Algorithms for solving boundary value problems in optimal control of seeded batch crystallization processes with temperature-dependent kinetics

Boundary value problems that arise when optimal control theory is applied to batch crystallization are complex and highly nonlinear, especially when crystallization kinetics are temperature-dependent, and conventional shooting methods sometimes fail to converge. In this work, three robust methods for solving such problems are presented and compared. The first method uses an approximation (neglecting the effect of the nucleated mass on the nucleation rate and the solution concentration) to simplify the problem. The second method uses a gradient-based algorithm to determine the optimal control input and the terminal constraints. The third method combines the first two, using the results from the approximation as an initial condition for the gradient-based algorithm. All three methods rely on a coordinate transformation of the population balance to enable explicit solutions to the population balance equations in the transformed domain and optimal control theory to provide the necessary condition of optimality. The methods are applied to study the trade-off between competing objectives (minimizing the number and the volume of the nucleated crystals) by constructing Pareto-optimal fronts. The results show that the first method (utilizing an approximation) is most efficient and introduces little error. Furthermore, using the result of this method as an initial condition for the gradient-based method drastically reduces the computation time compared to the gradient-based method alone. The proposed algorithms can be used to study the effect of crystallization kinetics on optimal control policies or develop advanced process control technologies due to their high computational efficiency.