10.2. Wave analogy for bubble rise velocity

Reffered from

Void fraction

Summary

  • Subject: Rise velocity in ellipsoidal bubble regime

  • Main conclusion: \(u^{2} = \frac{2 \sigma}{\rho d} \left( 1 + \frac{\Delta \rho g d^{2}}{4 \sigma} \right)\)

  • Key idea Potential theory of one-dimensional water wave describes the bubble rise velocity.

  • Reference:

The phase velocity of deep-water gravitational wave is given by

(10.21)\[ c_{p} = \sqrt{ \frac{g \lambda}{2 \pi} }\]

where \(\lambda\) is the wavelength. The rise velocity of a spherical-cap bubble was derived as (see Large bubbles in liquid)

(10.22)\[ u = \frac{2}{3} \sqrt{ g a } \sim \sqrt{ \frac{g d}{2} }\]

where \(d \sim 0.877 a\) was used. By setting \(\lambda = \pi d\), the phase velocity in Eq. (10.21) gives the velocity of spherical-cap bubble, that is,

(10.23)\[ c_{p} \xrightarrow[\lambda = \pi d]{} u\]

This relation is called wave analogy, in which the rise motion of a bubble is regarded as the propagation of disturbance as water wave. It is clear from Eq. (10.21) that the deep-water wave is dispersive. However, the spherical-cap bubble rises at a constant speed while maintaining its shape. This fact suggests that the bubble rise motion should be regarded as the water wave with the principal mode of \(\lambda = \pi d\).

The phase velocity of capillary-gravitational wave is given by (see Appendix Water wave of infinitesimal amplitude)

(10.24)\[ c_{p} = \sqrt{ \frac{2 \pi \sigma}{\rho \lambda} + \frac{g \lambda}{2 \pi} }\]

Mendelson [Men67] proposed the following velocity correlation for deformed bubbles by assuming the wave analogy:

(10.25)\[ u = \sqrt{ \frac{2 \sigma}{\rho d} + \frac{g d}{2} }\]

Tomiyama et al. [TKZS98] gave a physical interpretation of the wave analogy and replaced \(g\) with \(\Delta \rho g / \rho\) to make clear the buoyancy effect, i.e.,

(10.26)\[ u = \sqrt{ \frac{2 \sigma}{\rho d} + \frac{\Delta \rho g d}{2 \rho} }\]

Factorizing the R.H.S. by the factor \(2 \sigma / \rho d\) yields

(10.27)\[ u^{2} = \frac{2 \sigma}{\rho d} \left( 1 + \frac{\Delta \rho g d^{2}}{4 \sigma} \right)\]

Substituting this result into the force balance

(10.28)\[ C_{D} = \frac{4}{3} \frac{\Delta \rho g d}{\rho u^{2}}\]

gives

(10.29)\[ C_{D} = \frac{8}{3} \frac{\frac{\Delta \rho g d^{2}}{\sigma}}{\frac{\Delta \rho g d^{2}}{\sigma} + 4}\]

By defining the Eötvös number as

(10.30)\[ Eo = \frac{\Delta \rho g d^{2}}{\sigma}\]

we have

(10.31)\[ C_{D} = \frac{8}{3} \frac{Eo}{Eo + 4}\]

This drag correlation is applicable to a wide range of the bubble diameter, e.g. for air bubbles in water \(d\) larger than about 1 mm.

As can be seen in Eq. (10.26) the gravitational wave becomes dominant as \(d\) increases. For the limiting case of \(Eo \rightarrow \infty\), \(C_{D}\) becomes

(10.32)\[ C_{D} = \frac{8}{3}\]

which corresponds to the drag coefficient of spherical-cap bubble and the velocity of spherical-cap bubble recovers

(10.33)\[ u = \sqrt{ \frac{\Delta \rho g d}{2 \rho} }\]

For small \(Eo\), the capillary wave is dominant and

(10.34)\[ C_{D} = \frac{2}{3} Eo \left( 1 + \frac{Eo}{4} \right)^{-1}\]

can be approximated with the condition \(Eo/4 \ll 1\) as

(10.35)\[ C_{D} = \frac{2}{3} Eo \]

In the velocity form, this is of course

(10.36)\[ u = \sqrt{ \frac{2 \sigma}{\rho d} }\]

This represents that the rise velocity decreases with increasing bubble size, which agrees with experimental facts. The decreasing and increasing trends in Eqs. (10.36) and (10.33), respectively, indicate that there is a minimum velocity for a certain critical diameter, at which the dominant force changes. Differentiating Eq. (10.26) with respect to \(d\) and setting \(du/dd = 0\) yield the critical diameter

(10.37)\[ d_{c} = 2 \sqrt{ \frac{\sigma}{\Delta \rho g} }\]

The critical diameter is twice longer than the capillary length (\(l = \sqrt{\sigma / \Delta \rho g}\)). By substituting \(d_{c}\) into the velocity equation, we obtain

(10.38)\[ u_{c} = \sqrt{2} \left[ \frac{\Delta \rho g \sigma}{\rho^{2}} \right]^{1/4}\]

This velocity corresponds to that obtained with the Ishii-Chawla drag correlation [IC79]:

(10.39)\[ C_{D} = \frac{2}{3} \sqrt{Eo}\]

For an air-water system, \(d_{c}\) and \(u_{c}\) are about 5.5 mm and 0.23 m/s, respectively.