D E H
= system design = cost of heat exchanger
m n
= number of hot streams t o be cooled = number of cold streams t o be heated
0 p, s
t U w 7
a
-* T
=
literature Cited
cost of heater (plus cost of steam)
economic objective function steam pressure = total number of process streams = stream temperature = over-all heat transfer coefficient = flow rate = minimum allowable approach temperature difference = yearly equipment down time = termination of a stream = stream infeasibility = technological infeasibility = =
Hwa, C. S., “Mathematical Formulation and Optimization of Exchanger Networks Using Separable Programming,” A. I. Ch. E./I. Chem. E. joint meeting, London, 1965. Kesler, M. G., Parker, R. O., Chem. Ens. Progr. Symp. Sei-. 65, No. 92 (1969). Lawler, E. L., Wood, D. E., Operations Res. 11, No. 4, 699-719 (1966). Lee, K. F., Ph.D. thesis, University of Wisconsin, 1969. Masso, A. H., Rudd, D. F., A.I. Ch.E. J . 15, No. 1,lO-17 (1969). Rudd, D. F., A.I.Ch.E.J. 14, NO.2,343-9 (1968). Siirola, J. J., Powers, G. J., Rudd, D. F., A . I . Ch. E. J., in press, 1970. RECEIVED for review April 11, 1969 ACCEPTED November 28, 1969 Work financed in part by the National Science Foundation.
Near-Optimal Control by Trajectory Approximation Tubular Reactors with Axial Dispersion LeRoy 1. Lynn, Elliot S. Parkin, and Raymond 1. Zahradnik Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pa. 16215
A new method for obtaining near-optimal control policies is based on the use of weighted residual techniques to solve approximately the state and adjoint differential equations which result from application of Pontryagin’s maximum principle to the optimal control problem. The technique, trajectory approximation, is applied to the determination of optimal temperature profiles for a tubular reactor with axial dispersion, and the results are compared to numerical solutions obtained by quasilinearization. The results indicate that the method is computationally sound. In fact, it could lead to the realization of on-line near-optimal control for many chemical processes.
A method for computing near-optimal control policies could lead to real-time near-optimal control of chemical KEW
processes. Illustrative results are presented for near-optimal temperature profiles in a tubular reactor with axial dispersion. The mathematical formulation of optimal control policies is relatively advanced as a result of the pioneering iyork of Pontryagin et al. (1962). Considerable extensions and developments to the optimal control problem may be attributed to h t h a n s and Falb (1966), Fan (1966), Leitmann (1967), and Lapidus and Luus (1967), to mention but a few. However, the computational procedures proposed to implement this theory have, in general, been complex iterative schemes, requiring sophisticated methods of numerical analysis. A method for obtaining near-optimal control policies is presented here, that could considerably simplify and shorten the computational effort involved in determining optimal control policies for complicated nonlinear chemical processes. It is referred to as trajectory approximation. The method discussed is especially suitable for the determination of near-optimal control policies for processes in the 58
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VOL. 9 NO. 1 FEBRUARY 1970
chemical industry-for example, the computation of optimal temperature profiles for tubular reactors is of continuing interest. Solutions for various chemical kinetics in a tubular plug-flow reactor have been obtained by Bilous and Amundson (1956) and Siebenthal and Aris (1964). Lee (1964) has discussed the use of a gradient technique for determining optimal temperature profiles, and Flynn and Lapidus (1969) applied nonlinear programming to the same problem. Zahradnik and Parkin (1969) first applied the trajectory approximation algorithm t o the determination of nearoptimal temperature profiles for a tubular reactor without axial dispersion. They demonstrated that the two-point boundary value problem to be solved in the maximum principle formulation of the control problem can be reduced to a system of algebraic equations in a few variables, capable of being solved very rapidly on a digital computer. This method is a n equally effective way of computing near-optimal temperature profiles for a tubular reactor with axial dispersion. Results are obtained for the concentration profiles which are indistinguishable from the exact solutions over a wide range of axial I’eclet numbers.
Trajectory Approximation
A large class of optimal control problems can be formulated in terms of a dynamic system whose state is characterized b y a n n-dimensional state vector, z. The state vector is chosen so that the evolution of the system from some initial state, zo, a t time, t = 0, under the influence of a n ?-dimensional control effort, u,is described by the first-order vector differential equation
dx
dt = f h u )
(1)
with the boundary condition z(0) =
(2)
2,
It is desired to select u(2) from the class of piecewise continuous functions on the normalized time interval 0 _< t 1 so as to maximize a performance index, J , expressed as a function of the terminal values of the state variables as
