In this chapter, the governing equations needed to describe polydisperse multiphase flows are presented without a rigorous derivation from the microscale model. (See Chapter 4 for a complete derivation.) For clarity, the discussion of the governing equations in this chapter will be limited to particulate systems (e.g. crystallizers, fluidized beds, and aerosol processes). However, the reader familiar with disperse multiphase flow modeling will recognize that our comments hold in a much more general context. Indeed, the extension of the modeling concepts developed in this chapter to many other multiphase systems is straightforward, and will be discussed in later chapters.

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 NDF transport equation by integration over phase space. Finally, we briefly describe how turbulence modeling can be undertaken starting from the moment-transport equations.

Number-density functions (NDF)

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities).