Integral formulation of the population balance equation: Application to particulate systems with particle growth
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?
The cumulative quadrature method of moments (CQMOM) derived by Attarakih relies on formulating the governing population balance equation (PBE) in an integral form.
“In 2013, Attarakih [324] derived a cumulative QMOM scheme (CQMOM) that relies on formulating the governing PBE in an integral form.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [324]
CQMOM numerical results show excellent agreement with the available analytical solution for particle growth with a linear growth law and with a diffusion-limited one.
“Numerical results show an excellent agreement with the available analytical solution [301,324] for the cases of particle growth with a linear growth law, and with a diffusion-limited one.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [324]
In CQMOM, the low order moments are always realizable because the PD algorithm produces positive weights and abscissas.
“As a result, the low order moments are always realizable since the PD algorithm produces positive weights and abscissas.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [324]
Cites
Monotone Piecewise Cubic Interpolation
Fritsch, F. N., & Carlson, R. E. (1980) · SIAM Journal on Numerical Analysis · doi:10.1137/0717021 · Cited by 2,543 · see citing works
Isaacson, E., Keller, H. B., & Weiss, G. H. (1967) · Physics Today · doi:10.1063/1.3034415 · Cited by 2,278 · see citing works
Kurganov, A., & Tadmor, E. (2000) · Journal of Computational Physics · doi:10.1006/jcph.2000.6459 · Cited by 1,966 · see citing works
Miranker, W. L., Isaacson, E., & Keller, H. B. (1967) · Mathematics of Computation · doi:10.2307/2003280 · Cited by 1,877 · see citing works
Some problems in particle technology
Hulburt, H. M., & Katz, S. (1964) · Chemical Engineering Science · doi:10.1016/0009-2509(64)85047-8 · Cited by 1,670 · see citing works
Population Balances: Theory and Applications to Particulate Systems in Engineering
Ramkrishna, D. (2000) · Cited by 1,434 · see citing works
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
On the solution of population balance equations by discretization—I. A fixed pivot technique
Kumar, S., & Ramkrishna, D. (1996) · Chemical Engineering Science · doi:10.1016/0009-2509(96)88489-2 · Cited by 1,109 · see citing works
Numerical methods and software
Kahaner, D. K., Moler, C., & Nash, S. (1989) · Cited by 1,009 · see citing works
On the solution of population balance equations by discretization—II. A moving pivot technique
Kumar, S., & Ramkrishna, D. (1996) · Chemical Engineering Science · doi:10.1016/0009-2509(95)00355-x · Cited by 825 · see citing works
Solution of population balance equations using the direct quadrature method of moments
Marchisio, D. L., & Fox, R. O. (2005) · Journal of Aerosol Science · doi:10.1016/j.jaerosci.2004.07.009 · Cited by 803 · see citing works
Error Bounds in Equilibrium Statistical Mechanics
Gordon, R. G. (1968) · Journal of Mathematical Physics · doi:10.1063/1.1664624 · Cited by 453 · see citing works
Simulation of multicomponent aerosol dynamics
Gelbard, F., & Seinfeld, J. H. (1980) · Journal of Colloid and Interface Science · doi:10.1016/0021-9797(80)90587-1 · Cited by 446 · see citing works
Computational and experimental study of soot formation in a coflow, laminar diffusion flame
Smooke, M. D., McEnally, C. S., Pfefferle, L. D., Hall, R. J., & Colket, M. B. (1999) · Combustion and Flame · doi:10.1016/s0010-2180(98)00096-0 · Cited by 324 · see citing works
Simulation of multicomponent aerosol dynamics
Kim, Y. P., & Seinfeld, J. H. (1992) · Journal of Colloid and Interface Science · doi:10.1016/0021-9797(92)90432-l · Cited by 303 · see citing works
Numerical Methods and Software.
