Conclusions
SUBSCRIPTS
Combined bubble and foam fractionation operates successfully, with good stripping and enriching ratios obtainable. However. the incremental advantage of the elongated bubble section is small, a t least for the present system. Increasing the reflux ratio increases the degree of enrichment. Increasing the rate of gas flow improves the degree of stripping but decreases the degree of enrichment. Location of the feed inlet can be important under certain circumstances.
F
D
Nomenclature
C = solute concentration in liquid and collapsed foam streams, molarity D = tops flow rate, ml.,’min. F = feed flow rate. ml.,’m’n. G = gas flow rate. ml./min. h = height above gas bubbler, cm. Q = collapsed (air free) overhead flow rate, ml.,’min. R = reflux ratio T I * = bottoms flow rate, ml., min.
M’
= = =
tops feed bottoms
literature Cited
(1) Bikerman, J. J., “Foam,” p. 2, Reinhold, N e w York, 1953. (2) Brunner, C. A , , Lemlich, R.. Ind. Eng. Chem. Fundammtals 2, 297 (1963). (3) Lemlich, R., Lavi, E., Science 134, No. 3473, 191 (1961). (4) Leonard, R . .4., Lemlich. R., A.1.Ch.E. J . : in press. (5) Rubin, E. E., Gaden, E. L.? Jr., ”New Chemical Engineering Separation Techniques,“ H. M. Schoen, ed., Chap. 5, “Foam Separation,” p. 319, Interscience, New York, 1962. RECEIVED for review September 20, 1963 ACCEPTED July 20, 1964 b-ork supported by U. S. Public Health Service Research Grant WP-16 1.
PROCESS OPTIMIZATION BY NONLINEAR PROGRAMMING C A R L O S W .
D I
B E L L A A N D W I L L I A M
F. S T E V E N S
.Yorthroestern CniLerszty, Euanston, Ill.
A recently proposed nonlinear programming technique i s used to optimize a continuous chemical process. The problem i s to extremize a nonlinear criterion function, F(x,), subject to nonlinear constraints, G,(x,) = 0, 1 5 j 2 n; 1 5 i 5 rn n. The method consists of finding a feasible solution, and then applying the linear programming algorithm to the linearized mathematical model around successive feasible solutions until an optimum i s reached. The example solved i s of sufficient complexity to demonstrate the value of this method for chemical process optimization.
-one step in the process of arriving a t an engineering decision. If the optimization is to be done analytically. a t least two preliminary steps are necessary. First: the engineer must develop a mathematical model of his system. incorporating all that is known about the process into the resulting expressions. Second: he must select a measure of effectiveness, Lvhich is to be minimized or maximized by means of the optimization procedure. Both steps require an intimate knokvledge of the system under study and its relation to the rest of the company or the economy as a whole.
Mathematical
G,(xl",. . . .x,")
Let
We shall assume that the problem is completely deterministic a n d , therefore, is amenable to analytical solution. Mathematically. the nonlinear optimization problem for a complex chemical plant design may be represented in the following form. Maximize
aF
X?)
. . . .x,)
(1)
Subject to
Gr
(XI,
Lj
x2,
....x n )
5
xj
=
i
0 j
5; C;
=
=
1, 2, . . . .rn
=
w,
(11)
(X,
-
=
X,")
AYj
(12)
Now Equations 5, 6, and 7 may be rewritten as folloi+s Maximize n
(2)
1, 2, . . . .n
. . . bl")
(9)
ax5
and F(X1,
(XI",.
L'l, a constant
=
F - F" =
(3)
W3Axj 1-1
Subject to
and .x3
2 0
(4)
where F ( x , ) is the objective function (also known as the criterion function or cost function) to be extremized and is generally nonlinear. G,(x,) are the constraints and are also generally non2 , for linear. T h e equality sign could also be some or all of the m relations; usually m < n in the case of equality constraints. L , and C, are the lower and upper limits, respectively, on the variable x i . These limits arise from the physical limitations of equipment and materials.
and
s>
Discontinuities that may arise in any physical situation may be handled by smoothing over or by penalty functions. They are not considered in this report. Convergence of the optimization procedure to a global optimum is assured only under the very restrictive qualifications that all constraints are convex and the objective function is concave. If these conditions are satisfied. it is impossiblt to end u p in any place where the objective function cannot be improved by moving in a feasible direction (except a t the (optimum). In practice, few real problems meet these conditions. Most real systems are well behaved. however. a n d can be solved by multiple starting points and engineering judgment.
