Moment-transport equations for a PBE

for simplicity, let us consider a PBE with only particle length as the internal coordinate, for which the NDE is l(T x, L). The PBE reported in Eq. (2.14) then becomes 

In the equations below, for simplicity, we will assume that the particle-growth rate Gl is size-independent so that Gl. = Gl. However, in general, the functional dependence of Gl on size must be provided by the modeler in order to apply Eq. (2.19). 

The zeroth moment Wl.o is equivalent to the total number concentration N and its transport equation follows fromEq. (2.18)  

As it is possible to see, the drift term has disappeared since the continuous growth of particle size does not change the total number concentration (if Gl 0). However, N is influenced by the rate of formation of particles (e.g. nucleation), and the rates of aggregation and breakage, which cause appearance and disappearance of particles. These processes are all contained in the source term /tL.o The third-order moment mL,3 is related to the fraction of volume occupied by particles with respect to the suspending fluid and can be easily found fromEq. (2.18)  

It is interesting to note that, if the drift term is null (i.e. particles are not growing in size) and if there is no introduction of new particles into the system (i.e. the nucleation rate is null ), the right-hand side of Eq. (2.23) is null, resulting in a continuity equation for the particle phase of the form 

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