Modeling Cooling Batch Crystallization

Particulate processes are characterized by properties like the paxticle shape, size, surface area, mass, and product purity. In crystallization the particle size and total number of crystals vary with time. Thus, determining particle size distribution (PSD) is important in crystallization. A population balance formulation describes the process of crystal size distribution with time most effectively. Thus, modeling of a batch crystallizer involves use of population balances to model the crystal size 

Since both growth rate and nucleation rate are concentration dependent, the mass balance in terms of concentration of the solute in the solution is expressed as the following differential equation 

t) represents the population density of the crystals, the i-th moment of PSD is given by Equation 10.12. 

Each moment signifies a characteristic of the crystal [96], The zeroth moment corresponds to the particle number, the first moment corresponds to the particle size or shape, the second moment corresponds to its surface area, and the third moment corresponds to the particle volume. 

Since population balance equations are multi-dimensional, their calculations are tedious and hence a lot of research has been focused on the model order reduction methods. One of the most common and efficient reduction methods is the method of moments. There are other methods for solving PBEs such as the discretization methods [101], method of characteristics, successive approximation[102], etc. However, the method of moments is commonly used and is described below. 

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The following example derived from Yenkie et al. [98] illustrates the procedure for modeling and simulation of a cooling batch crystallizer. 

The purity of amino acids recovered by batch crystallization has been examined using L-isoleucine as a model system. The concentration of impurities in the feed solution were shown to affect crystal purity, as were variables that affect crystallization kinetics (e.g., agitation, precipitant addition rate, and cooling rate). 

The rate of cooling, or evaporation, or addition of diluent required to maintain specified conditions in a batch crystallizer often can be determined from a population-balance model. Moments of the population density function are used in the development of equations relating the control variable to time. As defined earlier, the moments are... 

Jones (1974) used the moment transformation of the population balance model to obtain a lumped parameter system representation of a batch crystallizer. This transformation facilitates the application of the continuous maximum principle to determine the cooling profile that maximizes the terminal size of the seed crystals. It was experimentally demonstrated that this strategy results in terminal seed size larger than that obtained using natural cooling or controlled cooling at constant nucleation rate. This method is limited in the sense that the objective function is restricted to some combination of the CSD moments. In addition, the moment equations do not close for cases in which the growth rate is more than linearly dependent on the crystal size or when fines destruction is... 

Monnier, O., Fevotte, G., Hoff, C. and Klein, J.P., 1997. Model identification of batch cooling crystallizations through calorimetry and image analysis. Chemical Engineering Science, 52, 1125-1139. 

Worlitschek, J. Mazzotti, M. Model-based optimization of particle size distribution in batch-cooling crystallization of paracetamol. Crystal Growth Design 2004, 4 (5), 891-903. 

Optimal control schemes have also been applied to batch cooling crystallization (Ajinkya and Ray, 1974 Morari, 1980 Chang and Epstein, 1982) and to the evaporative crystallization of sugar (Frew, 1973). A simplified model to obtain temperature-time trajectories for seeded batch cooling crystallization without to resorting to optimization techniques has been proposed by Rohani and Bourne (1990). 

S. Miller and J. Rawlings. Model identification and control strategies for batch cooling crystallizers. AIChE J., 40 1312-1327, 1994. 

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