A stochastic approach for the prediction of PSD in crystallization processes: Analytical solution for the asymptotic behavior and parameter estimation

Tronci, Stefania; Grosso, Massimiliano; Baratti, Roberto; Romagnoli, José A.

doi:10.1016/j.compchemeng.2011.01.007

Cited by 15 publications

(26 citation statements)

References 5 publications

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“…The ML has been already used to infer the parameters related to the process steady state 18 and it is demonstrated to give more efficient parame-ter estimation, that is, a minor number of experimental data are required to carry out an effective evaluation of the parameters, at least compared with the LS method. 26 is rigorous if the crystal observations are assumed to be independent, which seems a reasonable hypothesis: the measurements technique here performed ensures that the outcome of the single observation will not influence the other ones.…”

Section: Parameter Estimationmentioning

confidence: 99%

“…26 is rigorous if the crystal observations are assumed to be independent, which seems a reasonable hypothesis: the measurements technique here performed ensures that the outcome of the single observation will not influence the other ones. The ML has been already used to infer the parameters related to the process steady state 18 and it is demonstrated to give more efficient parame-ter estimation, that is, a minor number of experimental data are required to carry out an effective evaluation of the parameters, at least compared with the LS method. Indeed, since it achieves the Cramér-Rao lower bound, no asymptotically unbiased estimator has lower asymptotic mean squared error than the ML.…”

Section: Parameter Estimationmentioning

confidence: 99%

“…In particular, in the new formulation, the Gompertz 15 model is selected to describe the time evolution of the mean crystal size providing eventually a FPE with linear coefficients. 18 This policy can lead to a straightforward inference of the parameters, Correspondence concerning this article should be addressed to R. Baratti at baratti@dicm.unica.it. The analytical solution is obtained by exploiting the typical properties of the linear partial differential equations with linear coefficients or using the analogy with the Kalman filter, 17 in terms of the first two stochastic moments: mean size of crystals and variance of the PSD.…”

Section: Introductionmentioning

confidence: 99%

“…16,17 Furthermore, parameters estimation is performed by applying the maximum likelihood estimation method (ML) to obtain the parameters directly from the observed data. 18 This policy can lead to a straightforward inference of the parameters, without resorting to a preliminary estimation of the experimental PSD through, e.g., the development of a kernel based function. 14 In this way, the estimation error due by the introduction of the kernel basis function approximation is circumvented.…”

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: Parameter Estimationmentioning

confidence: 99%

Section: Parameter Estimationmentioning

confidence: 99%

Section: Introductionmentioning

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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“…By solving the Fokker–Planck equation, one can find that the asymptotic distribution is given by:

\begin{matrix} ψ_{A s} (x) = ℵ e x p true{\int_{0}^{x} - ϕ (ζ) d ζ true} & with \end{matrix} ϕ = - \frac{h (ζ) - D g' (ζ) g (ζ)}{D g^{2} (ζ)}

…”

Section: Stochastic Model For Csd In Linear Scalementioning

confidence: 99%

A generalized stochastic modelling approach for crystal size distribution in antisolvent crystallization operations

2016

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A generalized formulation for the development of stochastic models to predict the crystal size distribution (CSD) in antisolvent crystallization processes is proposed. Exploiting the result of the noise induced dynamic in stochastic processes, new results are provided to represent the CSD as function of the operational parameters. The generalized formulation enables the full description of the CSD using nonlinear drift term in conjunction with multiplicative noise, i.e., state dependent diffusion. For the first time a deterministic nonlinear differential equation to represent the mean and most probable size (mode) time evolution as function of the model parameters is provided. Furthermore, the analytical solution of the asymptotic probability distribution of the CSD can also be obtained. Finally, a global model formulation is finally presented by defining relationships between the model parameters and the operating conditions. Experimental results and validations are provided using the ternary system of NaCl, water, and ethanol

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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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A stochastic approach for the prediction of PSD in crystallization processes: Analytical solution for the asymptotic behavior and parameter estimation

Cited by 15 publications

References 5 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

A generalized stochastic modelling approach for crystal size distribution in antisolvent crystallization operations

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

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