literature Cited
Y
(1) Archer, D. H., Rothfus, R. R., Chem. Eng. Progr. Symp. Ser. 57, NO. 36, 2-19 (1961). (2) Berger, D. E., Campbell, G. G., Chem. Ens. Progr. 51, 348-52 (1 955). (3) Berger, D. E., Short, G. R., Znd. Eng. Chem. 48, 1027-30 (1 9 56). (4) Hoyt, P. R., Stanton, B. D., Petrol. Refiner 32, 115-19 (October 1953). (5) Oldenbourg, R. C., Sartorius, H., “The Dynamics of Automatic Control,” pp. 99-1 57, American Society of Mechanical Engineers, New York, 1948. (6) Rademacher, O., Rijnsdorp, J. E., “Dynamics and Control of Continuous Distillation Columns,” Proceedings of 5th World Petroleum Congress, Paper 5, Section VII, May 1959. (7) Rijnsdorp, J. E., Maarleveld, A., “Use of Electrical Analogues in the Study of the Dynamic Behavior and Control of Distillation Columns,” pp. A63 ff., Symposium on Instrumentation and Computation in Process Development and Plant Design, May 1959, Institution of Chemical Engineers, London. (8) Rose, A,, Johnson, C. L., Williams, T. J., Znd. Ens. Chem. 48, 1173-9 (1956). (9) Rose, Arthur, Sweeney, R. F., Schrodt, V. N., Zbid., 50, 737-40 (1958). (10) Rose, A,, Williams, T. J., Ibid., 47, 2284-9 (1955). (11) Rosenbrock, H. H., Brit. Chem. Ens. 3, 364-7, 432-5, 491-4 (1958). (12) Rosenbrock, H. H., Trans. Znst. Chem. Ens. London 35, 347-51 (19 5 7). (13) Rosenbrock, H. H., Tavendale, A. B., Storey, C., Challis, J. A,, “Transient Behavior of Multicomponent Distillation Columns,” Preprints of Papers, International Federation of Automatic Control Congress, Moscow, June 27 to July 7, 1960, pp. 1277-82, Butterworths, London, 1960. (14) Smith. 0. J. M., ZSA J . 6 , 28-33 (February 1959). (15) Williams, T. J., Znd. Ens. Chem. 50, 1214-22 (1958). (16) Williams, T. J., ZSA J . 9, 39-42 (July 1962). (17) Williams, T. J., Harnett, R. T., Chem. Eng. Progr. 53, 220-5 (19.57). (18) Williams, T. J., Harnett, R. T., Rose, A,, Znd. Ens. Chem. 48, 1008-19 (1956). (19) Williams, T. J., Min, H. S., I S A J . 6, 89-93 (September 1959). RECEIVED for review September 26, 1962 ACCEPTED March 8, 1963 Division of Industrial and Engineering Chemistry, 142nd meeting, ACS, Atlantic City, N. J., September 1962.
7 Figure 13. Uses of analyzer controller as secondary control element in column control
Upper.
Resetting of temperature set point
lower.
Fine tuning of flow control
PROCESS OPTIMIZATION B Y SEARCH TECHNIQUES D
.
M
.
H I MM ELB LA U
,
Department of Chemical Engineering, The University of Texas, Austin 12, Tex.
An adaptive search technique is proposed to solve the following type of problem: Maximize or minimize an objective function subject to linear or nonlinear constraints of the form G i ( X I , x2 , x,) = 0, 1 5 i 5 m; m < n. The objective function itself may b e linear, nonlinear, or expressed as some integral. An example application is presented for a simplified butane isomerization process.
..
HIS REPORT is concerned with the general problem of finding Ta solution to a set of m simultaneous equations with n unknowns, \\.here the n unknowns can be subject to constraints. Of special interest here is the case where the number of unknowns is greater than the number of independent equations, and consequently a “best solution” can be obtained only if a criterion of “best” is available as some objective function.
