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Agglomeration process modeling based on a PDE approximation of the safronov agglomeration equation / Andrey V. Bekker in Industrial & engineering chemistry research, Vol. 50 N° 6 (Mars 2011) 

Public ISBD [article]  in Industrial & engineering chemistry research > Vol. 50 N° 6 (Mars 2011) . - pp. 3464–3474 Titre : Agglomeration process modeling based on a PDE approximation of the safronov agglomeration equation Type de document : texte imprimé Auteurs : Andrey V. Bekker, Auteur ; Iztok Livk, Auteur Année de publication : 2011 Article en page(s) : pp. 3464–3474 Note générale : Chimie industrielle Langues : Anglais (eng) Mots-clés : Agglomeration  process Résumé : A one-dimensional dynamic partial differential equation (PDE) agglomeration model is derived based on the continuous Safronov agglomeration equation. A regularized PDE agglomeration model, represented by a set of convection−reaction−diffusion PDEs, can be solved within a standard adaptive-mesh implicit numerical framework that does not require additional quadrature assumptions to evaluate the aggregation integral. The PDE agglomeration model is solved numerically using a general Newton’s-method-based implicit Galerkin finite-element algorithm. The applied algorithm uses an automatic Gear-type time step and nonuniform adaptive-mesh strategies, which aids solution convergence. A numerical solution of the model for an agglomeration degree of 99.9% closely matches an asymptotic analytic solution of the original Safronov equation, which confirms the accuracy of the numerical procedure used. It is also shown that the number density function predicted by the new PDE agglomeration model satisfactorily agrees with the analytic solution of the Smoluchowski agglomeration equation. For small particle sizes and first and zeroth full moments, the two models give similar solutions. However, for larger particle sizes and the second full moment, the difference between the two models increases with increasing degree of agglomeration. Industrially important gibbsite agglomeration is used as a case study to demonstrate the application of the new numerical approach for agglomeration modeling. DEWEY : 660 ISSN : 0888-5885 En ligne : http://pubs.acs.org/doi/abs/10.1021/ie101933r [article] Agglomeration process modeling based on a PDE approximation of the safronov agglomeration equation [texte imprimé] / Andrey V. Bekker, Auteur ; Iztok Livk, Auteur . - 2011 . - pp. 3464–3474. Chimie industrielle Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 50 N° 6 (Mars 2011) . - pp. 3464–3474 Mots-clés : Agglomeration  process Résumé : A one-dimensional dynamic partial differential equation (PDE) agglomeration model is derived based on the continuous Safronov agglomeration equation. A regularized PDE agglomeration model, represented by a set of convection−reaction−diffusion PDEs, can be solved within a standard adaptive-mesh implicit numerical framework that does not require additional quadrature assumptions to evaluate the aggregation integral. The PDE agglomeration model is solved numerically using a general Newton’s-method-based implicit Galerkin finite-element algorithm. The applied algorithm uses an automatic Gear-type time step and nonuniform adaptive-mesh strategies, which aids solution convergence. A numerical solution of the model for an agglomeration degree of 99.9% closely matches an asymptotic analytic solution of the original Safronov equation, which confirms the accuracy of the numerical procedure used. It is also shown that the number density function predicted by the new PDE agglomeration model satisfactorily agrees with the analytic solution of the Smoluchowski agglomeration equation. For small particle sizes and first and zeroth full moments, the two models give similar solutions. However, for larger particle sizes and the second full moment, the difference between the two models increases with increasing degree of agglomeration. Industrially important gibbsite agglomeration is used as a case study to demonstrate the application of the new numerical approach for agglomeration modeling. DEWEY : 660 ISSN : 0888-5885 En ligne : http://pubs.acs.org/doi/abs/10.1021/ie101933r

An implicit FEM solution of a PBE model of gibbsite crystallization with constant and nonlinear kinetics / Andrey V. Bekker in Industrial & engineering chemistry research, Vol. 50 N° 8 (Avril 2011) 

Public ISBD [article]  in Industrial & engineering chemistry research > Vol. 50 N° 8 (Avril 2011) . - pp. 4641–4652 Titre : An implicit FEM solution of a PBE model of gibbsite crystallization with constant and nonlinear kinetics Type de document : texte imprimé Auteurs : Andrey V. Bekker, Auteur ; Iztok Livk, Auteur Année de publication : 2011 Article en page(s) : pp. 4641–4652 Note générale : Chimie industrielle Langues : Anglais (eng) Mots-clés : Crystallization  Nonlinear  kinetics Résumé : Numerical solution for a 1-D dynamic population balance crystallization model that includes gibbsite secondary nucleation and crystal growth kinetics was developed. The implemented numerical algorithm combines an implicit Galerkin formulation of the finite element method (FEM) with Newton iterations, variable Gear-type time-step and adaptive nonuniform mesh strategies. The numerical solution of the crystallization model is compared to the analytical solution derived for the case of constant gibbsite crystallization kinetics. For this case, it is shown that the numerical solution was considerably stabilized with the introduction of the artificial diffusion term and reduction of the relative error of Newton’s iterative step. Furthermore, the numerical algorithm is tested for the case of nonlinear gibbsite crystallization kinetics demonstrating its ability to deliver consistent solutions for both nucleation and crystal growth dominant cases. In each of the cases considered, the model solution, valid for an isothermal batch homogenously mixed crystallizer, predicted evolutions of the crystal size distribution (CSD) and relative supersaturation. Using the developed modeling technique, it is also shown that the initial seed loading strongly influences the shape of the product CSD, leading in some cases to multimodal CSDs. DEWEY : 660 ISSN : 0888-5885 En ligne : http://pubs.acs.org/doi/abs/10.1021/ie1019326 [article] An implicit FEM solution of a PBE model of gibbsite crystallization with constant and nonlinear kinetics [texte imprimé] / Andrey V. Bekker, Auteur ; Iztok Livk, Auteur . - 2011 . - pp. 4641–4652. Chimie industrielle Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 50 N° 8 (Avril 2011) . - pp. 4641–4652 Mots-clés : Crystallization  Nonlinear  kinetics Résumé : Numerical solution for a 1-D dynamic population balance crystallization model that includes gibbsite secondary nucleation and crystal growth kinetics was developed. The implemented numerical algorithm combines an implicit Galerkin formulation of the finite element method (FEM) with Newton iterations, variable Gear-type time-step and adaptive nonuniform mesh strategies. The numerical solution of the crystallization model is compared to the analytical solution derived for the case of constant gibbsite crystallization kinetics. For this case, it is shown that the numerical solution was considerably stabilized with the introduction of the artificial diffusion term and reduction of the relative error of Newton’s iterative step. Furthermore, the numerical algorithm is tested for the case of nonlinear gibbsite crystallization kinetics demonstrating its ability to deliver consistent solutions for both nucleation and crystal growth dominant cases. In each of the cases considered, the model solution, valid for an isothermal batch homogenously mixed crystallizer, predicted evolutions of the crystal size distribution (CSD) and relative supersaturation. Using the developed modeling technique, it is also shown that the initial seed loading strongly influences the shape of the product CSD, leading in some cases to multimodal CSDs. DEWEY : 660 ISSN : 0888-5885 En ligne : http://pubs.acs.org/doi/abs/10.1021/ie1019326

