Population balance equation

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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).