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Mass conservative solution of the population balance equation using the least-squares spectral element method / Zhengjie Zhu in Industrial & engineering chemistry research, Vol. 49 N° 13 (Juillet 2010) 

Public ISBD [article]  in Industrial & engineering chemistry research > Vol. 49 N° 13 (Juillet 2010) . - pp. 6204–6214 Titre : Mass conservative solution of the population balance equation using the least-squares spectral element method Type de document : texte imprimé Auteurs : Zhengjie Zhu, Auteur ; Carlos A. Dorao, Auteur ; Hugo A. Jakobsen, Auteur Année de publication : 2010 Article en page(s) : pp. 6204–6214 Note générale : Chemical engineering Langues : Anglais (eng) Mots-clés : Population  balance  equation  Mass  loss Résumé : In the standard formulation of the population balance equation that consists of breakage terms, significant loss of mass is observed for the dispersed phase. This mass loss is actually caused by the inexact conservation property reflected by many breakage kernels; hence, incorrect physical interpretations of the model simulations may be drawn. In this work, a constrained method is developed enforcing mass conservation. This numerical property is accomplished by adding an extra restriction to the original population balance equation in terms of the dispersed phase continuity equation through the Lagrange multipliers strategy. The discretized system resulting from applying the method to a two-phase population balance equation problem is symmetric and pseudopositive definite. Numerical experiments are carried out simulating the motion of a two-phase mixture passing through a 2D domain. The results obtained by the modified least-squares spectral element method show that the mass is conserved everywhere in the domain with high accuracy. En ligne : http://pubs.acs.org/doi/abs/10.1021/ie900710y [article] Mass conservative solution of the population balance equation using the least-squares spectral element method [texte imprimé] / Zhengjie Zhu, Auteur ; Carlos A. Dorao, Auteur ; Hugo A. Jakobsen, Auteur . - 2010 . - pp. 6204–6214. Chemical engineering Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 49 N° 13 (Juillet 2010) . - pp. 6204–6214 Mots-clés : Population  balance  equation  Mass  loss Résumé : In the standard formulation of the population balance equation that consists of breakage terms, significant loss of mass is observed for the dispersed phase. This mass loss is actually caused by the inexact conservation property reflected by many breakage kernels; hence, incorrect physical interpretations of the model simulations may be drawn. In this work, a constrained method is developed enforcing mass conservation. This numerical property is accomplished by adding an extra restriction to the original population balance equation in terms of the dispersed phase continuity equation through the Lagrange multipliers strategy. The discretized system resulting from applying the method to a two-phase population balance equation problem is symmetric and pseudopositive definite. Numerical experiments are carried out simulating the motion of a two-phase mixture passing through a 2D domain. The results obtained by the modified least-squares spectral element method show that the mass is conserved everywhere in the domain with high accuracy. En ligne : http://pubs.acs.org/doi/abs/10.1021/ie900710y

On the coupled solution of a combined population balance model using the least-squares spectral element method / Zhengjie Zhu in Industrial & engineering chemistry research, Vol. 48 N° 17 (Septembre 2009) 

Public ISBD [article]  in Industrial & engineering chemistry research > Vol. 48 N° 17 (Septembre 2009) . - pp. 7994–8006 Titre : On the coupled solution of a combined population balance model using the least-squares spectral element method Type de document : texte imprimé Auteurs : Zhengjie Zhu, Auteur ; Carlos A. Dorao, Auteur ; D. Lucas, Auteur Année de publication : 2009 Article en page(s) : pp. 7994–8006 Note générale : Chemical engineering Langues : Anglais (eng) Mots-clés : Cross-sectional  averaged  Population  balance  model  Flow  field  Spectral  element  method Résumé : In this work, a cross-sectional averaged two-fluid model combined with a population balance model is applied to simulate the flow field and the bubble size distributions in a two-phase bubble column. The Martinez-Bazan breakage kernel and a modified Prince and Blanch coalescence kernel have been chosen to describe bubble breakage and bubble coalescence, respectively. In the present study, we discuss the use of a higher-order spectral element method—the least-squares method—to compute the system of equations in a coupled manner. The least-squares method is highly accurate and has a number of advantages over the conventional numerical methods like the finite difference and finite volume methods. In contrast to the finite volume method, when desinging an overall solution algorithm, this least-squares method ensures that all the continuity equations are satisfied individually and it deals with both the convective and diffusive terms stably and accurately. The novel iterative algorithm solves the flow and the population balance equation in a coupled manner. The model has been validated against experimental data obtained for two-phase flow in a bubble column. The predicted bubble size distribution and other flow quantities are in good agreement with the experimental data. En ligne : http://pubs.acs.org/doi/abs/10.1021/ie900088q [article] On the coupled solution of a combined population balance model using the least-squares spectral element method [texte imprimé] / Zhengjie Zhu, Auteur ; Carlos A. Dorao, Auteur ; D. Lucas, Auteur . - 2009 . - pp. 7994–8006. Chemical engineering Langues : Anglais (eng) in Industrial & engineering chemistry research > Vol. 48 N° 17 (Septembre 2009) . - pp. 7994–8006 Mots-clés : Cross-sectional  averaged  Population  balance  model  Flow  field  Spectral  element  method Résumé : In this work, a cross-sectional averaged two-fluid model combined with a population balance model is applied to simulate the flow field and the bubble size distributions in a two-phase bubble column. The Martinez-Bazan breakage kernel and a modified Prince and Blanch coalescence kernel have been chosen to describe bubble breakage and bubble coalescence, respectively. In the present study, we discuss the use of a higher-order spectral element method—the least-squares method—to compute the system of equations in a coupled manner. The least-squares method is highly accurate and has a number of advantages over the conventional numerical methods like the finite difference and finite volume methods. In contrast to the finite volume method, when desinging an overall solution algorithm, this least-squares method ensures that all the continuity equations are satisfied individually and it deals with both the convective and diffusive terms stably and accurately. The novel iterative algorithm solves the flow and the population balance equation in a coupled manner. The model has been validated against experimental data obtained for two-phase flow in a bubble column. The predicted bubble size distribution and other flow quantities are in good agreement with the experimental data. En ligne : http://pubs.acs.org/doi/abs/10.1021/ie900088q