A discretized population balance for nucleation, growth, and aggregation
Abstract
Abstract The population balance for batch aggregation of particulate suspensions is recast in a form that may be solved simply and accurately. The transformed equation is deduced with the introduction of only one additional parameter, which is found to be a constant for all cases. The transformed equation is tested by comparison with some analytical solutions with which it is found to be in excellent agreement. In particular, the equation is shown to predict correctly the rate of change of total particle number and volume. Compatible descriptions of linear growth and nucleation are developed with similar success. The method is then applied to modeling the in vitro growth and aggregation of kidney stones (calcium oxalate monohydrate crystals). It is found that these phenomena are well described by McCabe's Δ L law, a size‐independent coalescence kernel, and first‐order kinetics. Simulated particle size distributions and their moments are in excellent agreement with the experimental results.
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Hounslow et al. [287] discretized the particle size domain of a PBE using a geometric progression with 3√2 intervals to simulate batches in-vitro kidney stone trials, considering aggregation, growth, and nucleation.
“In one of the earliest models, Hounslow et al. [287] discretized the particle size domain of a PBE using a geometric progression with 3 √ 2 intervals to simulate batches in-vitro kidney stone trials, considering aggregation, growth, and nucleation.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [287]
Lister et al. [289] extended and improved Hounslow et al. [287] using an adjustable discretization of the domain size of the form r_{i+1}/r_i = 2^{1/q}.
“Lister et al. [289] extended and improved the previous work of Hounslow et al. [287] using an adjustable discretization of the domain size, that assumes the form of r i + 1 / r i = 2 1 q .”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [287]
Hounslow et al. [287] compared numerical results with analytical solutions of Gelbard and Seinfeld [288] for constant and size-dependent coalescence kernels, then with experimental data, finding very good agreement.
“First, they compared numerical results with the analytical solutions provided by Gelbard and Seinfeld [288] for the case of a constant coalescence and a size-dependent coalescence kernel. Afterward, they confronted the numerical results with experimental data concerning particle size and moment distributions, finding a very good agreement.”
From Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review (2022) · cites this paper as [287]
Cites
Some problems in particle technology
Hulburt, H. M., & Katz, S. (1964) · Chemical Engineering Science · doi:10.1016/0009-2509(64)85047-8 · Cited by 1,670 · see citing works
Theory of Particulate Processes
(1971) · Elsevier eBooks · doi:10.1016/b978-0-12-579650-7.x5001-5 · Cited by 478 · see citing works
The Status of Population Balances
Ramkrishna, D. (1985) · Reviews in Chemical Engineering · doi:10.1515/revce.1985.3.1.49 · Cited by 393 · see citing works
Numerical solution of the dynamic equation for particulate systems
Gelbard, F., & Seinfeld, J. H. (1978) · Journal of Computational Physics · doi:10.1016/0021-9991(78)90058-x · Cited by 337 · see citing works
Calcium Metabolism in Renal Failure and Nephrolithiasis.
(1978) · Annals of Internal Medicine · doi:10.7326/0003-4819-88-3-443_2 · Cited by 290 · see citing works
Marchal, P., David, R., Klein, J. P., & Villermaux, J. (1988) · Chemical Engineering Science · doi:10.1016/0009-2509(88)87126-4 · Cited by 257 · see citing works
Ryall, R. L., Harnett, R. M., & Marshall, V. R. (1981) · Clinica Chimica Acta · doi:10.1016/0009-8981(81)90458-7 · Cited by 194 · see citing works
Self-citationCoalescence Model for Granulation
Kapur, P. C., & Fuerstenau, D. W. (1969) · Industrial & Engineering Chemistry Process Design and Development · doi:10.1021/i260029a010 · Cited by 138 · see citing works
Sastry, K. V. S. (1975) · International Journal of Mineral Processing · doi:10.1016/0301-7516(75)90021-6 · Cited by 111 · see citing works
Crystal Growth in Aqueous Solutions1: I—Theory
McCabe, W. L. (1929) · Industrial & Engineering Chemistry · doi:10.1021/ie50229a008 · Cited by 89 · see citing works
PELLETIZING KINETICS AND SIMULATION OF FULL SCALE BALLING CIRCUITS.
