Moment-transport equations for a GPBE

The very same approach as described above for the PBE can be applied to the GPBE (Eq. (2.16) on page 37). On applying the moment transform to Eq. (2.16) for the zeroth-order moment, the transport equation for the total number concentration is obtained  

This expression is the fundamental statement of the conservation of mass of the disperse phase. If the first-order particle-velocity moment transform is applied, we obtain 

Equation (2.28) is a transport equation for the mean particle velocity Up and has a structure very similar to the ones previously reported. The first term on the left-hand side represents accumulation, whereas the second one represents convection of velocity in physical space. The first term on the right-hand side represents the acceleration produced by a force acting on the particles, whereas the second term represents the change of velocity due to discrete events such as particle collisions. 

We note in passing that nearly all of the terms in Eq. (2.28) are unclosed at the moment level. We can also note that, for the case in which the particle density and size are constant, the relation = p kyL N will allow us to rewrite Eq. (2.28) as a balance for the disperse-phase momentum (ppUpUp)  

Likewise, the disperse-phase momentum is related to the particle mass by 

---