Population balance equation

Population balance equations (PBEs) have been introduced in several branches of modern science, mainly in Chemical Engineering,[1] to describe the evolution of a population of particles. This includes topics like crystallization,[2] leaching (metallurgy),[3][4] liquid–liquid extraction, gas-liquid dispersions like water electrolysis,[5] liquid-liquid reactions, comminution, aerosol engineering, biology (where the separate entities are cells based on their size or intracellular proteins[6]), polymerization, etc. Population balance equations can be said to be derived as an extension of the Smoluchowski coagulation equation which describes only the coalescence of particles. PBEs, more generally, define how populations of separate entities develop in specific properties over time. They are a set of Integro-partial differential equations which gives the mean-field behavior of a population of particles from the analysis of behavior of single particle in local conditions.[7] Particulate systems are characterized by the birth and death of particles. For example, consider precipitation process (formation of solid from liquid solution) which has the subprocesses nucleation, agglomeration, breakage, etc., that result in the increase or decrease of the number of particles of a particular radius (assuming formation of spherical particles). Population balance is nothing but a balance on the number of particles of a particular state (in this example, size).

  1. ^ Totis, Niccolò; Nieto, César; Küper, Armin; Vargas-García, César; Singh, Abhyudai; Waldherr, Steffen (April 2021). "A Population-Based Approach to Study the Effects of Growth and Division Rates on the Dynamics of Cell Size Statistics". IEEE Control Systems Letters. 5 (2): 725–730. doi:10.1109/LCSYS.2020.3005069. ISSN 2475-1456. S2CID 220606401.
  2. ^ Hulburt, H.M.; Katz, S. (August 1964). "Some problems in particle technology". Chemical Engineering Science. 19 (8): 555–574. doi:10.1016/0009-2509(64)85047-8.
  3. ^ Bortot Coelho, Fabrício Eduardo; Balarini, Julio Cézar; Araújo, Estêvão Magno Rodrigues; Miranda, Tânia Lúcia Santos; Peres, Antônio Eduardo Clark; Martins, Afonso Henriques; Salum, Adriane (June 2020). "A population balance approach to predict the performance of continuous leaching reactors: Model validation in a pilot plant using a roasted zinc concentrate". Hydrometallurgy. 194: 105301. Bibcode:2020HydMe.19405301B. doi:10.1016/j.hydromet.2020.105301. S2CID 216301270.
  4. ^ Coelho, Fabrício Eduardo Bortot; Balarini, Julio Cézar; Araújo, Estêvão Magno Rodrigues; Miranda, Tânia Lúcia Santos; Peres, Antônio Eduardo Clark; Martins, Afonso Henriques; Salum, Adriane (January 2018). "Roasted zinc concentrate leaching: Population balance modeling and validation". Hydrometallurgy. 175: 208–217. Bibcode:2018HydMe.175..208C. doi:10.1016/j.hydromet.2017.11.013.
  5. ^ Bisang J.M., Colli A.N. (2022). "Current and Potential Distribution in Two-Phase (Gas Evolving) Electrochemical Reactors by the Finite Volume Method". Journal of the Electrochemical Society. 169 (3): 034524. Bibcode:2022JElS..169c4524C. doi:10.1149/1945-7111/ac5d90. S2CID 247463029.
  6. ^ Alhuthali, Sakhr; Fadda, Sarah; Goey, Cher H.; Kontoravdi, Cleo (2017-01-01). "Multi-stage population balance model to understand the dynamics of fed-batch CHO cell culture". In Espuña, Antonio; Graells, Moisès; Puigjaner, Luis (eds.). 27th European Symposium on Computer Aided Process Engineering. 27 European Symposium on Computer Aided Process Engineering. Vol. 40. Elsevier. pp. 2821–2826. doi:10.1016/B978-0-444-63965-3.50472-4. ISBN 9780444639653. {{cite book}}: |work= ignored (help)
  7. ^ Ramkrishna, D.: Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, 2000