Fritsch, F. N., Kahaner, D. K., Moler, C., & Nash, S. (1990) · Mathematics of Computation · doi:10.2307/2008452 · Cited by 287 · see citing works
Xiong, Y., & Pratsinis, S. E. (1993) · Journal of Aerosol Science · doi:10.1016/0021-8502(93)90003-r · Cited by 254 · see citing works
Construction of Gauss-Christoffel quadrature formulas
Gautschi, W. (1968) · Mathematics of Computation · doi:10.1090/s0025-5718-1968-0228171-0 · Cited by 212 · see citing works
Techniques for the reconstruction of a distribution from a finite number of its moments
John, V., Angelov, I., Öncül, A. A., & Thévenin, D. (2007) · Chemical Engineering Science · doi:10.1016/j.ces.2007.02.041 · Cited by 199 · see citing works
Numerical simulations of gas–liquid mass transfer in bubble columns with a CFD–PBM coupled model
Wang, T., & Wang, J. (2007) · Chemical Engineering Science · doi:10.1016/j.ces.2007.08.033 · Cited by 194 · see citing works
Process systems modelling and applications in granulation: A review
Cameron, I. T., Wang, F. Y., Immanuel, C. D., & Stepanek, F. (2005) · Chemical Engineering Science · doi:10.1016/j.ces.2005.02.004 · Cited by 168 · see citing works
A quadrature-based third-order moment method for dilute gas-particle flows
Fox, R. O. (2008) · Journal of Computational Physics · doi:10.1016/j.jcp.2008.03.014 · Cited by 149 · see citing works
Diemer, R. B., & Olson, J. H. (2002) · Chemical Engineering Science · doi:10.1016/s0009-2509(02)00112-4 · Cited by 149 · see citing works
Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)
Attarakih, M. M., Drumm, C., & Bart, H.-J. (2009) · Chemical Engineering Science · doi:10.1016/j.ces.2008.05.006 · Cited by 141 · see citing works
Self-citationAamir, E., Nagy, Z. K., Rielly, C. D., Kleinert, T., & Judat, B. (2009) · Industrial & Engineering Chemistry Research · doi:10.1021/ie900430t · Cited by 129 · see citing works
Positive Definite and Definitizable Functions
Sasvári, Z. (1994) · Cited by 125 · see citing works
Droplet population balance modelling—hydrodynamics and mass transfer
Schmidt, S., Simon, M. C., Attarakih, M. M., Luis Lagar, G., & Bart, H. (2005) · Chemical Engineering Science · doi:10.1016/j.ces.2005.02.075 · Cited by 97 · see citing works
Self-citationAttarakih, M. M., Bart, H., & Faqir, N. M. (2005) · Chemical Engineering Science · doi:10.1016/j.ces.2004.12.055 · Cited by 92 · see citing works
Self-citationComparison of numerical methods for the simulation of dispersed phase systems
Motz, S., Mitrović, A., & Gilles, E. D. (2002) · Chemical Engineering Science · doi:10.1016/s0009-2509(02)00349-4 · Cited by 86 · see citing works
Robust control of particulate processes using uncertain population balances
Chiu, T. Y., & Christofides, P. D. (2000) · AIChE Journal · doi:10.1002/aic.690460207 · Cited by 77 · see citing works
One-Group Reduced Population Balance Model for CFD Simulation of a Pilot-Plant Extraction Column
Drumm, C., Attarakih, M. M., Hlawitschka, M. W., & Bart, H. (2010) · Industrial & Engineering Chemistry Research · doi:10.1021/ie901411e · Cited by 73 · see citing works
Self-citationRosner, D. E., McGraw, R., & Tandon, P. (2003) · Industrial & Engineering Chemistry Research · doi:10.1021/ie020627l · Cited by 70 · see citing works
Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments
Fox, R. O. (2008) · Industrial & Engineering Chemistry Research · doi:10.1021/ie801316d · Cited by 62 · see citing works
