were calculated. A third-degree polynomial is used for the interpolation of the equilibrium table (Table I). Convergence is assumed when A$3 is less than in the boundary value search. The accuracy of the Fibonaccian search is also set at searching in the range of 0.11 6 ~4 0.13. Table I1 lists the values of variables which give the optimal return. T h e maximum return for the recycle problem is 0.10189. Figure 2 is a flow diagram of the computer program. XI
