where f is the frequency of bubble formation and Q is the volumetric rate of gas flow in consistent units.
Equations (14-206) and (14-207) result from a balance of bubble buoyancy against interfacial tension. They include no inertia or viscosity effects. At low bubbling rates (<1/s), these equations are quite satisfactory. Van Krevelen and Hoftijzer [Chem. Eng. Prog., 46, 29 (1950)], Guyer and Peterhaus [Helv. Chim. Acta, 26, 1099 (1943)] and Wilkinson (op. cit.) report good agreement with Eq. (14-206) for water, transformer oil, ether, and carbon tetrachloride for vertically oriented orifices with 0.004 < D < 0.95 cm. If the orifice diameter becomes too large, the bubble diameter will be smaller than the orifice diameter, as predicted by Eq. (14-206), and instability results; consequently, stable, stationary bubbles cannot be produced. Kulkarni et al. (op. cit.) have discussed much more detailed models for predicting bubble size; however, the models are very difficult and tedious to use in practice. These more sophisticated models need to be considered if the added accuracy warrants the extra effort.
For bubbles formed in water, the orifice diameter that permits bubbles of about its own size is calculated as 0.66 cm. Davidson and Amick [AlChE J., 2, 337 (1956)] confirmed this estimate in their observation that stable bubbles in water were formed at a 0.64-cm orifice but could not be formed at a 0.79-cm orifice.
For very thin liquids, Eqs. (14-206) and (14-207) are expected to be valid up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. Inst. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci., 5, 127 (1956)] have shown that liquid viscosity has very little effect on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (1960)] and Krishnamurthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predicted by Eq. (14-206) for liquid viscosities above 1000 cP. In fact, Davidson et al. (op. cit.) found that their data agreed very well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes' law and the velocity of the bubble equator at break-off:
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