(167c) 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.