Modeling Hysteresis during Crystallization and Dissolution: Application to a Paracetamol-Ethanol System

Crystallization and dissolution processes are of great scientific and commercial interest. Significant efforts have been made in the past to develop mathematical models to describe these processes. In this work, observed hysteresis in particle counts during the crystallization and dissolution processes was used to develop and to test the mathematical models of crystallization and dissolution. Crystallization and dissolution experiments were performed with a system of paracetamol and ethanol. An undersaturated solution was first cooled at a particular rate causing crystallization and crystal growth. The solution was then reheated at the same rate to completely dissolve the generated particles. The particle counts and particle size distribution were measured online using a focus beam reflectance measurement (FBRM) probe. A hysteresis was observed in particle counts with respect to the solution temperature. It was also observed that this hysteresis was affected by the applied heating/cooling rates (0.3, 0.5, and 0.7 K/min) for of the solution. A systematic modeling framework based on the population balance equation (PBE) is developed for quantitatively capturing this hysteresis and the influence of cooling/heating rate on the hysteresis curve. A two-level approach was developed to simulate hysteresis: (a) PBE was solved using computationally efficient method of moments for the crystallization stage. This step was used to efficiently estimate values of parameters appearing in the model equations. (b) PBEs describing crystallization and dissolution were then solved using high resolution finite volume (HRFV) scheme coupled with the moving pivot method. The simulated hysteresis curve showed good agreement with the experimental data. Though simulated results over predicted the average particle (by 131%), the models were successful in qualitatively explaining the counterintuitive trends observed as the average particle size was tracked with time. The framework was thus shown to be a reliable and robust framework to model crystallization and dissolution processes. The developed approach, models, and results will be useful for simulating industrially relevant crystallization-dissolution processes.