Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Sintering of Particle Populations

Wright, D.; McGraw, Robert; Rosner, Daniel E.

doi:10.1006/jcis.2000.7409

Cited by 151 publications

(195 citation statements)

References 27 publications

(62 reference statements)

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“…This requires either some a priori assumptions regarding the PSDF or a suitable closure scheme. One closure scheme for the moment equations involves quadrature-based methods (McGraw 1997;Wright et al 2001;Marchisio and Fox 2005). Another type of closure is interpolative, as proposed by Frenklach and Harris (1987), hence the name MOMIC (Frenklach 2002).…”

Section: Methods Of Moments With Interpolative Closure (Momic)mentioning

confidence: 99%

“…As such, the DSM approach remains prohibitively expensive for turbulent combustion simulation. Finally, as a reasonable compromise between fidelity and computational efficiency, the method of moments (Dobbins and Mulholland 1984;Frenklach 1985;Frenklach and Harris 1987;McGraw 1997;Wright et al 2001;Frenklach 2002;Moody and Collins 2003;Lignell et al 2008;Mueller et al 2009a,b) describes the key soot variables and the size distribution information by solving for a subset of moments of the PSDF, which are transported along with the gas-phase species.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

Roy

Arias

Lecoustre

et al. 2014

Aerosol Science and Technology

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The method of moments with interpolative closure (MOMIC) for soot formation and growth provides a detailed modeling framework maintaining a good balance in generality, accuracy, robustness, and computational efficiency. This study presents several computational issues in the development and implementation of the MOMIC-based soot modeling for direct numerical simulations (DNS). The issues of concern include a wide dynamic range of numbers, choice of normalization, high effective Schmidt number of soot particles, and realizability of the soot particle size distribution function (PSDF). These problems are not unique to DNS, but they are often exacerbated by the high-order numerical schemes used in DNS. Four specific issues are discussed in this article: the treatment of soot diffusion, choice of interpolation scheme for MOMIC, an approach to deal with strongly oxidizing environments, and realizability of the PSDF. General, robust, and stable approaches are sought to address these issues, minimizing the use of ad hoc treatments such as clipping. The solutions proposed and demonstrated here are being applied to generate new physical insight into complex turbulence-chemistry-soot-radiation interactions in turbulent reacting flows using DNS.

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Section: Methods Of Moments With Interpolative Closure (Momic)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

Roy

Arias

Lecoustre

et al. 2014

Aerosol Science and Technology

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“…Using only one moment in size gives a simplified model which yields low computational cost but is not able to take into account the evaporation phenomenon since it provides no information on the mean size of the droplets. Therefore, some methods using several moments in size have been developed, allowing a more accurate description of the spray's physical behaviour (see for example [1][2][3][4]32,41]). The multi-fluid approach is based on a different point of view, which consists in first discretizing the size space before performing any average in size.…”

Section: Remark 12mentioning

confidence: 99%

“…In particular, breakup and coalescence cannot be taken into account in the standard two-fluid approach where only two moments in size of the distribution function are resolved (droplet number and droplet mass density). Some other Moment Methods have been developed (see for example [1,2,32,41]) and recently, Beck and Watkins proposed a real improvement on this kind of method, based on conservation equations for several droplet size moments allowing the development of spray submodels and the possibility to take into account some complex phenomena such as heat transfer, breakup and collision [3,4]. However, such a method still needs to presuppose a global shape for the whole size distribution of the spray, introducing some limitations.…”

Section: Introductionmentioning

confidence: 99%

A second-order multi-fluid model for evaporating sprays

Dufour¹,

Villedieu²

2005

ESAIM: M2AN

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Abstract. The aim of this paper is to present a method using both the ideas of sectional approach and moment methods in order to accurately simulate evaporation phenomena in gas-droplets flows. we propose an extension of this approach based on a more accurate representation of the droplet size number density in each section ensuring the exact conservation of two moments (as opposed to only one moment used in the classical approach). A corresponding second-order numerical scheme, with respect to space and droplet size variables, is also introduced and can be proved to be positive and to satisfy a maximum principle on the velocity and the mean droplet mass under a suitable CFL-like condition. Numerical simulations have been performed and the results confirm the accuracy of this new method even when a very coarse mesh for the droplet size variable (i.e.: a low number of sections) is used.Mathematics Subject Classification. 35Q35, 65Z05, 76T10.

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“…However, it is sometimes possible to solve the multivariate version of eq 6 for a selected set of multivariate moments by employing a nonlinear equation solver. 24 The success of such an approach will be dependent, in part, on whether the weights and abscissas are uniquely determined by the chosen set of moments. Indeed, unlike in the univariate case, there is no guarantee in the multivariate case that the weights will be non-negative and the abscissas realizable for a particular choice of moments.…”

mentioning

confidence: 99%

Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments

Fox

2008

Ind. Eng. Chem. Res.

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The direct quadrature method of moments (DQMOM) can be employed to close population balance equations (PBEs) governing a wide class of multivariate number density functions (NDFs). Such equations occur over a vast range of scientific applications, including aerosol science, kinetic theory, multiphase flows, turbulence modeling, and control theory, to name just a few. As the name implies, DQMOM uses quadrature weights and abscissas to approximate the moments of the NDF, and the number of quadrature nodes determines the accuracy of the closure. For nondegenerate univariate cases (i.e., a sufficiently smooth NDF), the N weights and N abscissas are uniquely determined by the first 2N non-negative integer moments of the NDF. Moreover, an efficient product-difference algorithm exists to compute the weights and abscissas from the moments. In contrast, for a d-dimensional NDF, a total of (1 + d)N multivariate moments are required to determine the weights and abscissas, and poor choices for the moment set can lead to nonunique abscissas and even negative weights. In this work, it is demonstrated that optimal moment sets exist for multivariate DQMOM when N ) nd quadrature nodes are employed to represent a d-dimensional NDF with n ) 1-3 and d ) 1-3. Moreover, this choice is independent of the source terms in the PBE governing the time evolution of the NDF. A multivariate Fokker-Planck equation is used to illustrate the numerical properties of the method for d ) 3 with n ) 2 and 3. The direct quadrature method of moments (DQMOM) can be employed to close population balance equations (PBEs) governing a wide class of multivariate number density functions (NDFs). Such equations occur over a vast range of scientific applications, including aerosol science, kinetic theory, multiphase flows, turbulence modeling, and control theory, to name just a few. As the name implies, DQMOM uses quadrature weights and abscissas to approximate the moments of the NDF, and the number of quadrature nodes determines the accuracy of the closure. For nondegenerate univariate cases (i.e., a sufficiently smooth NDF), the N weights and N abscissas are uniquely determined by the first 2N non-negative integer moments of the NDF. Moreover, an efficient product-difference algorithm exists to compute the weights and abscissas from the moments. In contrast, for a d-dimensional NDF, a total of (1 + d)N multivariate moments are required to determine the weights and abscissas, and poor choices for the moment set can lead to nonunique abscissas and even negative weights. In this work, it is demonstrated that optimal moment sets exist for multivariate DQMOM when N ) n d quadrature nodes are employed to represent a d-dimensional NDF with n ) 1-3 and d ) 1-3. Moreover, this choice is independent of the source terms in the PBE governing the time evolution of the NDF. A multivariate Fokker-Planck equation is used to illustrate the numerical properties of the method for d ) 3 with n ) 2 and 3. Disciplines Biological Engineering | Chemical Engineering

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Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Sintering of Particle Populations

Cited by 151 publications

References 27 publications

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

A second-order multi-fluid model for evaporating sprays

Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments

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