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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On the prediction of the phase distribution of bubbly flow in a horizontal pipe
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“Assuming that each bubble class travel at the common velocity as in DQMOM, the number density equation Kumar and Ramkrishna (1996) can be expressed as ∂ni ∂t + ∇ · (ugni) = BC + BB − DC − DB (36) In a form consistent with the variables used in two-fluid model, the transport equation in terms of…”
Method
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“[6, 7] compared experimental bulk polymerization results with two-dimensional fixed pivot technique [8] and Monte Carlo method [3].”
Result
Simulation of aggregating particles in complex flows by the lattice kinetic Monte Carlo method
Flamm, M. H., Sinno, T. R., & Diamond, S. L. (2011) · The Journal of Chemical Physics · doi:10.1063/1.3521395 · Cited by 30
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“This error could be reduced through either increasing the number of subdomains or by re;ning the grid on the upper size range (Kumar and Ramkrishna, 1996a), where the former one is adopted here (Mx = 50).”
BackgroundMethodInfluential
A moment projection method for population balance dynamics with a shrinkage term
Wu, S., Yapp, E. K. Y., Akroyd, J., Mosbach, S., Xu, R. Q., Yang, W., & Kraft, M. (2016) · Journal of Computational Physics · doi:10.1016/j.jcp.2016.10.030 · Cited by 30
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“Cheng et al. [10] applied the discretized population balance (DPB) scheme of Kumar and Ramkrishna [31] to investigate the precipitation of BaSO4 crystals in a stirred tank.”
Method
A new framework for population balance modeling of spray fluidized bed agglomeration
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Monitoring of Batch Industrial Crystallization with Growth, Nucleation, and Agglomeration. Part 1: Modeling with Method of Characteristics
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CFD modeling of the hydrodynamic characteristics of a bubble column in different flow regimes
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Mathematical Modeling of Bivariate Polymer Property Distributions Using 2D Probability Generating Functions, 1 – Numerical Inversion Methods
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Accurate and efficient solution of bivariate population balance equations using unstructured grids
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Modified Method of Characteristics for Simulating Microscale Energy Transport
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Dynamic evolution of the particle size distribution in suspension polymerization reactors: A comparative study on Monte Carlo and sectional grid methods
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“According to the original developments of Kumar and Ramkrishna (1996a), the droplet/particle volume domain was initially discretized into a number of NE elements (i.e., (V i , V i+1 ), i = 1, 2, . . . , NE v ) and the overall particle population (i.e., N i , i = 1, 2, . . . , NE v ) was assigned to…”
Method
Solution of bivariate population balance equations with high-order moment-conserving method of classes
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Solution of the bivariate dynamic population balance equation in batch particulate systems: Combined aggregation and breakage
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Influence of Dispersed Phase Viscosity on Drop Coalescence in Turbulent Flow
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CFD Modeling with Experimental Validation of the Internal Hydrodynamics in a Pilot-Scale Slurry Bubble Column Reactor
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“Assuming that each bubble class travels at the same mean algebraic velocity, the individual number density of a bubble class i can be expressed as (Kumar and Ramkrishna 1996): ∂ni ∂t +∇ uGnið Þ = X J RJ ! i =BC +BB −DC −DB (23) Where P J RJ ! i represents the net change in the number density…”
Background
Eulerian modeling of inertial and diffusional aerosol deposition in bent pipes
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“Moreover, the sectional formulation spanning many decades in droplet sizes allows for a straight-forward extension to include aerosol processes such as nucleation, condensation, evaporation and coagulation or break-up, as was done before (see [25,26,29] ).”
Background
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“The source term is discretized using a fixed pivot technique (Kumar and Ramkrishna, 1996) that allows the discrete equation to be internally consistent with two selected integral properties of the DSD.”
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Flow-Induced Aggregation and Breakup of Particle Clusters Controlled by Surface Nanoroughness
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Advances in numerical methods for the solution of population balance equations for disperse phase systems
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“The experimental hypothesis of a complex coupling between aggregation and breakage processes was confirmed by the numeric approach based on the discretized Kumar and Ramkrishna (1996) solving method of population balance.”
MethodInfluential
Modelling and simulation of suspension polymerization of vinyl chloride via population balance model
Bárkányi, Á., Németh, S., & Lakatos, B. G. (2013) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2013.06.008 · Cited by 25
“Among them are the methods of classes (Kumar & Ramkrishna, 1996a, 1996b; Vanni, 2000), the quadrature method of moments (Marchisio, Vigil, & Fox, 2003; McGraw & Wright, 2003), the direct quadrature method of moments (Marchisio & Fox, 2005) and the standard method of moments using the…”
Method
AN IMPROVED MOVING SECTIONAL AEROSOL MODEL OF SOOT FORMATION IN A PLUG FLOW REACTOR
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“The second method (Model II) uses the classical sectional description of the aerosol dynamics equations based on the fixed pivot approach (Kumar and Ramkrishna, 1996a; Park and Rogak, 2004).”
Method
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The ultimate goal of modeling—Simulation of system and plant performance
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“{ γixi + γi+1xi+1 = ν γi + γi+1 = 1 (15) Ramkrishna [50] has also reported the birth in class i due to coalescence in this way: Bi,agr = n ∑ k=0 n ∑ j=0 [ θ(xi−1 xj + xk xi)× (1− 1 2 δjk) ] × γi−1(xj + xk)β(xk, xj) 36α2g π2 f j fk (djdk) 3 + n ∑ k=0 n ∑ j=k [ θ(xi xj + xk xi+1)× (1− 1 2 δjk) ] × γi(xj + xk)β(xk, xj) 3…”
BackgroundInfluential
Experimental Characterization and Mathematical Modeling of Breakage of Needle-like Crystals in a Continuous Rotor-Stator Wet Mill
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“By using the concept of the combined Method of Characteristics with the discretization techniques proposed by Kumar and Ramkrishna (1996) the general PBE with growth and aggregation (equation 14) can be expressed as the following ODE system: ∑∑ = ≥≥ ≤≤ −−= +− M k kiki kji LLL kj kjkjkj i NNNN…”
BackgroundMethod
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Coalescence in Creaming Emulsions. Existence of a Pure Coalescence Zone
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Self-citation
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“Hereafter, they proposed an alternative discrete formulation of the PBE for the binary breakup, which is better than previous models made by Kumar and Ramkrishna [36].”
MethodInfluential
Self-similar inverse population balance modeling for turbulently prepared batch emulsions: Sensitivity to measurement errors
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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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