OPTIMAL DESIGN OF A SEQUENCE OF CO N T I N UOUS-FLO W STI R REDITAN K REACTORS WITH PRODUCT RECYCLE L. T . F A N ,
L. E. ERICKSON, R . W . SUCHER, AND G. S.
M A T H A D
Department of Chemical Engineering, Kansas State University, Manhattan, Kan.
The optimal design for carrying out a single reaction in a sequence of continuous-flow stirred-tank reactors with producir recycle is determined by using the discrete maximum principle for recycle processes. Relations are developed which enable the holding time and temperature for each reactor to be chosen such that the total profit is maximized. Computational procedures are discussed for a numerical example involving a first-order reversible reaction. The values of the state variables and decision variables are determined using a gradient search and interval halving technique until the recurrence relations are satisfied.
rate of reaction will be as large as possible (7). for a single reaction can be written as ( 7 , 2)
HE problem of optimization associated with a sequence of Tchemical reactors with no recycle was previously solved using dynamic progr,imming by Aris (7). Fan and Wang (2) studied this problem, applying the techniques based on a weakened form of the discrete maximum principle. I n this paper we consider the optimal policy associated with carrying out a single reaction in a sequence of continuous-flow stirredtank reactors with product recycle. T h e optimal policy involves the choice of holding time and temperature a t each stage in such a way that the final concentration results in the maximum profit from the operation of the system. Recurrence relations characterizing the optimal condition are first developed for a n A'-stage process. A first-order reversible reaction A2 A1 is then considered and a numerical example is presented.
2
where ai is the stoichiometric coefficient of At. (For example, AB = the reversible reaction A2 G AI may be written as A1 0.) If C," is the concentration of A , in the nth reactor and eln the holding time, a material balance for At about the nth reactor yields
-
CLn= Ctn-l
+ elnR(Cn; e?),
n = 1, 2,
. . .,N
(2)
where the rate of reaction is assumed only to be a function of the temperature, 02, and the concentration vector, C. [Sometimes the rate of reaction, R(Cn; e,"), is denoted by dCtn/dt, where the derivative signifies the change of concentration with time as determined in a batch reactor.] If the extent of reaction is defined as
Formulation of Problem
The problem is to find the optimal choice of holding time and temperature for carrying out a single reaction in a sequence of continuous-flow stirred-tank reactors, so that the total profit is maximized. The process conditions require that a part of the product from the last stage be recycled to the first stage. A schematic diagram of the process is given in Figure 1. I t will first be shown that, when only one reaction takes place, the temperature a t each stage should always be chosen so that the
Figure 1.
aiA i = 0
i=1
e
T 7
T h e equation
where Ci0 is a fixed reference concentration, Equation 2 may be written as ,yln
.....
= xln-I
+ BlnR(xln; W ) ,
n = 1, 2,
. . ., N
(4
1
XH
9-l
&
N
X"
-
9
X* r Continuous-flow stirred-tank reactor sequence with product recycle VOL. 4
NO. 4 O C T O B E R 1 9 6 5
431
Application of Discrete Maximum Principle
Equations 4 and 11 may be written in the following form (2) : Xln = ~ ( ~ ~ n -eln, 1 ; X2n
=
+
x2n--1
G(xln-l;
e,.)
(12)
eln, e2n)
(13)
The Hamiltonian function may be written as 4.
H" = zlnT(xln+;
Oln,
+ zzn[xzn--l + G(xln-l;
Ban)
Oln,
ea")] (14)
X12
Figure 2.
where zl" and
Gradient search
have the following recurrence relations:
ZZ'
2 2 design with a center point
T h e holding time,
eln,
is given by
Vn
eln = __
(5)
q+r
where V n is the volume of the nth reactor, q the flow rate of the feed to the system, and r the recycle rate. If X is the relative cost of the reactor volume, the profit may be written as N
P
= q(x1"
- Xlf) -
x nE= l vn
(6)
T h e mixing condition is given by and
(7)
z2.V
= 1
(18)
Thus, zgn = 1 and the Hamiltonian function becomes
and thus 4x1'
= (q
+ r) x10
(8)
rxls
Substitution of this into Equation 6 yields, upon rearrangement,
P
= (q
N
+ r)(x?
