JOHN J. EVANGELISTA STANLEY K A T 2
BEST TEMPERATURE SCHEDULES IN BATCH REACTORS Reaction temperature need not always be held constant over the duration of a batch; varying the temperature in a manner dependent on kinetics may lead to improved yield he best choice of temperature schedule in batch
Treactor systems has been the subject of considerable
study since the pioneering work of Bilous and Amundson (2). And the general mathematical questions left unanswered in the earlier studies seem all to have been resolved in the Aris application of the methods of dynamic programming to these problems ( I ) , and in the general control studies along the lines of the classical calculus of variations by Pontrjagin (6) and others. The numerical solution of particular problems remains, however, a formidable task, especially when bounds on the temperature need to be taken into account, and accordingly we offer here a compendium of such solutions carried out for some typical first-order reaction systems over broad ranges of kinetic parameters. Our solutions are all obtained numerically, by methods to be sketched below, and agree very well with the analytical results of Horn (4, 5) in those ranges where both apply. The calculations we present here are all for maximum yield in a specified batch time, and the results are presented in a form designed to show the corresponding raw material conversion and the variations in temperature that must be scheduled to achieve the highest yield. We hope that the results may serve as a page, so to speak, in a reactor designer's handbook, permitting him to find with a relatively small additional amount of work the answer to such optimization questions as depend on the economics of a particular process. Suitable crossplots will show the effect of varying batch time when the values of the kinetic parameters are known, as well as the sensitivity of the results to imperfect knowledge of these parameters. On this last point, the designer will want to consult as well the recent work of Ray and Aris (7) which, for much more general reaction mechanisms, discusses the dependence of the maximum reaction rate on the kinetic parameters. 24
INDUSTRIAL A N D ENGINEERING CHEMISTRY
The principal reaction systems studied here are simple consecutive M%p%\V (1) and parallel k y p
M h
h
\v
(2)
reactions, where raw material M makes product P and waste W according to the first-order rate constants k l , k 2 . For these systems, the nature of the solution to the maximum yield question depends, as one expects, on whether the main reaction has a higher or lower activation energy than the undesirable reaction. If the main reaction has the lower activation energy, there arises for each batch time a definite schedule of temperature that produces the highest yield; this yield can be made as near 10070 as desired by allowing longer and longer batch times. If, on the other hand, the main reaction has the higher activation energy, the best policy is, as one expects, to run at the highest allowable temperature; a definite ceiling on yield results, and there is, for the consecutive reaction systems, the chance to reach this ceiling by proper choice of the batch time. The corresponding results for the reversible reaction ki
M Sk2 P
(3)
are presented here as well for the sake of completeness, although, since there is only one independent reaction, the results are obtained without difficulties encountered in the consecutive and parallel reactions. A word may be in order on the numerical methods used in carrying out the search for best temperature schedules for the (complex) reaction systems (l), (2). These are essentially gradient searches in the space
For the parallel Reactions 2,
For the reversible Reaction 3,
Here the chemical symbols M and P of the reacting speck stand for their concentrations, and t is the time. Equations 5, 6, and 7 may be taken as describing the courses of the various reactions in batch, starting with pure raw material at concentration M f , perhaps tower with some inert material. From this point of view, the situation may be more conveniently described in terms of conversion of raw material of the temperature schedule, roughly along the lies suggested in a rather different context by Dreyfus (3) and others. They use the machinery of the classical calculus of variations, in that the appropriate gradients are calculated by solving the adjoints of the variational equations, but they do not directly call upon Pontrjagin's characterization of the solutions to these variational problems. These gradient search procedures are used here by way of numerical test, since, without enormously multiplying the computing time, they can be applied to much more complex reaction systems and to more subtle variational questions. They work well and economically hen, over the whole range of cases studied. The plan of this paper is as follows. Aftex making certain general formulations, we present the results for the different reaction mechanisms in turn, and then discuss the sensitivity of the results to uncertainties in our knowledge of the kinetic mechanism. The computing procedures are sketched in the Appendix.