Now let A*x, = .Ix, when AX, 2 0, and A - x j = -Atl \rhen AX] 5 0. Although this step essentially doubles the number of variables to be considered, it is necessary because the linear programming algorithm will handle only variables Ivhich are greater than or equal to !zero. Substituting in Equations 13 a n d 14 the problem becomes Maximize
Subject to : n ]
n
=1
Now restrictions of the following form may be added.
Nonlinear Programminsg Method
Griffith and Stewart ( 7 ) have presented the following technique for solving the nonlinear programming problem as given in Equations 1, 2. 3. a n d 4 by using a linear programming technique sequentially. T h e problem is first linearized in the region about some initial approximation of x j by expansion as a Taylor's series. ignoring terms of higher order than linear. T h e linearized problem then becomes Maximize
F
= F(Xl0... . . . X n " )
+
a n d M , is the maximum distance x j is allowed to move before reapproximating the problem. If any variable Y ] is close to its upper limit LT3, then (C; - xj0) will be approximately zero a n d A+x, will be small. T h e same reasoning holds for Y . Y ~ when ( x j o - L j ) is close to zero. Equations 16, 17, 18, 19. a n d 20 now define a system which can be solved by linear programming. After going through the AT2, linear programming routine. x j 3 is changed to . x I o 4-xj a n d the routine is repeated until no further_ change in x j is . indicated. This is the optimum.
+
Subject to
G , (.xl0,. . . ..no)
+
Chemical Process Example
i L, and
5
x,
5 u,
=
l,..m
j=l,..n
(7)
T h e design problem optimized in this report was first proposed by Williams and O t t o ( 3 ) . T h e Williams plant contains many of the characteristics of a typical chemical design problem a n d is realistic enough so that a steady-state optimization has meaning. VOL. 4
NO. 1
JANUARY
1965
17
1
I
I
I
f
T,
MCHANGER
e
REACTOR
Fp,*
-
DISTILLATION
HEAT
g-
CSTR
cooling water
v
&
DECANTER COLI"
t
Lc
I
I I
I
L
I Icwiing
Figure 1 shows a simplified block diagram of the process. T h e plant is to manufacture 40 million pounds of chemical P per year. T h e plant consists of a perfectly stirred reactor, a heat exchanger, a decanter, and a distillation column in series. There is recycle from the column reboiler to the reactor. T h e chemical reaction kinetics are complex. There are three second-order irreversible reactions. 1. A 2. C 3. P
+B +B +C
-
-+ -+
C reaction coefficient, k l P E reaction coefficient, k? G reaction coefficient, ka
+
Reactants A and B are fed separately to the reactor in pure form. Components C a n d E are intermediates and/or byproducts. They have no sales value but may be disposed of as plant fuel. By-product G is a heavy oil which must be disposed of as a waste material. T h e reaction coefficients may be expressed in the Arrhenius form:
kt where
=
F,,
- 0.1 FRE -
Fp = 0
3.
Material balance on component E
4.
Material balance on component P
5.
Material balance on component -4
= AIexp(--B,,'T)
X 109/hr., wt. fraction 12,000O Rankine (based on A or B) 2.5962 X 10I2/hr., wt. fraction 15,000° Rankine (based on B) 9.6283 X lOl5,'hr.) wt. fraction 20.000' Rankine (based on C) Z' = temperature (variable), O Rankine
A1
Gn
= 5.9755
F.4
B1 = A? = B2 = AS = BS =
6.
In addition the rate of reaction is negligible below 120' F., and undesirable decomposition occurs above 220' F., SO that the reactor temperature is constrained. 580 _< T
5
680
O
Rankine
0 (26)
=
0
(27)
=
0
(28)
Material balance on component B
F , L FR - F p - F , ~
7.
=
+ F,
Material balance on component C
(21)
Steady-State Balances
T h e following constraint equations are derived by making independent material balances across the system, by a constraint on separation efficiency in the distillation column, and by the definition of FR.
8.
Material balance on component G (29)
1, Over-all material balance G1
FA
+ FB -
FO
- Fp
- F,
=
0
(22)
2. Constraint on the separation efficiency of the distillation column (azeotrope formation) 18
l&EC PROCESS DESIGN A N D DEVELOPMENT
F,
=
0 (30)
T h e objective function is expressed as per cent return on investment.
% return
Table I. Initial Approximation, Improved Approximation, and “Optimal” Solution
+
F = 100 18400(0.30Fp 0.0068FD 0.02FA - 0.03Fb - 0.01FG) - 2.22FR (0.124)(8400)(0,30Fp f 0.0068FD) - 60 V p ] t =
600 V p
Variable
(31)
Thus. the mathematical model of the \Cilliams plant agrees with the nonlinear optimization problem description of Equations l : 2 > 3, and 4. T h e number of variables, n: is 1 2 : FA! FB2F , ? FG, FRAI F R B , FRcl FR,, F R p . F R s V : and 7‘. The number of equality constraints, rn: is 9. T h e reader interested in the complete development of the objective function is referred to \Villiams‘ original paper.
