A few words about phase-space integration

Computing moments from Eq. (4.39) basically involves integration over some of the independent variables in the NDF (i.e. Vp, p, Vf, and f) while keeping the others fixed (i.e. t and x). In order to facilitate this task, it is useful to review some basic rules of integration. 

The first term on the right-hand side of this equation is a total derivative, so it can be integrated formally to find 

The first term on the right-hand side of this equation is a total derivative, so it can be integrated formally. However, for internal coordinates the phase space does not usually extend to infinity. In order to see clearly what can happen, consider the case with a single internal coordinate that is bounded by zero and infinity  

The two terms on the right-hand side are the flux at infinity, which we can safely set to zero, and the flux at the origin. Depending on the forms of Gp)i and g, the flux at the origin need not be zero. For example, if g is nonzero when p = 0, then the product (Gp)in evaluated at p =0 would have to be zero in order for the flux term to cancel out. Unfortunately, there are important applications in which the flux term is nonzero, so one must pay attention to how the flux term is handled in the derivation of the moment-transport equations. For example, if p represents the surface of evaporating droplets and (Gp)i is constant (i.e. the evaporation rate is proportional to the surface area), then n will be nonzero at p = 0. Physically, the nonzero flux is due to the disappearance of droplets due to evaporation, and thus it cannot be neglected. 

In summary, computing the moment-transport equations starting from Eq. (4.39) involves integration over phase space using the mles described above for particular choices of g. In the following, we will assume that the flux term at the boundary of phase space can be neglected. However, the reader should keep in mind that this assumption must be verified for particular cases. 

---