An extended quadrature-based mass-velocity moment model for polydisperse bubbly flows
Abstract
Abstract Accurately predicting polydisperse bubbly flow is a nontrivial task due to the complexity of the bubble number density function (NDF) and the strong dependence of the instantaneous bubble velocity on the bubble size and shape. To describe polydisperse bubbly flow, a joint mass‐velocity NDF is adopted in this work. In the absence of mass transfer between phases and coalescence or breakage, the bubble mass is a conserved quantity from which the bubble size and shape can be found given the liquid pressure and surface tension. Quadrature‐based moment methods (QBMM) are applied to solve numerically the kinetic equation of the joint NDF using the extended quadrature method of moments (EQMOM) coupled with an open‐source incompressible Navier–Stokes solver for the liquid phase. Transport equations for the joint mass‐velocity moments are derived from a kinetic equation for the joint NDF and closure is attained using a monokinetic NDF valid in the limit of small bubble Stokes number. The integer moments with respect to mass are used to reconstruct the continuous univariate NDF with EQMOM, while the joint mass‐velocity moments are used to determine the bubble velocity as a continuous function of the bubble mass. The model is first applied to simulate a quasi‐2‐D bubble column with different aeration profiles and a narrow bubble size distribution in order to validate the approach with experimental data from the literature. Additional cases with a wide continuous bubble size distribution are used to show the ability of the modelling approach to describe polydisperse bubbly flows.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
EQMOM implemented in OpenFOAM for a quasi 2D bubble column yields numerical results agreeing well with experimental ones in time-averaged axial liquid velocity profile and time-averaged gas volume fraction.
“Yuan et al. [329] implemented the EQMOM in the open-source code OpenFOAM ® to simulate the fluid flow of a quasi 2D bubble column. Numerical results agree well with experimental ones, in terms of time-averaged axial liquid velocity profile and time-averaged gas volume fraction.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [329]
Cites
Description of aerosol dynamics by the quadrature method of moments
McGraw, R. (1997) · Aerosol Science and Technology · doi:10.1080/02786829708965471 · Cited by 1,128 · see citing works
Transverse migration of single bubbles in simple shear flows
Tomiyama, A., Tamai, H., Zun, I., & Hosokawa, S. (2002) · Chemical Engineering Science · doi:10.1016/s0009-2509(02)00085-4 · Cited by 1,085 · see citing works
Computational fluid dynamics of dispersed two-phase flows at high phase fractions
Rusche, H. (2003) · OpenGrey (Institut de l'Information Scientifique et Technique) · Cited by 1,024 · see citing works
The Motion of High-Reynolds-Number Bubbles in Inhomogeneous Flows
Magnaudet, J., & Eames, I. (2000) · Annual Review of Fluid Mechanics · doi:10.1146/annurev.fluid.32.1.659 · Cited by 759 · see citing works
Drag Coefficients of Single Bubbles under Normal and Micro Gravity Conditions
TOMIYAMA, A., KATAOKA, I., ZUN, I., & SAKAGUCHI, T. (1998) · JSME International Journal Series B · doi:10.1299/jsmeb.41.472 · Cited by 742 · see citing works
The force exerted on a body in inviscid unsteady non-uniform rotational flow
Auton, T. R., Hunt, J. C. R., & Prud'Homme, M. (1988) · Journal of Fluid Mechanics · doi:10.1017/s0022112088003246 · Cited by 650 · see citing works
Momentum and heat transfer in two-phase bubble flow—I. Theory
Yoshifusa, S. A. T. O., SADATOMI, M., & SEKOGUCHI, K. (1981) · International Journal of Multiphase Flow · doi:10.1016/0301-9322(81)90003-3 · Cited by 554 · see citing works
Shape of liquid drops moving in liquid media
Wellek, R. M., Agrawal, A. K., & Skelland, A. H. P. (1966) · AIChE Journal · doi:10.1002/aic.690120506 · Cited by 435 · see citing works
Computational Models for Polydisperse Particulate and Multiphase Systems
Marchisio, D. L., & Fox, R. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139016599 · Cited by 416 · see citing works
Self-citationComputational flow modelling and design of bubble column reactors
Joshi, J. B. (2001) · Chemical Engineering Science · doi:10.1016/s0009-2509(01)00273-1 · Cited by 392 · see citing works
CFD simulation of bubble column—An analysis of interphase forces and turbulence models
