The Gross Scale Averaged Two-Phase Transport Equations

6 The Gross Scale Averaged Two-Phase Transport Equations 

Given a tube with axis Oz (unit vector e ) in which a volume Vjt is limited by a boundary, A/ and cut by a cross section plane over area, A, as sketched in Fig. 3.7. The lateral dimensions of the control volume extend to the conduit walls. In this notation, njt is the outward directed unit vector normal to the interface of phase k. l] t, z) is the intersection of interface. A/, with the cross sectional plane, is the outward directed unit vector normal to the closed curve of phase k, z), in the cross section plane. 

The purpose of this derivation is to average the local, instantaneous balance equation over the variable cross-section area of a pipe occupied by the two phases, as sketched in Fig. 3.7. 

The local, instantaneous balance equation (1.3) is integrated over the cross section area, z), limited by the boundaries, Z/(f, z), with the other phase and with the pipe wall boundary, lw,k t, z)  

To proceed the limiting forms of the Leibnitz and the Gauss theorems, appropriate for two phase flow, are applied. These theorems are considered direct extensions of the single phase theorems examined in Sect. 1.2.6 so no further derivation is given here. In most reactor model formulations the pipe walls are supposed to be fixed and impermeable. In the limit z 0, the limiting form of the Leibnitz theorem for volume reduces to the following relation for area [47, 51]  

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