H. B. Matthews
Ph.D., University of Wisconsin-Madison, 1997
The population balance equation provides a convenient way to model particulate phase processes like crystallization. Using a population balance structure, this study identifies the kinetic models describing crystal nucleation and growth of the photochemical system in a 3-liter seeded, batch cooling crystallizer. A nonlinear optimization code maximizing the posterior density function of the parameters with respect to the data is used to infer the kinetic parameters from on-line measurements of liquid phase solute concentration and slurry turbidity. New models are identified to account for crystal habit dynamics and size-dependent nucleation. Linear 95 confidence intervals are determined to summarize the uncertainty in the optimal parameter estimates. In order to optimize the information content in the experimental data sets, two of the experiments are designed using optimal experimental design techniques.
Given the identified model, the optimization of the process is considered. Generally, the filtration of a crystallization slurry is facilitated when the particles are large and the size variance of the population is small. However, because seeded batch crystallizers tend to produce bi-modal size distributions consisting of a class of seeds and a class of nucleated crystals, the goals of large size and small variance are difficult to reconcile. Therefore, this study focuses on the effect of minimizing the final-time mass of crystals in the nucleated class relative to the mass of the seed crystals (). The objective is minimized with respect to a piecewise-linear temperature profile to be applied during the crystallization.
The optimal profile is determined using a nonlinear optimization that accounts for final-time and state constraints. This method guarantees that the solution adheres to realistic process limitations such as maximum cooling rates and yield constraints. The sensitivity of the optimal temperature profile to seed mass, run duration, and parameter uncertainty are analyzed. It is shown that increases in seed mass or run duration translate into improvements in the optimal value of the objective function.
Actual improvements in filtration resulting from implementation of two optimized input profiles are quantified experimentally by calculation of the average specific resistance of the filter cake. The filtration results for the optimal experiments are compared to filtration results from the model identification experiments. The slurries produced by the optimal profiles give the lowest resistance values recorded during the study and the total filtration times for the controlled runs are shorter despite higher solids densities. The optimal profile with larger seed load gives a cake resistance 25 lower than the best identification experiment.
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University of Wisconsin
Department of Chemical Engineering
Madison WI 53706