Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays

Laurent, Frédérique; Massot, Marc; Villedieu, Philippe

doi:10.1016/j.jcp.2003.08.026

Cited by 76 publications

(120 citation statements)

References 28 publications

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“…There exists a large variety of applications and methods such as multi-fluid models for sprays (see [28,29,27] extended from sectional models of Tambour, Greenberg et al and [34,35,6] for a review of recent results) or sectional approach for aerosols (see [45,25] and references therein), also called class methods in [5]. 2-to adopt a moment point of view and to close the set of evolution equations through quadrature methods such as QMOM, quadrature method of moment initiated in [38,47] or DQMOM, direct quadrature method of moment introduced by Marchisio and Fox (see [32] and the recent review in [16]).…”

mentioning

confidence: 99%

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Massot¹,

Laurent²,

Kah³

et al. 2010

SIAM J. Appl. Math.

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131

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Abstract. In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which include aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function NDF of particles/droplets which satisfies a population balance equation. (PBE, also called Williams equation in the spray community). In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinate, and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view. We obtain in the present contribution a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description which will combine both the flexibility of Eulerian models for which the size phase space is discretized into "sections" (i.e. size intervals) and the efficiency of Direct Quadrature Method of Moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi-dimensional configurations.

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mentioning

confidence: 99%

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Massot¹,

Laurent²,

Kah³

et al. 2010

SIAM J. Appl. Math.

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131

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“…f [17,27]. This closure aspect will be discussed in the next section when we will present the multi-fluid model (see also [24,25]). …”

Section: Semi-kinetic Modelmentioning

confidence: 99%

“…As it is noted in [24], such a choice is completely arbitrary and is related to the structure of the sectional approach which presupposes a shape of the distribution function inside a section which is independent of the state of the spray outside the section. We have to emphasize the fact that, as developed in [25] the multi-fluid model takes for size variable the radius of the droplets r. Hence, κ i is chosen constant as a function of the radius (see Rems. 1.4 and 1.5) and is determined using the conservation of the mass density.…”

Section: Multi-fluid Modelmentioning

confidence: 99%

“…The idea was to consider the dispersed phase as a set of continuous media called "fluids", each "fluid" corresponding to a statistical average between two fixed sizes: a section. More recently, Laurent and Massot provided a rigorous kinetic framework for dilute sprays in laminar flames [24] as well as for dense sprays with coalescence [25] and denoted this method as a "multi-fluid" one. This method was shown to apply to realistic cases with a limited number of sections if an appropriate discretization of the size phase space is used, combined with a particular treatment for the last section, involving an exponential approximation of the droplet density function [26].…”

Section: Introductionmentioning

confidence: 99%

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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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“…The work on multi-fluid models for laminar sprays was carried on by Laurent and Massot [128] who derived the sectional equations from the fundamental kinetic equation of Williams thus establishing the link between multi-fluid models and the PBE. Subsequently Laurent et al [129] extended the apprcoach to coalescing sprays, while Laurent et al [130] provided validation of the multi-fluid model with experiments. More recently, Fox et al [68] applied both DQMOM and the multifluid model to a laminar spray and compared results to a Lagrangian solver, indicating that both are viable alternatives to it.…”

Section: Sprays and Bubblesmentioning

confidence: 99%

Population balance modelling of polydispersed particles in reactive flows

Rigopoulos

2010

Progress in Energy and Combustion Science

135

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Polydispersed particles in reactive flows is a wide subject area encompassing a range of dispersed flows with particles, droplets or bubbles that are created, transported and possibly interact within a reactive flow environment -typical examples include soot formation, aerosols, precipitation and spray combustion. One way to treat such problems is to employ as a starting point the Newtonian equations of motion written in a Lagrangian framework for each individual particle and either solve them directly or derive probabilistic equations for the particle positions (in the case of turbulent flow). Another way is inherently statistical and begins by postulating a distribution of particles over the distributed properties, as well as space and time, the transport equation for this distribution being the core of this approach. This transport equation, usually referred to as population balance equation (PBE) or general dynamic equation (GDE), was initially developed and investigated mainly in the context of spatially homogeneous systems. In the recent years, a growth of research activity has seen this approach being applied to a variety of flow problems such as sooting flames and turbulent precipitation, but significant issues regarding its appropriate coupling with CFD pertain, especially in the case of turbulent flow. The objective of this review is to examine this body of research from a unified perspective, the potential and limits of the PBE approach to flow problems, its links with Lagrangian and multifluid approaches and the numerical methods employed for its solution. Particular emphasis is given to turbulent flows, where the extension of the PBE approach is met with challenging issues. Finally, applications including reactive precipitation, soot formation, nanoparticle synthesis, sprays, bubbles and coal burning are being reviewed from the PBE perspective. It is shown that popualtion balance methods have been applied to these fields in varying degrees of detail, and future prospects are discussed.

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Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays

Cited by 76 publications

References 28 publications

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

A second-order multi-fluid model for evaporating sprays

Population balance modelling of polydispersed particles in reactive flows

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