LINEAR TEMPERATURE CONTROL
IN BATCH REACTORS M . C. M I L L M A N AND STANLEY K A T 2 Department of Chemical Engineering, City Uniuersity of New York, New York, N . Y.
Mathematical models of batch reactors are studied to see how closely optimum temperature schedules can b e approximated b y conventional linear feedback controllers. Parallel and consecutive first-order reactions are investigated over a broad range of kinetic parameters. The optimal control settings are correlated with the kinetic parameters.
system operating to a fixed deadline, the choice of best as a choice of control schedule, or as a choice of control policy. If the state of the system is known a t some starting point, the schedule of control action from that starting point to the deadline may be determined all at once in such a way as to maximize some criterion of the over-all system performance. This question appears as the problem of Mayer, or of Bolza, in the classical calculus of variations, and modern ways of dealing with it are commonly associated with the name of Pontryagin (1962). T h e problem may be re-solved from any later starting points where the state of the system is known. Alternatively, if the state of the system is taken to be known at every point in time, the best control action a t any moment can be found as a function of the current state and the time remaining to deadline. This control policy question is commonly asked in the framework of Bellman's dynamic programming (1957). Now the control policy question is very much in the spirit of feedback control studies, while the search for a schedule is more in the nature of a feedforward control question. Still, the work of Rosonoer (1959) and others shows the two approaches to be substantially equivalent, at least to the extent that the noise present in actual systems may be ignored. However, methods of numerical solution cif real problems, whether grounded in the classical calculus of variations or in the newer techniques of dynamic programming., do not seem to be nearly economical enough to be applied on-line, in the control of real systems. Accordingly, the authors present here a series of case studies where, with the best control schedules being known from prior calculation, it is seen how well their effect can be approximated by conventional three-term controllers. The case studies consist of a survey of coiisecutive OR A
F control action can be thought of in two ways:
and parallel
P M k2
I W
first-order reactions, where raw material M makes product
P and waste W . T h e control action is the temperature, and the Drior calculations noted above are of those temnerature sequences that give the highest product yield in a batch of preset duration (Aris, 1961 ,; Horn, 1961a, b). T h e controllers are taken to measure product concentration and manipulate temperature. These case studies are too simple to be very realistic, but the results have some interest of
their own, and the methods may be applicable to more complex situations. The approach to the choice of conventional controllers that approximate the best control action is very pragmatic. Proportional control is tried by simply putting into the batch equations that temperature, u, is a linear function of product concentration, y,
(3)
u = c O + C 1 ' y
and finding, by gradient search, those values of GO, c1 which maximize the final yield. For proportional plus derivative control
(4) and so on. What success one may hope for in the enterprise may be estimated beforehand by seeing how well least squares analysis based upon Equations 3 and 4 and other types match the prior calculation of the best temperature schedules and the corresponding histories of composition. Good least squares constants are extremely helpful as starting estimates for the gradient search which must climb some very difficult surfaces. T h e gradients of yield with respect to trial values of the parameters co, cl, etc., needed for the numerical search are evaluated by taking the first variation of the batch performance equations and solving their adjoints. T h e general conclusion of these batch reactor studies is that a three-term controller can give essentially the maximum theoretical yield. I n the cases where proportional temperature control alone represents some improvement over best isothermal operation but still falls noticeably short of the theoretical highest yield, proportional plus derivative or proportional plus integral control comes very close to the theoretical maximum and it is hardly necessary to utilize the full strength of the three-term controller. I n the cases where proportional control alone comes close to the theoretical maximum, it is usually difficult and in some cases impossible to get much closer to the optimal yield by adding derivative or integral action. T h e detailed results are given in the following sections, and the numerical methods are sketched in an appendix. Questions of the general dynamical behavior of these systems, especially of their stability, are deferred for later report. Formulations
For both reaction systems, the rate constants are taken in the standard Arrhenius form
kl
A1 .
e-E1/R9,
kz = Az
.
(5)
where A1 and A2 are the frequency factors of the two reactions, VOL. 6
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El and E2 are their activation energies, R is the gas constant, and 0 is the absolute temperature. The kinetic equations for the two reaction systems in turn are then set down as follows. For the consecutive reactions,
are tried, where c 2 = c3 = c4 = 0 for proportional action, = c4 = 0 for proportional-derivative action, cz = 0 for proportional-integral action, and all five constants come into use for proportional-integral-derivative action. The question asked now is to maximize the yield in a given batch by optimal choice of the values of the controller constants of the particular controller used. The over-all batch time, t = tf, is fixed along with the kinetic parameters A I , A B ,El, and E B , a type of controller is chosen, and the problem is to
c3
For the parallel reactions,
Choose
cg, c1,
. . . . to makey/, =
, a maximum
subject to the condition 13 on the temperature
Here the chemical symbols M and P of the reacting species stand for their concentrations as well, and t is the time. Equations 6 and 7 describe the courses of the various reactions in batch, starting with pure raw material a t concentration M t , perhaps together with some inert. From this point of view, the situation may be conveniently described in terms of conversion of raw material
For E1 > E2, where the best choice of temperature schedule is Om, there is no particular feedback control question to be answered. However, for E1 < E2, where each determination o f a best temperature schedule and best controller settings calls for a heavy calculation, it will be convenient to carry out the calculation, and present the results, in terms of certain dimensionless quantities. Accordingly in place of time t the variable 7
is introduced.
(Y
P
(1 6)
t/t,
A
I 1
e - ~ l / ~ e
(17 )
Further, if the dimensionless parameters
and yield of product =
=
is introduced and in place of the temperature 0 the variable =
Y
(15)
=
(18)
EP/EI
and
M,
(9)
rather than in terms of concentrations M and P directly. There are in turn, for the consecutive reactions
are introduced, then the batch equations for the two reaction systems may be written as follows. For the consecutive reactions
(2 and for the parallel reactions
dx -= u dr
. (1 - x ) ;
dr = u dr
.
xI,=o
=
0
(20) (1
- x) - p u s ;
y,,=o = 0
For the parallel reactions
T h e optimal question initially asked of each of the batch Equations 10 and 1 1 is to maximize the yield in a given batch by choice of the temperature schedule-that is, the over-all batch time t = t f is fixed, along with the kinetic parameters A I , A P ,El, and E2 and the problem is to Choose B as a function o f t , 0
Choose u as a function of r, 0
< t < t,, to make yt = a maximum
l&EC PROCESS DESIGN A N D DEVELOPMENT
(22)
The upper bound on allowable temperatures now becomes an upper bound on allowable values of u, and accordingly Question 22 is asked subject to the condition
OSu
