Mass transfer to gas bubbles

Another interesting example is that of gas bubbles dispersed in a continuous liquid phase with which mass is exchanged. Also for this case the rate of change of the internal coordinates due to mass transfer is written starting from a simple mass balance for a single bubble. Following the standard notation for gas-liquid systems, the single-particle mass balance becomes 

Ap is the difference in material density between the liquid and gas phases. This situation is typically handled by describing the bubbles with a single internal coordinate (i.e. the equivalent-sphere diameter) and by introducing an aspect ratio, defined as the ratio between the minor and the major axes of the bubble. This aspect ratio E can be calculated by using the empirical equation proposed by Moore (1965) as a function of the Morton number E = 1/(1 + 0.043RCp Mo ). An alternative to this is the use of the correlation proposed by Wellek et al. (1966) for liquid-liquid droplets E = 1/(1 + 0.1613Eo° ), which is valid for Eo 40 and Mo 10 , whereas for Eo 40 and RCp 1.2 fluid particles are typically of spherical shape. Once the characteristic E value is known, the ratio of the real area of the bubble Ap and the area Aeq of a sphere with an equivalent volume can be calculated as follows  

Insofar as the mass-transfer coefficient for clean bubbles is concerned (see, for example, the review by Clift et al. (1978)), in the case of spherical bubbles moving under creeping (or Stokes) flow conditions, the following correlation has been proposed Sh = 1 + (1 + 0.564Pe ). For spherical particles with Rep 70 the Sherwood number can be expressed by the following relationship (Lochiel Calderbank, 1964)  

By imposing thermodynamic equilibrium (with the Clausius-Clapeyron law whose validity holds far away from the critical point) at the droplet surface, the conservation of mass and enthalpy (considering that changes in the droplet temperature must be related to the latent heat of evaporation) yields the following expressions  

The Sherwood number for this particular system can be calculated by using empirical correlations, such as the one proposed by Faeth and Fendell (Kuo, 1986)  

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