Gimbun, J., Nagy, Z. K., & Rielly, C. D. (2009) · Industrial & Engineering Chemistry Research · doi:10.1021/ie900548s · Cited by 61 · see citing works
Generalization and numerical investigation of QMOM
Grosch, R., Briesen, H., Marquardt, W., & Wulkow, M. (2006) · AIChE Journal · doi:10.1002/aic.11041 · Cited by 53 · see citing works
Solution of PBE by MOM in finite size domains
Strumendo, M., & Arastoopour, H. (2008) · Chemical Engineering Science · doi:10.1016/j.ces.2008.02.010 · Cited by 50 · see citing works
Lage, P. L. C. (2011) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2011.05.017 · Cited by 43 · see citing works
Lattice Boltzmann method for multi-dimensional population balance models in crystallization
Majumder, A., Kariwala, V., Ansumali, S., & Rajendran, A. (2011) · Chemical Engineering Science · doi:10.1016/j.ces.2011.04.041 · Cited by 42 · see citing works
de Souza, L. G. M., Janiga, G., John, V., & Thévenin, D. (2010) · Chemical Engineering Science · doi:10.1016/j.ces.2010.01.007 · Cited by 40 · see citing works
Majumder, A., Kariwala, V., Ansumali, S., & Rajendran, A. (2011) · Chemical Engineering Science · doi:10.1016/j.ces.2011.10.051 · Cited by 38 · see citing works
Piskunov, V. N., & Голубев, А. И. (2002) · Journal of Aerosol Science · doi:10.1016/s0021-8502(01)00073-8 · Cited by 34 · see citing works
Fast High-Resolution Method for Solving Multidimensional Population Balances in Crystallization
Majumder, A., Kariwala, V., Ansumali, S., & Rajendran, A. (2010) · Industrial & Engineering Chemistry Research · doi:10.1021/ie9016946 · Cited by 32 · see citing works
Droplets population balance in a rotating disc contactor: An inverse problem approach
Vikhansky, A., Kraft, M., Simon, M., Schmidt, S., & Bart, H. ‐. (2005) · AIChE Journal · doi:10.1002/aic.10735 · Cited by 30 · see citing works
An adaptive direct quadrature method of moment for population balance equations
Su, J., Gu, Z., Li, Y., Feng, S., & Xu, X. Y. (2008) · AIChE Journal · doi:10.1002/aic.11599 · Cited by 24 · see citing works
Strumendo, M., & Arastoopour, H. (2008) · Industrial & Engineering Chemistry Research · doi:10.1021/ie800272a · Cited by 23 · see citing works
Raikar, N. B., Bhatia, S. R., Malone, M. F., & Henson, M. A. (2006) · Chemical Engineering Science · doi:10.1016/j.ces.2006.08.022 · Cited by 23 · see citing works
Bruns, M. C., & Ezekoye, O. A. (2012) · Journal of Aerosol Science · doi:10.1016/j.jaerosci.2012.07.003 · Cited by 21 · see citing works
Recovery of Distributions via Moments
Mnatsakanov, R. M., & Hakobyan, A. S. (2009) · Lecture notes-monograph series · doi:10.1214/09-lnms5715 · Cited by 21 · see citing works
Mickler, M., Didas, S., Jaradat, M., Attarakih, M. M., & Bart, H. (2011) · Chemie Ingenieur Technik · doi:10.1002/cite.201000156 · Cited by 19 · see citing works
Self-citationInferential closed-loop control of particle size distribution for styrene emulsion polymerization
Abedini, H., & Shahrokhi, M. (2008) · Chemical Engineering Science · doi:10.1016/j.ces.2008.01.034 · Cited by 18 · see citing works
The dual-quadrature method of generalized moments using automatic integration packages
Fávero, J. L., & Lage, P. L. C. (2011) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2011.11.010 · Cited by 18 · see citing works
Pipeline agglomerator design as a model test case
Diemer, R. B., & Ehrman, S. H. (2005) · Powder Technology · doi:10.1016/j.powtec.2005.04.016 · Cited by 14 · see citing works
