Cited by 1,109Granular flow and fluidized bedsFluid Dynamics and MixingMineral Processing and Grinding

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On the solution of population balance equations by discretization—I. A fixed pivot technique

Kumar, Sanjeev, Ramkrishna, D.·1996·Chemical Engineering Science

DOI 10.1016/0009-2509(96)88489-2

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A discretized population balance for nucleation, growth, and aggregation
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The kinetics of cluster fragmentation and depolymerisation
Ziff, R. M., & McGrady, E. D. (1985) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/18/15/026 · Cited by 397
The Status of Population Balances
Ramkrishna, D. (1985) · Reviews in Chemical Engineering · doi:10.1515/revce.1985.3.1.49 · Cited by 393
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A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime
Landgrebe, J. D., & Pratsinis, S. E. (1990) · Journal of Colloid and Interface Science · doi:10.1016/0021-9797(90)90445-t · Cited by 287
Crystallization and precipitation engineering—I. An efficient method for solving population balance in crystallization with agglomeration
Marchal, P., David, R., Klein, J. P., & Villermaux, J. (1988) · Chemical Engineering Science · doi:10.1016/0009-2509(88)87126-4 · Cited by 257
New solutions to the fragmentation equation
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The general dynamic equation for aerosols. Theory and application to aerosol formation and growth
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Note on the Kinetics of Systems Manifesting Simultaneous Polymerization-Depolymerization Phenomena
Blatz, P. J., & Tobolsky, A. V. (1945) · The Journal of Physical Chemistry · doi:10.1021/j150440a004 · Cited by 169

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“In a recent paper, Kumar and Warnecke [5], referred to as part-I, convergence has been investigated for the fixed pivot technique [6].”
Background
Extended cell average technique for the solution of coagulation equation
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“(23) is quite similar to the mathematical definition in the CM in which the NDF is defined by [152] n where N i is the number of particles with diameter d i .”
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“…body of literature on the development of solution methods and the understanding of dynamics of population balance models ŽFriedlander, 1977; Gelbard and Seinfeld, 1978; Ramkrishna, 1985; Hill and Ng, 1996; Jerauld et al., 1983; Rawlings and Ray, 1987; Kumar and Ramkrishna, 1996a,b; Nicmanis and .”
Method
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“Different techniques to solve the aggregation population balance equation with one property coordinate have been proposed, including Monte Carlo methods [8], finite difference methods [9], finite element methods [10–13], finite volume schemes [14, 15], Galerkin’s method [16], method of moments [17–20], Adomian decompo…”
Method
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“…Tomiyama, Kataoka, Zun, & Sakaguchi, 1998), (b) bubble size distribution modeling (Hagesaether, Jakobsen, & Svendsen, 2002; Haringa et al., 2017; Kumar & Ramkrishna, 1996; Laakkonen, Moilanen, Alopaeus, & Aittamaa, 2007b), (c) modeling of bubble breakage and coalescence (Alopaeus, Koskinen, &…”
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MOMENT PRESERVING FINITE VOLUME SCHEMES FOR SOLVING POPULATION BALANCE EQUATIONS INCORPORATING AGGREGATION, BREAKAGE, GROWTH AND SOURCE TERMS
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“, 2009; Vishwanathan and Reitz, 2009, 2015), and comprehensive models (Frenklach and Wang, 1991; Kazakov and Frenklach, 1998; Kazakov et al., 1995; Kumar and Ramkrishna, 1996; Wen et al., 2005).”
Background
Dynamic prediction of the bivariate molecular weight–copolymer composition distribution using sectional-grid and stochastic numerical methods
Krallis, A., Meimaroglou, D., & Kiparissides, C. A. (2008) · Chemical Engineering Science · doi:10.1016/j.ces.2008.05.047 · Cited by 47
A conservative method for numerical solution of the population balance equation, and application to soot formation
Liu, A., & Rigopoulos, S. (2019) · Combustion and Flame · doi:10.1016/j.combustflame.2019.04.019 · Cited by 46
Particle Size Distributions via Mechanism-Enabled Population Balance Modeling
Handwerk, D. R., Shipman, P. D., Whitehead, C. B., Özkâr, S., & Finke, R. G. (2020) · The Journal of Physical Chemistry C · doi:10.1021/acs.jpcc.9b11239 · Cited by 45
Part III: Dynamic evolution of the particle size distribution in batch and continuous particulate processes: A Galerkin on finite elements approach
Roussos, A. I., Alexopoulos, A. H., & Kiparissides, C. A. (2005) · Chemical Engineering Science · doi:10.1016/j.ces.2005.06.021 · Cited by 46
A population model for crystallization processes using two independent particle properties
Gerstlauer, A., Mitrović, A., Motz, S., & Gilles, E. D. (2001) · Chemical Engineering Science · doi:10.1016/s0009-2509(00)00448-6 · Cited by 46
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
Modeling electrostatic separation for dehydration and desalination of crude oil in an industrial two-stage desalting plant
Aryafard, E., Farsi, M., Rahimpour, M. R., & Raeissi, S. (2015) · Journal of the Taiwan Institute of Chemical Engineers · doi:10.1016/j.jtice.2015.06.028 · Cited by 45
Numerical simulation of two-phase flow with bubble break-up and coalescence coupled with population balance modeling
Sattar, M. A., Naser, J., & Brooks, G. A. (2013) · Chemical Engineering and Processing - Process Intensification · doi:10.1016/j.cep.2013.05.006 · Cited by 46
Modeling of bubble size distribution in isothermal gas–liquid flows: Numerical assessment of population balance approaches
Cheung, S. C. P., Deju, L., Yeoh, G. H., & Tu, J. Y. (2013) · Nuclear Engineering and Design · doi:10.1016/j.nucengdes.2013.08.049 · Cited by 47
An accurate and efficient discrete formulation of aggregation population balance equation
Tsotsas, E., Kaur, G., & Kumar, J. (2016) · Kinetic and Related Models · doi:10.3934/krm.2016.9.373 · Cited by 45
“Several numerical schemes based on finite volume methods [3, 31, 7, 4, 8, 13], finite element methods [23, 21, 28], the method of moments [2, 20, 22, 1], sectional approaches [9, 11, 19, 32, 17, 18, 15, 12, 26, 16] have been developed for solving one dimensional aggregation population balance equation.”
Method
Simulation, estimation and control of size distribution in aerosol processes with simultaneous reaction, nucleation, condensation and coagulation
Kalani, A., & Christofides, P. D. (2002) · Computers & Chemical Engineering · doi:10.1016/s0098-1354(02)00032-7 · Cited by 45
“…is to discretize the population balance equation using finite difference/element methods (see, for example, Hounslow, 1990; Hill & Ng, 1995, 1996; Kumar & Ramkrishna, 1996a,b; Nicmanis & Hounslow, 1998; Mantzaris, Daoutidis & Srienc, 2001 and the review paper Ramkrishna, 1985), but these methods…”
Method
Experimental Investigation into Dynamics and Stability of Continuous MSMPR Agglomerative Precipitation of CaCO3 Crystals
Wójcik, J., & Jones, A. G. (1997) · Process Safety and Environmental Protection · doi:10.1205/026387697523516 · Cited by 45

