References and journal publications


This section summarizes the main references used in the development of the OpenQBMM package and aims at providing a compact set of references for OpenQBMM users. It includes both references from other research groups and from the OpenQBMM developers. Complete references for specific code components are provided in the header of the corresponding implementation and can also be seen in the code documentation.

  1. H. Dette, W.J. Studden, Matrix measures, moment spaces and Favard’s theorem for the interval [0,1] and [0,∞), Linear Algebra and Its Applications. 345 (2002) 169–193. doi:10.1016/S0024-3795(01)00493-1.
  2. R.O. Fox, Computational Models for Turbulent Reacting Flows, Cambridge University Press, 2003.
  3. F. Laurent, T.T. Nguyen, Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution, Journal of Computational Physics. 337 (2017) 309–338. doi:10.1016/
  4. E. Madadi-Kandjani, A. Passalacqua, An extended quadrature-based moment method with log-normal kernel density functions, Chemical Engineering Science. 131 (2015) 323–339. doi:10.1016/j.ces.2015.04.005.
  5. D.L. Marchisio, R.O. Fox, Computational models for polydisperse particulate and multiphase systems, Cambridge University Press, Cambridge, 2013.
  6. C. Yuan, R.O. Fox, Conditional quadrature method of moments for kinetic equations, Journal of Computational Physics. 230 (2011) 8216–8246. doi:10.1016/
  7. C. Yuan, B. Kong, A. Passalacqua, R.O. Fox, An extended quadrature-based mass-velocity moment model for polydisperse bubbly flows, The Canadian Journal of Chemical Engineering. 92 (2014) 2053–2066. doi:10.1002/cjce.22006.
  8. C. Yuan, F. Laurent, R.O. Fox, An extended quadrature method of moments for population balance equations, Journal of Aerosol Science. 51 (2012) 1–23. doi:10.1016/j.jaerosci.2012.04.003.
  9. V. Vikas, Z.J. Wang, A. Passalacqua, R.O. Fox, Realizable high-order finite-volume schemes for quadrature-based moment methods, Journal of Computational Physics. 230 (2011) 5328–5352. doi:10.1016/
  10. V. Vikas, C. Yuan, Z.J. Wang, R.O. Fox, Modeling of bubble-column flows with quadrature-based moment methods, Chemical Engineering Science. 66 (2011) 3058–3070. doi:10.1016/j.ces.2011.03.009.


This section lists a selection of the articles containing the theoretical foundations of OpenQBMM and illustrating some of the possible applications of the OpenQBMM framework.

Book chapters

  1. R.O. Fox, Quadrature-Based Moment Methods for Polydisperse Multiphase Flows, in: Stochastic Methods in Fluid Mechanics, Springer, Vienna, 2014: pp. 87–136. doi:10.1007/978-3-7091-1622-7_3.
  2. D.L. Marchisio, R.O. Fox, Reacting Flows and the Interaction between Turbulence and Chemistry, in: Reference Module in Chemistry, Molecular Sciences and Chemical Engineering, Elsevier, 2016. doi:10.1016/B978-0-12-409547-2.11526-4.
  3. R.O. Fox, Chapter One – Quadrature-Based Moment Methods for Multiphase Chemically Reacting Flows, in: A. Parente, J. De Wilde (Eds.), Advances in Chemical Engineering, Academic Press, 2018: pp. 1–50. doi:10.1016/bs.ache.2018.01.001.

Journal articles

  1. R.O. Fox, F. Laurent, A. Passalacqua, The generalized quadrature method of moments, Journal of Aerosol Science. 167 (2023) 106096. doi:10.1016/j.jaerosci.2022.106096.
  2. J.C. Heylmun, R.O. Fox, A. Passalacqua, A quadrature-based moment method for the evolution of the joint size-velocity number density function of a particle population, Computer Physics Communications. 267 (2021) 108072. doi:10.1016/j.cpc.2021.108072.
  3. J.C. Heylmun, B. Kong, A. Passalacqua, R.O. Fox, A quadrature-based moment method for polydisperse bubbly flows, Computer Physics Communications. 244 (2019) 187–204. doi:10.1016/j.cpc.2019.06.005.
  4. R.O. Fox, F. Laurent, A. Vié, Conditional hyperbolic quadrature method of moments for kinetic equations, Journal of Computational Physics. 365 (2018) 269–293. doi:10.1016/
  5. A. Passalacqua, F. Laurent, E. Madadi-Kandjani, J.C. Heylmun, R.O. Fox, An open-source quadrature-based population balance solver for OpenFOAM, Chemical Engineering Science. 176 (2018) 306–318. doi:10.1016/j.ces.2017.10.043.
  6. R.O. Fox, F. Laurent, A. Vié, Conditional hyperbolic quadrature method of moments for kinetic equations, Journal of Computational Physics. 365 (2018) 269–293. doi:10.1016/
  7. E. Askari, P. Proulx, A. Passalacqua, Modelling of Bubbly Flow Using CFD-PBM Solver in OpenFOAM: Study of Local Population Balance Models and Extended Quadrature Method of Moments Applications, ChemEngineering. 2 (2018) 8. doi:10.3390/chemengineering2010008.
  8. V. Quedeville, H. Ouazaite, B. Polizzi, R.O. Fox, P. Villedieu, P. Fede, F. Létisse, J. Morchain, A two-dimensional population balance model for cell growth including multiple uptake systems, Chemical Engineering Research and Design. 132 (2018) 966–981. doi:10.1016/j.cherd.2018.02.025.
  9. C. Chalons, R. Fox, F. Laurent, M. Massot, A. Vié, Multivariate Gaussian Extended Quadrature Method of Moments for Turbulent Disperse Multiphase Flow, Multiscale Model. Simul. 15 (2017) 1553–1583. doi:10.1137/16M109209X.
  10. E. Madadi-Kandjani, R.O. Fox, A. Passalacqua, Application of the Fokker-Planck molecular mixing model to turbulent scalar mixing using moment methods, Physics of Fluids. 29 (2017) 065109. doi:10.1063/1.4989421.
  11. B. Kong, R.O. Fox, H. Feng, J. Capecelatro, R. Patel, O. Desjardins, R.O. Fox, Euler–euler anisotropic gaussian mesoscale simulation of homogeneous cluster-induced gas–particle turbulence, AIChE Journal. 63 (n.d.) 2630–2643. doi:10.1002/aic.15686.
  12. B. Kong, R.O. Fox, A solution algorithm for fluid–particle flows across all flow regimes, Journal of Computational Physics. 344 (2017) 575–594. doi:10.1016/
  13. R.G. Patel, O. Desjardins, B. Kong, J. Capecelatro, R.O. Fox, Verification of Eulerian–Eulerian and Eulerian–Lagrangian simulations for turbulent fluid–particle flows, AIChE Journal. 63 (n.d.) 5396–5412. doi:10.1002/aic.15949.
  14. V. Raman, R.O. Fox, Modeling of Fine-Particle Formation in Turbulent Flames, Annual Review of Fluid Mechanics. 48 (2016) 159–190. doi:10.1146/annurev-fluid-122414-034306.
  15. A. Wick, T.-T. Nguyen, F. Laurent, R.O. Fox, H. Pitsch, Modeling soot oxidation with the Extended Quadrature Method of Moments, Proceedings of the Combustion Institute. 36 (2017) 789–797. doi:10.1016/j.proci.2016.08.004.
  16. T.T. Nguyen, F. Laurent, R.O. Fox, M. Massot, Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes, Journal of Computational Physics. 325 (2016) 129–156. doi:10.1016/