Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
reasonable to expect this solution to be the only one to do so.
Once the type of control has been determined, the numerical values of the switching times are found from the process equations and the maximum principle. The residence time 8 for the new steady state is found by integrating the process equations from x(0) = 1.3 to x(1) = 0.333, giving 0 = 1.11538. The intervals a-b and b-d of velocities u* and u*, respectively, are determined by the total residence time and by the total equipment length: (b-a) + (d-b) = 0 and u* (b-a) + U. (d-b) = 1,giving b-a = 0.9872 and d-b = 0.1282. The exit time c of the step change is found from H = 0 over the inverval b-d I c ( x ( l )- xd)x(l)’ d7
+ I d(x(1) - xd)x(l)’ d7 = 0
giving c = 1.06 after some numerical calculations. The new value of P may then be found by performing the integration in eq 3 from b to d. An approximate value may be found by taking a linear relationship between x(1) and 7 over b-c and c-d, giving P = 6.05 X lo4. This is almost a tenfold improvement over the results given by Seinfeld et al. They used fixed coordinates, and had to solve the problem numerically. While they correctly give a “bang-bang’’ solution for the non-singular period, they incorrectly assume a priori one constant value for the final singular control. Their numerical method is based
195
upon “a systematic search over a number of predetermined control values until the value of P can no longer be decreased”. These predetermined control values each last for an arbitrary but predetermined time interval. Unless one knows beforehand the correct kind of final singular control including the switching times, it seems likely that their numerical method will have problems converging to the correct solution, and that it may in general converge to the best constant value of the control if that value is among those predetermined to be used in the search. Thus, in their paper Seinfeld et al. also treat a case with two state variables (concentration and temperature in an adiabatic reactor) and identical performance equation. While they give as their solution one switch to the best constant value, the optimal solution will again be an endlessly periodic control. In fact, with integral square deviations of the state variables as profit function, it is easily seen from the maximum principle developed here that a periodic singular control will be optimal in general when the disturbance is a step change (Lovland, 1977). Literature Cited Lovland, J., paper presented at 4th International Congress in Scandinavia on Chemical Engineering, Copenhagen, April 1977. Seinfeld, J. H.. Gavalas, G. R., Hwang, M., Ind. Eng. Chem. Fundam., 9, 651 (1970).
Jorgen Lovland
Department of Chemical Engineering Norway Institute of Technology Trondheim, Norway
Control of Plug-Flow Tubular Reactors by Variation of Flow Rate Sir: We appreciate Lovland’s interest and comments on our paper. The optimal control policy is, in fact, a periodic “bang-bang’’ control rather than a singular control with a single switch. However, as we will show, Lovland’s solution procedure is not constructive and does not lead to his result. 1. Preferred Method Before we discuss Lovland’s procedure, we wish to consider first the results in our paper. We applied the direct search method for a first-order approximation instead of solving the necessary condition of the two-point boundary-value problem. The actual numerical computation reported was based on 20 switching intervals (A0) over 0~[0,3].Unfortunately, both the fixed size of A0 and the numerical integration of the performance index (via the trapezoidal rule) resulted in numerical errors such that the optimality of the solution obtained was obscured. To avoid numerical errors the performance index (PZ) can be integrated analytically with parameterized switching time (Os), and then the PI can be minimized with respect to 19~. We now demonstrate this. From the system equation
01711 we have, from the method of characteristics
-
+ 1 + A(70)
E = 26/29 (-0.897)
(5)
Based on the above, we can compute the optimal switching time (8,) for the case of u* = 1 and u, = 26/29. This situation is depicted in Figures 1(A) and (B). By knowing d-b = 29/26 = l / u * and c-b = (1- b)/u*, we can integrate the PI analytically
PI =
1
T
Ix(0,l) - xd)*dO= x d [ x ( O , l )- 1/3j2 d0 =
)
3 In (2a(c-d)
+ 3)
+
+
1
+ A(8) 020
‘0)
where Tf(70)is the actual exit time of a fluid particle that entered the reactor at T ~ i.e. , (Tf(TO) - T ~ is) its residence time. The corresponding feed-forward control to a step change in the input concentration of A(0) is given, from eq 4 and xd = 113
0
