Population balance equation

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[7]

${\displaystyle {\frac {d}{dt}}\int _{\Omega _{x}(t)}dV_{x}\int _{\Omega _{r}(t)}dV_{r}\,f(\mathbf {x} ,\mathbf {r} ,t)=\int _{\Omega _{x}(t)}dV_{x}\int _{\Omega _{r}(t)}dV_{r}\,h(\mathbf {x} ,\mathbf {r} ,\mathbf {Y} ,t)}$

This is a generalized form of PBE.[7]

Monte Carlo methods [8] ,[9] discretization methods[10] and moment methods [8][9][11][12][13][14] are mainly used to solve these equations. The choice depends on the application and computing infrastructure.[1]