# The Breakup and Coalescence of Bubbles Considering Interphase Turbulence Transfer in Bubbly Flows

The

Breakup and Coalescence of Bubbles Considering Interphase Turbulence Transfer in

Bubbly Flows

S. Azizi, Y.M. Lau, M. Schubert

Helmholtz-Zentrum Dresden-Rossendorf,

Institute of Fluid Dynamics

Bautzner LandstraÃŸe 400, 01328 Dresden, Germany

Tel. +49 351

260 3765,Fax: +49 351 260 2383,Email: s.azizi@hzdr.de.

**Introduction**

The

ability to accurately predict the bubble size distribution in bubble column

reactors is a requirement for any process design as well as for scale-up. The

bubble size distribution depends mainly on the magnitude of bubble breakup and

coalescence. Several breakup and coalescence models have been developed

assuming different driving mechanisms, such as turbulence dissipation and shear

rate of the liquid phase. An overview of these models is given by Liao and

Lucas (2009, 2010). The proposed breakup and coalescence models contain

turbulence contributions in breakup and coalescence of the bubbles and also the

relative velocity of the bubbles. The realistic expressionfor the mentioned

terms is missing for the implementation of the breakup and coalescence models

due to poor knowledge on the turbulence behavior of the bubbly flows. Here, bubble-liquid

turbulence interactions of the bubbly flows are considered to predictparticipating

turbulence energy in breakup and also relative velocity of the bubbles at

coalescence of the bubbles.Â

** **

**Bubble
breakup phenomenon **

A

bubble with an arbitrary shape inside the liquid flow is considered having a velocity

of *v _{b}* split into mean velocity of

*v*and

_{mb}fluctuating velocity of

*v*(both based on the center of mass

_{b}?movement with respect to a fixed coordinate system). The surrounding liquid of the

bubble moves with a mean velocity

*v*and a fluctuating

_{ml }velocity

*v*due to the turbulent nature of the liquid flow. If

_{l}?the liquid eddyis smaller than the bubble size, it hits the bubble surface from

a random direction with the fluctuating velocity component of the liquid (

*v*).

_{l}?Accordingly, the velocity difference at different sides of the bubble deforms

the bubble. For large eddies hitting the bubble, the bubble is transported by

the eddy and the turbulent kinetic energy is added to the bubble.

The

deformation of the bubble is defined as a change of bubble surface regarding to

change of bubble diameter. The deformation energyeither breaks the bubble or is

saved as elastic energy in the bubble.The elastic energy of the bubble is

converted to turbulent kinetic energy of the bubble in order to reach to a stable

state and vice versa. The turbulent kinetic energy of bubbles is also transported

to the liquid phase. A reasonable approximation is provided by the assumption

that all turbulent energy lost by the bubble due to drag is converted into

turbulent kinetic energy of the liquid in the wake of the bubble (Roland Rzehak

and Eckhard Krepper, 2013).

The

breakup criterion of the bubble is satisfied, if the required stress for

breakup of the bubble is produced due to bubble bombarding with liquid eddies. The

turbulence field of the dispersed bubbles is considered as the liquid turbulence

transfer to the bubble in form of elastic energy (assuming the bubble does not

break). Afterward, the saved elastic energy is converted into turbulent kinetic

energy of the bubble that can be return to the liquid phase when the

fluctuating velocity component of bubbles moves the bubble at the downstream.

Knowing the turbulence behavior of the dispersed bubbles, the breakup models

can be re-written in order to consider breakup mechanisms based on turbulence

of liquid and bubbles.

Here,

the breakup model of Martinez-Bazan (1999) is extended considering the new

turbulence feature of the bubbly flow.The turbulent stress for the breakup of

the bubble is the summation of the bubble turbulent stress and liquid turbulent

stress:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (1)

whereas,

the definition of the kinetic energy (*k=v? ^{2}/2*) is considered:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2)

The

minimum energy needed to deform a bubble of size *d *depends on surface

tension (*σ*) and accordingly, the stress is:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (3)

Martinez-Bazen

(1990) postulated that the larger difference between the gradient of stress

produced by the turbulent fluctuations on the surface of the bubble *τ _{t}*

and the restoring stress caused by surface tension

*τ*, the

_{s}larger is the probability that the bubble will break in a certain time. On the

other hand, mentioned that the breakup frequency decreases to a zero limit

value,ifthis difference of the stresses vanishes. Thus, the bubble breakup time

is estimated as:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (4)

where

*u _{b}* is the characteristic velocity of the bubble breakup

process. The new characteristic velocity of the breakup process is replaced in the

Martinez-Bazen model (1990) based on the new turbulence feature of the bubbly

flow that considers the kinetic turbulence energy conservation between bubbles

and liquid flow.

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (5)

The breakup frequency

is given as:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (6)

** **

**Bubble
coalescence phenomenon **

Similar

to the previous section, with using the new turbulence feature of the

bubble-liquid flows, the coalescence model considering the collision frequency

by Kennard (1938) (Eq. 7) and the coalescence efficiency by Sovova (1981) (Eq.

8) are used to extendthe coalescence kernel.

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (7)

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (8)

where

*V* is volume of colliding bubbles.

The

coalescence kernel of bubbles (Eq. 9) requires the relative velocity of the

bubbles. The relative velocity of the bubbles can be estimated from the fluctuating

velocityof each bubble immersed in the liquid flow that can be provided by

solving the turbulent kinetic energy conservation equation for the dispersed

bubbles instead of using a ratioof the dissipation kinetic energy in liquid

phase.

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (9)

In

conclusion, this work presents the distribution of the turbulence within bubble

and the liquid phase that predicts the real contribution of the turbulence

energy for breakup and coalescence of the bubbles. The interphase turbulence

transfer between the liquid phase and the bubbles is postulated as available

turbulence energy for breakup of the bubble and the relative

velocity of the bubbles in the collision is based on turbulence information of

the bubbles.

**References**

Y.X. Liao, D. Lucas,

A literature review of theoretical models for drop and bubble breakup in

turbulent dispersions. Chemical Engineering Science 64 (2009) 3389-3406.

Y.X. Liao, D. Lucas,

A literature review on mechanisms and models for the coalescence process of

fluid particles.Chemical Engineering Science 65 (2010) 2851-2864.

Roland Rzehak and

Eckhard Krepper, Bubble-induced turbulence: Comparison of CFD models, Nuclear

Engineering and Design 258 (2013), 57-65.

C. Martinez-Bazen, C.

Artinez-Bazen, J. L. Montanes, J. C.,Lasheras, On the breakup of an air bubble

injected into a fully developed turbulent flow, Part 1. Breakup frequency, Journal

of Fluid Mechanics 401 (1999), 157-182.

E.H. Kennard, Kinetic

Theory of Gases. McGraw-Hill, NewYork (1938).

H. SovovÃ¡ , J. ProchÃ¡zka,

Breakage and coalescence of drops in a batch stirred vessel-I Comparison of

continuous and discrete models, Chemical Engineering Science 36 (1981),

163-171.