Comparison of FEM and DPB numerical methodologies for dynamic modeling of isothermal batch gibbsite crystallization / Andrey V. Bekker in Industrial & engineering chemistry research, Vol. 50 N° 7 (Avril 2011) 

Public ISBD [article]  in Industrial & engineering chemistry research > Vol. 50 N° 7 (Avril 2011) . - pp. 3994-4002 Titre : Comparison of FEM and DPB numerical methodologies for dynamic modeling of isothermal batch gibbsite crystallization Type de document : texte imprimé Auteurs : Andrey V. Bekker, Auteur ; Tian S. Li, Auteur ; Iztok Livk, Auteur Année de publication : 2011 Article en page(s) : pp. 3994-4002 Note générale : Chimie industrielle Langues : Anglais (eng) Mots-clés : Crystallization  Batchwise  Modeling  Dynamic  model Résumé : A population balance equation based dynamic gibbsite crystallization model, incorporating crystal growth, nucleation, and agglomeration, was solved using two different numerical techniques, namely, a fully implicit finite element method (FEM) and an explicit discretized population balance (DPB) numerical scheme. Unlike previous crystallization modeling approaches, the agglomeration model in the FEM framework was formulated based on the Safronov agglomeration equation and its partial differential equation (PDE) approximation [Bekker, A. V.; Livk, I. Agglomeration process modeling based on a PDE approximation of the Safronov agglomeration equation. Ind. Eng. Chem. Res. 2011, in press. The FEM numerical solution is implemented using a fully implicit Newton's method Galerkin finite element algorithm, which applies automatic Gear-type time-step and nonuniform adaptive mesh strategies to help optimal solution convergence. The dynamic FEM and DPB model predictions of isothermal gibbsite crystallization, using estimated kinetic parameters, are validated against experimental crystallization data that were generated under process conditions relevant to Bayer gibbsite crystallization. Additional modeling results obtained for a modified set of kinetic parameters, prolonged crystallization times, and more complex crystal size distributions show that the novel FEM-based crystallization modeling framework offers a more accurate and computationally efficient model solution than that based on the DPB approach. DEWEY : 660 ISSN : 0888-5885 En ligne : http://cat.inist.fr/?aModele=afficheN&cpsidt=24027646 [article] Comparison of FEM and DPB numerical methodologies for dynamic modeling of isothermal batch gibbsite crystallization [texte imprimé] / Andrey V. Bekker, Auteur ; Tian S. Li, Auteur ; Iztok Livk, Auteur . - 2011 . - pp. 3994-4002. Chimie industrielle Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 50 N° 7 (Avril 2011) . - pp. 3994-4002 Mots-clés : Crystallization  Batchwise  Modeling  Dynamic  model Résumé : A population balance equation based dynamic gibbsite crystallization model, incorporating crystal growth, nucleation, and agglomeration, was solved using two different numerical techniques, namely, a fully implicit finite element method (FEM) and an explicit discretized population balance (DPB) numerical scheme. Unlike previous crystallization modeling approaches, the agglomeration model in the FEM framework was formulated based on the Safronov agglomeration equation and its partial differential equation (PDE) approximation [Bekker, A. V.; Livk, I. Agglomeration process modeling based on a PDE approximation of the Safronov agglomeration equation. Ind. Eng. Chem. Res. 2011, in press. The FEM numerical solution is implemented using a fully implicit Newton's method Galerkin finite element algorithm, which applies automatic Gear-type time-step and nonuniform adaptive mesh strategies to help optimal solution convergence. The dynamic FEM and DPB model predictions of isothermal gibbsite crystallization, using estimated kinetic parameters, are validated against experimental crystallization data that were generated under process conditions relevant to Bayer gibbsite crystallization. Additional modeling results obtained for a modified set of kinetic parameters, prolonged crystallization times, and more complex crystal size distributions show that the novel FEM-based crystallization modeling framework offers a more accurate and computationally efficient model solution than that based on the DPB approach. DEWEY : 660 ISSN : 0888-5885 En ligne : http://cat.inist.fr/?aModele=afficheN&cpsidt=24027646