Stochastic approach for the calculation of anti-solvent addition policies in crystallization operations: An application to a bench-scale semi-batch crystallizer

This work discusses a stochastic approach for the calculation of robust anti-solvent addition policies for controlling the mean crystal size (MCS) and prediction of the crystal size distribution (CSD) in semi-batch crystallization operations. The proposed strategy is based on a structured-like population balance where uncertainties associated with the start-up condition and random fluctuations along the semi-batch operation can be taken into account in a very natural way. To include and quantify the effect of the uncertainties, a deterministic crystal growth model is embedded into a Fokker–Planck equation (FPE) resulting in a stochastic model for the MCS dynamics. This approach uses the generalized logistic equation (GLE) that has an adequate mathematical structure that suits the dynamic characteristic of the crystal growth. The numerical solution of the resulting FPE provides the most likely MCS evolution for a given anti-solvent flow-rate. The effects of the anti-solvent are incorporated into the parameters of the FPE and they are computed as linear piece-wise interpolating functions of the anti-solvent flow-rate. The strategy uses a PI-like regulator in closed-loop mode with the FPE to calculate the anti-solvent addition flow-rates for different set-point targets in the MCS. To validate the model and assess the merits of the proposed strategy, the crystallization of sodium chloride in water using ethanol as anti-solvent is performed in an experimental bench-scale semi-batch crystallizer. The implementation of the calculated anti-solvent policies resulted in a good control of the MCS despite modelling mismatch and uncertainties present during the crystallization operation.