Determination of Start-up Conditions for Chemical Reactor Stability SIR: I n a recent letter concerning niy suggestion (Luus, 197 1) that through proportional control t'he unstable state in the system considered by Hail (l9i0) could be made stable quit,e easily has motivated H a n (1971) to make statements which have no scientific backing. I wish to show quant'itatirely how easily the objective caii be achieved and t,hereby answer Han's criticism of my "incorrect" statement'. At the ijanie time the caclulatioiis will show the comparison bet'weeii the actual stabilit'y region and the stability region as determined by Hail. Proportional control caii be used to control the temperature ill the continuous stirred-tank reactor considered by Hail (19i0) by inserting a cooling coil into the reactor and inakiiig the flow rate of t,he coolant proportional to the temperature deviation from the desired level. Let us take the desired temperature to be 595" R to correspond to the temperature of the unstable steady stat'e. Let us assume t'hat when t'he valve is half open, the overall heat transfer term for the coil is 2250 13tui'hr OR and let us use the same values for the other parameters as used by Han. Then, if the valve opening is regulated in proportion to the deviation from t'he desired temperature, the equations describing the change in concentration and t,emperature, respect'ively, become
-dT_ - 0.833(525 - 2') de
+ C.~(800)(7.08X 10'O) - 1.25u(T - 523)
exp
(
T
)
where the control u is given by u
=
(t" 1
+ k(T - 595)
if k(T - 595) > 1 if Ik(T - 595)l 5 1 if k ( T - 595) < 1
The upper limit of control IL = 2.0 corresponds to having the valve completely open, aiid the lower limit u = 0 correspoiids to the shut position. Also, eq 1 and 2 reduce ideiit,ically to Han's eq 32 and 33 if h: is taken to be zero. In eq 1 and 2 the same notation and units are used as used by Han, so that the reader may make direct comparison. Equations 1 and 2 Ivith k = 0.1 were integrated with fire difererit initial conditions to give the phase plane trajectories shown in Figure 1. It is immediately seen that the steady state poilit, (C, = 0.276, T = 595) is stable. Also, from the same figure we may readily determine the region of st'ability for the initial states since, in autonomous system the trajectories cannot intersect. It should also be noted t'hat with k = 0.1 the steady state caii be reached without overshoot by a judicious choice of initial states. By taking k = 0, numerous initial conditions yielded the phase plane shown in Figure 2. Also shown in the figure in dashed lines is the region 461 < T < 598, 0.2iO < CA < 0.698 which was determined by H a n as the region for initial values to ensure stability. I t is thus readily seen that part of that region allows the system to go to the upper steady state and
284 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1 9 7 2
+
'q------
T
IO
TEMPERATURE, T
Figure 1 .
Phase plane of CSTR with
TEMPERATURE. T
Figure 2.
k
=
0.1
-
Phase plane of CSTR with
k
= 0.0
therefore constitutes uustable startiiig points. Furthermore, a phase plane allows one actually t'o see how the steady stat,e is approached so t'hat the relative stabilitj- of a starting point may be determined. This knowledge is of great importance if there is any uncert'ainty iii the parameters. The second point raised by Hail (19il) is complet,ely irrelevant, since if further decomposition of products would take place, then the equations describing the kinetics \v.ould have to be changed and his Figure 5 would be completely meaningless the way it' stands. I n the determination of start-up conditions for a chemical reactor, Hail has oversimplified the problem as simply determination of stability only aiid has neglected the more iniportant aspect of controllability and the means of determining the optimal control. Ai little recoiisiderat'ioii of his results would have slio~vnthat to run the reactor only 4 R" above the inlet temperature is not a practical situation, and should have brought forth a t least some ideas of how to operate the reactor without restricting oneself to operate it in an openloop fashion a t the low-conversion steady state. literature Cited
LUUS,It., 1x0. ENG.CHF;M., FUNDAM. 10, 322 (1971). Han, c. D., IND. ENG.CHEM.,F u N D . i M . 9, 634 ( 1 Y i O ) . Han, c. L),, I S D . ESG. C H I : M . , F U S D . 4 M . 10, 5% (1971).
Rein Luus Department of Chemical Engineering Cniversit y of Toronto Toronto b , Ontario, Canada