x =
-M
Mf
MI
and yield of product Y=-
P
(9)
Md
rather than in terms of the concentrations M and P directly. We have, in turn,for the consecutive Reactions 1, Xl+O
=0
for the parallel Reactions 2,
General Formulations
For all the reaction systems (l), (2), (3), we take the rate constants in the standard ArrheniF form
kl = Ale-EIIm k, = &-EdR@ (4) where AI, A2 are the frequency factors of the two r e m tions, El, E, their activation energies, R the gas constant, and 0 the absolute temperature. We may then set down the kinetic equations for the different reaction systems in turn as follows. For the consecutive Reactions 1,
and for the reversible Reactions 3, taking account of the fact that x = y,
du = A le -EilB4(1 - y ) - &-E*/Ba
dt
Y
Ylr-0
= 0 (12)
The question we ask of each of the batch Equations 10, 11, and 12 in turn is to maximize the yield in a given batch by choice of the temperature schedule. That is, we fix an over-all batch time t , along with the kinetic parametem As As, El, E*, and then choose 0 as a function t 6 t,tomake oft,whercO
E%but before going to the k e d batch time question in Equations 10 and 13, we formulate a subsidiary problem: that of maximizing the yield for a fixed conversion. We may pose this question formally by taking from Equation 10 the working equation
=0
For the parallel Reactions 2 and in place of Equation 13 then choose 0 as a function of x, 0 x x,, to make
<
E,. We note h t that, fium Equations 10, the final conversion x, is always increased by raising the temperature 0 in any part of its range; and since the relation in Equation 32 is based on -9 = e,, it gives in fact the highest xI attainable for a given t, x, =
we lind, on solving Equation 25, the highest final yield p, in the form y, =
1 - (1 - x,) rC { (1 -
y , = -(1
- x,)
}
- x,)
In (1
for c # 1
(28)
for c = 1
The general shape of the curve of Equations 28 is shown in Figure 1. Each curve has a maximum at the place
and the corresponding maximum value ofy,
is, for each c, the abwlute maximum of the attainable yields. We can now bring time back to the problem of Equations 25 and 26 by solving Equation 10 with 8 = 8. We set cI = Ala-E'/Rh"
c, = Ay-EdBm
(31)
so that c = cJci. Then the batch time t, needed to reach a given conversion x,may be written down as ti =
- -1I n C1
(1
-
x,)
The time tr incnases steadily with xr and takes the value
when xr is at the maximum given in Equations 29.
1
-
#-"If
(34)
We can take advantage of this fact to construct a solution to Equations 10 and 13 by positioning ourselves suitably on the curve given hy Equations 28 (Figure 1). If the assigned t , falls below (or at) the value given in Equations 33, then the highest possible conversion for that t,, given by Equation 34,falls below (or at) the value given in Equations 29, that is, on the near side of the maximum of Equations 28. The highest yield y, can accordingly be found by entering Equations 28 with the x, of Equation 34, and will be attained by running at the If, on the other highest allowable temperature e = e,. hand, the assigned t , falls above the value given in Equations 33, then the highest possible conversion, given by Equation 34, falls above the value given in Equations 29, that is, on the far side ofthe maximum of Equations 28. This highest conversion will no longer give the highest yield, which will instead be the absolute maximum of Equations 30, attainable for example, by running at 0 = 0, up to the time given in Equations 33, and then completing the batch at e = 0, so as to quench all reactions. I t develops from all this that, when E1 > EX,we may ask a sharper question than Equation 13 of the batch Equations 10. That is, we may choose both the batch time, t,, and the temperature schedule e(& 0 6 t 6 t , to make Ylc-I
(35)
a maximum. The highest yield will be the absolute maximum given in Equations 30, and may be attained, for example, with f I given hy Equations 33 and with
e
=
er.
The foregoing is all for E1 > E, and the argumenta made there cannot be carried over to the case El < &. Here we go to the dimensionless formulation of Equations 19 and 22, and pMceed numerically according to the methods &etched in the Appendiu. The main result of the calculations is embodied in Figure 2, which shows contoum of highest final yield y , plotted in the plane of the dimensionless batch timtkinetic parame m a, B of Equations 17 and 18. Each determination of the highest linal yield y , carries with it the determination of a best temperature schedule, and a typical schedule of this kind appeara in Figure 3. It takes the VOL 60 N R 3 MARCH 1968
27
maximum allowable value -9 = e,, or u = y, for some initial time, and then decreases steadily to some clearly defined minimum, not zero, at the end of the batch. The corresponding best yields turn out not to depend appreciably on y as long as y > 20, and the contours in Figure 2 are all drawn for this range of y. The parameter @ contains, of course, a knowledge of the batch time t,, as we3l as of the p d y kinetic parameters. One may see how the highest attainable yield y, increases with f ,by making an appropriate cmssplot from Figurr 2. Such crussplots bear on the economic determination of batch h e in particular casen, and a trpical one, for illustrative values of the kinetic parameters AI, As, El, EI, is shown in Figure 4. The increase in yield with increased batch time is made +le by a decrease in the m i n i u m temperature needed to construct the best schedule, and information about this minimum temperature appeara in Figure 5. Also, for its bearing on the economics of separation and repromaing, information about the raw material conversion x, attained with the best temperature schedule is given in Figure 6. imFinally, R word may be in order on how big provement in yield free manipulation of the temperature schedule permits over the best yield attainable at any single temperature. This says something about how sensitive the yield is to changes in the best temperature schedule, and also mves to indicate the parameter ranges whve it might be worth looking for some economic advantage. No dramatic improvements were found, the increase in y, for all cases studied l+g between 0.005 and 0.015, the upper end of the range being approached for 2 < OL < 4,0.2 < 19< 5.
