Bubble boundary, resistance

The second term in Equations (1) and (2) accounts for diffu-sional transfer across the bubble boundary. (A factor e /(1+e p is sometimes (e.g. 49) included in the bracket of Eq. 2 o account for the dense phase diffusional resistance.) There is some question (30) of the extent to which there is interference between the bulk flow and diffusion terms. Nevertheless, most experimental evidence suggests that the two terms are additive and that the diffusional term is described by the penetration theory. With these changes, and including a small enhancement factor for bubble interaction. Sit and Grace (35) have recommended the following equations as being in best accord with existing experimental data ... 

Resistance functions have been evaluated in numerical compu-tations15831 for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels. 

The resistance to mass transfer within a slug in a liquid of low viscosity has been measured by Filla et ai (F5), who found that kA) was approximately proportional to the square root of the diffusivity within the bubble, p, as predicted by the thin concentration boundary layer approximation. In addition, kA JA was independent of slug length for 1 < L/D < 2.5. 

The mass transfer resistance at a liquid-vapor interface results from two resistances, the liquid boundary layer and the gas boundary layer. In conditions involving water and sparingly soluble gases, such as occurs here, the liquid-phase resistance is almost always predominant [71]. For this reason, equation (16) involves only k, the mass transfer coefficient across the liquid boundary, and a, which is the gas bubble surface area per unit volume of liquid. Often, as here, those factors cannot be estimated individually, so k is treated as a single parameter. 

The Davidson and Harrison approach concentrates solely on the resistance at the bubble/cloud boundary (or bubble/dense phase boundary for a < 1). The transfer coefficient, referred to bubble surface area, is... 

For most practical conditions, a comparison of k and k from Equations (4) and (5) would suggest that the principal resistance to transfer resides at the outer cloud boundary. However, when (a), (b) and (c) are taken into account, this is no longer the case. In fact, experimental evidence (e.g. 30,31,32) indicates strongly that the principal resistance is at the bubble/ cloud interface. With this in mind, it is probably more sensible to include the cloud with the dense phase (as in the Orcutt (23, 27) models) rather than with the bubbles (as in the Partridge and Rowe (37) model) if a two-phase representation is to be adopted (see Figure 1). If three-phase models are used, then Equations (2) and (5) appear to be a poor basis for prediction. Fortunately the errors go in opposite directions. Equation (2) overpredicting the bubble/cloud transfer coefficient, while Equation (5) underestimates the cloud/emulsion transfer coefficient. This probably accounts for the fact that the Kunii and Levenspiel model (19) can give reasonable predictions in specific instances (e.g.20),... 

The region of saturated boiling is followed by that of convective evaporation. With the increasing vapour content the heat transfer from the wall to the fluid improves. The thermal resistance of the boundary layer decreases in comparison to the thermal resistances in nucleate boiling. Likewise, the wall temperature drops, cf. Fig. 4.53, so that only a few or no bubbles are formed at the wall. The heat transfer is predominantly or exclusively determined by evaporation at the phase boundary between the liquid at the wall and the vapour in the core flow. 

For two-phase systems, mixing promotes faster mass transfer by creating higher interfacial area due to smaller bubbles or drops. Turbulence also helps reduce the boundary-layer resistance around drop or bubble surfaces, leading to faster mass transfer. 

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