Time evolution of psd in crystallization operations: an analytical solution based on Ornstein-Uhlenbeck process

A new formulation of the recent stochastic approach for the description of the particle-size distribution (PSD) time evolution in antisolvent crystal-growth processes is presented. In this new approach, the crystals size is modeled as a random variable driven by a Gompertz growth term and the corresponding Fokker-Planck equation is carried out. This proposed formulation, allows an analytical solution to describe the time evolution of the PSD as a function of the model parameters. The analytical solution is obtained by exploiting the typical properties of linear partial differential equations with linear coefficients, and using the analogy with Kalman filter, in terms of the first two stochastic moments: mean and variance of the PSD. Furthermore, an alternative way for the parameters estimation based on the maximum likelihood estimation is also introduced. Validations against experimental data are provided for the NaCl-water-ethanol antisolvent crystallization system.