A mechanistic model for particle breakage within agitated crystallization systems

Throughout the field of particulate processing the application of modelling techniques to predict, optimize, or otherwise enhance processes is widespread. Perhaps the most common technique when it comes to modelling particu late populations themselves and how they interact with the physical systems they are contained in is the Population Balance concept formalized by Ran dolph and Larson (1971). Through this concept it is possible to formalize mathematical models of general particulate mechanisms such as dissolution, growth, nucleation, agglomeration, and breakage. However, while the math ematical formalization of the general models is well understood in terms of their functional forms (Hounslow, Pearson and Instone (2001); Kumar (2006); Kumar and Kumar (2013); Qamar and Warnecke (2007); Qamar et al. (2009), for example) a challenge remains in developing robust phys ical models for each mechanism that can accurately describe experimental observations. Presented in this work is a new breakage kinetic that incorporates fun damental hydrodynamic factors of the agitation vessel itself, and material response characteristics of the crystal materiel undergoing agitation. Com bined, these factors provide a realistic breakage model for pharmaceutical particulate processes involving typical agitation techniques. Ultimately, this kinetic serves to expand the state-of-the-art for modelling agitated crystal lization processes, and provides the foundations for further understanding in crystal-crystallizer interactions. For the hydrodynamic factors, the role of boundary layer effects on par ticle trajectories and impact rates was investigated (Tyrrell et al., 2018). Through the use of shadowgraphy imaging it was shown that there exists a critical Reynolds threshold, below which collision between particles and an mpeller blade is unlikely. Furthermore, those particles that do collide ex perience only a fraction of the nominal impeller tip-speed. Thus, this gives credit to the presence of a squeeze film boundary layer cushioning impacts at the impeller and around probes/baffles. As a result, the actual impact rate of particles with a typical crystallization system is often much lower than expected as the hydrodynamic conditions serve to protect the crystals from collision events. For the material response factors, the breakage characteristics of crystals when impacting a target surface, such as a stainless steel impeller blade, were investigated (Tyrrell and Frawley, 2018). Crystals were accelerated to wards a target and impacted at various speeds and across a range of crystal sizes. This allowed construction of a failure probability heatmap, outlin ing the probability of damage occurring to the crystal after an impact had occurred. From this, the probability of failure for any crystal size and veloc ity pair was extracted and compared to theoretical forms for the expected failure probability distributions. Good agreement was found between the proposed failure model and the observed failure rates. Lastly, the hydrodynamic and material response characteristics were com bined using a physically relevant rate expression based on the characteristic circulation time of the system. The resulting expression forms a new break age model for particulate processing. To validate this model, Particle Size Distributions (PSDs) of three Active Pharmaceutical Ingredients (APIs) were gathered after two separate agitation experiments. A Genetic Algo rithm (GA) was then used to investigate if the proposed hydrodynamic and material response factors could accurately explain the process outputs. Overall, it was found that the model performed well in parameterizing for the output PSDs, returning realistic parameters for both the hydrodynamic and material response terms in the model.