The methods that apply NLP solvers can be separated into two groups: there are the sequential and the simultaneous strategies. The solution strategy suitable for the global optimization is based on a proposal that discretizes the differential equations by means of central finite differences. This results in a system of highly nonlinear equations, thus it is possible to approach the optimization problem in one level (simultaneous solution). Usually this kind of optimization requires two stage flow information (sequential strategy).
The constraints and the objective function are solved with a combination of Simulated Annealing algorithm (SA) in Wolfram Mathematica 6.0® and a reduced space interior point algorithm. The role of the simulated annealing is to generate a set of random continuous variables, and to allow simultaneously for wrong-way movements, proving convergence to the global optimum. In a summarized manner, the method simply discretizes the dependent variable in n arbitrary collocations points or mesh. Depending on the mesh size, the number of variables and equations varies. Clearly as the number of points in the mesh increases, the computational time increases too and the optimization problem becomes bigger, however in all the studied cases in this work, the CPU time didn't exceeded 120 seconds on small workstations.
The optimization with differential equations doesn't become a trivial problem because there are three important problems to overcome: (i) it is a strongly Non Linear Problem (NLP) - No Convexes, (ii) there exist several hundred of variables to optimize, depending of the number of discretization, and (iii) there are mathematical stability problems due to the exothermic reaction.
o-xylene oxidation to phthalic anhydride is considered as an illustrative example. The kinetic used correspond to an expression of Finlayson: extended o-xylene oxidation in the presence of side reactions.This system is strongly exothermic and the undesired reactions are favored at high temperatures.
To check the convergence of the proposed solution strategy, several initial configurations are tested. First a feasible configuration: all optimization constraints (model equation and safe restrictions) are satisfied. This configuration is obtained by the solution of the model with trial and error, using simple simulation, without any optimization algorithm. Such feasible solution is used as initialization for the optimization problem. Afterwards simulations are run where unfeasible solutions are obtained since optimization constraints are not satisfied, mainly that the maximum allowable operating temperature reestriction is violated.
The results of the optimization are discussed. Pre- and post optimal sensitivity analysis is performed. We present a brief comparison of the optimizations with both objective functions and the final configurations.