Beyond log‐normal distributions: Hermite spectra for solving population balances
Hamilton, R. A., Curtis, J. S., & Ramkrishna, D. (2003) · AIChE Journal · doi:10.1002/aic.690490909 · Cited by 14 · see citing works
Gomes, L. N., Guimarães, M. L., Lopes, J. C. B., Madureira, C. M. N., Stichlmair, J., & Cruz-Pinto, J. J. C. (2004) · Industrial & Engineering Chemistry Research · doi:10.1021/ie030277i · Cited by 6 · see citing works
Rosner, D. E. (2006) · International Journal of Chemical Reactor Engineering · doi:10.2202/1542-6580.1269 · Cited by 6 · see citing works
Attarakih, M. M., & Bart, H. (2012) · Computer-aided chemical engineering/Computer aided chemical engineering · doi:10.1016/b978-0-444-59506-5.50057-2 · Cited by 5 · see citing works
Self-citationIntegral Formulation of the Population Balance Equation using the Cumulative QMOM
Attarakih, M. M., Jaradat, M., Hlawitschka, M. W., Bart, H. ‐., & Kuhnert, J. (2011) · Computer-aided chemical engineering/Computer aided chemical engineering · doi:10.1016/b978-0-444-53711-9.50017-1 · Cited by 4 · see citing works
Self-citation
Flagged first, then by citations
3 of the 60 references couldn’t be matched to a known paper.
Cited by
Sorted by citations
Kamp, J., Villwock, J., & Kraume, M. (2016) · Reviews in Chemical Engineering · doi:10.1515/revce-2015-0071 · Cited by 184 · see citing works
Ridder, B. J., Majumder, A., & Nagy, Z. K. (2014) · Industrial & Engineering Chemistry Research · doi:10.1021/ie402806n · Cited by 64 · see citing works
Attarakih, M. M., Hlawitschka, M. W., Abu-Khader, M. M., Alzyod, S., & Bart, H. (2015) · Applied Mathematical Modelling · doi:10.1016/j.apm.2015.04.006 · Cited by 57 · see citing works
Self-citationAkbari, V., Borhani, T. N. G., Shamiri, A., & Shafeeyan, M. S. (2024) · Experimental and Computational Multiphase Flow · doi:10.1007/s42757-023-0158-x · Cited by 54 · see citing works
Quadrature-based moment methods for the population balance equation: An algorithm review
Li, D., Li, Z., & Gao, Z. (2018) · Chinese Journal of Chemical Engineering · doi:10.1016/j.cjche.2018.11.028 · Cited by 52 · see citing works
Models for the Numerical Simulation of Bubble Columns: A Review
Mühlbauer, A., Hlawitschka, M. W., & Bart, H. (2019) · Chemie Ingenieur Technik · doi:10.1002/cite.201900109 · Cited by 49 · see citing works
Modeling 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
Population Balances for Extraction Column Simulations—An Overview
Bart, H., Jildeh, H. B., & Attarakih, M. M. (2019) · Solvent Extraction and Ion Exchange · doi:10.1080/07366299.2019.1691136 · Cited by 36 · see citing works
Self-citationPan, H., Liu, Q., & Luo, Z. (2018) · Powder Technology · doi:10.1016/j.powtec.2018.01.014 · Cited by 29 · see citing works
Attarakih, M. M., & Bart, H. (2014) · Chemical Engineering Science · doi:10.1016/j.ces.2013.12.031 · Cited by 28 · see citing works
Self-citationAkbari, V., Borhani, T. N. G., Aramesh, R., Hamid, M. K. A., Shamiri, A., & Hussain, M. A. (2015) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2015.07.001 · Cited by 22 · see citing works
Asadollahzadeh, M., Torkaman, R., & Torab‐Mostaedi, M. (2022) · Process Safety and Environmental Protection · doi:10.1016/j.cherd.2022.01.009 · Cited by 17 · see citing works
Attarakih, M., Bart, H., & Abu-Khader, M. M. (2019) · Chemical Engineering Science · doi:10.1016/j.ces.2019.115168 · Cited by 15 · see citing works
