Chemical Process Optimization Using Penalty Functions Byron S. Gottfried, Paul R. Bruggink, and Eldon R. Harwood Gulf Research & Development Co., Pittsburgh, Pa.
15230
A penalty function technique can be used t o optimize continuous, nonlinear, chemical process problems of moderate dimensionality. The method i s applied to the optimal separation of four isomers of a gasoline blending stock. Optimal solutions are presented, and the characteristics of the method are discussed. The method i s efficient and troublefree for this example. It has t w o generally favorable characteristics compared with other nonlinear programming techniques: The constraints need not be satisfied until the end of the computation, which implies that recycle loops need not be closed, product specifications need not be satisfied, etc.; and the algorithm i s noniterative in the sense that a trial-and-error procedure, which may or may not converge, is not utilized.
RECENT
ADVANCES in mathematical programming techniques and the ubiquitv of digital computers have created great interest in the optimization of complex chemical processes. Many different approaches to this subject have been proposed by various authors, and a unified theory has not yet emerged. Consequently, a review of the recent literature can be most confusing, because of the wide variety of methods suggested for seemingly similar problems. Some of the mathematical techniques of current interest are presented in recent textbooks (Gue and Thomas, 1968; Wilde and Beightler, 1967). However, these books present only methodology; suggestions are not offered as to which techniques are most applicable to chemical process problems. Many process optimization problems of current industrial interest have as their purpose the economic evaluation of proposed processing schemes. Such problems usually involve an objective function which represents some economic criterion, and a set of constraints which represent over-all engineering design relationships. A detailed description of the physical and chemical phenomena which occur within the individual plant components usually is not included in such models. Typically, problems of this type are described by continuous, nonlinear mathematical models of moderate dimensionality (perhaps three to 20 decision variables), containing both equality and inequality constraints. Moreover. recycle streams are invariably present. Problems of this nature can sometimes be simplified by structural decomposition techniques, such as those described by Rudd and Watson (1968). In practice, however, the use of these techniques requires considerable attention to each individual problem prior to optimization. This entails an appreciable time expenditure. The realities
of industrial economics are such that answers must be forthcoming quickly and easily. Thus a need exists for one or more mathematical techniques which will allow process optimization problems t o be solved with minimum effort, from a standpoint of process analysis as well as computational efficiency. A class of techniques which frequently satisfies these requirements is nonlinear programming. This in itself, however, is a very broad area. Griffith and Stewart (1961), DiBella and Stevens (1965), and Prabhakar (1968) have proposed the use of iterative linear programming for solving process optimization problems. whereas Avriel and Wilde (1967) and Blau and Wilde (1969) have suggested the use of geometric programming. Other nonlinear programming techniques have been recommended by Goldfarb and Lapidus (1968), Gurel and Lapidus (1968), and Lee (1969). Additional applications of nonlinear programming to chemical process optimization problems are scattered through various technical journals. We have found a nonlinear programming technique utilizing penalty functions particularly effective for optimizing continuous, nonlinear problems of moderate dimensionality. Use of this method for solving chemical process problems was suggested, although very briefly, by Weisman, Wood, and Rivlin (1965). An elementary process example is also presented by Bracken and McCormick (1968). However, neither of these references places sufficient emphasis on the power of this approach for optimizing chemical process models of the type described earlier. In this paper, we apply the method to a C8 isomer separation process involving seven independent variables and four constraints. Optimal solutions are presented, the relative merits of the technique are discussed, and suggestions are offered for subsequent research in this area. Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4,1970
581
Description of Algorithm
The strategy employed in penalty-function optimization consists of transforming a constrained optimization problem into an equivalent unconstrained problem. The unconstrained function. called the modified objective function. is constructed in such a manner that one pays a “penalty” for violating any constraints that may be present. Under certain conditions, the modified objective function also contains “barriers” which prevent the optimizer from overstepping the bounds on the independent variables. Once the modified objective function has been formed. the problem is solved by optimizing a sequence of unconstrained problems, paying a successively higher penalty for each problem. The actual optimization is accomplished using any one of several known hill-climbing procedures. Modified Objective Function. The original problem is to optimize some continuous, differentiable objective function y (X),subject to the set of constraints
g ( X ) = 0. j = 1. 2 . . . .