296
l&EC PROCESS DESIGN A N D DEVELOPMENT
While a wide choice of methods of solution is available when all the equations and the objective function are linear, the same cannot be said if the equations and/or the objective function are nonlinear. Certainly there is ample room in this area for new suggestions as to iterative or other techniques to resolve such difficult problems with simplicity and speed. A new method of handling this problem is described, that of a n
adaptive search technique, and a n example application is given for the optimization of a simplified butane isomerization process.
tinuous processes. To indicate the character and scope of direct search, consider the following problem : Minimize @;
Theoretical
F , ( x , :x ? . . . S C )
@ =
(6)
i=l
The problem of interest is deterministic and is related to the usual nonlinear programming problem, but it is not quite the same in all the details. A mathematical statement of the problem is: Maximize an objective function of one of these forms: n s1 =
(1)
CIX,
]=1
sz
=
2
&(XI,
xz.. . x m )
(2)
k=l
(3)
S1 represents a linear combination, S p a nonlinear combination, and Sa a n averaging (over time, for example) of some linear or nonlinear combination of the variables involved in a particular problem. I t is entirely possible to combine SI, s*,and Sf in a single expression of some form. The constraints of the system are: Gi
(XI,
xz
.
x,)
=
1
0
5i5
rn;
rn
- Hooke and Jeeves is a clever logical scheme of search which can be modified in the details to suit personal fancies as to the best way to proceed. Wood (5) has given flo\v diagrams and described a n IBM 704 program for direct search. Mugele (3) recently proposed a n IBM search routine. As an illustration of search in t\vo dimensions (so that the results can be diagrammed), examine Figure 1. T ~ v ovariables are involved, and the objective function is shown by the contour lines forming a “hill.” The constraints are also indicated. Once the search starts it improves the objective function on each trial until no further improvement can be obtained ; then the step size is reduced. If a forbidden region
Search Technique For Optimization
If a n individual operator had instruments to read the variables x , which describe the process conditions, and if he knew the mathematical model, Equations 4 and 5 (possibly in some nonmathematical form), and the objective function, he might determine those condiiions which met the objective and a t the same time satisfied the constraints by systematic trial and error in changing the process conditions. H e would say that the process operation was optimum if the adjustment of any controlled variable did lone of two things : Decreased the objective function Violated one of the constraints in Equation 5-i.e., ceeded, for example, a temperature or pressure limit
x2
ex-
If the operator has only one or two variables to work with, he can probably actually find the optimum conditions for operation, but if a large number of variables are involved, the analysis becomes practically hopeless. Then a search procedure, as r u n o n a computer, can be substituted for the human search to obtain the desired information. Hooke and Jeeves (2) have proposed a procedure called “direct search” for use in statistical analysis and optimization which seems to be well suited for use in optimization of con-
/
XI Figure 1.
Illustration of the path of search VOL. 2
NO. 4
OCTOBER 1 9 6 3
297
COOLANT Table I. Mathematical Model of the Problem Material balance
F = P + R Reactor actual conversion definition
- Xtr
c = Xtn --
1 - xtt
CATALYST AICI,
Reactor expected conversion definition
ABSORBENT
c * =- x XI6 - x1, Reactor chemical transformation
HCI
RECYCLE n- C 4
8
I
-
PRODUCT i -C,
cF
=
0.925 - 0.001OT
xte =
Experimental conversion
WASTE
@ Figure 2. process
P
Temperature dependence of equilibrium concentration
=
K c K ~ K r K(1.70) ~
=
0.04621 ( T - 120)
KF
=
740 7
Kr
=
T f 14.7 ___
Temperature factor
Flow diagram of the butane isomerization
In K T Space velocity factor
is encountered, the search moves along the edge of this region. The dotted lines in Figure 1 show steps which do not improve the objective function (Le., failures) and result in withdrawal back to the base point for that particular direction.
