Bubbles boundary-layer theory

Fig. 5.29 Drag coefficients for bubbles in pure systems predictions of numerical, Galerkin, and boundary layer theories compared with selected experimental results.

The thin concentration boundary layer approximation, Eq. (3-51), has also been solved for bubbles k = 0) using surface velocities from the Galerkin method (B3) and from boundary layer theory (El5, W8). The Galerkin method agrees with the numerical calculations only over a small range of Re (L7). Boundary layer theory yields... 

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... 

Figure 10-12. The shape of a rising gas bubble as a function of the Weber number, as predicted by small-deformation, boundary-layer theory. We = 0,0.25, 0.5.

The drop may conserve its spherical form until Re = 300 [94]. Since usually the boundary layer on a drop or a bubble is considerably thinner than on a solid sphere, one can use methods based on the boundary layer theory even for 50 < Re < 300. By using these methods, the following formula was obtained in [94] for the drag coefficient for Re 1 ... 

Schlichting H. (1955) Boundary-layer theory. McGraw-HUl Book Co., New York, USA. Shaikh A, Al-Dahhan M. (2010) A new methodology for hydrodynamic similarity in bubble columns. Can. J. Chem. Eng., 88 503-517. 

We also need to develop the theories for hquid film coefficient to use in the aforementioned equations. For drops that are close to spherical, without separation, Levich (1962) assumed that the concentration boundary layer developed as the bubble interface moved from the top to the bottom of a spherical bubble. Then, it is possible to use the concepts applied in Section 8.C and some relations for the streamlines around a bubble to determine Kl. ... 

Carbon Monoxide Oxidation. Analysis of the carbon monoxide oxidation in the boundary layer of a char particle shows the possibility for the existence of multiple steady states (54-58). The importance of these at AFBC conditions is uncertain. From the theory one can also calculate that CO will bum near the surface of a particle for large particles but will react outside the boundary layer for small particles, in qualitative agreement with experimental observations. Quantitative agreement with theory would not be expected, since the theoretical calculations, are based on the use of global kinetics for CO oxidation. Hydroxyl radicals are the principal oxidant for carbon monoxide and it can be shown (73) that their concentration is lowered by radical recombination on surfaces within a fluidized bed. It is therefore expected that the CO oxidation rates in the dense phase of fluidized beds will be suppressed to levels considerably below those in the bubble phase. This expectation is supported by studies of combustion of propane in fluidized beds, where it was observed that ignition and combustion took place primarily in the bubble phase (74). More attention needs to be given to the effect of bed solids on gas phase reactions occuring in fluidized reactors. 

If one considers, that kt was defined according to the Two-film theory [594, 327) by kt X D/6, see Fig. 4.1 (D is the diffusivity of the gas in the liquid S is the thickness of the liquid-side boundary layer), a strong dependence kt = /(boundary layer not only depends upon the viscosity of the liquid, but also upon the bubble diameter di,. 

In his theory of coalescence Marrucci [364] concerned himself with the kinematics of the thinning process (drainage), which the liquid lamella between neighboring gas bubbles is subjected to, until it bursts. The reason for the stretching of both boundary layers of the film is due to the pressure difference between the liquid in the film and that in the bulk. They are in equilibrium with the difference in surface tension between the film and the liquid bulk ... 

Foundation of the Theory of Diffusion Boundary Layer And Dynamic Adsorption La yer of Moving Bubbles... 

In connection with the development of the theory of convective diffusion in liquids the foundation of the theory of diffusion boundary layers and dynamic adsorption layers are given by Levich (1962) in his works on physico-chemical hydrodynamics. A variety of problems of convective diffusion in liquids was solved which are of essential interest for the description of different heterogeneous processes in liquids the rate of which is limited by diffusion kinetics. In connection with the objectives of the present chapter, only a general approach to problems of diffusion boundary layers and their concrete results (Levich 1962) are reported. These are of direct interest for the theory of dynamic adsorption layers of bubble. 

In the dynamic adsorption layer theory of Deqaguin-Dukhin, the hydrodynamic field of a bubble can be assumed to be known as first approximate, while the more difficult stagnant cap problem has still to be solved. For the solution of this hydrodynamic problem unusual and very difficult boundary conditions exist which are very inconvenient even after essential simplifications. The hydrodynamic field of a bubble is studied imder the assumption that the stagnant cap is completely immobilised and any motion of the surface beyond the stagnant cap is ignored. Since the description of the stagnant cap is to a large extent a hydrodynamic problem, it has received less attention (cf. Section 8.7). 

Theory of Dynamic Adsorption and Diffusion Boundary Layers of a Bubble WITH Pe 1, Re 1 and Weak Surface Retardation... 

For the first time Mileva (1990) has considered the effect of a hydrodynamic boundary layer on the elementary act of inertia-free microflotation based on a mobile bubble surface free of an adsorption layer and at high Reynolds numbers. The velocity distribution is a potential one, Eq. (8.117), with an additional contribution of the velocity differential along the cross-section of the boundary layer, Eq. (8.127). The difference between the velocity distribution along the bubble surface and the potential distribution is given by Eq. (8.128). In contrast, Mileva (1990) used the formulas of a related theory by Moore (1963) which are the solution of the same Eq. (8.122) under the same boundary condition (8.118). 

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). 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

---