. 1
I, 1 < n
toward the constrained optimum, the g.(X) tend to be satisfied, and the region boundary is approached gradually from within if the optimum lies on the boundary-as the unconstrained. modified objective function z(X, A:. A?) is optimized with respect to X for increasing values of and A?. Although the above concepts are intuitively appealing, there is reason to question whether or not the optimum of z(X, hi. A?) with respect t o X does coincide with a constrained optimum of the original problem. Fiacco and McCormick (1968) answer this question in the affirmative, providing that y(X) and g,(X), j = 1, 2, . . . , m , are continuous and t h a t X l k and X a are positive, monotonically increasing sequences. Specifically, Fiacco and McCormick show that
(1)
and
g,(X) S O , j = 2 + 1 , 1 + 2 , . . . . , m
(2)
The g,(X) are also assumed to be continuous and differentiable. I n addition, the independent variables are assumed to be bounded-Le.,
a5X5-b
(3)
In general, the modified objective function z(Xi can be formed from the original nonlinear programming problem as follows:
z(X,A I ,
A,) = y(X)
*{
XlUl(X)
+ (11x2) fJ?(X) }
(4)
where
and
where Xi.represents the optimal solution vector of z(X, ilk, A?,$), and X“ is the desired policy vector of the original optimization problem. (Fiacco and McCormick precede their proof with a rather esoteric set of hypotheses which. from an engineering viewpoint, simply implies that the problem be physically meaningful and well posed.) Moreover, the optimum will be a global optimum if the problem satisfies certain convexity requirements, a condition rarely attainable in process optimization problems. The modified objective function need not contain both r 1 ( X ) and c.(X) for all problems. Consider, for example, an unconstrained optimization problem with bounds on the independent variables. For such a problem one cannot define u1(X), so that one has simply
z(X,A,) The u l ( X ) and u 2 ( X ) terms are nonnegative functions called exterior and interior penalty functions respectively, and X I and are positive scalars called the penalty coefficients. The a i are positive scale factors chosen to avoid domination of the constraint system by any one constraint or group of constraints. T h e 6 , factors, which apply only to inequality constraints, are defined such that
= y(X) i ( 1 1 A L )
582
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
(11)
This is referred to as a pure interior point problem. On the other hand, suppose one wishes t o solve an optimization problem containing both functional constraints and bounds, using a n optimization technique which is self-bounding-Le., a hill-climbing procedure constructed in such a manner that the independent variable bounds are not violated. Under these circumstances, the modified objective function becomes
z(X,A,)
Finally; the plus sign in Equation 4 is chosen for a minimization problem, whereas the minus sign is selected for a maximization problem. The bracketed term in Equation 4 represents the penalty which one pays for either violating one or more constraints or moving too close to the region boundary. Increasing the penalty coefficients allows one to move closer to the region boundary. but a higher “price” is paid for constraint violation. The penalty term and the original objective function are combined in such a manner that the policy vector moves toward an optimal feasible point-Le., y(X) moves
cr?(X)
= y(X)
*
AlUl(X)
(12)
which represents a pure exterior point problem. The more general case expressed by Equation 4 is called a mixed interior-exterior problem. Such problems are more difficult to solve, computationally. than pure interior or pure exterior problems. Over-all Strategy. Having formulated the modified objective function in a manner consistent with the given problem, the following strategy can be employed. Choose initial values for h l and A ? (assuming a mixed interior-exterior problem), say and bI, and optimize z(X, A l l , with respect to X. The numerical values for hli and h.; must be chosen carefully, since values that are too large or too small can cause some of the terms in the modified objective function to overshadow the
