(351f) A New Global Stability Result for Isothermal Msmpr Crystallizers

The operation of isothermal mixed suspension mixed product removal (MSMPR) crystallizers involves two degrees of freedom: the inlet concentration and the residence time of the reactor. The kinetics of crystallization in turn is typically solely dependent on the concentration and temperature in the crystallizer. It thus follows that the steady state of the MSMPR is a consequence of the operating conditions and the physical properties of the system. Since the underlying crystallization kinetics are typically highly nonlinear (and in a feedback loop with the solution concentration), the steady state of an MSMPR can exhibit complex, nonlinear behavior. For a simple system that exhibits only homogeneous nucleation, there exists a unique steady state that is either asymptotically stable or exhibits stable limit cycle oscillations [1,2]. Clearly, avoiding the unstable behavior is desirable when the crystallization process is designed. 

The main contribution of this work is to derive an analytical expression for the global stability of the crystallizer. The analytical condition is expressed as an explicit function of the physical properties and control variables. To arrive at this stability condition, the crystallizer is described using a population balance equation model that is reduced using the method of moments [3]. Convergence theory ensures the existence and uniqueness of a stable equilibrium point that all the solutions to a finite set of ordinary differential equations converge to [4]. We apply the Routh-Hurwitz criterion to derive the condition that ensures global exponential convergence. The resulting stability criterion provides sufficient, but not necessary, conditions that eliminate self-sustained oscillatory behavior in the crystallizer. Contrary to previous studies [1,2] our result is a global stability analysis and therefore does not require knowledge of the steady state point nor full kinetic information; on the contrary it simply requires knowledge of an upper bound of the nucleation rate. 

The stability criterion can be thought of as a boundary curve in a two-dimensional plot with horizontal axis as inlet solute concentration and vertical axis as residence time is presented. Any combination of inlet concentration and residence time in the area above the curve gives sustained oscillations. On the contrary, the area below the boundary curve results in stable operations. As the inlet concentration decreases and gets further away from the unstable boundary, the oscillations die out faster. On the other hand, if the inlet concentration increases to the boundary, it takes longer time to dampen oscillations. Given a longer residence time, we need to have the feed solute with a smaller inlet concentration to maintain the process at stable operation. 

Using kinetic data from the crystallization of Aspirin from ethanol [5] we compare our analytical stability criterion to numerical simulations and find that the analytical criterion provides a conservative estimate of the boundary between the stable and unstable region in the above mentioned two-dimensional plot.

References:

1. Lakatos B. G., Sapundzhiev T. J. and Garside J., Stability and dynamics of isothermal CMSMPR crystallizers, Chemical Engineering Science, 62: 4348-4364, 2007.
2. Jerauld G. R., Vasatis Y. and Doherty M. F., Simple conditions for the appearance of sustained oscillations in continuous crystallizers, Chemical Engineering Science, 38(10): 1675-1681, 1983.
3. Randolph, A. D., Larson, M. A., Theory of Particulate Process: Analysis and Techniques of Continuous Crystallization, 2nd ed.; Academic Press: Toronto, Canada, 1988.
4. Lohmiller W. and Slotine J. J. E., On contraction analysis for non-linear systems, Automatica, 34(6): 683-696, 1998
5. Griffin D. W., Mellichamp D. A. and Doherty M. F., Reducing the mean size of API crystals by continuous manufacturing with product classification and recycle, Chemical Engineering Science, 65: 5770-5780, 2010.