Present work focuses on the development of a computationally efficient tool to convert a CLD to a PSD. Conversion of CLDs to PSDs is usually done in a two step process: 1) Generation of a matrix that converts the PSD of a population of particles with given shape into the corresponding CLD using a 3 dimensional geometric model. 2) Computation of the solution of the resulting linear matrix equation for the PSD [1]. Since the inverse problem is usually mathematically ill-posed, the method of constrained least squares minimization (CLSM) used for the inversion requires regularization. The selection of the optimum regularization parameter is computationally expensive due to the requirement of huge number of repetitive minimizations to generate L-curve that forms a basis for the selection of an optimal regularization parameter.
In the present work we propose the use of a direct analytic model of the L-curve for Tikhonov regularization parameter selection proposed by Mc Carthy [2]. Implementation of a CLSM along with the direct analytic model reduces the computational time significantly
The tool developed is expected to speed up the process of the inversion of the CLD to PSDs which in turn are used in the extraction of the crystallization kinetics using the inverse problem approach of Mahoney et al (2002). A hybrid Population balance [3] - data driven modeling approach is underdevelopment for modeling the crystallization process.
References
1. Jorg Worlitschek (2003), “Monitoring, Modeling and Optimization of batch cooling crystallization”(PhD. Thesis), Zurich, Swiss Federal Institute of Technology.
2. P J Mc Carthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Problems, 19 (2003) 643-663.
3. Alan W. Mahoney, Francis J. Doyle III, Doraiswami Ramkrishna, Inverse problems in population balances: Growth and nucleation from dynamic data, AIChE J., 48(5), 981-990 (2002)