stream are assumed known. The object is t o determine the recycle rate and the steady-state valve settings-Le., the stream splits shown in Figure 1--such that the per cent return on investment before taxes is maximized. The per cent return (mathematically, the objective function) is given by
remaining terms, which in turn can cause the optimization to go astray. After z(X, A l l , Xil) is optimized, the values for A, and X r are increased to X I ? and A D , z(X, A I ? , A L ? ) is optimized, and so on. T h e procedure is repeated sequentially, for increasing values of A I and A?, until the A's have become sufficiently large and the constraints are satisfied to within some c tolerance. Each successive optimization is started from the previous solution point. By solving a sequence of optimization problems in this manner, the desired optimunl is approached gradually in small steps. The sequence of small steps, rather than one large step, is required to prevent one or more terms in the modified objective function from dominating the remaining terms. Since the modified objective function is an unconstrained function, the actual optimization can be accomplished by any one of a number of known hill-climbing procedures. Two methods which we have found to be particularly effective are the pattern search of Hooke and Jeeves (1961), and the variable metric technique described by Fletcher and Powell (1963). Several more recent variable metric methods are described by Pearson (1969). Programming and implementation of these techniques on a computer are not difficult. Thus one of the two principal advantages of penalty function optimization becomes apparent-that unconstrained problems are much easier to solve than constrained problems. The other principal advantage is that the original constraints need not be satisfied during the course of the optimization.
y = d)I(X) 'd)?(X)
(13)
where bl(X) and $?(X) are nonlinear functions of X , as described in the Appendix, and X is a seven-dimensional policy vector whose components represent the recycle rate and the various valve settings. The problem is constrained by two nonlinear equalities and two nonlinear inequalities. written functionally as
&(XI &(XI
=0 =0
(14)
50 &(X) 5 0
g I(X)
These expressions represent closure of the recycle stream, specified flow rate of the overhead stream leaving the crystallization unit, and upper and lower bounds on the rate a t which A3 leaves the top of the second fractionation tower. The explicit constraining equations are given in the Appendix. Finally, the policy vector is bounded by the numerical values shown in Table I. Optimal Solution to Problem. This problem has been solved from ten randomly generated starting points, within the bounds given in Table I, to increase the likelihood
Application to Ce Isomer Separation Process
T o demonstrate the applicability of this algorithm to a typical process optimization problem, consider the C8 isomer separation process shown in Figure 1. I t is desired to determine the optimal operating policy for this process, which involves the separation of isomers A1 and A3 from a mixture of A l , A2, A3, and A4. Each of isomers A1 and A3 can be separated from the mixture and sold as a single component. The remaining mixture can be used as gasoline blending stock, with a portion of the mixture isomerized into an equilibrium mixture of A l , A2, A3, and A4 and then recycled. I n addition to the Ca isomers, C- and CS- compounds are present in varying amounts. The feed rate, the feed composition, and the composition of the exit stream from each unit relative to each entering
Table I. Upper and Lower Bounds of independent Variables tower Bounds
Upper Bounds
a, = 10
bi = 5000 b, = 1 b,= 1 bi = 1 b, = 1 be = 1 b- = 1
a, = ai = a4 = a, = as = a-=
0 0 0 0 0 0
F = 1349 M lbiday
-
GASOLINE ELENDINZ STOCK
[I-X,) II-Xg)
7 x6 i
I
-
x5
\
, lSOYERlZ4TlON
UNIT
UNIT
n(l-XS) x4
(I-Xp)
MIXED ISOMER FEED
-FUEL
'7
-iXT4L:IZ4TlON
XZ
,,.--,
(I-Xq)
yT- I
RECYCLE (EPUILIBRIUM MIXTURE A I, A 2 , A 3 , A 4 )
L
- 0
i
GAS L I G H T AROMATICS HEAVY AROM AT I C s
-$;
A3
F"
- g: r
P
$5
(I-X3)
XI
x3 HEAVY A R O M AT ICs
K I
Figure 1 .
