The goal of this chapter is to provide a relatively rigorous derivation of the generalized population-balance equation (GPBE) starting from the microscale description of a disperse multiphase system. We begin by defining the number-density function (NDF) for a system of discrete particles using a probability-density-function (PDF) approach. Once the NDF has been defined, we proceed to the derivation of the GPBE by introducing the concept of conditional expected values. The latter contain the mesoscale representation of the microscale physics and thus contain the mesoscale models needed to close the GPBE. Next, we provide a detailed explanation of how the transport equations are found from the GPBE for selected multivariate moments of interest in later chapters. The chapter concludes with a short description of moment closures in the context of moment-transport equations.

Particle-based definition of the NDF

In the particle-based definition of the NDF, we begin at the microscale and write a dynamic equation for the rate of change of the disperse-phase particle properties at the mesoscale. The simplest system, which we consider first, is a collection of interacting particles in a vacuum wherein the particles interact through collisions and short-range forces. Such a system is referred to as a granular system. We then consider a disperse two-phase system, wherein the particles are dispersed in a fluid.