(25d) Population Balance Model Comparison of Growth, Secondary Nucleation By Attrition and Ripening

Secondary nucleation is a phenomenon present everywhere in nature and of fundamental importance for crystallization processes [1,2]. There are two families of mechanisms of secondary nucleation: the first occurs at the interface of the seed crystal and the solution, and it is a combination of an activated process and the effect of shear, i.e. surface- or shear secondary nucleation. The second family results from mechanical collision happening to the seed crystal, i.e. secondary nucleation by attrition. Attrition is the mechanism through which fragments form after an impact of a crystal with a stirrer, other crystals or the wall of the reactor. Those fragments, if small enough, can be considered secondary nuclei. For both families, there is not a clear understanding of the governing physics, and secondary nucleation rates are usually given by an empirical relationship depending on the extent of mixing, on the amount of crystals suspended and on the supersaturation level [1].

Methods and Models

In this contribution, we have extended the well-established mechanistic attrition model developed by Gahn and Mersmann [3] to two different population balance equation (PBE) models, which are used to simulate secondary nucleation processes. The traditional approach describes the formation of attrition fragments due to collisions as a secondary nucleation rate, which is included in the model as a boundary condition (we call it nucleation model). In the alternative approach, the formation of attrition fragments is described as a breakage expression and the growth rate is the result of size-dependent solubility (we call it breakage model) [4,5]. Conversely to the traditional model, this formulation takes into account the size distribution of attrition fragments and their evolution due to growth, see the figure for a schematic representation of the two models. The two approaches have specific challenges and they have different pros and cons.

In both models, the number of parameters to estimate has been limited by using expressions, where the physical and mechanical properties of the system, together with operative conditions, have been employed. In particular, for the breakage frequency, a physical model suitable for PBE has been derived and used in the simulations. The traditional attrition distribution of Mersmann has been adapted to a mechanistic breakage daughter distribution, that we were able to use in a PBE. For the secondary nucleation rate, a new expression has been derived and then compared with standard ones[5].

Results

The physical-based model of Mersmann was successfully integrated into two different population balance models to describe secondary nucleation by attrition for a population of crystals. In order to compare the models, two simulations of a batch, isothermal system have been performed. In both cases, the simulated time was very long, thus resulting in an almost complete depletion of supersaturation. The two models lead to very similar results in the region where supersaturation is high enough to ensure growth and secondary nucleation. However, the breakage model can describe the mechanism of Ostwald ripening, as a result of the embedded size-dependent solubility, when supersaturation is very close to 1, and the simulated time is very long.

The two models are solved in two different numerical ways since the population balance in the case of the breakage model is more complex and numerically intensive than the implementation of the boundary condition, which is the only necessary thing to solve for the nucleation model. This required the implementation of two different numerical schemes: the nucleation model has been solved by following a finite volume scheme with boundary conditions for nucleation. The attrition model has been solved by discretizing the growth part with a finite volume scheme and the breakage term with the fixed pivot technique [6,7].

Conclusions

The two models are very similar in the growth regime, thus where secondary nucleation and growth are the dominant phenomena. At extremely low values of supersaturation, thanks to size-dependent solubility, the first model yields to further development of the crystal population, e.g., Ostwald ripening and aging. The main result is that secondary nucleation by attrition can be described as a birth/death term or as a source term according to the final application of the model. Since the two approaches have very different computational intensities, one can choose the right model based on the objective of the simulation study.

Acknowledgments

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement No 2-73959-18.

References

1. Agrawal, S. G., & Paterson, A. H. J. (2015). Chemical Engineering Communications, 202(5), 698–706.
2. Mersmann, A. (Ed.). (2001). Crystallization technology handbook. CRC Press.
3. Gahn, C., & Mersmann, A. (1999). Chemical Engineering Science, 54(9), 1273–1282.
4. Iggland, M., & Mazzotti, M. (2011). Crystal Growth and Design, 11(10), 4611–4622.
5. Bosetti, L., & Mazzotti, M. (2019) Crystal Growth & Design1, 307-319.
6. LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems (Vol. 31). Cambridge university press.
7. Kumar, S., & Ramkrishna, D. (1996) Chemical Engineering Science, 51(8), 1311–1332.

Figure. Results of an isothermal desupersaturation experiment. On the l.h.s., schematic representation of the two models. On the center, the evolution of supersaturation (green), and the increase of the zeroth moment (blue for breakage model, red for nucleation model) corresponds to the formation of attrition fragments: in this region, the two models are almost indistinguishable. On the r.h.s., evolution of the initial population with a zoom on the population of fines, where the difference in the mechanisms of formation of nuclei is visible.