Journal of The Society of Instrument and Control Engineers

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A large number of industrial chemical processes require devices interconnected in a complex way. In solving the optimization problems of such processes, the amount of numerical calculation necessitates the use of fast computers and computing methods of optimization. Various methods can be used to solve numerically the nonlinear, multivariable optimization problems. The most useful methods are the nonlinear programming, the continuous and discrete maximum principles, the dynamic programming and the various versions of gradient methods.
Generally, the nonlinear programming is useful for solving nonlinear problems with a small number of stages. The computational difficulty in applying the continuous and discrete maximum principles to complex optimization problems is that the resulting equations form two-point boundary value problems. The dynamic programming can overcome the boundary value difficulty, but is limited to the optimization problems of not more than three state variables, because of the dimensionality difficulty.
The gradient methods are considered to be very powerful for optimizing nonlinear complex processes with many variables. This paper describes the application of gradient methods to the optimization problems in chemical processes. A basic concept of gradient methods is explained and the computational principles and procedures of the methods are presented for the multi-stage and continuous processes.
The optimum solvent distribution in a multi-stage cross-current extraction with recycle is obtained by the first-order and quadratic convergence gradient methods. The first order gradient method for continuous systems is applied to optimize the temperature profile of a tubular reactor and the operating pressure of a batch reaction with terminal constraints. Some extensions of the quadratic convergence gradient methods to function space are discussed.

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