**Vertical
stability of bubble chain**

** **

1Ruzicka M.C., 1Drahos
J., 2Thomas N.H.

*1
**Institute of Chemical
Process Fundamentals, Czech Academy of Sciences, Rozvojova 135,*

*16502
Prague, Czech Republic *(ruzicka@icpf.cas.cz)

*2
**FRED Ltd., Aston Science
Park, Birmingham B7 4BJ, UK *(Neale@Thomas.net)

It
seems that we lack detailed knowledge about dynamic behaviour of
hydrodynamically interacting particles arranged in one-dimensional arrays at
intermediate Reynolds number, of which a bubble chain rising through a liquid
is a typical example (Harper 1997; Yuan & Prosperetti 1994). Despite the
fact that lines of bubbles are commonly encountered in carbonated beverages
practically everyday (e.g. Liger-Belair & Jeandet 2002), their stability
has not been studied in detail yet. This contribution addresses the stability
of one-dimensional array of uniformly spaced bubbles with respect to small
displacements from their equilibrium positions along the vertical.

The
system is described by a relatively simple force-law model (Ruzicka 2000), a
set of ODE's, with leading-order forces of viscous and inviscid origin relevant
at *Re *= 50-200. Besides the local forces acting between the nearest
neighbours, coupling between distant bubbles is modelled by a linear
superposition of the pairwise forces.

The
governing equations are linearized around the equilibrium point of uniform
steady spacing, which yields several linear force components. The stability
features of these components, both individual and in combinations, are
determined by evaluation of the eigenvalues of the corresponding Jacobians. It
is shown that the main destabilizing force is the shielding force, which is the
distance-dependent component of the drag force that reflect the wake
interaction between bubbles. Further, it is found that the distant coupling has
an adverse effect on the chain stability.

As
the next step, the continuum limit of the above discrete chain is taken. This
corresponds to a look at the chain from a larger distance, i.e. employing
larger length scales. The stability is investigated for the corresponding
force-law PDE, which describes bubbles continuously distributed along one
spatial co-ordinate with the uniform base state. The stability results obtained
for the discrete and continuous chains are compared and discussed.

Finally,
the continuous momentum equation is supplied with the mass equation and the
resulting set is discussed in the context of modelling one-dimensional two-phase
flows. Since all the terms can be easily traced back to the original pairwise
forces, we can see where the constitutive parameters of the system (e.g. bulk
elasticity) originate from.

Harper, J.F.,
1997. Bubbles rising in line: why is the first approximation so bad? J. Fluid
Mech. 351, 289-300.

Liger-Belair, G.
& Jeandet, P. 2002 Effervescence in a glass of champagne: A bubble story.
Europhysics News 33 (1), 10-14.

Ruzicka, MC,
2000. On bubbles rising in line. Int. J. Multiphase Flow 26, 1141-1181.

Ruzicka, MC,
2001. Vertical stability of bubble chain. (in preparation)

Yuan, H.,
Prosperetti, A., 1994. On the in-line motion of two spherical bubbles in a
viscous fluid J. Fluid Mech. 278, 325-349.

* *

*Acknowledgement*

Supported
by GACR (Grant No. 104/01/0547).

**Can
viscosity stabilize uniform bubble bed?**

** **

1Ruzicka M.C., 1Drahos
J., 2Thomas N.H.

*1
**Institute of Chemical
Process Fundamentals, Czech Academy of Sciences, Rozvojova 135, 16502*

*Prague,
Czech Republic *(ruzicka@icpf.cas.cz)

*2
**FRED Ltd., Aston Science
Park, Birmingham B7 4BJ, UK *(Neale@Thomas.net)

Chemical engineers commonly produce bubbly
mixtures in their equipments called 'bubble columns' to contact the gas and
liquid phases to promote transport and reaction phenomena (Deckwer 1992; Kastanek et al. 1993). There are two basic flow regimes
in bubble columns, uniform (homogeneous) and non-uniform (heterogeneous), which
differ from one another in occurrence of large-scale motions called
'circulations' (Zahradnik et al. 1997). The circulations are absent in the
former and present in the latter. The uniform regime is stable with respect to
voidage and velocity disturbances within certain limits of the operational
parameters, for instance, column geometry and dimensions, gas flow rate,
physico-chemical properties of the phases, pressure, temperature, etc. Beyond
these limits, it becomes unstable and turns into the non-uniform regime
(Ruzicka et al. 2001).

The
available experimental data spread over the literature strongly indicate that
the viscosity has an adverse effect on the stability of the uniform bubble bed
(e.g. Wilkinson et al. 1992). This common opinion is based on the fact that
viscosity decreases the gas holdup and tends to deteriorate the uniformity by
promoting bubble coalescence, which leads to polydispersity of bubble sizes,
hence velocities. Moreover, it seems that it is almost impossible to produce
uniform beds at large viscosities (Zahradnik et al. 1997).

The
goal of this study is to determine experimentally the effect of the liquid
viscosity on the uniform regime stability, which has not been done yet. Uniform
bubble beds of different viscosities (water + glycerol, 1-20 mPas) are
generated in a column, with the gas input as the control parameter. The
critical gas input where the transition to the non-uniform regime occurs is
found by the drift-flux plot method (Wallis 1969). The value of the critical
gas input is the measure of the uniform regime stability. The critical values
are plotted against the viscosity.

The
experiments are currently under way. The data collected so far clearly indicate
that small viscosity can stabilize the uniform bed, which is in contrast with
the common expectation, but is in accord with the prediction of the recent
stability theory (Ruzicka & Thomas 2003). On the other hand, larger
viscosities have apparent destabilizing effect, as believed.

Deckwer, W. D.
(1992) Bubble column reactors. J. Wiley, Chichester.

Kastanek, F.
Zahradnik, J., Kratochvil, J. & Cermak, J. (1993) Chemical reactors for
gas-liquid systems. E. Horwood, Chichester.

Ruzicka, M.C.
& Thomas, N.H. (2003) Buoyancy-driven instability of bubbly layers: analogy
with thermal convection Int. J. Multiphase Flow, in press.

Ruzicka, M.,
Zahradnik, J., Drahos, J. & Thomas, N. H. (2001) Homogeneous-heterogeneous
regime transition in bubble columns. Chem. Eng. Sci. 56, 4609-4626.

Wallis, G. B.
(1969) One-dimensional two-phase flow. McGraw-Hill, New York.

Wilkinson, P. M.,
Spek, A. P. & Dierendonck, L. L. van (1992) Design parameters estimation
for scale-up of high-pressure bubble columns. AIChE J. 38, 544-554.

Zahradnik, J.,
Fialova, M., Ruzicka, M., Drahos, J., Kastanek, F. & Thomas, N. H. (1997)
Duality of the gas-liquid flow regimes in bubble column reactors. Chem. Eng.
Sci. 52, 3811-3826.

* *

*Acknowledgement*

Supported
by GACR (Grant No. 104/01/0547).