Tabib, M., Roy, S. A., & Joshi, J. B. (2007) · Chemical Engineering Journal · doi:10.1016/j.cej.2007.09.015 · Cited by 358 · see citing works
Averaged Equations for Two‐Phase Flows
Drew, D. A., & Segel, L. A. (1971) · Studies in Applied Mathematics · doi:10.1002/sapm1971503205 · Cited by 271 · see citing works
Conditional quadrature method of moments for kinetic equations
Yuan, C., & Fox, R. O. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.07.020 · Cited by 232 · see citing works
Self-citationAn extended quadrature method of moments for population balance equations
Yuan, C., Laurent, F., & Fox, R. O. (2012) · Journal of Aerosol Science · doi:10.1016/j.jaerosci.2012.04.003 · Cited by 225 · see citing works
Self-citationCFD simulation of bubble columns incorporating population balance modeling
Bhole, M. R., Joshi, J. B., & Ramkrishna, D. (2008) · Chemical Engineering Science · doi:10.1016/j.ces.2008.01.013 · Cited by 203 · see citing works
Numerical simulation of the dynamics of two-phase gas—liquid flows in bubble columns
Lapin, A., & Lübbert, A. (1994) · Chemical Engineering Science · doi:10.1016/0009-2509(94)e0121-6 · Cited by 181 · see citing works
Lateral Migration of Single Bubbles Due to the Presence of Wall
Hosokawa, S., Tomiyama, A., Misaki, S., & Hamada, T. (2002) · doi:10.1115/fedsm2002-31148 · Cited by 169 · see citing works
On the role of the lateral lift force in poly-dispersed bubbly flows
Lucas, D., & Tomiyama, A. (2011) · International Journal of Multiphase Flow · doi:10.1016/j.ijmultiphaseflow.2011.05.009 · Cited by 129 · see citing works
Void fraction disturbances in a uniform bubbly fluid
Biesheuvel, A., & Gorissen, W. C. M. (1990) · International Journal of Multiphase Flow · doi:10.1016/0301-9322(90)90055-n · Cited by 101 · see citing works
Realizable high-order finite-volume schemes for quadrature-based moment methods
Vikas, V., Wang, Z. J., Passalacqua, A., & Fox, R. O. (2011) · Journal of Computational Physics · doi:10.1016/j.jcp.2011.03.038 · Cited by 98 · see citing works
Self-citationLaurent, F., Massot, M., & Villedieu, P. (2003) · Journal of Computational Physics · doi:10.1016/j.jcp.2003.08.026 · Cited by 96 · see citing works
FLUID MECHANICAL ASPECTS OF THE GAS-LIFT TECHNIQUE
Guet, S., & Ooms, G. (2005) · Annual Review of Fluid Mechanics · doi:10.1146/annurev.fluid.38.061505.093942 · Cited by 92 · see citing works
Numerical aspects of an algorithm for the Eulerian simulation of two‐phase flows
Oliveira, P. J., & Issa, R. I. (2003) · International Journal for Numerical Methods in Fluids · doi:10.1002/fld.508 · Cited by 79 · see citing works
Buffo, A., & Marchisio, D. L. (2014) · Reviews in Chemical Engineering · doi:10.1515/revce-2013-0015 · Cited by 55 · see citing works
Linear stability analysis of a two-fluid model for air–water bubble columns
Monahan, S. M., & Fox, R. O. (2007) · Chemical Engineering Science · doi:10.1016/j.ces.2007.03.021 · Cited by 55 · see citing works
Self-citationPang, M. J., & Wei, J. J. (2011) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2011.03.046 · Cited by 42 · see citing works
Ziegenhein, T., Rzehak, R., Krepper, E., & Lucas, D. (2013) · Chemie Ingenieur Technik · doi:10.1002/cite.201200223 · Cited by 38 · see citing works
Vikas, V., Wang, Z. J., & Fox, R. O. (2013) · Journal of Computational Physics · doi:10.1016/j.jcp.2013.05.002 · Cited by 29 · see citing works
Self-citationBubble columns : Structures or stability?
Harteveld, W. K. (2005) · Research Repository (Delft University of Technology) · Cited by 23 · see citing works
Modeling of bubble-column flows with quadrature-based moment methods
Vikas, V., Yuan, C., Wang, Z. J., & Fox, R. O. (2011) · Chemical Engineering Science · doi:10.1016/j.ces.2011.03.009 · Cited by 23 · see citing works
Self-citationMonahan, S. M., & Fox, R. O. (2009) · Industrial & Engineering Chemistry Research · doi:10.1021/ie801202x · Cited by 11 · see citing works
Self-citationQuadrature-Based Moment Methods for Polydisperse Multiphase Flows
Fox, R. O. (2013) · Courses and lectures · doi:10.1007/978-3-7091-1622-7_3 · Cited by 8 · see citing works
Self-citation
Flagged first, then by citations
Cited by
Sorted by citations
Nguyen, T. T., Laurent, F., Fox, R. O., & Massot, M. (2016) · Journal of Computational Physics · doi:10.1016/j.jcp.2016.08.017 · Cited by 102 · see citing works
Self-citationShiea, M., Buffo, A., Vanni, M., & Marchisio, D. L. (2020) · Annual Review of Chemical and Biomolecular Engineering · doi:10.1146/annurev-chembioeng-092319-075814 · Cited by 74 · see citing works
