Parameter Estimation of Preferential Crystallization Using a Detailed Mathematical Model

Separation of chiral compounds is of large interest because most of the organic molecules are chiral and usually only one of the enantiomers exhibits the desired properties with regard to therapeutic activities or metabolism, whereas the other enantiomer may be inactive or may even cause some undesired effect. For this reason, enantiomeric separations have become increasingly important and their application ranges from the pharmaceutical to the agricultural industry. An attractive separation technique for chiral separation is the so-called preferential crystallization[1], where homochiral crystals are seeded to a supersaturated solution of the racemic mixture, and only one of the enantiomers is preferentially produced. It is important to note that this process must be stopped before the equilibrium is reached; the preferred enantiomer of a high purity can be obtained only before the other (counter) enantiomer starts to crystallize. Therefore, an accurate prediction method is in crucial need for efficient design and operation of this process. Recently Elsner et al.[2] have proposed a detailed physical model which takes into account the size-dependent crystal growth as well as the primary and secondary nucleation, which closely reflects the underlying thermodynamics of the crystallization process. Due to the complexity of this mathematical model, however, it is necessary to resort to a numerical parameter estimation technique to identify some of the model parameters. Furthermore, we require that both enantiomers have identical parameter values, adhering closely to the underlying physics; this poses a further challenge in obtaining a good fit to the experimental data. In this study, we carry out a simple batch experiment changing the seed mass amounts, where the mass fractions of the both enantiomers in the liquid phase are measured. For the multiple experimental data sets obtained, an efficient parameter estimation technique is applied based on a Newton-based optimization approach. Here, the population balance equation is turned into a set of ordinary differential equation using the quadrature method of moments (QMOM)[3]. The QMOM model is then discretized into a large number of nonlinear algebraic constraints. Finally, the parameter estimation problem for the multiple experimental data sets is formulated as an optimization problem constrained by the nonlinear algebraic constraints. We show that this problem can be solved efficiently for the multiple data sets, and the mathematical model with the obtained parameters can be a powerful tool for process optimization. [1] Elsner, M.P., Ziomek, G., Seidel-Morgenstern, A.: Simultaneous preferential crystallization in a coupled, batch operation mode - Part I: Theoretical analysis and optimization, Chemical Engineering Science, 62, 4760-4769 (2007) [2] Elsner, M.P., Seidel-Morgenstern, A : One suitable kinetic model for enantioselective crystallization of theonine in water, Crystal Growth & Design (in preparation) [3] McGraw, R: Description of Aerosol Dynamics by the Quadrature Method of Moments, Aerosol Science and Technology, 27, 255-265(1997) © 2008 American Institute of Chemical Engineers [accessed November 17, 2008