Cg
isomer separation process
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 583
~
~~
~
~~
~~~~
~~
Table II. Optimal Solutions from Ten Randomly Generated Starting Points R u n No. Variable
3' XI X2
xi X4
Xi
XS XUO
IPL IT EOBJ
1
2
3
4
5
6
7
8
9
10
25.6 1834 0.338 0.385 1.000 0 1.000 0.749 6 x 10.' 12 914 6265
25.6 1839 0.005 0.611 1.000 0 1.000 0.750 5x 13 548 3942
25.6 1839 0.013 0.606 1.000 0 1.000 0.750 4 x 10." 8 628 4317
25.6 1839 0.003 0.613 1.000 0 1,000 0.750 2 x 10 ' 22 1204 8459
25.6 1836 0.198 0.480 1.000 0 1.000 0.750 3 x 10 12 712 5014
25.6 1839 0.003 0.613 1.000 0 1.000 0.750 9 x 10 " 11 972 6612
25.6 1839 0.011 0.608 1.000
25.5 1829 0.739 0.110 1.000 0 1.000 0.748 3 x 10 19 554 4126
25.6 1834 0.347 0.378 1.000 0 1.000 0.749 8 x 10 ' 21 612 4615
25.6 1835 0.263 0.436 1.000 0 1.000 0.749 3 x 10 -' 8 598 4143
of finding the global optimum if several local optima had been obtained. The pattern search of Hooke and Jeeves (1961) was used to maximize the modified objective function. Since the pattern search algorithm is self-bounding, the modified objective function was constructed in accordance with Equation 12. The ten optimal solutions are listed in Table 11. Also listed are the sum of the squares of the unscaled constraints , number of penalty levels which are not satisfied ( u ~ ) the required ( I P L ) ,the number of major search moves ( I T ) , and the number of functional evaluations ( I O B J ) for each solution. The optimal per cent return on investment is approximately the same (y,,, 25.6) for each run. The policy is not unique, however, as is seen from the varying distribution of fresh feed and recycle sent to the fractionation and crystallization units. The fact that different policies result in essentially the same optimum is not surprising, since the recycle stream is an equilibrium mixture whose composition is very similar to that of the fresh
0 1.000 0.750 3 x 10 15 852 6066
feed stream. Other runs, made with cost parameters different from those shown in the Appendix, revealed several distinctly different local optima. Table I1 also reveals that the amount of computation required to obtain an optimum depends upon the starting point. This is seen from the numbers of penalty levels. major search moves, and functional evaluations required for each problem. All ten problems were run in less than 3 minutes on an IBM 360175, however, so that the computational effort is not excessive. (This timing can be altered by changing the numerical values for certain convergence criteria within the computer program.) Characteristics of Method. Of greater interest than the answers obtained for this particular problem are the method itself and its general applicability t o chemical process problems. To gain further insight into these matters, a computational history of run 2 is presented in Table 111. In this table, one can see the maximum value of the modified objective function and the corresponding
-
Table Ill. Penalty level Variable
0"
185.7 185.7 2779 0.823 0.815 0.704 0.993 0.963 0.843 1.5 x loJ 2.6 x 10' 5.2 x 10' -6.9 x 10'
2
Y XI
X?
X, XI Xi
Xfi X: gl
gi gi gi (YI
U,
ai a4
x
2.6 x 10'
UG
u1
IT IOBJ
1
633.5 633.5 5000 1.000 1.000 1.ooo
1.000 0.613 0.493 5.0 x 10' 3.4 x 10' 1.2 x 10' -1.4 x 10' 0.49 3.6 2.0 47 2.3 x 10 -' 2.7 x 101.4 x 10' 22 157
2
3
4
5
6
633.3 633.5 5000 1.000 1.000 1.000 1.000 0.613 0.493 5.0 x 10" 3.4 x 10' 1.2 x 10' -1.4 x 10.' 0.44 4.5 2.1 74 1.8 x l o . * 2.7 x 10: 1.4 x 10: 4 57
631.5 633.5 5000 1.000 1.000 1.000 1.000 0.613 0.493 5.0 x 10' 3.4 x loL 1.2 x 10' -1.4 x 10' 0.44 4.5 2.1 74 1.5 x 10 2.7 x 10; 1.4 x 10; 4 57
617.1 633.5 5000 1.000 1.000 1.000 1.000 0.613 0.493 5.0 x 10' 3.4 x lo1 1.2 x 10' -1.4 x 10' 0.44 4.5 2.1 74 1.2 x 10 2.7 x 10; 1.4 x 10' 4 57
501.9 633.5 5000 1.ooo 1.000 1.000 1.ooo 0.613 0.493 5.0 x 10' 3.4 x 10' 1.2 x 10' -1.4 x 10' 0.44 4.5 2.1 74 9.4 x 10 '> 2.7 x 10: 1.4 x 10' 4 57
91.5 383.6 1752 1.000 1.000 1.000 1.000 0.613 0.493 1.8 x 10' 3.4 x lo? 4.5 x 10' -6.2 x 10' 0.44 4.5 2.1 74 7.6 x 10 ' 3.4 x 10" 3.9 x 10" 28 227
Starting point. ' Optimal solution.
