Stochastic Approach for the Prediction of PSD in Crystallization Processes: Formulation and Comparative Assessment of Different Stochastic Models

Abstract

A stochastic formulation for the description of antisolvent mediated crystal growth processes is discussed. In the proposed approach, the crystal size growth dynamics is driven by a deterministic growth factor coupled to a stochastic component. The evolution in time of the particle size distribution (PSD) is then described in terms of a Fokker-Planck equation. In this work, we investigate and assess comparatively the performance of the FPE approach to model the crystal size distribution based on different expressions for the stochastic component. In particular, we investigate the one-dimensional Fokker-Planck equation with a nonlinear diffusion coefficient to represent the crystal growth process. Validations against experimental data are presented for the NaCl water ethanol antisolvent crystallization system. It is shown that the stochastic model better suited to describe the experiments is given by the Geometric Brownian Motion (GBM), which gives an excellent agreement, with the experiments for a wide range of process conditions (i.e., antisolvent feed rate)

Time evolution of the PSD in crystallization operations: An analytical solution based on Ornstein‐Uhlenbeck process

“…It should be reminded that the GBM assumption is consistent with the law of proportionate effect (LPE), 19,20 which assesses that the rate of growth is proportional to the linear-size times a random number, thereby making growth rate size-dependent In order to take into account the growth fluctuations and the unknown dynamics not captured by the deterministic term, a random component can be introduced as a geometric Brownian motion 14 GBM, where the intensity of the fluctuations depends linearly on the crystal size.…”

“…19 The temperature in the bath is measured using an RTD probe, which is wired up to a slave temperature control system capable of heating and cooling. 19 The temperature in the bath is measured using an RTD probe, which is wired up to a slave temperature control system capable of heating and cooling.…”

“…Following the procedure outlined by Grosso et al 14,19 the calibration of the proposed models is carried out separately for every run. The set of parameters y ¼ [log 10 (D), r, K] is inferred using two different procedures, i.e., least-squares method (LS) on the kernel-based density function estimation and maximum likelihood estimation method (ML).…”

“…By performing a suitable variable transformation, the problem can be reformulated as an Ornstein-Uhlenbeck process, 16 and the analytical solution is then obtained to describe the time evolution of the PSD as a function of the model parameters. 19 To evaluate the reliability of the proposed formulation, validations against experimental data for the NaCl-waterethanol antisolvent crystallization system and the comparison with the previous model 14, 19 is presented. In fact, assuming a Gaussian initial PDF at time t 0 , the transient PDF will remain Gaussian at any time t ¼ t 0 , and, therefore, it could be characterized by the first two moments.…”

Image-Based Multiresolution-ANN Approach for Online Particle Size Characterization

“…The available experimental data used as targets for ANN weight estimation are the manually measured mean size and CSD standard deviation (obtained from the best global model using nonlinear Fokker−Planck Equation 32 ). They have been divided into two sets: (i) training set used for estimating the model weight and (ii) test set used to assess the prediction capability of the developed neural network (unseen data).…”