The DOI matches its Crossref record; 21 of the 21 references it cites were found in a registry or an index.
On the solution of population balance equations by discretization—I. A fixed pivot technique
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“Kumar and Ramkrishna [292] presented a new framework for the discretization of PBEs, considering particle populations in discrete size ranges to be concentrated only in representative volumes. One of the main advantages of this approach is that the coarseness of the discretization can be varied freely, without any specific geometric type, thus having a general and flexible pattern.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [292]
In the citing paper
“The technique has been applied to binary and multiple breakages with power-law breakage rates, aggregation with constant, sum, and product kernel, and simultaneous breakage and aggregation. Comparison between analytical [293-295] and numerical results show good agreement.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [292]
Citations
Cites
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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
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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
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“27), we gather some useful definitions and theorems from Linz (1975), Hundsdorfer & Verwer (2003) and Kumar et al.”
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BackgroundChallenges in Polymerization Reactor Modeling and Optimization: A Population Balance Perspective
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“…of the PSD for highly aggregating processes, numerical instabilities for growth-dominated processes, increased stiffness of the system of DAEs for processes involving rapid particle nucleation, and domain errors for high-order aggregation kernels (Kumar and Ramkrishna, 1996a,b ; Dafniotis, 1996).”
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“An adaptive refinement technique, as described by Kumar and Ramkrishna (1996), could increase accuracy without significant computational cost in future model implementations.”
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Particle size distribution by design
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“The fixed-pivot technique of Kumar and Ramkrishna (1996) is used for discretizing the coagulation term.”
MethodEffective rates of coalescence in oil–water dispersions under constant shear
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“There have been several numerical studes reported in the literature for the solution of the population alance equation (Kumar and Ramkrishna, 1996; Alexopoulos t al., 2004), whereas studies that consider PBE coupled with CFD”
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Analytical time-stepping solution of the discretized population balance equation
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Dynamic model for simulating transient behaviour of rotary drum granulation loop
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“The cell average scheme has shown higher accuracy and faster solution compared to other frequently used sectional methods as the Hounslow scheme (Hounslow et al., 1988) and the Fixed Pivot scheme (Kumar and Ramkrishna, 1996).”
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“…kernels), including the finite discretized population balances, extensions of the quadrature method of moments, method of weighted residuals, high-resolution finite volume methods, etc. (e.g., developed in Kumar and Ramkrishna, 1996; Koren, 1993; Marchisio and Fox, 2005, and many more).”
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“The discretization of coagulation integrals appearing in PBEs is a difficult task for which the fixed pivot technique presented by Kumar and Ramakrishna [29] has been implemented [30].”
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“The continuous PBE was discretized using the Fixed Pivot technique [31].”
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Metadata
- Authors
- Kumar, S., & Ramkrishna, D.
- Affiliations
- Purdue University West Lafayette (US)
- Corresponding author
- Ramkrishna, D.
- Venue
- Chemical Engineering Science
- Publisher
- Elsevier BV
- Published
- 1996-04
- Volume
- 51
- Issue
- 8
- Pages
- 1311-1332
- ISSN
- 0009-2509 (ISSN-L)
- 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
- 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
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 ›
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