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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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Sectional representations for simulating aerosol dynamics
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
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The Status of Population Balances
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A discrete-sectional model for particulate production by gas-phase chemical reaction and aerosol coagulation in the free-molecular regime
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Crystallization and precipitation engineering—I. An efficient method for solving population balance in crystallization with agglomeration
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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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“The population balance equation (PBE) for particles with both aggregation and breakup is given as (Kumar & Ramkrishna, 1996) ( , , ) − ( , , ) [ ( , ) − 3 ] − ( ( , , ) ) = 1 2 ∫ 0 ( − ′, , ) ( ′, ) ( − ′, , ′ ) ′ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ I:…”
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Background
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“(16) is an integro-differential equation and was solved by discretizing the size domain using the ﬁ xed pivot technique (Kumar and Ramkrishna, 1996).”
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“The problem is more serious for the collision–coagulation processes, which need to be handled with advanced numerical techniques (e.g., Drake, 1972; Tzivion et al., 1987; Landgrebe and Pratsinis, 1990; Chen and Lamb, 1994; Kumar and Ramkrishna, 1996b).”
Background
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Modified method of characteristics for solving population balance equations
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Numerical simulation of multi-size bubbly flow in a continuous casting mold using an inhomogeneous multiple size group model
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“Liang et al. (2016) used a different kind of sectioning of the PBM for aggregation of particles as, namely, the one proposed by Kumar and Ramkrishna (1996) to predict aggregation behavior of submicron-sized particles of praseodymium-doped zirconium silicate in aqueous suspension.”
Enhancing CO2 capture efficiency: Computational fluid dynamics investigation of gas-liquid vortex reactor configurations for process intensification
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“The problem is therefore characterized by a set of Nsec differential equations that has been fully described by Kumar and Ramkrishna [19].”
Microgel Formation in Emulsion Polymerization
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“…a pure 137Cs aerosol with volume 2x. Coagulation rates of the aerosols can be calculated using a monovariate population balance (PB) equation (Kumar & Ramkrishna, 1996): ∑ ∑= ⎛ ⎝ − ⎞ ⎠ − ≤ + ≤ ≥ = − + dN dt δ η α β N N α β N N1 1 2 k l m x x x x l m lm Kr lm lm lm l m l M kl kl k l , 1 k l m…”
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A quasi-Monte Carlo based flocculation model for fine-grained cohesive sediments in aquatic environments
Shen, X., Lin, M., Zhu, Y., Ha, H. K., Fettweis, M., Hou, T., Toorman, E. A., Maa, J. P. ‐., & Zhang, J. (2021) · Water Research · doi:10.1016/j.watres.2021.116953 · Cited by 17

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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