Combined quadrature method of moments and method of characteristics approach for efficient solution of population balance models for dynamic modeling and crystal size distribution control of crystallization processes

The paper presents a novel methodology for the estimation of the shape of the crystal size distribution (CSD) during a crystallization process. The approach, based on a combination of the quadrature method of moments (QMOM) and the method of characteristics (MOCH), provides a computationally efficient solution of the population balance equation (PBE) and hence a fast prediction of the dynamic evolution of the CSD for an entire batch. Furthermore, under the assumption that for supersaturation-controlled crystallization the main phenomenon is growth, an analytical CSD estimator is derived for generic size-dependent growth kinetics. These approaches are evaluated for the crystallization of potassium alum in water. The model parameters are identified based on industrial experimental data, obtained using an efficient implementation of supersaturation control. The proposed methods are able to predict and reconstruct the dynamic evolution of the CSD during the batch. The QMOM-MOCH solution approach is evaluated in a model based dynamic optimization study, which aims to obtain the optimal temperature profiles required to achieve desired target CSDs. The technique can serve as a soft sensor for predicting the CSD, or as a computationally efficient algorithm for off-line design or on-line adaptation of operating policies based on knowledge of the full CSD data.