584
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
solution vector after each penalty level-i.e., after each of the sequences of unconstrained optimizations. Also shown are the values of the constraints, their scale factors, the penalty coefficient, and other related parameters. From the first two rows in Table 111, one can see that the penalty coefficient is initially chosen such that )(X), rather than cl(X),dominates the modified objective function. Experience has shown this frequently (but not always) to be desirable. Our computer program has a rather complicated set of criteria for choosing the initial value for A ; these computational details are not discussed in this paper. The sequence of policy vectors shows that movement in the hyperspace is not obtained during each penalty level-for example, in penalty levels 2, 3, 4, and 5, the penalty coefficient has not yet become sufficiently large to cause movement toward a feasible region. Considerable movement is observed in penalty levels 6 through 10. During the last few penalty levels, there continues to be some movement, although slight. z(X) and u0(X) decrease monotonically after the first penalty level [z(X)will increase monotonically in a minimization problem]. This in an important characteristic of the algorithm. The strategy is designed to reduce m ( X ) monotonically without iteration in the sense that a trialand-error procedure which may or may not converge is not utilized. This is a distinct advantage over many nonlinear programming algorithms, including iterative linear programming, which may suffer from lack of convergence. Another important characteristic of the method is that the constraints can be violated during the course of the optimization. This implies that recycle loops need not be closed, and that product specifications need not be satisfied, until the final optimum has been obtained. 'The significance of this feature becomes apparent when one considers the many alternative schemes which have been proposed, and the computational effort involved. in satisfying the constraints a t each step of an optimization.
Finally, the use of constraint scaling factors is not necessary to the success of the method. However, judicious choice of these parameters increases the computational efficiency of the algorithm for many problems (Keefer and Gottfried, 1970). Penalty function methods are not free of drawbacks. Their success is often sensitive to the choice of numerical values for certain parameters, such as the penalty coefficient. Furthermore, the presence of constraints causes the modified objective function to take on ridge-like convolutions that can tax the ability of even the most advanced hill-climbing procedures. Nonetheless, penalty function techniques can be very useful for problems of the type presented herein. Their attractive features seem to warrant additional investigation in applying these methods to chemical process problems. In particular, problems of higher dimensionality which contain more constraints, such as problems with multiple recycle loops, problems involving unknown concentrations and operating conditions, and problems which include additional engineering fundamentals, are fruitful areas for future research. Appendix. Octane Isomer Separation Process
Four isomers, A l , A2, A3, and A4, of a C, compound A are fed into a separation plant, as shown in Figure 1. Briefly, A1 and A3 can either be separated out and sold as separate products or used as gasoline blending stock; A2 and A4 can either be used as gasoline blending stock or isomerized into an equilibrium mixture of Al, A2, A3, and A4. Also present in varying quantities are C;- and C g - hydrocarbon mixtures. The object of the analysis is to determine the recycle rate and the steadystate valve settings such that return on investment is maximized. The inlet flow rate, the feed Composition, and the composition of the exit stream from each unit relative to each entering stream are assumed known.
