The motion of a large gas bubble rising through liquid flowing in a tube

Abstract

The theory presented here describes the motion of a large gas bubble rising through upward-flowing liquid in a tube. The basis of the theory is that the liquid motion round the bubble is inviscid, with an initial distribution of vorticity which depends on the velocity profile in the liquid above the bubble. Approximate solutions are given for both laminar and turbulent velocity profiles and have the form \begin{equation} U_s = U_c+(gD)^{\frac{1}{2}}\phi(U_c/(gD)^{\frac{1}{2}}), \end{equation}Us being the bubble velocity, Uc the liquid velocity at the tube axis, g the acceleration due to gravity, and D the tube diameter; ϕ indicates a functional relationship the form of which depends upon the shape of the velocity profile. With a turbulent velocity profile, a good approximation to (1) which is suitable for many practical purposes is \begin{equation} U_s = U_s + U_{s0}, \end{equation}Us0 being the bubble velocity in stagnant liquid. Published data for turbulent flow are known to agree with (2), so that in this case the theory supports a well-known empirical result. Our laminar flow experiments confirm the validity of (1) for low liquid velocities.

Velocity of long bubbles transported by a liquid in horizontal channel

“…The flow is supposed to be inviscid and rotational, the upstream vorticity distribution being determined by a given velocity profile. Collins et al [2] have obtained expressions for the bubble velocity, which agree well with the experimental results and the results of Bendiksen taken into account the influence of the surface tension. In the horizontal case, the theoretical analysis of Benjamin [4] showed that, the drift velocity is equal to V ∞ = 0.542 √ gD in a horizontal tube and V ∞ = 0.5 √ 2ga in a channel (D is the tube diameter and 2a is the height of the channel).…”

“…It is appropriate particularly for the flows in which exists an interface where viscous shearing can be neglected due to the fact that the vorticity diffusion closes to the interface remains negligible. It is the case for axis-symmetric bubbles in vertical pipes which was treated successfully by Collins et al [2] and by Bendiksen [3].…”

“…The influence of the liquid motion on the bubble velocity was analyzed theoretically by Collins et al [2] and Bendiksen [3] for axis-symmetrical bubble in vertical pipes. The flow is supposed to be inviscid and rotational, the upstream vorticity distribution being determined by a given velocity profile.…”

Movement of Two Consecutive Taylor Bubbles in Vertical Pipes

“…Nicklin et al (1962), Grace and Clift (1979), and Fabre and Liné (1992), among others, obtained these values experimentally. Collins et al (1978) arrived at very close values. Polonsky et al (1999) carried out simultaneous measurements of the velocity profiles ahead of the Taylor bubble using particle image velocimetry (PIV) and of the bubble propagation velocity in undeveloped laminar and turbulent pipe flow.…”

Effect of length-to-diameter ratio on axial velocity and hydrodynamic entrance length in air-water twophase flow in vertical pipes

“…Another group of authors [10][11][12][13] have stated that for a fully developed gas-liquid two-phase upward slug flow, the translational propagating velocity is independent of the length of the Taylor bubble. The intricacies of obtaining fully developed gas-liquid flow relies on bubbles expansion caused by the pressure changes along the tubing as it rises from the bottom, leading to increase in bubble volume thereby affecting its motion.…”