Aspect ratio distribution and chord length distribution driven modeling of crystallization of two-dimensional crystals for real-time model-based applications

Two-dimensional (2D) crystals, for which the shape is described by two linear sizes, are common in fine chemical and pharmaceutical industries. Since the crystal size and shape are directly related to the performance of active pharmaceutical ingredients, the simultaneous size and shape distribution control is of paramount importance in pharmaceutical crystallization engineering. To efficiently achieve simultaneous size and shape control often requires model-based control strategies; however, the increased computational cost of the process simulation and the substantial differences between the simulated and measurable quantities make the implementation of model-based control approaches challenging. This paper addresses the important problem of the real-time simulation of the most likely measurable chord length distribution (CLD) and aspect ratio distribution (ARD) as well as the concentration variations during the crystallization of 2D needle-shaped crystals. This enables the application of focused beam reflectance measurement (FBRM) and particle vision and microscopy (PVM), two routinely applied probes, as quantitative direct feedback control tools. Artificial neural network (ANN)-based FBRM and PVM soft-sensors are developed, which enable the direct and fast transformation of 2D crystal size distribution (CSD) to CLD and ARD on arbitrary 2D grids. The training data for the ANN are generated by a first principle, geometrical model-based simulation of FBRM and PVM for high aspect ratio crystals, although the ANN approach is applicable for any simulated or experimental training data sets. It is also demonstrated that the in situ imaging-based shape measurement underestimates the real aspect ratio (AR) of crystals, for which a simple correction is proposed. From the model-equation solution perspective, the soft-sensors require full population balance solution. The 2D high-resolution finite volume method is applied to simulate the full 2D CSD, which is an accurate, stable, but computationally expensive technique. The real-time applicability is achieved through various implementation improvements including grid optimization and data-type optimized hybrid central processing unit-graphical processing unit calculations.