Moment-transport equation from GPBE

The primary purpose of this chapter is to introduce the key concepts and notation needed to develop models for polydisperse multiphase flows. We thus begin with a general discussion of the number-density function (NDF) in its various forms, followed by example transport equations for the NDF with known (PBE) and computed (GPBE) particle velocity. These transport equations are written in terms of averaged quantities whose precise definitions will be presented in Chapter 4. We then consider the moment-transport equations that are derived from the NDE transport equation by integration over phase space. Einally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations. 

For all other cases, it will be necessary to solve the moment-transport equations derived from the GPBE as described in Chapter 4. In Chapter 8 the numerical algorithms used to find approximation solutions to the GPBE using quadrature-based moment methods are presented in detail. 

As before, the exact solution to Eq. (8.3) is n(t, x, v, s) = n(0, x - vt, v, s), from which we can observe that. y is indeed passive because it is simply carried along with velocity v. Or, in other words, the free-transport term in the GPBE can be solved separately for each value of. y. However, the moment-transport equations are coupled because they depend on both k and 1. Nevertheless, we can observe that if we consider a set of moments with k fixed, but I free, it is possible to use QBMM to represent the unclosed moments. For example, with A = 2 the moment-transport equations are... 

In this appendix we discuss the soiution of the moment-transport equations found from the GPBE using kinetics-based finite-voiume methods (KBFVM) in muitipie spatiai dimensions. As in Section 8.2, the discussion focuses on time advancement using a singie-stage Euier expiicit time-integration scheme. Readers interested in more detaiis on finite-voiume methods are referred to Leveque (2002). 

In Chapter 4 the GPBE is derived, highlighting the closures that must be introduced for the passage from the microscale to the mesoscale model. This chapter also contains an overview of the mathematical steps needed to derive the transport equations for the moments of the NDF from the GPBE. The resulting moment-closure problem is also throughly discussed. 

---