Time-Averaged Navier-Stokes Equations as Basic Equations for the Flow of the Mixed Phase

Abstract

The Navier-Stokes equations and the equations of continuity are intergrated with respect to time. When the time-averaging procedure is put into the spatial differentiation, it is necessary to introduce surface-residual terms due to the moving boundary of the two phases.
The surface-residual terms for the substantial derivative terms in the Navier-Stokes equations and in the equations of continuity are naturally eliminated in carrying out proposed mathematical treatment.
The interaction term between the two phases consists of the surface-residual terms of the stress tensor and the static pressure.
Basic equations are considered for the flow of the gas and liquid phases in a bubble column which is long enough to have a constant flow pattern in the axial direction. The following results are obtained. 1. The meaning of the turbulent viscosity in a bubble column is expressed by a modified Boussinesq equation. 2. Axial gradient of static pressure is shown to be constant in the radial direction. 3. Calculated value of the mean gas hold-up by the use of the axial gradient of the static pressure at the wall is evaluated. The error of this value is less than 2 % for the reported data on bubble columns with the internal circulating flow.
The proposed equations can similarly be applied to other apparatus with two phase flow, such as fluid beds and extraction columns.