Numerical approximations of a population balance model for coupled batch preferential crystallizers

Qamar, Shamsul; Angelov, I.; Elsner, M. P.; Ashfaq, Asia; Seidel‐Morgenstern, A.; Warnecke, Gerald

doi:10.1016/j.apnum.2008.03.033

Cited by 11 publications

(8 citation statements)

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“…Population balance models have been applied in the past decades to a number of industrialscale applications involving aerosols and flame synthesis of materials [14,60,113,122,159], crystallization [9,22,27,32,86,98,108], polymerization/depolymerization [2, 41-43, 57, 58, 63, 144, 158], emulsions [21,47,151,152], twin-screw granulation [10,11,15,50,51,65,121,123,155] and sprayed fluidized bed granulation [6,53,54,84,85,106,149,161,166]. What these problems share in common is that the central unknown is the distribution of a distributed property, usually 'size'.…”

Section: Introductionmentioning

confidence: 99%

Challenges and opportunities concerning numerical solutions for population balances: a critical review

Singh

Ranade²,

Shardt³

et al. 2022

J. Phys. A: Math. Theor.

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Population balance models are tools for the study of dispersed systems, such as granular materials, polymers, colloids and aerosols. They are applied with increasing frequency across a wide range of disciplines, including chemical engineering, aerosol physics, astrophysics, polymer science, pharmaceutical sciences, and mathematical biology. Population balance models are used to track particle properties and their changes due to aggregation, fragmentation, nucleation and growth, processes that directly affect the distribution of particle sizes. The population balance equation is an integro-partial differential equation whose domain is the line of positive real numbers. This poses challenges for the stability and accuracy of the numerical methods used to solve for size distribution function and in response to these challenges several different methodologies have been developed in the literature. This review provides a critical presentation of the state of the art in numerical approaches for solving these complex models with emphasis in the algorithmic details that distinguish each methodology. The review covers finite volume methods, Monte Carlo method and sectional methods; the method of moments, another important numerical methodology, is not covered in this review.

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Section: Introductionmentioning

confidence: 99%

Challenges and opportunities concerning numerical solutions for population balances: a critical review

Singh

Ranade²,

Shardt³

et al. 2022

J. Phys. A: Math. Theor.

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“…(2) the method of characteristics [4,5]; (3) the method of weight residuals/orthogonal collocation [6,7]; (4) the Monte Carlo method [8,9]; (5) the finite-difference schemes/discrete population balances [10]. The review by Ramkrishna [2] is helpful in this regard.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev Spectral Collocation Method for Population Balance Equation in Crystallization

Ruan

2019

Mathematics

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The population balance equation (PBE) is the main governing equation for modeling dynamic crystallization behavior. In the view of mathematics, PBE is a convection–reaction equation whose strong hyperbolic property may challenge numerical methods. In order to weaken the hyperbolic property of PBE, a diffusive term was added in this work. Here, the Chebyshev spectral collocation method was introduced to solve the PBE and to achieve accurate crystal size distribution (CSD). Three numerical examples are presented, namely size-independent growth, size-dependent growth in a batch process, and with nucleation, and size-dependent growth in a continuous process. Through comparing the results with the numerical results obtained via the second-order upwind method and the HR-van method, the high accuracy of Chebyshev spectral collocation method was proven. Moreover, the diffusive term is also discussed in three numerical examples. The results show that, in the case of size-independent growth (PBE is a convection equation), the diffusive term should be added, and the coefficient of the diffusive term is recommended as 2G × 10−3 to G × 10−2, where G is the crystal growth rate.

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“…In both the thermal and attrition-driven examples, the near-equilibrium conditions of these processes ensured that unwanted primary nucleation of the nonseeded enantiomer did not occur. This process of coupled preferential crystallization represents an operationally far easier process than kinetically driven preferential crystallization techniques …”

Section: Introductionmentioning

confidence: 99%

Resolution of Omeprazole Using Coupled Preferential Crystallization: Efficient Separation of a Nonracemizable Conglomerate Salt under Near-Equilibrium Conditions

Hein

Cao

Meijden

et al. 2013

Org. Process Res. Dev.

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A facile and efficient resolution of omeprazole as the monopotassium salt diethanol solvate using coupled preferential crystallization has been developed. This approach uses small perturbations in solution-phase concentration to control the competing process of selective crystal growth while suppressing unwanted primary nucleation. The result is a selective crystallization technique that replaces the traditional batch-type isolation with a continuous process amenable to scale, which provides easy access to enantiopure conglomerate crystals.

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Numerical approximations of a population balance model for coupled batch preferential crystallizers

Cited by 11 publications

References 10 publications

Challenges and opportunities concerning numerical solutions for population balances: a critical review

Challenges and opportunities concerning numerical solutions for population balances: a critical review

Chebyshev Spectral Collocation Method for Population Balance Equation in Crystallization

Resolution of Omeprazole Using Coupled Preferential Crystallization: Efficient Separation of a Nonracemizable Conglomerate Salt under Near-Equilibrium Conditions

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