Solutions to inversion problems in preferential crystallization of enantiomersaPart II: Batch crystallization in two coupled vessels

In this series of two papers, we investigate inversion techniques for models describing the crystallization of conglomerate forming enantiomers, with application for preferential crystallization. In Part I, population balance and moment model equations for crystallization in a single vessel were analyzed and inverted. Here, in Part II, a configuration consisting of two crystallizers coupled by the exchange of crystal-free liquid is considered. A main problem addressed in this series concerns the use of previous results on orbital flatness of a model for single substance crystallization. An analysis presented in Part I for the single crystallizer moment model, which is a single-input system, is extended in this paper to show that also the moment model for the coupled crystallizer configuration, which is a multi-input system, is not orbitally flat. Despite the absence of orbital flatness, it is possible to obtain inversion results for the simultaneous realization of crystal size distributions (CSDs) in both vessels. The techniques we use build on a time transformation introduced in previous work. We also investigate the effect of idealizations, which are motivated by the design principles of the coupled crystallizer configuration. In combination with idealizing assumptions that were introduced in Part I, solutions to inversion problems are greatly simplified. In particular, the resulting idealized systems are orbitally flat. We extend a numerical example presented in Part I to the coupled crystallizer configuration. In this example we compare the exact and simplified solution techniques and demonstrate how the idealizing assumptions can be justified a posteriori.