A stochastic formulation for the description of the crystal size distribution in antisolvent crystallization processes

Grosso, Massimiliano; Galán, Omar; Baratti, Roberto; Romagnoli, JA

doi:10.1002/aic.12130

Cited by 20 publications

(51 citation statements)

References 30 publications

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“…In the following, the stochastic model based on the logistic equation introduced in Grosso et al 14,19 is briefly summarized, hereafter referred as the LG model. In such formulation the deterministic growth model is assumed to follow a Logistic equation.…”

Section: Drift Term Based On the Logistic Equation (Lg)mentioning

confidence: 99%

“…14,16 The corresponding FPE becomes @Wðy; tÞ @t ¼ D @ 2 Wðy; tÞ @y 2 À @ @y Wðy; tÞ ry 1 À y K h i 14,16 The corresponding FPE becomes @Wðy; tÞ @t ¼ D @ 2 Wðy; tÞ @y 2 À @ @y Wðy; tÞ ry 1 À y K h i…”

Section: Drift Term Based On the Logistic Equation (Lg)mentioning

confidence: 99%

“…14 In fact, it allows capturing the occurrence of distortions from the ideal symmetric case in the PSD solution W(y,t) for any time.…”

Section: Drift Term Based On the Logistic Equation (Lg)mentioning

confidence: 99%

“…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).…”

Section: Parameter Estimationmentioning

confidence: 99%

“…[8][9][10][11][12][13] Along this way, Grosso et al 14 proposed a simple model based on the Fokker-Plank equation (FPE) approach to model crystallization systems characterized by the particle-size distribution (PSD). [8][9][10][11][12][13] Along this way, Grosso et al 14 proposed a simple model based on the Fokker-Plank equation (FPE) approach to model crystallization systems characterized by the particle-size distribution (PSD).…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2012

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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. © 2012 American Institute of Chemical Engineers AIChE J, 2012

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Section: Drift Term Based On the Logistic Equation (Lg)mentioning

confidence: 99%

Section: Drift Term Based On the Logistic Equation (Lg)mentioning

confidence: 99%

“…14 In fact, it allows capturing the occurrence of distortions from the ideal symmetric case in the PSD solution W(y,t) for any time.…”

Section: Drift Term Based On the Logistic Equation (Lg)mentioning

confidence: 99%

Section: Parameter Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2012

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Stochastic approach for the prediction of PSD in nonisothermal antisolvent crystallization processes

et al. 2013

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A stochastic formulation for the description of cooling-antisolvent mediated crystal growth processes based on the Fok- ker-Planck equation is discussed. Previous results are further extended to include not only the additional degree of free- dom (temperature) in the approach, but also to formulate the model parameters dependencies with the input manipulated variables (antisolvent flow rate and temperature) toward a global model to be used within all possible operating regimes. The obtained global models are used to define, for the first time, an operating map of the crystalliza- tion process, where asymptotic isomean and isovariance curves are reported in an antisolvent flow-rate-temperature plane. Input multiplicities are identified and validated both numerically and experimentally for the NaCl-water-ethanol nonisothermal antisolvent crystallization system

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On‐line control of crystal properties in nonisothermal antisolvent crystallization

et al. 2015

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The issues regarding the design and implementation of on-line optimal control strategies of crystal properties in noniso- thermal antisolvent crystallization processes to control particles’ mean size and standard deviation are dealt. The one- dimensional Fokker–Planck equation is used to represent the dynamic characteristics of the crystal growth and generate iso-mean and iso-standard deviation curves. Using controllability tools it is demonstrated that the system is ill condi- tioned in the whole operational range, posing limitations on the achievable control performance. To circumvent the problem, a control strategy is formulated by pairing crystals’ mean size with antisolvent feed rate and manipulating temperature to control the standard deviation. A novel digital image-texturing analysis approach is discussed and imple- mented to track crystals’ size distribution along the experiment and providing the on-line information for further feed- back control action. Subsequently, alternative control strategies are implemented and tested to achieve a desired crystal size distribution

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A stochastic formulation for the description of the crystal size distribution in antisolvent crystallization processes

Cited by 20 publications

References 30 publications

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

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

Stochastic approach for the prediction of PSD in nonisothermal antisolvent crystallization processes

On‐line control of crystal properties in nonisothermal antisolvent crystallization

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