Self-citationTwo Analytical Approaches for Solution of Population Balance Equations: Particle Breakage Process
Hasseine, A., Senouci, S., Attarakih, M. M., & Bart, H. (2015) · Chemical Engineering & Technology · doi:10.1002/ceat.201400769 · Cited by 14 · see citing works
Self-citationAlzyod, S., Attarakih, M. M., & Bart, H. (2016) · Process Safety and Environmental Protection · doi:10.1016/j.cherd.2016.09.024 · Cited by 13 · see citing works
Self-citationA meshless Radial Basis Method (RBM) for solving the detailed population balance equation
Alzyod, S., & Charton, S. (2020) · Chemical Engineering Science · doi:10.1016/j.ces.2020.115973 · Cited by 12 · see citing works
On The Semi-Analytical Solution of Integro-Partial Differential Equations
Hasseine, A., Attarakih, M. M., Belarbi, R., & Bart, H. J. (2017) · Energy Procedia · doi:10.1016/j.egypro.2017.11.222 · Cited by 9 · see citing works
Self-citationPlana‐Fattori, A., Chantoiseau, E., Doursat, C., & Flick, D. (2013) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2013.01.020 · Cited by 4 · see citing works
Roy, N., Dürr, R., Bück, A., Kumar, J., & Sundar, S. (2021) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2021.08.015 · Cited by 4 · see citing works
Attarakih, M. M., & Bart, H. (2014) · Computer-aided chemical engineering/Computer aided chemical engineering · doi:10.1016/b978-0-444-63456-6.50151-4 · Cited by 3 · see citing works
Self-citationdos Santos, F. P., Lage, P. L. C., Fávero, J. L., & Senocak, I. (2019) · The Canadian Journal of Chemical Engineering · doi:10.1002/cjce.23697 · Cited by 2 · see citing works
InvQMOM: A Simple Inversion Method that Works
Attarakih, M. M., Fricke, A., & Bart, H. (2021) · Computer-aided chemical engineering/Computer aided chemical engineering · doi:10.1016/b978-0-323-88506-5.50085-1 · Cited by 1 · see citing works
Self-citationChaiken, J., Goodisman, J., & Kornilov, O. (2015) · The Journal of Physical Chemistry A · doi:10.1021/acs.jpca.5b00408 · Cited by 1 · see citing works
SQMOM Hydrodynamic Modelling of Kühni Columns for Reactive Extraction
Korb, C., Alzyod, S., Attarakih, M. M., & Bart, H. (2017) · Chemie Ingenieur Technik · doi:10.1002/cite.201700048 · Cited by 1 · see citing works
Self-citationGoodisman, J., & Chaiken, J. (2016) · International Journal of Chemical Kinetics · doi:10.1002/kin.21010
- Authors
- Attarakih, M.
- Affiliations
- University of Jordan (JO)
- Corresponding author
- Attarakih, Menwer
- ORCID
- Attarakih, Menwer
- Venue
- Computers & Chemical Engineering
- Publisher
- Elsevier BV
- Published
- 2013-01
- Volume
- 48
- Pages
- 1-13
- ISSN
- 0098-1354 (ISSN-L)
- OpenAlex
- W1990753109
- Cited by
- 25
- References
- 60
- Citations by year
- 2024: 1 · 2022: 2 · 2021: 2 · 2020: 1 · 2019: 4 · 2018: 2 · 2017: 2 · 2016: 3 · 2015: 4 · 2014: 3 · 2013: 1
- FWCI
- 2.54
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Computers & Chemical Engineering (publishedVersion)
- Funders
- Deutsche Forschungsgemeinschaft
- Topics
- Physical Sciences › Engineering › Computational Mechanics › Granular flow and fluidized beds
- Physical Sciences › Engineering › Biomedical Engineering › Fluid Dynamics and Mixing
- Physical Sciences › Engineering › Ocean Engineering › Particle Dynamics in Fluid Flows
- Keywords
- population balance equation, integral formulation, closure rules, meshless methods, particle growth, particulate systems, quadrature method of moments
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
