Linear stability analysis for the effect of the lift force in a bubble column

The role of the lift force for the stability of a homogeneous bubble column is investigated. Instabilities caused by the lift force may be one important reason for the transition from homogeneous to heterogeneous bubble column. The lift force acts on rising bubbles in lateral direction, when gradients of the liquid velocity are present. Non-uniform liquid velocity fields may be induced, if the gas fraction is not equally distributed, e.g. caused by local disturbances. For regions with locally increased gas volume fraction the liquid is accelerated in upwards direction and following a gradient of the liquid velocity occurs. This causes a lateral redistribution of the gas bubbles. Depending on the sign of the lift force coefficient this can act to smooth the spatial bubble distribution or to increase the initial disturbances. It is obvious, that a positive lift coefficient stabilizes the flow, while a negative coefficient leads to unstable gas fraction distributions, and thus it favours the appearance of a heterogeneous bubble column regime. According to the well know correlation obtained by Tomiyama, the lift force coefficient is positive for small bubbles (for air-water at ambient conditions for bubbles with an equivalent diameter < 5.8 mm) and change to negative values for large bubbles. Thus this mechanism may be a possible explanation for the experimentally observed ‘pure heterogeneous’ regime, which occurs when large bubbles are injected. In this case the bubble column is heterogeneous also for small gas volume flow rates.
The above mentioned is feedback mechanism is studied by the means of a linear stability analysis. In addition to the lift force, the turbulent dispersion force is considered, which has always a stabilizing effect, i.e. it partially compensates the destabilisation induced by a negative lift coefficient. The stability analysis for a mono-dispersed system nevertheless showed, that influence of the lift force is much larger, compared to the influence of the turbulent dispersion force, if only bubble induced turbulence is considered. Thus the stability condition is practically the positive sign of the lift force coefficient. The analysis was then extended to two bubbles classes - one being small enough to have a positive lift force coefficient, the other being large enough to have a negative coefficient. The result of the analysis is a condition for the minimum fraction of small bubbles needed for stability. Finally a generalized criterion for N bubble classes and for a continuous bubble size distribution is given.