(159d) Revisiting the Mathematical Description of Continuous Crystallizers

Keywords: Crystallization, Nucleation, Modeling, Population Balance Equation, Steady State, Stability, Continuous Manufacturing, Polymorphism

Main body

We report a novel mathematical approach to analyzing continuous crystallizers, which are widely used across industries, as they enable the manufacture of crystalline materials under well-controlled conditions [1]. In particular, we develop a general framework to assess the stability of its steady states based on the birth rate ν, which denotes how many secondary nuclei a crystal forms throughout its lifetime in the crystallizer. A stable steady state with crystals is defined as follows:

• ν(c_ss) = 1; the birth rate at the steady state concentration must equal one.
• dν/dc > 0; the derivative of the birth rate must be positive.
• c₀ > c_ss; the steady state must maintain a positive suspension density.

These conditions enable the steady state analysis under general conditions, i.e., for arbitrary rate expressions of crystal growth and secondary nucleation, for size-dependent crystal growth and withdrawal, and for growth rate dispersion. The stability analysis further applies to scenarios with steady-state multiplicity.

The application of this theory to compounds with multiple solid forms, i.e., polymorphic and chiral compounds, is of particular interest. The analysis of the polymorphic steady states agrees with and generalizes the existing approaches that have been based on the method of moments [2], and it explicitly accounts for the occurrence of cross-nucleation among polymorphs. Concerning chiral compounds that are crystallized from a racemic feed, enantiopure steady states are found to be unstable with respect to the racemic steady state, hence explaining the experimental challenges in the enantiopure crystallization of chiral compounds.

References

[1] LT Deck and M Mazzotti: General stability analysis of the steady states in the continuous mixed-suspension crystallizer. Chemical Engineering Journal (2024), 483, 148721.

[2] TC Farmer, CL Carpenter, MF Doherty: Polymorph selection by continuous crystallization. AIChE Journal (2016), 62, 3505–3514.