Buoyancy-driven bubbly flows: scaling of velocities in bubble columns operated in the heterogeneous regime

Abstract

The hydrodynamics of bubble columns in the heterogeneous regime is revisited. Focusing on air–water systems at large aspect ratio, we show from dimensional analysis that buoyancy equilibrates inertia, and that velocities scale as $(gD\varepsilon )^{1/2}$ , where $D$ is the bubble column diameter, $\varepsilon$ the void fraction and $g$ the gravitational acceleration. From new experiments in a $0.4$ m diameter column with ${{O}}(10^3)$ particle Reynolds number bubbles and from a detailed analysis of published data, we confirm the self-organization prevailing in the heterogeneous regime, and that the liquid flow rate is only set by the column diameter $D$ . Besides, direct liquid and gas velocity measurements demonstrate that the relative velocity increases above the terminal velocity $U_T$ in the heterogeneous regime, and that it tends to ${\sim }2.4 U_T$ at very large gas superficial velocities $V_{sg}$ . The proposed velocity scaling is shown to hold for liquid and gas mean velocities and for their standard deviations. Furthermore, it is found to be valid over a wide range of conditions, corresponding to Froude numbers $Fr=V_{sg}/(gD)^{1/2}$ from 0.02 to 0.5. Then, the relevance of this scaling for coalescing media is discussed. Moreover, following the successful prediction of the void fraction with a Zuber & Findlay approach at the beginning of the heterogeneous regime, we show how the void fraction is correlated with $Fr$ . Further investigations are finally suggested to connect the increase in relative velocity with meso-scale structures known to exist in the heterogeneous regime.