The population balance equation (PBE) is a balance equation based on the number density function f (?; x, t), where ? and x are internal and external coordinates, respectively. PBE accounts for the spatial and temporal evolutions of the number density function in a single control volume. Depending on the system of interest, the number density function f(?; x, t) may have only one internal coordinate (i.e., particle size) or multiple coordinates, such as particle size and surface area (March- isio et al. 2003a). Here we consider only a univariate system with the particle size (?) being the only internal coordinate.

For an inhomogeneous particulate system, the general governing equation becomes

The terms on the left-hand side are the accumulation term, convective term with respect to the external coordinate, convective term with respect to the internal

д?.

coordinate, and diffusive term, respectively. In the third term, is the flux in ?-space (Marchisio et al. 2003a) or, in other words, the growth rate of the internal variable ? (e.g., size). v_{p} and D_{pt} are particle-phase velocity and turbulent diffusivity, respectively, which generally are functions of time, location, and internal coordinates.

The source term h(?; x, t) on the right-hand side accounts for the net rate of introduction of new particles into the system. It assumes that aggregation/coalescence and breakage are the only mechanisms causing birth and death of particles or droplets in the system. The aggregation/coalescence source term could be written in the form of the right-hand side of the classical Smoluchowski equation (Smoluchowski 1917):

On the right-hand side of Eq. (4.2), the first term accounts for birth of particles with size ? due to aggregation or coalescence of two smaller particles with size ? — n and n; the second term represents the death of particles with size ? due to aggregation or coalescence with particles of all other sizes. в is the aggregation kernel, which gives the frequency that particles of size ? — n and n collide to form particles of size ?. Aggregation/coalescence usually depends on particle-particle interactions, local shear rate, and fluid-particle properties.

Similarly, the net rate of introduction of new particles of size ? into the system due to breakage can be defined as

where a is the breakage kernel which gives the rate of breakage of a particle of a certain size and b(? in) is the daughter-size distribution function on breakage of particles of size X (Marchisio et al. 2003a; Marchisio and Fox 2005).