History of Run 2 Penalty Level
7
35.5 45.9 1807 0 0.628 1.000 0 1.000 0.649 2.3 x 10' 3.5 0.7 -1.6 x 10' 0.56 3.7 2.6 54 6.0 x 10 * 5.5 x 10' 1.7 x 10' 270 1761
8
27.0 28.3 1839 0.004
0.613 1.000 0 1.000 0.737 32 0.01 0.002 -1.6 x 10' 0.52 13 19 19 4.8 x l o - ' 1.0 x 10' 2.8 x 10' 50 354
9
10
11
12
13b
25.8 26.0 1839 0.013 0.607 1.000 0 1.000 0.748 4.0 -0.001 0.003 -1.6 x 10' 0.52 8.9 7.1 7.1 3.9 x 10 16 4.4 24 186
25.6 25.7 1839 0.007 0.611 1.000 0 1.000 0.750 0.49 0.004 0.003 -1.6 x 10' 0.52 3.2 2.5 2.5 0.31 0.24 0.067 102 691
25.6 25.6 1839 0.005 0.611 1.000 0 1.000 0.750 0.070 -0.004 -0.44 -1.6 X 10' 0.52 1.6 1.3 1.3 2.5 5.0 x 10 1.4 x 10 20 165
25.6 25.6 1839 0.005 0.611 1.000 0 1.000 0.750 9.6 x 0 ' 9.6 x O J -0.52 -1.6 x 0' 0.52 1.6 1.3 1.3 20 9.4 x 10 ' 2.8 x 10 8 87
25.6 25.6 1839 0.005 0.611 1.000 0 1.000 0.750 1.5 x 10 1.6 x 10 -0.54 -1.6 x lo2 0.52 1.6 1.3 1.3 160 4.8 x 10 '' 7.3 x 10 E 8 86
'
'
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
585
The unknown quantities in this problem are the recycle rate and the fractional flow streams shown in Figure 1. The following material balance equations apply:
Constraints. The unknown recycle rate, xI, must match the calculated recycle rate, R . This can be expressed an a n equality constraint
g,=xi-R=O The flow rate of the crystallization unit overhead steam is assumed to be fixed. This piece of information enters the model as a n additional equality constraint: g?= P - CO = 0 Finally, the rate a t which A3 is produced from the fractionation unit must lie between specified upper and lower bounds: g i = T2,4:3- 274 5 0 gj = 110 - T2-4.4 5 0 Thus the model contains two equality constraints, two inequality constraints, and seven dependent variables, as defined by the above expressions. The objective is t o maximize per cent return on investment before taxes. The following items constitute the total plant investment. FRACTIONATION TOWERS I f ;= 7.4 (FF)"' + 2.3 ( B l ) " ' 586
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
CRYSTALLIZATION UNIT (INCLUDING ROYALTY) 1c -- i ~ . ( c O ) "+' 4.5(CO) -7
ISOMERIZATION UNIT (INCLUDINGCATALYSTA N D ROYALTY) I I = 6.4(F1Id' + 0.06(FI) + 50 UTILITIES(STEAM, POWER, WATER) I L= 0.00956(FF) + 0.002775(B1) + 0.441(CO) + O.O5(FI)
TANKAGE (CHARGETANK, A1 PRODUCT STORAGE. A3 PRODUCT STORAGE) A N D YARDLINES I T = 0.2935 (F + CO + T2) ADDITIONAL OFF-SITESINVESTMENT 1,)= 0.1[7.4(FF)"'+ 75.O(C0j0' + 6.4(FI)"+ 11-+ I T ]
TOTAL PLANT INVESTMENT I = IF
+ I ( + 1 1 + I [ + 17 + I o
WORKING CAPITAL WC = 9.75(T2) + 5.25(CO) The net return is the return from sales less the costs. These items are determined as follows: (DAILYBASIS) RETURNFROM SALES S = 65(T2) + 30(CO) + [26.7(FG) - 8 . 9 ( F G ~ i )+] 17.9(IP( ;-) + 18.8(1(9+ + B2,9+) + 5 . 8 ( 1 P ~ ( , ) COSTOF CHARGE C ( = 24.7(F) + 4 6 . 9 ( F I ~) MANUFACTURING EXPENSE(LABOR,FUEL, UTILITIES, CHEMICALS, CATALYST) C \ I = 816 + 0.028(FF) + 0.0081(B1) + 3.2(CO) + 1.15(FI)
IOBJ = number of functional evaluations I P = output from isomerization unit, M lb/day I P L = number of penalty levels IT = number of major search moves I = number of equality constraints m = total number of constraints n = number of independent variables
P = desired flow rate of crystallization unit overhead stream, M lbiday R = calculated recycle rate, M lb/day s = return from sales, M $ / d a y T1 = overhead from first fractionation tower, M lb/ day
T2 = overhead from second fractionation tower, M
wc X
Xa X"
x, x?
