The effects of bubbles on the volume fluxes and the pressure gradients in unsteady and non-uniform flow of liquids

Abstract

In this paper the motion of a single bubble or particle in an accelerating liquid flow is analysed using a generalized force equation. The motion of the bubble relative to the fluid gives rise to a drift flux of liquid which affects the mean flow field of the liquid. This flux is calculated in terms of the effective or added mass coefficient of the bubble, C m, which is equal to 1 2 for a small bubble at high Reynolds number. By analysis of the flow in terms of the three flow fields associated with the interstitial liquid, the displaced liquid and the bubble itself, we obtain a rational method for calculating the forces acting on the bubble, the mass conservation equations and the pressure field in the liquid. For a bubbly flow with low void fraction this form of the mass conservation equations reduces to an expression for liquid and gas superficial velocity as a function of relative velocity (the slip measured relative to the interstitial velocity, to be defined) and void fraction, where the constants are defined in terms of C m. Unlike the commonly used relation of Zuber & Findlay (1965), our expression can be generalized to non-uniform flows. Our predictions (using C m = 1 2 ) agree with Zuber & Findlays' empirical equations for low void fraction ϵ in a vertical pipe. They do not, however, agree for high values of ϵ. A new analysis of the mean pressure field in disperse two-phase flow is presented. While the bubbles respond to the interstitial velocity and pressure field, the pressure of interest is usually the average over the whole liquid volume. We show how these are related. The models developed here are applied to the practically important flow of air bubbles in an inclined nozzle, and are compared with recent laboratory measurements.