Authors: Kumar, S., & Ramkrishna, D.
Affiliations: Purdue University West Lafayette (US)
Corresponding author: Ramkrishna, D.
ORCID: Kumar, Sanjeev
Ramkrishna, D.
Venue: Chemical Engineering Science
Publisher: Elsevier BV
Published: 1996-04
Volume: 51
Issue: 8
Pages: 1311-1332
ISSN: 0009-2509 (ISSN-L)
DOI: 10.1016/0009-2509(96)88489-2
OpenAlex: W4249613604
Cited by: 1,109
References: 21
Citations by year: 2026: 55 · 2025: 50 · 2024: 36 · 2023: 48 · 2022: 47 · 2021: 61 · 2020: 52 · 2019: 67 · 2018: 51 · 2017: 48 · 2016: 44 · 2015: 43 · 2014: 55 · 2013: 57 · 2012: 29
FWCI: 7.75
Type: journal article
Language: en
Open access: closed
Licence: www.elsevier.com/tdm/userlicense/1.0/ (tdm)
www.elsevier.com/legal/tdmrep-license (tdm)
Locations: Chemical Engineering Science (publishedVersion)
Topics: Physical Sciences › Engineering › Computational Mechanics › Granular flow and fluidized beds
Physical Sciences › Chemical Engineering › Fluid Flow and Transfer Processes › Fluid Dynamics and Mixing
Physical Sciences › Engineering › Mechanical Engineering › Mineral Processing and Grinding
Keywords: population balance equation, discretization, aggregation, breakage, mesh refinement, representative volume element, aerosol dynamics

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On the solution of population balance equations by discretization—I. A fixed pivot technique

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