Population balance equation

Population balance equations (PBEs) have been introduced in several branches of modern science, mainly in branches with particulate entities.^[1] This includes topics like crystallization, liquid-liquid extraction, gas-liquid dispersions, liquid-liquid reactions, communition, aerosol engineering, biology (where the separate entities are cells), polymerization, etc. Population balance equations can be said to derived from the Boltzmann equation which describes a gas particle behavior. PBEs define how populations of separate entities develop in specific properties over time. They are a set of Integro-partial differential equations which gives the behavior of a population of particles from the analysis of behavior of single particle in local conditions.^[2] 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).

In demography the process of population change can be expressed in the balancing of the population equation. Example. p(2000) = P(1990) + B(1990-2000) - D(1990-2000) + I(1990-2000) + E(1990-2000) N.B : I= Immigration and E= Emigration, used for international migration.

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^[2]

$\frac{d}{dt} \int_{\Omega_x(t)} dV_x \int_{\Omega_r(t)} dV_r\,f(\bold{x},\bold{r},t) = \int_{\Omega_x(t)} dV_x \int_{\Omega_r(t)} dV_r\,h(\bold{x},\bold{r},\bold{Y},t)$

This is a generalized form of PBE.^[2]

Solution to PBE

Monte Carlo methods, discretization methods and moment methods ^{[3][4] [5]}are mainly used to solve these equations. The choice depends on the application and computing infrastructure.

References

1. http://como.cheng.cam.ac.uk/index.php?Page=Research&Section=Algorithms&Subsection=PBM
2. Ramkrishna, D.: Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, 2000
3. Description of Aerosol Dynamics by the Quadrature Method of Moments, Robert McGrawa, Aerosol Science and Technology, Volume 27, Issue 2, 1997, pages 255-265
4. 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.
5. 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.