Moment closures for the GPBE

In this chapter, we have introduced the basic steps needed to derive the mesoscale model starting from a microscale description of the polydisperse multiphase flow. The first step is to identify a set of microscale variables, denoted here by (X  

The second example is the case in which the disperse-phase velocity is equal to the conditional expected disperse-phase velocity given the other mesoscale variables Vp = (Upl p, Vf, f). In this case, fluctuations in the disperse phase are slaved to the fluid-phase fluctuations and can depend on the particle internal coordinates (e.g. particle size). The NDF in this case is n(Vp, p, Vf, f) = n( p, Vf, f)(5(Vp - Up p, Vf, f and the GPBE reduces to 

The only exception is for the fluid mass seen by the particle, which was denoted earlier in this chapter as ff2-In the monokinetic fluid limit, we will let ff2 = where fp2 is the particle mass. 

A popular method for closing a system of moment-transport equations is to assume a functional form for the NDF in terms of the mesoscale variables. Preferably, the parameters of the functional form can be written in closed form in terms of a few lower-order moments. It is then possible to solve only the transport equations for the lower-order moments which are needed in order to determine the parameters in the presumed NDF. The functional form of the NDF is then known, and can be used to evaluate the integrals appearing in the moment-transport equations. As an example, consider a case in which the velocity NDF is assumed to be Gaussian  

In order to increase the number of degrees of freedom in a systematic manner, a functional expansion can be used to represent the NDF (Grad, 1949b). Using the velocity distribution as an example, the formal expansion is 

---

Moment closures for the GPBE Quadrature-based moment methods... 

---