In the previous chapter, turbulence modeling issues have been discussed in the context of engineering computations. In this chapter, we will employ the concept of volume averaging to tackle scale disparity in complex transport phenomena. Specifically, the formulation of the macroscopic transport equations will be presented. This approach has its roots in the analysis of multiphase flow, encountered in a wide variety of engineering problems involving porous media, particle suspensions, solute sedimentation, energy conversion, chemical reaction, etc. Of these applications, the study of macroscopic momentum transport in porous media has received much attention. Theoretical analysis and experimental verification in this area is abundant in the literature (Drew 1983, Ghaddar 1995, Slattery 1967, Whitaker 1967).

We will use materials solidification as an example to discuss the macroscopic transport equations of mass, momentum, energy, and species. Some issues that have not been addressed adequately in the literature (Beckermann and Viskanta 1993, Ganesan and Poirier 1990) will be pointed out. Since macroscopic transport equations are derived upon the classical microscopic transport equations, a brief discussion of the microscopic equations is presented first.

Microscopic Transport Equations

Solids, liquids, and gases are composed of distinct molecules or atoms. However, in many engineering models materials are conveniently treated as continuous media instead of individual molecules or atoms. This is because engineers are mostly concerned about the averaged features of materials, represented by such quantities as density, velocity, pressure, temperature, and so on, which vary continuously in space and time.