ABSTRACT
In dealing with a mixture of droplets or particles, it is impractical to solve for the fluid properties at every point in the mixture. Thus one is driven to consider the average properties in a volume containing many particles as discussed in Chapter 6. For example, the concept of bulk density as the mass of the phase per unit volume of mixture is an average property. A formal approach is now sought to express the conservation laws for the continuous phase in terms of average properties. The approach used here differs from the approach used by Anderson and
Jackson (1967) in which a monotone decreasing, differentiable weighting function had to be introduced to ensure the differentiability of the volume averages. The introduction of a weighting function is not required here. Consider a mixture of fluid and particles enclosed in the spherical volume
shown in Figure B.1. The averaging volume ( ) is composed of the volume of the continuous phase () and the volume occupied by the dispersed phase (). The volume has to be large enough such that a small increase in the volume will not affect the value of the average; that is, the same value for bulk density would be obtained if the volume were slightly changed. Still, the volume has to be small compared to the system dimensions or it will not be possible to write differential equations for the conservation laws. The formalism used for volume averaging can be found in Slattery (1972).
Based on the volume over which averages are taken, there are two types of averages to be considered. Let be some property of the continuous phase. The phase average of is the average over the volume of the continuous phase and is defined as
