A Simpler Method for Developing Batch Crystallization Recipes

Batch crystallization is an important unit operation, particularly for the manufacture of high value-added products such as pharmaceuticals and fine/specialty chemicals. When crystallization is operated batchwise, it is necessary to specify discrete parameters including seed mass, average seed size, batch time and the initial and final batch concentration, as well as one continuous (path) variable, which is related to the saturation concentration versus time over the course of the batch. Collectively, this information is called the “recipe” for batch operation.

Many papers in the literature have discussed the optimization of batch crystallization processes. Typically, experiments are conducted to determine the parameters of a model for the kinetics of nucleation and growth, and then a continuous optimization routine is employed to determine the optimal saturation concentration trajectory. The drawbacks of this approach are that it may be time-consuming and the kinetic model may not be robust to changes in parameters such as vessel size, stirring rate, etc. Other authors have suggested that it may be nearly optimal to operate batch crystallization processes with a constant supersaturation [1]. Since simple crystallization models assume that crystal growth rate depends only on supersaturation, this implies that the crystal growth rate would also be constant. However implementing a truly constant supersaturation policy requires either a kinetic model or online supersaturation measurement, which also presents technical challenges.

In 1971, Mullin and Nyvlt [2] showed that with certain assumptions it is possible to calculate analytically the concentration trajectory corresponding to a constant linear crystal growth rate without the benefit of a kinetic model. The resulting trajectory is a cubic polynomial which is consistent with intuition because if seed crystals grow at a constant linear rate, then their mass will increase with time raised to the third power. Thus the saturation concentration should be decreased at the same rate.

Although the assumptions that underlie the Mullin-Nyvlt trajectory are seldom strictly satisfied, the Mullin-Nyvlt trajectory performs much better than other trajectories (linear or natural cooling) that can be implemented without the benefit of a kinetic model. In spite of this, most researchers that calculate optimal trajectories compare the results of the optimal trajectories with the results of a linear or natural cooling trajectory. The comparison typically causes the optimal trajectory to appear to be quite attractive. However the benefit obtained by undertaking the time-consuming process of determining a kinetic model and optimizing it remains uncertain because researchers have not compared the result of the optimization with the best achievable result that does not depend on a kinetic model or optimization, namely the Mullin-Nyvlt cubic trajectory. Furthermore, existing reports are based on studies of particular solute-solvent systems, which makes it difficult to compare results and draw general conclusions.

To address these deficiencies, we have developed a new generic dimensionless model for a batch crystallization process. We used the model to compare the results of the optimal trajectory with the results from a constant growth rate trajectory, the Mullin-Nyvlt cubic trajectory and a linear trajectory. We find that the linear trajectory is poor and causes excessive nucleation at the beginning of the batch. By contrast, the Mullin-Nyvlt cubic trajectory performs nearly as well as the optimal trajectory in most cases. In situations where the Mullin-Nyvlt trajectory gives an inadequate performance, a much greater improvement in performance can be achieved by manipulating seed properties rather than by optimizing the saturation concentration trajectory.

References

1.    Fujiwara M, Nagy ZK, Chew JW, Braatz RD. First-principles and direct design approaches for the control of pharmaceutical crystallization. J. Process Contr. 2005; 15:493-504.

2.    Mullin JW, Nývlt J. Programmed cooling of batch crystallizers. Chem. Eng. Sci. 1971; 26: 369-377.