x.1 x4
xi xi;
x; y z N
INVESTMENT-BASED EXPENSE C I = 0.15(1)
6 B A
TOTAL EXPENSE c = c c + C\I + CI
UII
PER CENT RETURNON INVESTMENT BEFORE TAXES
UI
(YEARLY BASIS) ? / =[(100)(365)(S- C ) ] / ( I + WC) = @i(X)/$?(X) Careful reflection reveals both the numerator and the denominator of the objective function to be complicated, nonlinear functions of the decision variables (XI, x,, . . . ,
x-i .
Nornencla ture
a = lower bound on policy vector B1 = bottoms from first fractionation tower, M lblday B2 = bottoms from second fractionation tower, M lblday b = upper bound on policy vector C = total expense, M $/day CO = overhead from crystallization unit, M lb/day CR = raffinate from crystallization unit, M lbiday F = mixed isomer feed, 1349 M lb/day FC = feed to crystallization unit, M lb/day F F = feed to fractionation unit, M lbiday FG = feed to gasoline blending station, M lbiday F I = feed to isomerization unit, M lb/day g = constraining equation I = investment, M $
U?
lb / day = working capital, M $ = policy vector = optimal policy vector for hth penalty level = optimal policy vector for given problem = unknown recycle rate, M lbiday = fraction of mixed isomer feed stream sent to fractionation unit = fraction of recycle stream sent to fractionation unit = fraction of overhead from first fractionation tower sent to crystallization unit and gasoline blending station = fraction of crystallizationigasoline feed from first fractionation unit sent t o gasoline blending station = fraction of crystallization/ gasoline fresh feed sent to gasoline blending station = fraction of raffinate from crystallization unit sent to isomerization unit = objective function (5 return on investment. yearly basis) = modified objective function constraint scale factor = = inequality constraint "on-off' switch = theoretical overhead from crystallization unit = penalty coefficient = sum of squares of unscaled constraints not satisfied (obtained from Equation 5 by setting each LY, = 1) = penalty function defined by Equation 5 = barrier function defined by Equation 6
SUBSCRIPTS A1 A2 A3 A4
c-
= = = = =
= H =
cs.
1
=
J = k =
isomer A1 isomer A2 isomer A3 isomer A4 light aromatics heavy aromatics hydrogen ith independent variable j t h constraint kth penalty level
Literature Cited
Avriel, M., Wilde, D. J., IND. ENG. CHEM. PROCESS DES.DEVELOP. 6,256-63 (1967). Blau, G. E., Wilde, D. J.. Can. J . Chem. Eng. 47, 31726 (1969). Bracken, J., McCormick, G. P., "Selected Applications of Nonlinear Programming," pp. 37-45, Wiley, New York, 1968. DiBella, C. W., Stevens, W. S., IND.ENG.CHEM.PROCESS DES. DEVELOP. 4,16-19 (1965). Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
587
Fiacco, A. V., McCormick, G. P., “Nonlinear Programming. Sequential Unconstrained Minimization Techniques,’’ pp. 60-1, Wiley, New York, 1968. Fletcher, R., Powell, M. J. D., Computer J . 6, 163-8 (1963-4). Goldfarb, D., Lapidus, L., Ind. Eng. Chem. Fundam. 7, 142-51 (1968). Griffith, R. E., Stewart, R. A., Management Sei. 7, 37992 (1961). Gue, R. L., Thomas, M. E., “Mathematical Methods in Operations Research,” Macmillan, Xew York, 1968. Gurel, O., Lapidus, L., Ind. Eng. Chem. Fundam. 7, 61722 (1968). Hooke, R. J., Jeeves, T. A., J . Assoc. Comp. Mach. 8, 212-29 (1961). Keefer, D. L., Gottfried, B. S., A . I . I . E . Trans., in press, 1970.
