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 Landstraße 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. 

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_mb and
fluctuating velocity of v_b? (both based on the center of mass
movement with respect to a fixed coordinate system). The surrounding liquid of the
bubble moves with a mean velocity v_ml and a fluctuating
velocity v_l? due to the turbulent nature of the liquid flow. If
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:

                                                                                                   (1)

whereas,
the definition of the kinetic energy (k=v?²/2) is considered:

                                                                                                                       (2)

The
minimum energy needed to deform a bubble of size d depends on surface
tension (σ) and accordingly, the stress is:

                                                                                                                                   (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_s, the
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:

                                                                                                                                     (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.

                                                                                                       (5)

The breakup frequency
is given as:

                                                                                                  (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.

                                                                                                    (7)

                                                                                           (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.

                                (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. Sovová , J. Procházka,
Breakage and coalescence of drops in a batch stirred vessel-I Comparison of
continuous and discrete models, Chemical Engineering Science 36 (1981),
163-171.