8 - Ellipsoidal model of the rise of a Taylor bubble in a round tube

Summary

The rise velocity of long gas bubbles (Taylor bubbles) in round tubes is modeled by an ovary ellipsoidal cap bubble rising in an irrotational flow of a viscous liquid. The analysis leads to an expression for the rise velocity that depends on the aspect ratio of the model ellipsoid and the Reynolds and Eötvös numbers. The aspect ratio of the best ellipsoid is selected to give the same rise velocity as the Taylor bubble at given values of the Eötvös and Reynolds numbers. The analysis leads to a prediction of the shape of the ovary ellipsoid that rises with same velocity as that of the Taylor bubble.

Introduction

The correlations given by Viana et al. (2003) convert all the published data on the normalized rise velocity Fr = U/(gD)^1/2 into analytic expressions for the Froude velocity versus buoyancy Reynolds number, Re_G = [D³g (ρ_l – ρ_g)ρ_l]^1/2/μ for fixed ranges of the Eötvös number, Eö = gρ_lD²/σ, where D is the pipe diameter, ρ_l, ρ_g, and σ are densities and surface tension. Their plots give rise to power laws in Eö; the compositions of these separate power laws emerge as bipower laws for two separate flow regions for large- and small-buoyancy Reynolds numbers.