Lee, E. S., A.1.Ch.E. J . 15,393-400 (1969). Pearson, J. D., Computer J . 12, 171-8 (1969). Prabhakar. T., Ind. Eng. Chem. Fundam. 7, 626-32 (1968). Rudd, D. F., Watson, C. C., “Strategy of Process Engineering.” pp. 251-73, 282-8, Wiley, New York. 1968. Weisman, J., Wood, C. F., Rivlin, L., Chem. Eng. Progr. S y n p . Ser. 61, 50-63 (1965). Wilde, D. J., Chem. Eng. Progr. 61, 86-93 (1965). Wilde, D. J., Beightler. C. S., “Foundations of Optimization,” Prentice-Hall, Englewood Cliffs, N. J.. 1967.
RECEIVED for review September 22, 1969 ACCEPTED May 21, 1970
Influences of Catalyst Formulation and Poisoning on the Activity and Die-off of l o w Tempera tu re Shift Catalysts John S. Campbell Agricultural Division, Imperial Chemical Industries, Ltd., Billingham, Teesside, England The use of copper catalysts for the low temperature shift reaction has become much more widespread in the last decade in hydrogen and ammonia synthesis gas production.
This has led to a general awareness among plant operators of the problem of catalyst die-off. Die-off i s shown to be the consequence of two factors: thermal sintering, which can be reduced and, in some cases, eliminated by correct catalyst formulation methods; a n d poisoning b y small concentrations of impurities such as sulfur and chlorine, carried in the gas stream. The importance of the method of catalyst manufacture on subsequent activity and stability i s outlined; with a well-formulated catalyst, not maltreated, poisoning
i s the maior cause of loss of activity on the commercial scale. The reaction kinetics, using a commercially available catalyst, have been examined and a design equation
is proposed. The activity of the catalyst, in the absence of a diffusion limitation, is directly proportional to the copper area and initial activity increases with increasing copper content. The use of hydrogen in steam for reduction of the catalyst i s discussed.
THE
great majority of recent ammonia and hydrogen plants employ the low temperature water gas shift reaction in conjunction with methanation for carbon monoxide removal. This combination has come about for economic reasons and is advantageous because, a t the reaction temperature obtained with a commercial iron oxide-chromia catalyst, the reaction is equilibrium-limited. Since there is no volume change and the reaction is exothermic, the equilibrium conversion of carbon monoxide can be increased only by decreasing the temperature. The variation of the equilibrium concentration of carbon monoxide with temperature for typical shift exit conditions is shown in Figure 1. The widespread use of low temperature shift catalysts during the last decade [although they have been known since the later twenties (Larson, 1931a,b; 1947)] has led 588
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Y, No. 4,
1970
to a general awareness among plant operators of the problem of catalyst die-off. This paper shows that die-off is the consequence of two factors: thermal sintering, which can be reduced and in some cases eliminated by correct catalyst formulation methods, and poisoning, by small concentrations of impurities, such as sulfur and chlorine, carried in the gas stream. The importance of the method of catalyst manufacture in subsequent activity and stability is outlined. The structural problems that have to be taken into consideration prior to deciding on a method of formulation are discussed. With a well-formulated catalyst. provided it is not maltreated, poisoning is the major cause of loss of activity. The function of each of the components in IC1 and Katalco catalyst 52-1 (subject of catalyst and process patents and patent applications in many countries) has
