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).