Boundary layer mass transfer bubbles

LAMINAR BOUNDARY LAYER MASS TRANSFER AROUND SOLID SPHERES, GAS BUBBLES, AND OTHER SUBMERGED OBJECTS... 

One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

Answer For boundary layer mass transfer across gas-liquid interfaces, X = and y =. In the laminar flow regime, 2 = 5. This problem is analogous to one where the bubble is stationary and a liquid flows past the submerged object at intermediate Reynolds numbers. 

Air Sparging Gas sparging or injection of air bubbles has been effectively used to reduce concentration polarization and enhance mass transfer. " The secondary flows around bubbles promote mixing and reduce the thickness of the concentration polarization boundary layer. When the bubble diameter exceeds that of the membrane (tubular or hollow fiber), slugs are then formed further increase in bubble diameter has no effect on flux improvement. Large slugs can displace most of the boundary layer and cause the pressure to pulsate. This results in enhancing the flux. 

In a number of refining reactions where bubbles are formed by passing an inert gas through a liquid metal, the removal of impurities from the metal is accomplished by transfer across a boundary layer in the metal to the rising gas bubbles. The mass transfer coefficient can be calculated in this case by the use of the Calderbank equation (1968)... 

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 film and boundary layer theories presuppose steady transport, and can therefore not be used in situations where material collects in a volume element, thus leading to a change in the concentration with time. In many mass transfer apparatus fluids come into contact with each other or with a solid material for such a short period of time that a steady state cannot be reached. When air bubbles, for example, rise in water, the water will only evaporate into the bubbles where it is contact with them. The contact time with water which surrounds the bubble is roughly the same as that required for the bubble to move one diameter further. Therefore at a certain position mass is transferred momentarily. The penetration theory was developed by Higbie in 1935 [1.31] for the scenario described here of momentary mass transfer. He showed that the mass transfer coefficient is inversely proportional to the square root of the contact (residence) time and is given by... 

The target quantity of the gassing process is the absorption rate in the gas/liquid (G/L) system. It is directly proportional to the interfacial area between the gas phase and the liquid phase. The limiting factor is the diffusion of the dissolved gas through the liquid-side of the boundary layer, which can only be affected by its thickness to a limited extent. A substantial intensification of mass transfer is only possible by increasing the G/L interfacial area gas sparging by means of stirrers, nozzles, sintered or perforated plates etc. should therefore effect a dispersion of the gas into fine bubbles. 

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. 

The solution of hydrodynamic problems for an arbitrary straining linear shear flow (Gkm = Gmk) past a solid particle, drop, or bubble in the Stokes approximation (as Re -> 0) is given in Section 2.5. In the diffusion boundary layer approximation, the corresponding problems of convective mass transfer at high Peclet numbers were considered in [27, 164, 353]. In Table 4.4, the mean Sherwood numbers obtained in these papers are shown. 

The basic classical theories, such as the film, boundary layer,transient film, and penetration hypotheses are obviously outside the scope of this chapter, but the reader is assumed to be familiar with their basic concepts. Harriott s (H8) recent review on mass transfer to interfaces is recommended in this connection. An excellent treatise on the motion of drops and bubbles in fluid media is found in Levich s Physicochemical Hydrodynamics (L8, Ch. 8). 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

The mass transfer boundary layer is very thin relative to the bubble diameter. Hence, a locally flat description is appropriate at high Schmidt numbers. If necessary, analysis is restricted to the front hemisphere of the bubble with respect to the approaching fluid to justify this claim. 

Tangential Velocity Component vg within the Mass Transfer Boundary Layer Creeping and Potential Flow around a Gas Bubble... 

Unlike creeping flow about a solid sphere, the r9 component of the rate-of-strain tensor vanishes at the gas-liquid interface, as expected for zero shear, but the simple velocity gradient (dvg/dr)r R is not zero. The fluid dynamics boundary conditions require that [(Sy/dt)rg]r=R = 0- The leading term in the polynomial expansion for vg, given by (11-126), is most important for flow around a bubble, but this term vanishes for a no-slip interface when the solid sphere is stationary. For creeping flow around a gas bubble, the tangential velocity component within the mass transfer boundary layer is approximated as... 

BOUNDARY LAYER SOLUTION OF THE MASS TRANSFER EQUATION AROUND A GAS BUBBLE... 

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