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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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Gelbard, F., Tambour, Y., & Seinfeld, J. H. (1980) · Journal of Colloid and Interface Science · doi:10.1016/0021-9797(80)90394-x · Cited by 607
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
Ziff, R. M. (1991) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/24/12/020 · Cited by 187
The general dynamic equation for aerosols. Theory and application to aerosol formation and growth
Gelbard, F., & Seinfeld, J. H. (1979) · Journal of Colloid and Interface Science · doi:10.1016/0021-9797(79)90289-3 · Cited by 170
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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“Thus, the sectional approach of Kumar and Ramkrishna (1996) was used in this study to discretize the integral terms of Eq.”
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“In addition, many authors proposed different schemes to solve the aggregation PBE for granule size distribution including finite volume schemes [22, 31, 35, 37, 38, 40, 42], least square methods [3, 49], method of moments [1], stochastic methods [24], cell average techniques [21, 38] and fixed pivot techniques [20, 23…”
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“The present sectional method formulation closely follows the approach presented by Kumar and Ramkrishna (1996).”
Method
A comparative study of the fixed pivot technique and finite volume schemes for multi-dimensional breakage population balances
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“To solve this issue, (Kumar and Ramkrishna, 1996) showed 308 that some conditions can lead to the conservation of several quantities using only a reduced 309 set of pivots.”
Background
Accuracy and optimal sampling in M onte C arlo solution of population balance equations
Yu, X., Hounslow, M. J., & Reynolds, G. (2015) · AIChE Journal · doi:10.1002/aic.14837 · Cited by 12
“(2) Randomly select particle mj3 (row j3 of the array).”
BackgroundMethod
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An Analysis of Uncertainty Propagation Methods Applied to Breakage Population Balance
Bhonsale, S. S., Telen, D., Stokbroekx, B., & van Impe, J. F. M. (2018) · Processes · doi:10.3390/pr6120255 · Cited by 12
“The fixed pivot technique (FPT) of Kumar and Ramakrishna [21] is used in this study.”
Method
Efficient modeling of distributions of polymer properties using probability generating functions and parallel computing
Asteasuain, M. (2019) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2019.06.009 · Cited by 12
Comparison of a new mass-concentration, chain-reaction model with the population-balance model for early- and late-stage aggregation of shattered graphene oxide nanoparticles
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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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