Li, D., Gao, Z., Buffo, A., Podgórska, W., & Marchisio, D. L. (2016) · AIChE Journal · doi:10.1002/aic.15557 · Cited by 65 · see citing works
Santarelli, C., & Fröhlich, J. (2016) · International Journal of Multiphase Flow · doi:10.1016/j.ijmultiphaseflow.2016.01.004 · Cited by 63 · see citing works
Ferraro, F., Russo, C., Schmitz, R., Hasse, C., & Sirignano, M. (2020) · Fuel · doi:10.1016/j.fuel.2020.119353 · Cited by 56 · see citing works
Kong, B., Fox, R. O., Feng, H., Capecelatro, J., Patel, R. G., Desjardins, O., & Fox, R. O. (2017) · AIChE Journal · doi:10.1002/aic.15686 · Cited by 46 · see citing works
Self-citationModeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review
Luzi, G., & McHardy, C. (2022) · Energies · doi:10.3390/en15113966 · Cited by 44 · see citing works
A quadrature-based moment method for polydisperse bubbly flows
Heylmun, J. C., Kong, B., Passalacqua, A., & Fox, R. O. (2019) · Computer Physics Communications · doi:10.1016/j.cpc.2019.06.005 · Cited by 30 · see citing works
Self-citationDürr, R., & Bück, A. (2020) · Processes · doi:10.3390/pr8040414 · Cited by 28 · see citing works
Askari, E., Proulx, P., & Passalacqua, A. (2018) · ChemEngineering · doi:10.3390/chemengineering2010008 · Cited by 24 · see citing works
Self-citationPeña-Monferrer, C., Passalacqua, A., Chiva, S., & Muñoz-Cobo, J. L. (2016) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2016.03.006 · Cited by 20 · see citing works
Self-citationA moment-based kinetic theory model for polydisperse gas–particle flows
Kong, B., & Fox, R. O. (2019) · Powder Technology · doi:10.1016/j.powtec.2019.04.031 · Cited by 18 · see citing works
Self-citationLi, D., Marchisio, D. L., Hasse, C., & Lucas, D. (2019) · AIChE Journal · doi:10.1002/aic.16732 · Cited by 17 · see citing works
Schäfer, J., Hlawitschka, M. W., Attarakih, M. M., & Bart, H. (2019) · AIChE Journal · doi:10.1002/aic.16694 · Cited by 16 · see citing works
Pollack, M., Salenbauch, S., Marchisio, D. L., & Hasse, C. (2015) · Journal of Aerosol Science · doi:10.1016/j.jaerosci.2015.10.008 · Cited by 13 · see citing works
twoWayGPBEFoam: An open-source Eulerian QBMM solver for monokinetic bubbly flows
Li, D., Marchisio, D. L., Hasse, C., & Lucas, D. (2019) · Computer Physics Communications · doi:10.1016/j.cpc.2019.107036 · Cited by 12 · see citing works
Heylmun, J. C., Fox, R. O., & Passalacqua, A. (2021) · Computer Physics Communications · doi:10.1016/j.cpc.2021.108072 · Cited by 9 · see citing works
Self-citationModeling Dilute Gas–Solid Flows Using a Polykinetic Moment Method Approach
Dunn, D. M., & Squires, K. D. (2015) · Journal of Fluids Engineering · doi:10.1115/1.4031687 · Cited by 8 · see citing works
Quadrature-Based Moment Methods for Multiphase Chemically Reacting Flows
Fox, R. O. (2018) · Advances in chemical engineering · doi:10.1016/bs.ache.2018.01.001 · Cited by 8 · see citing works
Self-citationZero-flux approximations for multivariate quadrature-based moment methods
Pollack, M., Pütz, M., Marchisio, D. L., Oevermann, M., & Hasse, C. (2019) · Journal of Computational Physics · doi:10.1016/j.jcp.2019.108879 · Cited by 3 · see citing works
Cheng, T., Zhang, Y., Hong, Q., Zhang, D., & Kong, B. (2025) · Chemical Engineering Science · doi:10.1016/j.ces.2025.122759
Self-citationNyande, B. W., & Raponi, A. (2026) · Chemical Engineering Science · doi:10.1016/j.ces.2026.125072
- Authors
- Yuan, C., Kong, B., Passalacqua, A., & Fox, R. O.
- Affiliations
- Iowa State University (US)
- Venue
- The Canadian Journal of Chemical Engineering
- Publisher
- Wiley
- Published
- 2014-10
- Volume
- 92
- Issue
- 12
- Pages
- 2053-2066
- ISSN
- 0008-4034 (ISSN-L)
- 1939-019X
- OpenAlex
- W1998655311
- Cited by
- 23
- References
- 32
- Citations by year
- 2026: 1 · 2025: 1 · 2023: 1 · 2022: 1 · 2021: 1 · 2020: 3 · 2019: 6 · 2018: 2 · 2017: 1 · 2016: 4 · 2015: 2
- FWCI
- 1.21
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- The Canadian Journal of Chemical Engineering (publishedVersion)
- Topics
- Physical Sciences › Engineering › Biomedical Engineering › Fluid Dynamics and Mixing
- Physical Sciences › Environmental Science › Water Science and Technology › Minerals Flotation and Separation Techniques
- Physical Sciences › Engineering › Ocean Engineering › Particle Dynamics in Fluid Flows
- Keywords
- quadrature-based moment methods, bubble size distribution, bubble velocity, kinetic equations, Stokes number, bubble column, two-phase flow modeling
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
