Population balance equation

Population balance equations (PBEs) have been introduced in several branches of modern science, mainly in Chemical Engineering, to describe the evolution of a population of particles. This includes topics like crystallization[1], leaching (metallurgy)[2][3], liquid–liquid extraction, gas-liquid dispersions, liquid-liquid reactions, comminution, aerosol engineering, biology (where the separate entities are cells based on their size or intracellular proteins[4]), 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.[5] 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).

Formulation of PBE

Consider the average number of particles with particle properties denoted by a particle state vector (x,r) (where x corresponds to particle properties like size, density, etc. also known as internal coordinates and, r corresponds to spatial position or external coordinates) dispersed in a continuous phase defined by a phase vector Y(r,t) (which again is a function of all such vectors which denote the phase properties at various locations) is denoted by f(x,r,t). Hence it gives the particle characteristics in property and space domains. Let h(x,r,Y,t) denote the birth rate of particles per unit volume of particle state space, so the number conservation can be written as[5]

This is a generalized form of PBE.[5]

Solution to PBE

Monte Carlo methods [6] ,[7] discretization methods and moment methods [6][7][8][9][10][11] are mainly used to solve these equations. The choice depends on the application and computing infrastructure.

gollark: Yes, it's popular.
gollark: Node's method of async with tons of callbacks was awful. Promises are okay, but async requiring sprinkling of magic syntax bits everywhere is beelike.
gollark: Just because JS happened to be picked for clients. though. And now with WASM tons of stuff can do that.
gollark: And it just has such weird inconsistency. Also bad async handling.
gollark: Also the ecosystem gets deprecated every month or so.

References
  1. 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.
  2. 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. doi:10.1016/j.hydromet.2020.105301.
  3. 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. doi:10.1016/j.hydromet.2017.11.013.
  4. Alhuthali, Sakhr; Fadda, Sarah; Goey, Cher H.; Kontoravdi, Cleo (2017-01-01), Espuña, Antonio; Graells, Moisès; Puigjaner, Luis (eds.), "27th European Symposium on Computer Aided Process Engineering", Computer Aided Chemical Engineering, 27 European Symposium on Computer Aided Process Engineering, Elsevier, 40, pp. 2821–2826, doi:10.1016/B978-0-444-63965-3.50472-4, ISBN 9780444639653 |chapter= ignored (help)
  5. Ramkrishna, D.: Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, 2000
  6. Hashemian, N.; Armaou, A. (2016). "Simulation, model-reduction and state estimation of a two-component coagulation process". AIChE Journal. 62 (5): 1557–1567. doi:10.1002/aic.15146.
  7. Hashemian, N.; Ghanaatpishe, M.; Armaou, A. (2016). Development of a reduced order model for bi-component granulation processes via laguerre polynomials. Proceedings of the American Control Conference. pp. 3668–3673. doi:10.1109/ACC.2016.7525483. ISBN 978-1-4673-8682-1.
  8. Description of Aerosol Dynamics by the Quadrature Method of Moments, Robert McGrawa, Aerosol Science and Technology, Volume 27, Issue 2, 1997, pages 255-265
  9. Yu, M., Lin, J., and Chan, T. (2008). A New Moment Method for Solving the Coagulation Equation for Particles in Brownian Motion. Aerosol Sci. Technol., 42(9):705–713.
  10. Marchisio, D. L., and Fox, R. O. (2005). Solution of Population Balance Equa- tions Using the Direct Quadrature Method of Moments. J. Aerosol Sci., 36(1):43–73.
  11. Andalibi, M. Reza; Kumar, Abhishek; Srinivasan, Bhuvanesh; Bowen, Paul; Scrivener, Karen; Ludwig, Christian; Testino, Andrea (2018). "On the mesoscale mechanism of synthetic calcium–silicate–hydrate precipitation: a population balance modeling approach". Journal of Materials Chemistry A. 6 (2): 363–373. doi:10.1039/C7TA08784E. ISSN 2050-7488.
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