0
0
0.2
0.4
0.8
0.6
1.0
I, dln@ni@nlasi lime &we 3. A lvpiuJ rcocrionr
OPrinuJ tmprafwt &-hedub fa cmuwkiu
1.o
s
.E 0.8
3
L
0.6
1 r
e 0.4
0.2 0 1
IO
IM
lo.m I o o . 0
1 0
We develop here the results for the parallel R e a d o m 2, taking the variational problem in the form of Equations l l and 13 when E1 > Ea, and in the form of Equations 20 and 22 when E1 < &. We begin with the case EI > Et, and develop, 6% working relationships between the p m e t batch time t, and the final conversion x , and yield y, that are developed in the batch. We may solve the equation for x in Equations 11 to give
where, in the integral, B is to be thought of as a function of t. Also, we may combine the equations for x and y to give
28
INDUSTRIAL AND ENQINEERING CHEMISTRY
1.01 0.01
I
0.I
1.o
I
IO
- UI
I
1.0 0 01
0.1
O
1.o
~
~
10
and solve this in Yy
(37)
=
where, in this integral, 0 is to be thought of as a function of x. Now in Equation 36, we see that any increase in e in any part of its range increases the integrand, and so incream x,. Also in Equation 37 we may see that (since 4 > E*) any increase in e in any part ofits range increaw this integrand as well. We conclude that any increase in e must increase y y since it increases both the integrand in Equation 37 and the upper limit. Accordingly we may say that the solution d Equations 11 and 13 is attained by taking 0 = e, the highest allowable temperature. Defining CI and CI by Equations 31, we find for the conversion 1 c-(c'+Nu,
- -
I-
and for the yield
This maximum yield has, as we see, a definite asymptote, not unity, as the batch time t,increasa. The foregoing is for E1 > %, and the arguments do not carry over to the case E1 < 4. We go to the dimensionless formulation in Equations 20 and 22, and proceed numerically according to the methods sketched in the Appendix. The main result of the calculation is embodied in Figure 7, which shows contours of the highest yield y y in the plane of the parameters a, 9, of Equations 17 and 18. As for consecutive reactions, each determination of the highest yield yy carries with it the determination of a best temperature schedule, and a typical schedule of this kind appears for p a d e l reactions in Figure 8. It shows here a steady increase, as one expects, from a clearly defined minimum, not zero, at the outset, to the highest allowable value, e = e,, for an interval at the end of the batch. As for the consecutive reactions in Figure 2, the yield contours of Figure 7 are drawn for the range of y, y > 20, where the yields no longer depend appreciably on y.
s i 0.8
tIx2eEl (UNO
for = 1.0
I
I
4
= 2.0
1,
0 I
IO
100
Im
1o.m
100,wo
Ant, dimonrionlesr #roup for limo
Fime 9. A typical cww fa best yirld as. b&k tinu fapmdld rm'nu V O L 6 0 NO. 3 M A R C H 1 9 6 8
29
Figure 9 shows a typical crossplot fmm Figure 7, where, for illustrative values of AI, As, El, E*, one may see how y, tends toward 1 as I, grows lager. This increase in yield is accompanied, as for consecutive reactions, by a decrease in the minimum temperahue needed to construct the best schedule, and information about this minimum temperature appears in Figure 10. Figure 11 shows the raw material conversion x, attained with the best temperature schedule. Finally, the increase in yield obtained by use of the best temperature schedule, over the best yield attainable at any single temperature, was found to lie between 0.013 and 0.042, with values near the latter being obtained for the range 1.4 < a < 3, 6 < 0.7. I t may be noted, in conclusion, that apart from the reversal in shape of the beat temperature schedule, the general results for yield in parallel reactions are not dramatically different from the results for consecutive reactions presented earlier, although the picture for convemion is, of course, very Werent. Reversible Reactions
We develop here for completeness the results for the reversible Reactions 3. This system, since it involves only a single independent reaction, presents an wentially simpler problem than the consecutive or parallel Reactions 1 and 2. If we take the problem in the form of Equations 12 and 13, we find the solution simply by maximizing the right-hand side of Equation 12. Accordingly, when the batch time t, is increased, the new best temperature schedule is simply a continuation ofthe old, and does not have to be recomputed from the beginning as for the more complicated reaction systems. For the case El > E,, an elementary examination of Equation 12 shows that, as long as the right-hand side is A#-EJ"y
A 18 -EdRO
(1 -Y)