Pressure factor
Optimization of a Butane Isomerization Process
Total variables = 1 5 Total equations = 10
T o illustrate the actual application of the search process described above, take as a n example a mathematical model of a n n-butane isomerization process containing 10 independent equations and 15 unknowns. This example was chosen because the process has previously been described by Stout (4) and is readily available in the literature. Figure 2 shows a simplified block diagram of the actual process. ,4 vaporized feed stream containing n-butane and some isobutane is mixed with dry HC1 gas and introduced into the reactor. The reactor, shown as one block, actually consists of several reactors in parallel. After reaction, an absorber removes the small amounts of volatilized AlC13 which lvould otherwise foul subsequent equipment. Then follow several separation stages including a stripping column for the HC1 gas and fractionation units, which separate the n- from the isobutane. All these are shown as one block. The exact setup has been described by Gunness ( 7 ) . The details of the analysis of the problem and the method of obtaining the mathematical model have been given by Stout (4). Naturally, this development is the most difficult phase of the optimization analysis and is not to be passed over lightly. However, the concern here is with the technique of solution once the model is established; hence only a brief explanation of the derivation of the equations and constraints is given. To state the problem as simply as possible, Stout made a number of approximations which d o not detract from the method of optimization here, since it will handle considerably more complex statements of functional relationships than are actually used. The physical basis and mathematical statement of the 10 governing equations are shown in Table I. There is no reason why some of the equations cannot be combined together in the actual solution of the problem to reduce both the number of equations and unknowns sought. However, no matter 298
I&EC
P R O C E S S DESIGN A N D DEVELOPMENT
264.7
Table II. Objective Functions and Variables to Be Determined Given variables xii,
7
Variables to be determined F: T : SI:e*,
C,
K T , KT, K F , Kc: R, P,xis, xin
Objective functions Maximize instantaneous profit rate
D
=
apP - I?FF4- DER - Do
Maximize average profit rate
Constraints 180 0 0
< T < 230
3 R
(1) Gunness, R. C., in “Progress in Petroleum Technology,” Advan. Chem. Ser. No. 5 , pp. 109-19, American Chemical Society, Washington, D. C., 1951. (2) Hooke, R., Jeeves, T. A., J . Assoc. Computing Mach. 8, 212 (1961). (3)’-Mugele,R. A., ZBMSystems J. 1, 2 (1962). (4) Stout, T. M., Trans. A m . Znst. Elec. Engrs. 7 7 , Pt. 11, 640 (1959).
= number of functions in Eq. 6 = number of catalyst changes = product rate, bbl./day
= recycle rate, bbl./day = objective function defined by Eq. 1, 2, or 3
S, T
= reactor severity factor = temperature, F.
vi
= upper limit of constraint on variable
of variables or constants
literature Cited
xi
= number of simultaneous equations = number of unknowns = number of variables in Eq. 1, 2, and 3
S
= identification
(5) Wood, C. F., Westinghouse Research Laboratories, Sci. Paper 6-4121011-P1,Oct. 12,7960. \
,
RECEIVED for review November 13, 1962 ACCEPTED March 29, 1963
xi
OPTIMAL ADIABATIC BED REACTORS FOR SULFUR DIOXIDE WITH COLD SHOT COOLING K U N G - Y O U
L E E A N D R U T H E R F O R D A R l S
Department of Chemical Engineering, Uniuersity of Minnesota, Minneapolis 14, Minn.
To prove whether the dynamic programming scheme for the optimal design of a multibed adiabatic converter is feasible, the oxidation of sulfur dioxide is studied in detail. The computational problems and their solutions are described and results using the Calderbank kinetic equation are given.
THE general scheme for
the optimal design of multibed adiabatic reactors by the technique of dynamic programming has been known for some time ( 7 ) . When the cooling between stages is by heat interchanger, there are some analytical results that link with other approaches (4, 7), but when the cooling is accomplished by bypassing of cold feed it is necessary to use numerical computations from the very beginning. To see if the scheme that had been outlined was really practical, the case of sulfur dioxide oxidation was studied in detail. I t was found that the problems of search and storage could be overcome and results were obtained for two- and three-bed converters under various economic conditions. Since the scheme has been discussed fully (7), we do not give a full description here. I t may help, however, to outline the salient ideas. We are considering the design of a reactor with AT adiabatic beds and for the sake of definiteness take N = 3 (Figure 1). T h e reactant stream of total mass flow rate, G, is divided into two parts, a fraction XSG going to a preheater, where its tem300
l & E C P R O C E S S DESIGN A N D DEVELOPMENT
perature is raised from TOto Ts,and the remainder. (1 - &)G, serving as a bypass cooling stream. The composition of the stream is defined by the conversion, g (defined more precisely later), and in bed 3: containing a weight of catalyst 71’3, the conversion increases from g3 to g3’. Since the reaction is exothermic, the temperature will increase from T3 to T, and if this is too close to equilibrium it should be cooled before entering bed 2 by mixing it with a fraction ( A ? - A,) of the original feed; the temperature and conversion fall to T S and
