Dynamic batch crystallization model

Batch crystallizers are often used in situations in which production quantities are small or special handling of the chemicals is required. In the manufacture of speciality chemicals, for example, it is economically beneficial to perform the crystallization stage in some optimal manner. In order to design an optimal control strategy to maximize crystallizer performance, a dynamic model that can accurately simulate crystallizer behavior is required. Unfortunately, the precise details of crystallization growth and nucleation rates are unknown. This lack of fundamental knowledge suggests that a reliable method of model identification is needed. 

The dynamic model used in predicting the transient behavior of isothermal batch crystallizers is well developed. Randolph and Larson (5) and Hulburt and Katz (6) offer a complete discussion of the theoretical development of the population balance approach. A summary of the set of equations used in this analysis is given below. 

An alternative scheme, proposed by Garside et al. (16,17), uses the dynamic desupersaturation data from a batch crystallization experiment. After formulating a solute mass balance, where mass deposition due to nucleation was negligible, expressions are derived to calculate g and kg in Equation 3 explicitly. Estimates of the first and second derivatives of the transient desupersaturation curve at time zero are required. The disadvantages of this scheme are that numerical differentiation of experimental data is quite inaccurate due to measurement noise, the nucleation parameters are not estimated, and the analysis is invalid if nucleation rates are significant. Other drawbacks of both methods are that they are limited to specific model formulations, i.e., growth and nucleation rate forms and crystallizer configurations. 

The process dynamic model of a batch crystalliser is straightforward, fully described by the energy, mass and population balances. However, the dynamic of the crystal size distribution can be ignored if a batch is initially fed with seeds closely sized between two adjacent sieve sizes. General equations and constraints are developed for anhydrous salts. Additional equations are required to describe other transformations as in the case of hydrates and organic compounds. The subscript f and the superscript in the following equations denotes feed and saturation, respectively. The rate change in ... 

The design and implementation of control systems for both batch and continuously operated industrial crystallisers can be achieved by mathematical and physical structured models for the process dynamic behaviour and from on-line measurements of the crystal distribution (CSD). 

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