Optimal control and simulation of multidimensional crystallization processes

Abstract

The optimal batch control of a multidimensional crystallization process is investigated. A high resolution algorithm is used to simulate the multidimensional crystal size distribution under the operations defined by two optimal control trajectories. It is shown that a subtle change in the optimal control objective can have a very large effect on the crystal size and shape distribution of the product crystals. The effect of spatial variation is investigated using a compartmental model. The effect of differing numbers of compartments on the size and shape distribution of the product crystals is investigated. It is shown that the crystal size distribution can be very different along the height of the crystallizer and that a solution concentration gradient exists due to imperfect mixing. The nucleation rate can be significantly larger at the bottom of the crystallizer and the growth rate can be much larger at the top. The high resolution method provides high simulation accuracy and fast speed, with the ability to solve large numbers of highly nonlinear coupled multidimensional partial differential equations over a wide range of length scales. A parallel programming implementation results in simulation times that are short enough for using the simulation program to compute optimal control trajectories.

Introduction

Crystallization from solution is an industrially important unit operation due to its ability to provide high purity separations. For efficient downstream operations and product effectiveness, controlling the crystal size and shape distribution can be critically important. This is especially true for the multidimensional crystals produced in the pharmaceutical and photographic industries.

This paper investigates the effectiveness of optimal control for providing the desired product characteristics for a batch crystallization process with multiple growth axes. A high resolution algorithm accurately simulates the multidimensional population balance equation along the optimal control trajectories. While not well known in the crystallization or control communities, high resolution algorithms have been used in the computational physics community for a wide variety of applications (Harten, 1983, LeVeque, 1992, LeVeque, 1997, Osher and Chakravarthy, 1984, Sweby, 1984, Yang, Huang and Tsuei, 1995). High resolution algorithms can provide second-order accuracy without the spurious oscillations that naive second-order methods usually exhibit, while also reducing numerical diffusion inherited by first-order methods. The wide range of length scales inherent in crystallization processes makes these numerical issues especially critical.

This paper also investigates the effect of spatial variations on the multidimensional crystal size distribution. In batch crystallization, the crystal product characteristics are determined by the seed characteristics, the supersaturation profile, and the mixing conditions. The compartmental modeling approach can be used to take imperfect mixing into account (Braatz and Hasebe, in press, Kramer, Dijkstra, Neumann, Meadhra and van Rosmalen, 1996). The crystallizer is subdivided into a finite number of smaller sections (called compartments). Perfect mixing is assumed in each compartment. Each compartment has input and output streams that share flows with its neighbors. The quantity of crystals and the crystal characteristics in the input and output streams are governed by the local hydrodynamic conditions. The compartment model enables a much more accurate modeling of the formation of new crystals, which is key to quantifying the quality of the crystal size distribution of the final product. This approach is computationally feasible using today's computer hardware, whereas the full solution of the fluid and particle momentum equations is too computationally intensive, at least for design or control purposes in which multiple simulations are required.

The paper is organized as follows. The model of a KH₂PO₄–H₂O crystallizer is presented first, then the optimal control formulation for two-dimensional crystallization is introduced, followed by a summary of the compartmental modeling approach and the high resolution method, and the results, discussion, and conclusions.