Volume-averaging procedure

In the case of two immiscible fluids, a characteristic phase indicator function, Xj, may be defined that is equal to 1 In one of the phases and 0 in the other phase. Then Xi and nj are related by nidi = VXi analogue to the relations used In standard volume averaging procedures [54] [164]. An averaged representation of this relation may be given as f mSida = —Vai... 

Proceeding in the same way as explained presenting the volume averaging procedure, we obtain the following result... 

As distinct from the time and volume averaging procedures, the application of the ensemble averaging operator is not restricted by any space- and time scales. The ensemble averaging is thus considered the fundamental method [58, 59]. 

The time after volume averaging procedure can be applied under a unified set of conditions denoting the sum of the two sub-sets of requirements formulated in sects 3.4.1 and 3.4.2 for the pure volume and time averaging procedures to handle the scale disparity in a proper manner. 

The 2D steady-state two-fluid model presented in this section is based on the early work by Torvik and Svendsen [22], Svendsen et al [21], and Jakobsen [12], The two-fluid model was derived based on the time-after volume averaging procedure, described in sect 3.4.4. 

In a continuous stirred tank separator (CSTS) employed in particulate systems, there are no spatial gradients inside the separator therefore we need to integrate either equation (6.2.51b) or equation (6.2.51c) over the total volume V of the separator. To that end, we may employ the type of relations (6.2.4b-d) utilized in volume-averaging procedures. For example, the second term, V- Upn rp)j, on the left-hand side of (6.2.51b) will lead to... 

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