ECCOMAS Congress 2016
Moment methods for the accurate description of soot dynamics: mathematical
modeling and realizable numerical schemes
1, 2*
Frédérique LAURENT
1, 2
1, 2
, Tan-Trung NGUYEN , Benedetta Franzelli
3
1, 2
, Marc MASSOT
1, 2,
, Rodney O. FOX
1
Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay
Grande Voie des Vignes, 92295 Châtenay-Malabry cedex, France
[frederique.laurent,tan-trung.nguyen,benedetta.franzelli,marc.massot]@centralesupelec.fr
2
Fédération de Mathématiques de l'Ecole Centrale Paris
FR CNRS 3487
Grande Voie des Vignes, 92295 Châtenay-Malabry cedex, France
[frederique.laurent,tan-trung.nguyen,marc.massot]@centralesupelec.fr
3
Department of Chemical and Biological Engineering
Iowa State University
2114 Sweeney Hall, Ames, IA 50011-2230, USA
[email protected]
ABSTRACT
The evolution of a population of fine, that is non-inertial, particles in a carrier fluid can be described by a
population balance equation (PBE). Soot is an example of such particles, for which it is important to predict
global quantities such as the total mass or volume fraction, but also their size distribution represented by
their number density function (NDF). The PBE is then a transport equation for the NDF of the soot
particles, describing their nucleation, spatial transport, surface growth, oxidation, aggregation and
fragmentation.
When considering the monovariate case, where the properties of the particles are described by only one
variable, different kinds of methods are available in the literature to solve the PBE. The first is based on a
discretization of the size variable and the transport of the total mass density of the particles inside each size
interval, referred to as a section. The sectional method can be derived from the PBE, with a closure based on
a reconstruction of the NDF inside each section [1]. The accuracy of sectional methods depends on the
number of sections, which can lead to a high computational cost for high accuracy. Moment methods were
therefore developed to lower the cost. These consist in transporting a finite set of the NDF size-moments. In
our work, the moment equations are closed through a continuous reconstruction of the NDF [2], allowing the
treatment of particle loss due to oxidation, which requires a point-wise evaluation of the NDF [3]. Moment
methods require moment realizability, i.e., all moments must remain moments of a positive distribution.
Adapted numerical schemes are then developed to ensure realizability. Next, hybrid methods combining
sectional and moment methods are considered. Initially developed for sprays [4], they employ a few
moments inside each section in such a way that the realizability constraint is quite simple and the NDF is
locally better described than with sectional methods, using a smaller number of sections. Finally, we develop
realizable numerical schemes to solve the moment equations and compare all methods for selected 1-D
flame configurations.
References
[1] K. Netzell, H. Lehtiniemi, and F. Mauss. Calculating the soot particle size distribution function in
turbulent diffusion flames using a sectional method. Proc. of Comb. Institute, 31:677–674, 2007.
[2] C. Yuan, F. Laurent, and R. O. Fox. An extended quadrature method of moments for population balance
equations. Journal of Aerosol Science, 51:1–23, 2012.
[3] M. Massot, F. Laurent, D. Kah, and S. De Chaisemartin. A robust moment method for evaluation of the
disappearance rate of evaporating sprays. SIAM J. Appl. Math., 70:3203–3234, 2010.
[4] F. Laurent, A. Sibra, and F. Doisneau. Two-size moment eulerian multi-fluid model: a flexible and
realizable high-fidelity description of polydisperse moderately dense evaporating sprays. submitted, available
on HAL: https://hal.archives-ouvertes.fr/hal-01169730, 2015.
Powered by TCPDF (www.tcpdf.org)