- x10)
- x nE= l
V"
(9)
H
=
zlnT(xlndl;
+
Ol",
xan-l
+ G(xln-';
eln,
Ban)
(19)
According to the weak form of the discrete maximum principle, the optimal choice of the temperature at each stage, if the temperature is not constrained, will be found where (2, 3)
From Equation 4, one obtains (20)
.V
This gives or
bR(xln; 02') bO2n
oln
(Zln
+ 1) = 0
Since the holding time, O l n , is not equal to zero, and since (z1' 1) = 0 leads to a trivial solution, one concludes that
+
Substituting this into Equation 9 yields
E eln[R(xln;ean) - XI
q + r
n=l
If a new state variable, X2n
=
N
- P- -
X2n-1
x2,
is defined as
+ eln[R(xln;
e,") - XI; X,O
=
o
(11)
it can be seen that X2N
=
P ~
q + r T h e problem is thus transformed into one of maximizing xzN by proper choice of eln and Osn, n = 1, 2, . . ., A' with the conditions xl' = n; xao = 0. 432
l & E C PROCESS D E S I G N A N D DEVELOPMENT
Equation 22 may be used to determine the optimal temperature a t each stage. I t has been pointed out ( 7 ) that Equation 22 leads to the selection of temperatures which make the rate of reaction as large as possible a t each stage. If this maximum rate of reaction is represented by R ( x l n ) ,Equations 4, 11, 12, and 13 may be rewritten as x l n = xln--l X2n
= X2n--1 xln
xZn =
=
xZn-l
+
OlnR(x1Q)
+ eln[R(xln)- X I
T(xln-l;
(24) (25)
01.)
+ G(xln-';
(23)
eln)
(26)
T h e problem is thus reduced to one in which we are to find a sequence of eln, n = I , 2, . . ., N , so as to maximize xi". I t is seen that the problem has now been reduced to a onedimensional problem and the available technique (2) may be used in seeking the optimal solution. LVhen the recurrence relation characterizing the optimal condition for one-dimensional processes [Equations 17 to 19, p . 30, ( Z ) ] is used, the following equations are obtained
Y-'=
-
yN(l
- yN)
+P
-~ y.V
[
r
(2)'
q__r
yJ
+
(I - y")P+l 1 - y'
+
and
(33) where
PP
(P
=
Solving Equation 27 for 23 give xln--l = xln
eln
[:;in, R(xin)
R(xln)
R(xln+')
(k20)P
ln y n = X-
co
j
1
(klO)P+'
and
and substituting it into Equation
+
+ 1)P+'
; n = 1,2, . "9
N
From this set of equations the optimal values of yn, n = 1, 2, . . ., N may be calculated. From Equation 23, the optimal holding times may be recovered. T h e optimal temperature may be obtained from
-1
bxln
(29) Solving Equation 28 for 01" and substituting it into Equation 23 give xl.\r-l
=
X1.V
which results when Equation 22 is solved for etn. When r = 0, the results reduce to the known case previously given by Fan and Wang ( 2 ) .
7
r
Numerical Example
'
T h e following kinetic data were employed for a first-order
I
kl
reversible reaction, AZS AI:
L
k2
E1 = 9.2 kcal./gram mole EZ = 12.5 kcal./gram mole klo = 105.4 min.-' k20 = 107.3 m i n . 3 CzO = 1 gram mole/cc. Clo = 0 gram mole/cc.
\Vhen Equations 29 and 30 are used, the values of the state variable 2-1, resulting from the optimal policy, may be calculated by trial and error. T h e optimal holding time for each reactor may be recovered from Equations 27 and 28. First-Order Reversible Reaction
We seek the optimal temperatures and holding times to maximize the profit in a three-tank reactor sequence with product recycle for which
K o w consider the first-order reversible reaction AZ A1 ivhich may be written as A1 - A2 = 0. T h e reference concentrations for A1 and A2 are Clo/Co = 0 and Czo/Co = 1, where CO = Cl0 CZ?. Following the notation of ( Z ) , and using Equations 29 and 30 together with the system equations, the following set of equations representing the optimal condition are obtained.
q = l
+
Xlf =
0.00
From the given data, we have
p = - E1 = 2.79 Ez - E1 n = 1,
Table 1. Recycle Rate, r 0.0 0.2 0.4 0.6 0.8 1. o
. . ., N - 1
(31)
Optimal Values of State and Decision Variables for Various Amounts of Product Recycle for Extent of Reaction X10
0,0000 0,1049 0.1757 0.2255 0.2687 0.3005
Holding Time, Min.
X = 0.005
Temperature,
X 11
X12
XI3
81'
el2
'A3
0.4195 0,4301 0.4251 0.4368 0.4459 0.4553
0.5614 0,5605 0.5356 0.5369 0.5370 0.5394
0.6375 0.6294 0.6148 0.6095 0.6045 0.6011
2.059 1.957 1.317 1.282 1.224 1.218
4.551 3.999 2.500 2.301 2.098 1.998
7.146
380.1
5 . 747 .
