考量初級成核之晶種與非晶種批次結晶製程之最適化

Abstract

本研究開發了一個框架來解決晶種和非晶種批次結晶系統最適化問題。在本研究中，應用最佳控制理論(optimal control theory)來求解考量初級成核之多目標批次結晶製程最適化問題，本論文陳述並求解晶種與非晶種之最適化問題，同時也追蹤成核晶體的質量，並將其包含在質量平衡中。然而由此產生的兩點邊界值問題(two-point boundary value problem)非常複雜，狀態(states)及協態(costates)變數的導數表達式為高度非線性的，因此傳統的打靶法(shooting method)往往難以收斂。在含有晶種的案例中，可透過假設在哈密頓(Hamiltonian)方程式中成核的質量相較於晶種成長的質量是可忽略不計的，來簡化數學計算。而在非晶種的案例中，可應用基於梯度(gradient-based)的演算法來有效地求解兩點邊界值問題。 本論文以氨苄青黴素(ampicillin)在水中析出結晶為個案研究，展示本論文中所提出的最佳化方法，進而求得最佳控制輸入曲線(如過飽和度軌跡、pH值軌跡)。本研究亦透過建構柏拉圖最佳解前緣(Pareto-optimal front)來分析不同目標函數間的競爭權衡關係，如最小化成核晶體數量、成核質量，或最大化晶體之加權平均大小。本研究所提出之演算法快速且穩健，或許可適用於模型基底線上控制。除此之外，此演算法未來也可用於解決更複雜之批次結晶系統之最適化問題。

This work develops a framework for solving optimization problems for both seeded and unseeded batch crystallization systems. In this work, optimal control theory (OCT) is applied to solve multi-objective optimization problems for batch crystallization processes with primary nucleation. Optimization problems for both seeded and unseeded cases are stated and solved. The mass of the nucleated crystals is tracked and included in the material balance. The resulting two-point boundary-value problems (TPBVPs) are difficult to solve because the expressions for the derivatives of the states and costates are highly non-linear. Conventional shooting methods usually fail to converge. In seeded case, the nucleus-grown mass in the Hamiltonian equation is assumed to be negligible compared to the seed-grown mass, which simplifies the mathematics. In unseeded case, a gradient-based algorithm is applied to solve the TPBVP efficiently. A case study of ampicillin crystallized from water illustrates the method developed in this work. Optimal control input profiles (e.g. supersaturation trajectories and pH trajectories) and the complete product CSD are determined. Furthermore, the Pareto-optimal fronts are constructed to analyze the trade-off between the competing objectives of minimizing the number of nucleated crystals and the nucleated mass or the weight mean size. The algorithm is found to be both fast and robust, suggesting that it might be suitable for online model-based control. Moreover, the algorithm proposed in this work might be useful for solving even more complicated optimization problems for batch crystallization systems with complex kinetics.