Comparison of objective functions for batch crystallization using a simple process model and Pontryagin's minimum principle

- Optimal, nearly-analytical trajectories for batch crystallization are determined.
- Nine objective functions are considered for three case study systems.
- Process model includes only crystal nucleation and simple growth.
- Nearly-analytical results facilitate comparison of objectives.

In this contribution different objective functions based on the moments of the product crystal size distribution are compared using optimal control theory to solve for the optimal batch trajectory for each objective. For a simple crystallization process model with only nucleation and ordinary crystal growth, and neglecting the contribution of the nucleated mass to the nucleation rate and material balance, mostly analytic expressions are obtained for the optimal control vector. Different objective functions lead to different final values for the costates, which lead to different sets of coupled differential and algebraic equations which must be solved to determine the values of constants numerically. The results of nine different objective functions for three crystal systems are presented. The objective functions based on the lower moment of the nucleated crystals lead to late-growth trajectories while the objective functions based on the higher moment of the nucleated crystals lead to early-growth trajectories, consistent with previous findings. The effect of seed loading is also investigated.