FA FB FD Fc FRA FRB FRC FRE V
FR FRP T
S
7;return
Initial Guess 10.000 40,000 20.000 5,000 6,000 35,000 10.000 15.000 150 100.000 6,000 61 0 3 x 109
Improzed Initial Guess
“Optimal” Solution
11,540 31 -230 36.010
13.546 31,523 36 6g7 3,609 18,187 60.915 3,331 60.542 60 157,391 10,817 656 100 72 75
2.010
8.820 39.910 2.360 31.660 223 92.640 7 890 610 1 2 x 104 =18
Units Lb./hr Lb./hr. Lb./hr. Lb./hr. Lb. /hr. Lb. /hr. Lb /hr. Lb./hr,
cu. ft.
Lb./hr. Lb./hr. O Rankine c /C
Method of Solution
T h e Williams plant problem is optimized by applying a modified steepest descent technique to the constraints, and then using a linear programming technique to improve the value of the objective function. First a n ‘.arbitrary” set of values is picked for the variables.. Then the steepest descent routine is applied in order to bring the variables .’close” to satisfying the set of constraint equations. T h e objective function and the constraints are linearized next, a n d the objective function is improved in a small region about the set of variables by the method of linear programming. T h e variables are changed and the entire process is repeated. In the steepest descent routine, we are concerned only with improving the initial guess. ( x 1 O > . , .xl2’)> to the equality constraint sct: G,(.ul,.. x , ? ) i= 1 . . . .9>and the objective function is ignored for the time being. T h e steepest descent method may be formulated mathematicallv as a problem of minimizing the sum of the squares of the residual errors-i.e., Given G,
G, Find a set of
Y]
(XI
. . . . XI?)
=
i
0
=
1, . . . .9
(32)
such that 9
S
Gt2 =
:=
o
(33)
,=l
Obviously. if a set of x I can be found to satisfy Equation 33, this set \\ill also satisfv the constraint equations. Now the slope of S is determined in respect to each variable. T h e n each variable is given a small increment proportional to a n d opposite to its slope. T h e slopes are evaluated as follows.
T h e steepest descent technique is generally considered to be one of the better methods available for solving simultaneous algebraic equations ( 2 ) Actually, in the problem a t hand, it is not necessary to use the steepest descent technique. Sound engineering judgment \+auld give a n initial guess which is close enough to satisfying the constraints so that ihe linearizing a n d linear programming methods could be applied directly. A Fortran I1 program was written for a n I B M 709 computer. T h e program applied the methods of steepest descent a n d linear programming to Equations 21 through 31 which define the \Villiams system. ‘The initial guess chosen for a starting point is given in Table I T h e value of S i s very high, which indicates a poor initial g,uess. T h e poor guess was chosen to show the convergence of the method T h e steepest descent
program was used to decrease S to approximately lo4, a t which point the linear programming program was applied. It was found that a set of variables which yielded a value of S of this order of magnitude differed from a “true” solution by no more than + l % . In view of the poor precision of most measuring devices below this figure, it was decided that the linear programming program could be applied to a solution set with this error. T h e improved initial guess also i s given in Table I. T h e value of the reactor temperature, T,was he13 constant Lvhile applying the steepest descent method because of the relatively narrow limits imposed on T. It was found that S could be held to lo3 or less by the proper choice of the negative “costs” associated with the artificial variables added to the linear programming tableau, and by choosing rather small values of M , [Equation 181. This increases the amount of computer time required but has the advantage that every linear programming solution has meaning. T h e final per cent return was found t o be 72.757,. T h e complete solution is given in Table I. There were strong indications that the objective function could be improved further. T h e amount of improvement of the objective function with each iteration was decreasing, however, and any increases in the objective function would be a t the expense of progressively larger amounts of computer time. Furthermore, the solution was tending toward an area where the very high throughput in the reactor could cause pumping costs to be higher than estimated and could make temperature control difficult. There are 12 variables in the problem. But for the linear programming routine this number must be doubled to prevent dealing with negative variables. There are nine artificial variables contributed from the nine equality constraints, and there are 12 slack variables from the upper- a n d lowerbound inequality constraints. Thus there are 45 variables and 21 equations in the linearized linear programming matrix. T h e Simplex method was used to solve the linear programming problem and required about 11 seconds per pass, exclusive of write-out and compiling time. Approximately 600 passes through the linear programming routine were made between the improved initial guess and the “optimal” solution. Much time was used in exploring the computational intricacies of the method. Any efficiently programmed routine would locate the optimum in a small fraction of the time actually used. Conclusions
