(424d) Modular Optimization and Optimal Control of Polymerization Reactions

Modeling and simulation of polymerization reactions range from studies of elemental kinetic phenomena to real-time plant simulations. Where formerly monomer conversion and property averages had been studied, now product properties like full molecular weight distributions, copolymer composition, branching degree and network densities are under consideration.

Because of the numerical complexity, even off-line optimization and optimal control of such properties is still challenging. An optimal control problem for polymerization might consist of several objectives and controls requiring the solution of high-dimensional population balances. If controls like recipe start-up, feed strategies, reactor cooling or time-optimality are applied in one single problem set-up, the number of variables can easily increase to several dozens. Fortunately and in contrast to classical parameter identification of kinetic rate constants, for control purposes parameters and controls have not to be unique - but optimal ?only?. This requires special numerical algorithms efficient enough to ensure fast convergence, but capable of dealing with overestimated least-square problems.

The program package Predici has been used for more than 12 years for the modeling of polymerization kinetics. The core feature of Predici and its underlying Galerkin h-p-method - the computation of full chain-length distribution - has been extended to additional distributed properties such as mentioned above. The question is to make use of this complexity when it comes to optimization without change of model structure and definition (since models are permanently changed and improved, an optimization procedure should leave the model itself unchanged). Whereas for classical optimization problems special algorithms are available - mostly based on first and second derivatives of a differentiable objective function - can be used as general purpose solver, in the context of full chain-length distributions a computation of higher order derivatives is prohibitive. Therefore in Predici the basic parameter estimation tool has been extended by additional conditions to allow for optimal control problems. The idea is to take the unchanged model set-up and automatically add special objective functions and weightings by an automatic preprocessing. This can be used for many kinds of objectives, even in connection to direct rheology computations (for mostly linear polymer chains) which are available employing the full information of the molecular weight distribution. After a successful optimization run, the model is - again automatically - reduced to its core.

The talk presents this modular approach which might be exemplary for comparable tools. It is shown how general parameters and controls can be optimized vs. a set of most typical conditions. In particular an updated Gauss-Newton method, special aspects of weighting and the appropriate formulation of objective functions is discussed.