Generalization and numerical investigation of QMOM

Grosch, R.; Briesen, Heiko; Marquardt, Wolfgang; Wulkow, Michael

doi:10.1002/aic.11041

Cited by 48 publications

(40 citation statements)

References 38 publications

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“…For case I, although the values of the condition numbers are close for all methods, DQMoMa has shown to be . This behavior was also found by Grosch et al (2006), who compared methods using monomial and Laguerre generalized moments. For the finite domain problems, the condition number values are 3 2 10 10 − times smaller for the Legendre generalized moment methods.…”

Section: Robustness Analysismentioning

confidence: 53%

“…On the other hand, Grosch et al (2006) presented a generalized framework for the quadrature method of moments that is based on the concepts of generalized moments and coordinate transformations. In analogy to the QMoM, this framework transforms the PBE using the generalized moment integral operator, system of equations (QMoM-DAE) by solving the QMoM moment equations simultaneously with:…”

Section: Introductionmentioning

confidence: 99%

“…The framework developed by Grosch et al (2006) also includes the DQMoM developed by Marchisio and Fox (2005), the QMoM developed by McGraw (1997) and their variations with generalized moments (DQMoGeM). Grosch et al (2006) tested these variants of QMoM by comparing their solutions of mono-variate PBE problems in the semi-infinite domain to those obtained by the Parsival solver (Wulkow et al, 2001), which ensures a specified tolerance by using a hpadaptive discontinuous finite elements method. Grosch et al (2006) concluded that the methods based on Laguerre generalized moments were only marginally more robust than the corresponding methods using the monomial moments.…”

Section: Introductionmentioning

confidence: 99%

“…Grosch et al (2006) tested these variants of QMoM by comparing their solutions of mono-variate PBE problems in the semi-infinite domain to those obtained by the Parsival solver (Wulkow et al, 2001), which ensures a specified tolerance by using a hpadaptive discontinuous finite elements method. Grosch et al (2006) concluded that the methods based on Laguerre generalized moments were only marginally more robust than the corresponding methods using the monomial moments. They recommended the usage of the DQMoM formulation derived using the monomial moment integral operator and solved in terms of the quadrature weights and abscissas because of its good robustness and low computational cost.…”

Section: Introductionmentioning

confidence: 99%

“…Although Grosch et al (2006) did analyse the numerical behavior of quadrature moment methods using generalized moments, the analysis was limited to Laguerre generalized moments due to the choice of population balance problems in semi-infinite domains. However, there is an increasing interest in moment methods for problems in finite domains (Strumendo andArastoopour, 2008, Lage, 2011).…”

Section: Introductionmentioning

confidence: 99%

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Numerical aspects of direct quadrature-based moment methods for solving the population balance equation

Santos

Lage

Fontes³

2013

Braz. J. Chem. Eng.

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-Direct-quadrature generalized moment based methods were analysed in terms of accuracy, computational cost and robustness for the solution of the population balance problems in the [0, ) ∞ and [0,1] domains. The minimum condition number of the coefficient matrix of their linear system of equations was obtained by global optimization. An heuristic scaling rule from the literature was also evaluated. The results indicate that the methods based on Legendre generalized moments are the most robust for the finite domain problems, while the DQMoM formulation that solves for the abscissas and weights using the heuristic scaling rule is the best for the infinite domain problems.

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Section: Robustness Analysismentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Numerical aspects of direct quadrature-based moment methods for solving the population balance equation

Santos

Lage

Fontes³

2013

Braz. J. Chem. Eng.

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Dynamic simulation of crystallization processes: Adaptive finite element collocation method

Ulbert¹,

Lakatos²

2007

AIChE Journal

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in Wiley InterScience (www.interscience.wiley.com).An adaptive orthogonal collocation on finite elements method with adaptively varied upper bound of the relevant size interval is developed for numerical solution of population balance equation of crystallizers. Nucleation producing monosized and heterosized nuclei, size-dependent crystal growth, seeding, classified product removal and fines removal with dissolution are included into the model. Adaptation of the number and length, as well as the distribution of finite elements over the variable length computational interval is carried out forming a number of adaptation rules, based on ordering the finite elements of size coordinate according to the maxima of first derivatives of the population density function. The approximation is obtained using the Lagrange interpolation polynomials. The method is used for solving the mixed set of nonlinear ordinary and partial differential equations, forming a detailed dynamical model of continuous crystallizers with product classification and/or fines removal. The program can be used efficiently for simulation of stationary and dynamic processes of crystallization systems, computing either transients or long-time oscillating steady states generated by different nonlinear phenomena, and internal and external feedbacks. AIChE J, 53: 3089-3107, 2007 Keywords: crystallization, population balance model, numerical solution, orthogonal collocation on finite elements, dynamic simulation American Institute of Chemical Engineers IntroductionThe population balance model is an adequate mathematical description of crystallization processes. This model consists of a mixed set of ordinary and partial integrodifferential equations even in the simplest case of MSMPR (mixed suspension, mixed product removal) crystallizers, and the crucial point of using this modeling approach is the numerical solution of the population balance equation.A number of numerical methods have been proposed for solving this equation, focusing on different modeling aspects of disperse systems, among which the weighted residual methods have received considerable attention. Subramanian and Ramkrishna, 1 considering microbial populations, were the first to use the method of weighted residuals for population balance equation, showing that the Laguerre functions were suitable trail functions on semi-infinite interval [0, 1). Singh and Ramkrishna 2 proposed problem-specific polynomials to obtain solutions to dynamic population balance problems. Lakatos et al. 3 applied the generalized Laguerre functions for simulation of batch crystallization processes, solving the population balance equation by means of orthogonal collocation, and making possible of modifying the distribution of collocation points along the size coordinate by varying the scale factor. Witkowski and Rawlings, 4 investigating identification and control issues of solution crystallizers, applied the Laguerre polynomials for computing the crystal-size distribution.Correspondence concerning this article shou...

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Population balance modeling of bubbly flows considering the hydrodynamics and thermomechanical processes

Cheung

Yeoh

2008

AIChE Journal

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in Wiley InterScience (www.interscience.wiley.com).Three-dimensional two-fluid model and population balance equation is presented to treat the complex hydrodynamics and thermomechanical processes under various bubbly flow conditions. The class method, realized by the MUSIG model, alongside with suitable bubble coalescence and bubble breakage kernels is adopted. Homogeneous MUSIG model predictions have shown to yield good agreement against isothermal bubbly flow measurements. Subcooled boiling flow is further modeled through the use of class method with an improved wall heat partition model. Against experimental data, numerical results also showed good agreement for the local Sauter mean bubble diameter, void fraction, and interfacial area concentration profiles. Inhomogeneous MUSIG model is applied to investigate transition bubbly-to-slug flow. Better prediction of bubble diameter is accomplished, especially capturing separation of small and large bubbles. Weakness exists nonetheless in the interfacial forces model. Work is in progress through the consideration of swarm and cluster bubbles instead of isolated spherical bubble calibration.

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Generalization and numerical investigation of QMOM

Cited by 48 publications

References 38 publications

Numerical aspects of direct quadrature-based moment methods for solving the population balance equation

Numerical aspects of direct quadrature-based moment methods for solving the population balance equation

Dynamic simulation of crystallization processes: Adaptive finite element collocation method

Population balance modeling of bubbly flows considering the hydrodynamics and thermomechanical processes

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