Population balance equation


Population balance equations 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, leaching, liquid–liquid extraction, gas-liquid dispersions, liquid-liquid reactions, comminution, aerosol engineering, biology, 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.
Particulate systems are characterized by the birth and death of particles. For example, consider precipitation process which has the subprocesses nucleation, agglomeration, breakage, etc., that result in the increase or decrease of the number of particles of a particular radius. Population balance is nothing but a balance on the number of particles of a particular state.

Formulation of PBE

Consider the average number of particles with particle properties denoted by a particle state vector dispersed in a continuous phase defined by a phase vector Y is denoted by f. Hence it gives the particle characteristics in property and space domains. Let h denote the birth rate of particles per unit volume of particle state space, so the number conservation can be written as
This is a generalized form of PBE.

Solution to PBE

s
, discretization methods and moment methods are mainly used to solve these equations. The choice depends on the application and computing infrastructure.