# Population Balance Equation

The population balance equation (PBE) is a balance equation based on the number density function * f (?;* x,

*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.*

**t),**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

*are particle-phase velocity and turbulent diffusivity, respectively, which generally are functions of time, location, and internal coordinates.*

**D**_{pt}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

*is the daughter-size distribution function on breakage of particles of size X (Marchisio et al. 2003a; Marchisio and Fox 2005).*

**b(? in)**