Batterham, R., Hall, J., & Barton, G. J. (1981) · Cited by 74 · see citing works
Hartel, R. W., & Randolph, A. D. (1986) · AIChE Journal · doi:10.1002/aic.690320716 · Cited by 66 · see citing works
Drach, G. W., Randolph, A. D., & Miller, J. D. (1978) · The Journal of Urology · doi:10.1016/s0022-5347(17)57395-4 · Cited by 38 · see citing works
Ryall, R. G., Ryall, R. L., & Marshall, V. R. (1986) · Journal of Crystal Growth · doi:10.1016/0022-0248(86)90373-8 · Cited by 9 · see citing works
Self-citationModelling the Formation of Urinary StonesPreprint
Hounslow, M., Ryall, R., & Marshall, V. R. (1988) · Chemeca 88: Australia's Bicentennial International Conference for the Process Industries; Preprints of Papers · Cited by 9 · see citing works
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On the solution of population balance equations by discretization—I. A fixed pivot technique
Kumar, S., & Ramkrishna, D. (1996) · Chemical Engineering Science · doi:10.1016/0009-2509(96)88489-2 · Cited by 1,109 · see citing works
Quadrature method of moments for population-balance equations
Marchisio, D. L., Pikturna, J. T., Fox, R. O., Vigil, R. D., & Barresi, A. A. (2003) · AIChE Journal · doi:10.1002/aic.690490517 · Cited by 461 · see citing works
Adjustable discretized population balance for growth and aggregation
Lister, J. D., Smit, D. J., & Hounslow, M. J. (1995) · AIChE Journal · doi:10.1002/aic.690410317 · Cited by 366 · see citing works
Self-citationApproximate Population Balance Equations for Aggregation–Breakage Processes
Vanni, M. (2000) · Journal of Colloid and Interface Science · doi:10.1006/jcis.1999.6571 · Cited by 244 · see citing works
Buhse, T., Cruz, J.-M., Noble-Terán, M. E., Hochberg, D., Ribó, J. M., Crusats, J., & Micheau, J.-C. (2021) · Chemical Reviews · doi:10.1021/acs.chemrev.0c00819 · Cited by 189 · see citing works
Gupta, A., & Roy, S. (2013) · Chemical Engineering Journal · doi:10.1016/j.cej.2012.11.012 · Cited by 125 · see citing works
Attarakih, M. M., Bart, H.-J., & Faqir, N. M. (2004) · Chemical Engineering Science · doi:10.1016/j.ces.2004.03.005 · Cited by 93 · see citing works
Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review
Luzi, G., & McHardy, C. (2022) · Energies · doi:10.3390/en15113966 · Cited by 44 · see citing works
- Authors
- Hounslow, M. J., Ryall, R. L., & Marshall, V. R.
- Affiliations
- The University of Adelaide (AU)
- Flinders Medical Centre (AU)
- ORCID
- Hounslow, M. J.
- Venue
- AIChE Journal
- Publisher
- Wiley
- Published
- 1988-11
- Volume
- 34
- Issue
- 11
- Pages
- 1821-1832
- ISSN
- 0001-1541 (ISSN-L)
- 1547-5905
- OpenAlex
- W2127025276
- Cited by
- 995
- References
- 15
- Citations by year
- 2026: 23 · 2025: 20 · 2024: 25 · 2023: 19 · 2022: 25 · 2021: 42 · 2020: 38 · 2019: 43 · 2018: 34 · 2017: 32 · 2016: 27 · 2015: 36 · 2014: 27 · 2013: 36 · 2012: 33
- FWCI
- 2.61
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- AIChE Journal (publishedVersion)
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Crystallization and Solubility Studies
- Physical Sciences › Engineering › Ocean Engineering › Particle Dynamics in Fluid Flows
- Physical Sciences › Engineering › Computational Mechanics › Granular flow and fluidized beds
- Keywords
- population balance, aggregation, nucleation, particle growth, particle suspensions, particle size distribution, coalescence kernel, calcium oxalate monohydrate crystals, kidney stones
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