178 4.
5.346 4.538 3.938 3.424
379.2 374.5 372.4 369.1
O K .
8Z2
ea3
336.6
316.4
117 . 6-
345.2 343.9 343.8 343.2
118 A
- _ - . I
323.2 324.1 325.6 326.9 ~
VOL.
4 NO. 4
OCTOBER 1965
~~
433
46001
optimal conditions by numerically perturbing the results slightly-Le., by means of numerical experimentation. Conclusions
0.500
0.450
I 0
L
09
0,Z
Recycle
0.6
0.8
IP
Rote
Figure 3. Optimal profit for first-order reversible reaction as a function of product recycle rate with relative reactor cost as parameter
The discrete maximum principle for recycle processes may be used to determine the optimal design for carrying out a single reaction in a sequence of continuous-flow stirred-tank reactors with product recycle. T h e method allows the design variables (reactor temperature and holding time in this paper) to be chosen such that the maximum profit is obtained. This paper illustrates how the gradient search technique may be employed to obtain numerical solutions of the difference equations which result when the discrete maximum principle is employed. Nomenclature = chemical species, dimensionless = concentration of A , , gram moles/cc.
T h e following values of the relative reactor cost, A, and the recycle rate, r , were considered :
h = 0.005, 0.010, 0.015
r = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 For the case of no recycle, the procedure previously given was used. For the case involving recycle the procedure was somewhat more involved. Values of xi2 and el2 were first assumed. T h e corresponding value of R(xI2)was then calculated. From this value together with the assumed values, xi1 was calculated using Equation 23. T h e corresponding value of xi0 was then computed using Equation 31. T h e mixing operator, Equation 33, was then used to recover the value of xI3. With these values, Equation 32 was used to compute x l z . This procedure was repeated until the calculated values of x? agreed with the assumed value within allowable error limits. The search technique used was a combination of a gradient search technique (4) and an interval halving technique. A 22 design with a center point was used for the gradient search, the initial guesses of xi2 and being used as the center point (see Figure 2). The absolute value of the difference between the assumed x? and the calculated x12 was to be minimized. If this value (at the center point) was minimum with respect to the corner points, the interval considered was halved; if not, a new center point was computed by proceeding along the path of steepest descent as determined by the gradient. This procedure was repeated until the absolute value of the difference between the assumed xI2 and the calculated xi2 was within allowable error limits. Some of the results for this example are shown in Table I and the profit as a function of the recycle rate is plotted in Figure 3 with h as a parameter. The results obtained can be verified to be at least the locally
434
l & E C PROCESS D E S I G N A N D DEVELOPMENT
= reference concentration of A , , gram moles/cc. = activation energy, kcal./gram mole = operating function, dimensionless
H = Hamiltonian function, dimensionless k,o = reaction rate constant, mim-1 P = profit, cc./min. q = feed rate, cc./min. Q = constant, min.-’ r = recycle rate, cc./min. = reaction rate function, m i n . 3 R ( x l ; e2) R(x1) = maximum reaction rate function, min.-’ t = time, min. T = operating function, dimensionless V = reactor volume, cc. x t = ith state variable, dimensionless yn
= x1n/co
zi
= covariant variable. dimensionless
a i = stoichiometric coefficient of A , , dimensionless
el e2
= holding time, min. = temperature of ieactor, = relative reactor cost
O
K.
h SUPERSCRIPTS 1, 2, . . , n = stage number SUBSCRIPTS 1 , 2, , , . , i = component of state or control vector
literature Cited
(1) Ark, Rutherford, “Optimal Design of Chemical Reactors,” Academic Press, New York, 1961. (2) Fan, L. T., LVang, C., “The Discrete Maximum Principle. A Study of Multistage Systems Optimization,” Wiley, New I’ork, 1964. (3) Horn, F., Jackson, R., IND. ENG. CHEM.FUNDAMENTALS 4, 110 (1965). (4) Ralston, A., Wolf, H. S., “Mathematical Methods for Digital Computers,” LViley, New York, 1962. for review February 1, 1965 RECEIVED ACCEPTEDMay 13, 1965
Division of Industrial and Engineering Chemistry, 149th Meeting, ACS, Detroit, Mich., April 1965. Work partly aided by a grant from the Office of Saline Water, U. S. Department of the Interior.
