Bubble diffusion boundary layer

Diffusion Boundary Layer Near the Surface of a Drop (Bubble)... 

For a viscous flow past drops (bubbles) and for an ideal fluid flow past solid particles, we have m = 1. For a laminar viscous flow past smooth solid particles, one usually has m = 2 there also exist some examples in which m = 3 [166]. It follows from the preceding that in the leading approximation, the tangential velocity vv (4.6.19) in the diffusion boundary layer near the drop surface is constant and is equal to the fluid velocity on the interface, whereas in the diffusion boundary layer close to the surface of a solid particle, the tangential velocity in the leading approximation depends on the distance to the surface linearly (sometimes, even quadratically) and is zero on the surface. 

A method for solving three-dimensional problems on the diffusion boundary layer based on a three-dimensional analog of the stream function, was proposed in [348, 350]. In [27, 166, 353], this method was used for studying mass exchange between spherical particles, drops, and bubbles and three-dimensional shear flow. 

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. 

In the diffusion boundary layer approximation with allowance for the corrections (with respect to the Reynolds number) to the potential flow past the bubble, one can obtain the following two-term expansion of the dimensionless total flux I ... 

The diffusion boundary layer in problem (4.12.1), (4.12.2) is first adjacent to the particle surface and then rapidly spreads over the flow region with the subsequent exponential relaxation to a steady state. The characteristic relaxation time rr is of the order of Pe-2 3 for a solid particle and of the order of Pe-1 for bubbles and drops of moderate viscosity. 

Table 4.8 presents a comparison of the mean Sherwood numbers calculated according to Eq. (4.12.3) with available data for various flows past spherical drops, bubbles, and solid particles at high Peclet numbers (in this table, we use the abbreviation DBLA for diffusion boundary layer approximation ). 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

In Table 5.1, the maximum error of formulas (5.3.8) and (5.3.9) are shown in the entire range of the parameter ky for six different kinds of spherical particles, drops, or bubbles. All these estimates were found by comparison with the closed-form solution of problem (5.3.1), (5.3.2) obtained in the diffusion boundary layer approximation [363]. 

It was shown in [349] that at high Peclet numbers (in the diffusion boundary layer approximation), by solving the corresponding nonlinear problem on transient mass exchange between drops or bubbles and the flow, one obtains the following expression for the mean Sherwood number ... 

Supply of surfactant to the leading surface of the bubble and withdrawal of desorbing surfactant into the bulk from the lower half is governed by diffusion and leads to the formation of a so-called diffusion boundary layer adjacent to the surface. Its thickness Sj, is much smaller than the bubble radius a. ... 

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. 

The formula of Hadamard and Rybczynski are also valid for the "moving bubble" problem with Ti Ti. Using the Hadamard-Rybczynski velocity field, it is easy to show that the difference between the tangential component of the velocity in the diffusion boundary layer and the surface velocity field is negligible. This is the reason why the reduction of equation (8.8) to variables 0, P leads to a coefficient on the right hand side which independent of T = xsin 0,... 

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

The surface concentration T, like the surface tension a, is an appropriate characteristic value. In deriving the result, take the diffusion boundary layer thickness on the nose of the bubble to be given in order of magnitude by S a/ u a/D). ... 

The existence of a thin diffusion boundary layer near the bubble surface allows us to find an approximate solution of the formulated problem. Let us use the method of integral relations, which boils down to selecting a diffusion layer of thickness (5 J in the liquid around the bubble, with the assumption that the change of concentration of the dissolved component from up to p j occurs in this layer. Then following conditions should be satisfied ... 

Here Mi follows certain cumbersome equation,including f(k). The approximation Ja l corresponds to the case of a thin thermal boundary layer around the growing bubble. Since, for polymeric solutions Le 1, the condition of small thickness of the diffusion boundary layer is satisfied in this situation as well. 

At faceted interface by capture of melt-solution droplets (or gas bubbles) from the diffusion boundary layer... 

Figure 5.18 shows the only reliable Nui c data available near the critical Reynolds number (XI). Since the data were taken with a side support, there is some effect on the separation and transition angles. Thus the values of Nuj are probably subject to error (R2, R3) although the trend with Re should be correct. At Re = 0.87 x 10 the Shi variation is similar to that shown at lower Re in Fig. 5.17. At Re = 1.76 x 10 the critical transition has already occurred, with the separation bubble accounting for the minimum in Nuj c at 0 — 110°. The maximum in Nuj at 0 = 125° reflects the increased transfer rate in the attached turbulent boundary layer. The local minimum at 0 = 160° is due to final separation. These angles do not agree exactly with those in Fig. 5.11 because of the crossflow support and the fact that angular diffusion shifts the...

For Re < 110 the wake is closed and laminar as discussed above. Transfer over the front portion of the cap is again described by Eq. (8-20). Transfer from the base occurs by diffusion into the wake fluid as it moves along the bubble base, producing a concentration boundary layer. The solute in this... 

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. 

Transfer from a gas phase to a microorganism occurs according to the following mechanisms (1) transport by convection in the gas bubble (2) diffusion through the gas boundary layer in the vicinity of the gas-liquid interface (3)... 

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