T h e optimization technique studied was found to be a rapid method for finding a n improved solution vector. T h e method VOL. 4
NO. 1
JANUARY
1965
19
produced intermediate nonoptimal solution vectors a n d corresponding values of the objective function. I t would be of great value in both the design and operation of a chemical plant. In design, variations in uncontrollable parameters such as raw material costs, product price, etc., could be considered a n d a design with complete flexibility produced. When the plant is actually operating and a change in one or more of the parameters is encountered, it is necessary to change only those values in the program and then reoptimize. T h e method presented is simply another variation on the many organized sequential search techniques proposed recently. As such, it is susceptible to the same shortcomings of other methods-namely, more than one extremum, concave constraint space, convex objective function, etc. Fortunately, in most real systems, these problems are nonexistent or can be handled easily by multiple starting points. T h e advantages of the method lie in the speed of solution, the relative ease of programming, and, most importantly, its ability to handle nonlinear equality constraints. Although many nonlinear programming methods have been proposed in recent years. very little investigation has been made of their relatiLe computational efficiency or practical worth. T h e present paper is a step in the desired direction, presenting a new combination of two known methods and indicating the practical use of the combined method. Nomenclature
A,
= Arrhenius reaction rate pre-exponential factor for ith
B,
= Arrhenius reaction
FA
=
reaction rate exponential factor for ith reaction flow rate of reactant A to reactor, Ib./hr.
flow rate of reactant B to reactor? Ib./hr. flow rate of portion of column bottoms sent to plant fuel?Ib./hr. flow rate of G from decanter (to waste), lb.,'hr. flow rate of product P from column, 40 million 1b.j year = 4763 lb.;hr. flow rate of A from reactor, lb.,'hr, flow rate of B from reactor, lb.,'hr. flow rate of C from reactor, lb., hr. flow rate of E from reactor, lb.;hr. total flow rate from reactor, lb./hr. flow rate of P from reactor. lb.;hr. reaction coefficient of ith reaction lower limit on j t h variable maximum distance t i is allowed to change in each linear programming iteration molecular weight of B: 100 molecular weight of C: 200 molecular weight of G. 300 molecular weight of E , 200 error criterion; sum of squared residuals reactor temperature, O Rankine upper limit on j t h variable reactor volume, cu. ft. density of reactor solution, 50 lb.,'cu. ft:
literature Cited
(1) Griffith, R. E., Stewart, R. A , , .Mana,Tement Sci'.7 , 379-92 (July 1961). (2). Naphthali, L. M., "Iteration Methods for Non-linear Equations," Preprint 145, A.1.Ch.E. AMeeting,Chicago, December 1962. (3) LYilliams, T. J.! Otto, R. E., A.I.E.E. Trans. 7 9 (Communications and Electronics), 458-73 (November 1960).
RECEIVED for review Xovember 26, 1963 ACCEPTED February 3, 1964
F U N D A M E N T A L S T U D I E S ON A N O D I C PROTECT ION Carbon Steel in Sulfuric Acid Z
.
A
.
F0 R 0 U L I S
,
General Engineertng Diviszon, Esso Research and Engineering Co., 'Madison. .V. J .
The anodic polarization behavior of carbon steel in sulfuric acid was investigated b y potentiostatic techniques in acid concentrations from 0.5 to 96.4% H2S04 and in the temperature range - 1 .O" to 88" C. An increase in acid concentration up to approximately 50 to 60% leads to a displacement of the polarization curve toward more cathodic potentials and higher currents. A further increase up to 96.4% HzS04 displaces the polarization curve toward more cathodic potentials and lower currents. An increase in temperature displaces the polarization curve toward higher currents, and decreases the passive potential range. Particular emphasis was placed on interpretation from an engineering viewpoint of the anodic polarization parameters such as critical current, passive current, etc., as affected b y changing acid concentration and temperature.
anodic passivation of metals and alloys, although known T f o r many years, has only recently been applied commercially to solve certain corrosion a n d contamination problems (6, 8. 72). This report presents the results of a program initiated to evaluate the feasibility of using anodic protection with a commercial carbon steel in various sulfuric acid concentrations. T h e results of a previous laboratory program on anodic passivation of Alloy 20 in H2SOawere presented in another report (3). HE
20
l&EC PROCESS DESIGN A N D DEVELOPMENT
Experimental
T h e experimental technique used has been described (
