(612b) Population Balance Modeling and Optimization of an Integrated Batch Crystallizer – Wet Mill System for Crystal Size Distribution Control
Through the crystal size is often the most critical property of the end-product (keeping in mind that the purity, yield and crystal form specifications must be respected), the control of crystallization process in many practical case consists on controlling the product CSD. In crystallization systemsâ engineering, from the perspective of product size, the general objective is the production of large, uniform crystals  in order to facilitate the downstream operations. However, the production of small, uniform crystals is also of great interest due to the increased specific surface and bioavailability.
Since the crystallization is governed by the simultaneously ongoing nucleation and growth, which are non-linear functions of concentration, controlling the relative rate of these generally raises complex control issues. Generally, model predictive control (MPC) is employed to overcome the process nonlinearityâs while respecting various constraints. In contrast with the continuous tank systems, the control of batch operation is sometimes more challenging due to the fact that the initial and final system states are far from each other, during which transition the process nonlinearityâs has more significant effects . Out of the model based techniques, with the quick spread of process analytical technologies the model free control methods, such as the direct nucleation control (DNC) and supersaturation control (SSC) . The batch crystallization control, regardless on the way of implementations, in most cases relies in the manipulation of applied temperature profile, which influences the crystallization kinetics through the supersaturation. In crystallization practice the attainable particle size domain is always delimited by the crystallization kinetics .
In order to widen the attainable crystal size domain, other manipulated variables affecting crystal size are also implied, such as using antisolvent or, more often, applying combined antisolvent-temperature variations for supersaturation generation. Out of the nucleation and growth, the secondary operations, like the agglomeration and growth are also promising sup-processes to be controlled from the point of view of crystal size control . The seed addition is a very popular startup procedure of batch crystallizers, which also presents high potential of tailoring the product CSD. Growth rate modifier additives were also successfully applied to adjust the crystal size and shape during the crystallization.
If the CSD cannot be obtained directly in the crystallization, secondary operations, such as granulation and milling are required to adjust the size of crystals properly. This however is not an advantageous solution as involves extra operations in the production line and, moreover, the quality of crystals not reaches those that obtained in direct crystallization. The straightforward question arises if the crystallization and milling could be coupled with each other for the sake of production uniform crystals in lower size domains? There is a forward looking experimental work analyzing a continuous crystallizer-wet mill cascade system  which was also successfully extended to continuous wet mill based direct nucleation control for cooling crystallization . Others attempted to model and simulate the attainable crystal size distribution in a very similar setup.
For the mathematical description of particulate systems the population balance modelling (PBM) is the widely accepted and applied approach . Since the introduction of PBMâs a large number of publications was addressed to this topic. The PBMâs, from the most basic form of nucleation and growth assuming one crystal dimension have been extended to multidimensional cases and involving additional crystallization mechanisms such as fragmentation, agglomeration, crystal-solution heat transfer  etc. From process engineering perspective, the numerical solution of the PB equations (PBE) is the bottleneck of crystallization simulation and is often more challenging that the derivation of the PBM. Numerous methods have been addressed to PBE solution from the moment based methods, through Monte Carlo simulations and method of characteristics to discretization based techniques. To improve the real time applicability of full PBM based simulations, the solution time is usually improved by various techniques from using adaptive grid, through application of parallel and GPU calculations to coordinate system transformations.
Despite of the fact that PBM modelling was successfully applied for various crystallization systems and that using a recycle wet-mill can considerably widen the attainable crystal size domain of batch crystallizers a study for the optimization of such a promising integrated system has not been published yet. The first objective of this work is to develop a PB based model for a batch crystallizer-wet mill system, involving primary nucleation, growth and dissolution for the crystallizer and fragmentation and attrition for the wet mill and to solve the equations efficiently and accurately . The second objective of the study is to analyze and optimize the system through numerical experimentation. The authors address the current study as a quality-by-design (QbD) approach driven deeper understanding effort of simultaneous role of system integration and hidden potential of process optimization.
The optimization results indicate that the target crystal size distribution can be successfully achieved in the cases when the pure crystallizer cannot even approximate the shape of target CSD. The optimized system in all cases presents similar operation pattern, which can be summarized in 4 point:
1) The crystallizer is cooled quickly to produces some crystals
2) The pump is turned on to transport the crystals to the external wet mill (the mill is not running yet)
3) The pump is turned off (the crystallizer and wet mill is decoupled from each other). During this, the crystallizer is heated up to the solubility temperature (complete dissolution occurs) while the wet mill operates with high rotation speed (the existing crystals are milled).
4) When the crystals are sufficiently milled, the system is started to be cooled in controlled manner while the seeds are added dynamically through the recirculation stream.
Consequently, the optimal operation of integrated crystallizer-external wet mill system utilizes the strengths of optimal dynamic seeding and in situ seed generation, making unnecessary the time and high accuracy demanding seed preparation step.
Nevertheless, the system model was not validated for a particular system, in which context the results are not valid for any practical crystallization. Although, real crystallization kinetics and a realistic breakage model was used. Indeed, if the nucleation, growth, dissolution and breakage parameters would be change the attainable crystal size domain would certainly be modified but the contribution of this work, the optimal trajectories of temperature, flowrate and wet mill rotation speed, would remain the same.
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