A least-squares method with direct minimization for the solution of the breakage–coalescence population balance equation

Zhu, Zhengjie; Dorao, Carlos Alberto; Jakobsen, Hugo A.

doi:10.1016/j.matcom.2008.05.001

Cited by 27 publications

(29 citation statements)

References 12 publications

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“…In a number of publications Dorao and Jakobsen [8,[30][31][32][33], and Zhu et al [34] investigated the least-squares method for the solution of relatively simple PB problems in which an analytical solution was available in most of the numerical test cases. The least-squares method has been applied to more comprehensive PB problems, e.g.…”

Section: A N U S C R I P Tmentioning

confidence: 99%

“…The sensitivity to the Newton and Picard iteration techniques for a least-squares solution of an analytical (manufactured) PB problem were also investigated by Zhu et al [34]. These authors [i.e.…”

Section: A N U S C R I P Tmentioning

confidence: 99%

“…In this way the source term g is defined such that the PB equation is fulfilled. Following this approach, Zhu et al [34] obtained the following analytical PB equation:…”

Section: A N U S C R I P Tmentioning

confidence: 99%

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Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

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The behavior of dispersed systems such as gas-liquids or liquid-liquid systems depends on the characteristics of the dispersed phase. The population balance (PB) equation is encountered in numerous engineering disciplines in order to describe complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behavior of the system. In the present study, the orthogonal collocation, Galerkin and least-squares methods have been adopted to solve a non-linear PB equation which consists of both breakage and coalescence terms. The performance of the methods is demonstrated by comparing the numerical solution results with the (manufactured) analytical solution of the problem. For the least-squares method, the choice of linearization technique influences the numerical performance, whereas the Galerkin and orthogonal collocation methods obtain the same numerical accuracy for both the Picard and Newton iteration techniques. The least-squares method suffers from lack of convergency using the Picard method. On the other hand, if the Newton method is employed, the least-squares method obtains the same accuracy as the Galerkin and orthogonal collocation methods.

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Section: A N U S C R I P Tmentioning

confidence: 99%

Section: A N U S C R I P Tmentioning

confidence: 99%

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Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

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“…kv; v′ e À t coal =t cont (12) The coalescence time and the contact time in Eq. (12) are average values.…”

Section: Breakage and Coalescence Modelsmentioning

confidence: 99%

“…A number of studies used the principles of least squares for solving the inverse problem, e.g., in [12] the authors used a combined method involving the least squares method and the direct minimization method to solve PBE and reported that a fast convergence was obtained. In the present paper, a different method from those mentioned above is considered, based on the use of a genetic algorithm (GA), to determine coalescence parameters.…”

Section: Introductionmentioning

confidence: 99%

Estimation of Coalescence Parameters in an Agitated Extraction Column Using a Hybrid Algorithm

Kabouche¹,

Méniai²,

Hasseine³

2011

Chem Eng & Technol

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A method based on a genetic algorithm (GA) combined with Levenberg-Marquardt (LM) method applicable to the population balance model for coalescence parameter estimation in a liquid-liquid biphasic system is presented. The toluene/water system in a rotating disk contactor was taken as an example. Estimation methods for such a problem are often based on deterministic optimization models that are rather instable and divergent around a local minimum. To overcome these limitations, the present study involves the introduction of a semi-stochastic method that is able to provide at first the estimation of coalescence parameters from the GA based on an inverse approach, exploiting the principle of GA. The LM algorithm was applied to ensure that the results are not restricted to a local minimum.

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On the solution of the dynamic population balance model describing emulsification: Evaluation of weighted residual methods

et al. 2013

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Numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau and least‐squares, are evaluated for the solution of transient population balance (PB) models describing liquid–liquid emulsification systems in stirred batch vessels. The numerical solution techniques are compared based on (i) a breakage dominated system with experimental data available, and (ii) a breakage–coalescence test case. Two numerical approaches are studied for the transient term: (i) time‐differencing by a low order finite difference approximation, and (ii) the spectral‐element technique. Both approaches use spectral approximations in the phase space dimension. Based on a residual measure, computational costs, and implementation complexity the combined finite difference–spectral approach is recommended above the spectral‐in‐time‐spectral‐in‐space approach. Within this recommended solution framework, it is not necessary to use a more mathematical complex spectral method than the orthogonal collocation technique.

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A least-squares method with direct minimization for the solution of the breakage–coalescence population balance equation

Cited by 27 publications

References 12 publications

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Estimation of Coalescence Parameters in an Agitated Extraction Column Using a Hybrid Algorithm

On the solution of the dynamic population balance model describing emulsification: Evaluation of weighted residual methods

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