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
Hounslow, M. J., Ryall, R. L., & Marshall, V. R. (1988) · AIChE Journal · doi:10.1002/aic.690341108 · Cited by 995
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
Self-citation
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
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
Self-citation
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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“Sectional methods, including the fixed pivot and moving pivot techniques developed by Kumar and Ramkrishna (1996), divide the particle size range into discrete sections, offering good accuracy at higher computational cost.”
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“…for the contribution of nucleation based on A4 dimerization, PAH condensation based on the collisions of soot particles and A4 molecules, coagulation based on [32], and surface chemistry (growth and oxidation of particles) through the H-Abstraction C 2 H 2 - Addition (HACA) mechanism [33, 34].”
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“Several approximate resolution methods, such as weighted residuals [1 - 7], successive approximations [12], finite elements techniques [9, 13 - 16], finite volume scheme [17 - 22], fixed and moving pivot approaches [23], Monte Carlo methods [24 - 26] and method of moments with these several extensions such as QMOM, DQ…”
Method
Developing a digital twin of a polymerization reaction for process optimization
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Numerical Solution Methods
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Fragmentation of polymer‐bridged silica flocs by high shear impact: experiment and population balance modelling
Au, P., & Leong, Y. (2015) · Asia-Pacific Journal of Chemical Engineering · doi:10.1002/apj.1889 · Cited by 2
Numerical Modeling and Analysis of Gas Entrainment for the Ventilated Cavity in Vertical Pipe
Xiang, M., Jiang, Z., Zhang, W., & Tu, J. Y. (2014) · Chinese Journal of Chemical Engineering · doi:10.1016/s1004-9541(14)60033-1 · Cited by 2
Prediction of Bivariate Molecular Property Distributions in Free-Radical Polymerization Systems Using Monte Carlo and Sectional Grid Methods
Meimaroglou, D., Krallis, A., & Kiparissides, C. A. (2008) · Chemical Product and Process Modeling · doi:10.2202/1934-2659.1221 · Cited by 2
“Moreover, based on the original work of Kumar and Ramkrishna (1996), a two-dimensional fixed pivot technique (2D-FPT) is developed for solving the resulting system of bivariate PBEs to calculate the bivariate LCB-MW and CCMW distributions, in free-radical polymerizations, up to very high monomer…”
Method
Crystallizers: CFD–PBE modeling
Yang, C., & Mao, Z. (2014) · Elsevier eBooks · doi:10.1016/b978-0-08-099919-7.00006-8 · Cited by 2
Sensitivity analysis of multi-regime population balance model for control of multiple particulate properties in granulation
Févotte, F., & Doyle, F. J. (2005) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2005.1470080 · Cited by 2
“In each bin, the particles are assumed to be uniformly distributed — in contrast to the majority of earlier methods, such as the pivot techniques presented in [9] and [10], where the granules are assumed to be concentrated at a single point in each bin.”
Result
Multi-scale modeling for prediction of distributed cellular properties in response to substrate spatial gradients in a continuously run microreactor
Fernandes, R. L., Krühne, U., Ingmar Nopens, A., Jensen, A. D., & Gernaey, K. V. (2012) · Computer-aided chemical engineering/Computer aided chemical engineering · doi:10.1016/b978-0-444-59507-2.50101-3 · Cited by 2
Comparison of the Accuracy and Performance of Different Numbers of Classes in Discretised Solution Method for Population Balance Model
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“Zhenliang Li,1,2 Zhien Zhou,2 Sheng Zhang,2 and Hongqiang Jiang3 1College of Urban Construction and Environmental Engineering, Chongqing University, Chongqing 400044, China 2Chongqing Research Academy of Environmental Sciences, Chongqing 401147, China 3State Environmental Protection Key Laboratory of Environmental Pla…”
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Tanaka, M., Hayashi, K., & Tomiyama, A. (2009) · TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series B · doi:10.1299/kikaib.75.759_2182 · Cited by 2
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“We adopted here the Fixed-Pivot Technique [20] to obtain the numerical solution of the SCE.”
Method
Variational iteration method and projection method solution of the spatially distributed population balance equation
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“The discrete form of the PBE with the FP scheme is then written as (Kumar and Ramkrishna, 1996)”
BackgroundMethodInfluential
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Nonlinear control of particulate processes with input constraints
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Size Enlargement
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“The theory and solution methods for general PBEs are detailed in several references, such as Hulburt and Katz [123], Randolph [124], Ramkrishna [125,126], Kumar and Ramkr-ishna [127,128].”
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“The population balances were solved using the discretization method of Kumar and Ramkrishna [ 41 ]. using 30 pivots.”
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“…PAH condensation, which accounts for collisions be-tween soot particles and A4 molecules; coagulation, modeled according to the work of Kumar et al. [36]; and surface chemistry, encompassing particle growth and oxidation through the H-Abstraction C 2 H 2 -Addition (HACA) mechanism [37, 38].”
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Integrators of several orders in time to study the evolution of an aerosol by coagulation
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“…be challenging and has motivated several researchers in this area for decades to develop specialised algorithms for solving PBE, for example Ramkrishna (1981), Hounslow et al.(1988), Litster et al. (1995), Kumar and Ramkrishna (1996), Nicmanis and Hounslow (1996), and Mahoney and Ramkrishna (2002).”
Background
On the Numerical Study of Bubbly Wakes Generated by Ventilated Cavity Using Population Balance Approach
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Aggregation and breakage rates in the flocculation of estuarine cohesive sediments
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“There are quite a few sectional methods developed aiming to alleviate the nonposi‐ tiveness and nonconservation problems found in earlier works [13], and to obtain better efficiency and flexibility, such as the fixed/moving pivot method [14, 15] and cell average method [16, 17].”
Method
Population balance model and calibration method for simulating the time evolution of floc size distribution of activated sludge flocculation
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Population balance modeling of synthesis of nanoparticles in aerosol flame reactors
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“1) was numerically solved by approximating the integral expression using the fixed pivot technique with 100 unequally-spaced node points [32].”
Method
A new approach to analyze the equilibrium and transient behaviors of particulate systems and the subsequent application to multiphase fluid systems
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Steady state population balance modelling of precipitation processes: Nucleation, growth and size-dependent agglomeration
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“19 can be found in Kumar and Ramkrishna (1996).”
Background
Multiphase Phenomena and Design of Liquid-Liquid Stirred Tanks
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Preliminary group classification and exact solutions of Smoluchowski equation with a source
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A bi-variate sectional moment projection method for particle population balance dynamics
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Adaptive moving pivot technique for growth dominated population balance equations
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Discretized n-Dimensional Population Balance for Agglomeration
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A combinatorial tool for monitoring flocculation processes: Using non-invasive measurements and hybrid deep learning assisted modelling
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Modeling Frameworks of Complex Separation Systems
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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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