Diffusion boundary layer approximation

Diffusion boundary layer approximation. Now let us take into account the fact that common fluids are characterized by large Schmidt numbers Sc. Obviously, by substituting the leading term of the expansion of v as z -4 0 into (3.2.8) and (3.2.9), one can readily obtain the asymptotics of these formulas as Sc -4 oo. By using (3.2.5) and (3.2.8) and carrying out some transformation, we obtain the dimensionless concentration... 

Diffusion boundary layer approximation. For x = 0(1), the concentration mostly varies on the initial interval in a thin diffusion boundary layer near the free boundary of the film. In this region we expand the transverse coordinate according to the rule... 

The comparison of this formula with (3.4.11) shows that the diffusion boundary layer approximation can be used in the region x < 0.1 Pe. 

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 solution of the corresponding mass exchange problem for a circular cylinder and an arbitrary shear flow was obtained in [353] in the diffusion boundary layer approximation. It was shown that an increase in the absolute value of the angular velocity Cl of the shear flow results in a small decrease in the intensity of mass and heat transfer between the cylinder and the ambient... 

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). 

Concentration distribution is described by the steady-state equation (3.1.1) and boundary conditions (3.1.2) and (3.1.5), where = Y is the distance from plate surface. In the diffusion boundary layer approximation, the exact analytical... 

The surface concentration can be calculated by the formulae s = C (1 -j/joo)-The results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of power-law surface reaction for n = 1 /2 and n = 2 was indicated in [133]. The maximum inaccuracy of formula (5.1.7) in these cases is about 10% for any value of the surface reaction rate constant. 

Let us investigate steady-state convective diffusion on the surface of a flat plate in a longitudinal translational flow of a viscous incompressible fluid at high Reynolds numbers (the Blasius flow). We assume that mass transfer is accompanied by a volume reaction. In the diffusion boundary layer approximation, the concentration distribution is described by the equation... 

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]. 

Diffusion boundary layer approximation. For high Peclet numbers, problem (5.6.1) was investigated in [236], The solution was obtained by using the diffusion boundary layer method, and the following formula was derived for the mean Sherwood number ... 

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 ... 

Let us assume that the concentration of the absorbed substance on the film surface is constant and is equal to C = Cs and that a pure fluid with zero concentration is supplied through the cross-section with the coordinate X=0. We restrict our consideration to the case of high Peclet numbers, when the diffusion along the film can be neglected. In the diffusion boundary layer approximation (that is, when only the leading term V Umax of the expansion of the fluid velocity near the free boundary is considered), the concentration distribution inside the film is described under the above assumptions by the following equation and boundary conditions ... 

Now let us consider mass transfer from a solid wall to a fluid film. We assume that the concentration on the surface of the plate is constant and is equal to Cs and that a pure fluid is supplied through the input cross-section. In the diffusion boundary layer approximation, the velocity profile near the surface of the plate can be approximated by the expression... 

Convective mass and heat transfer to a plate in a longitudinal flow of a non-Newtonian fluid was considered in [443]. By solving the corresponding problem in the diffusion boundary layer approximation (at high Peclet numbers), we arrive at the following expression for the dimensionless diffusion flux ... 

Gupalo, Yu. P., Polyanin, A. D., Ryazantsev, Yu. S., and Sergeev, Yu. A., Convective diffusion to a drop under arbitrary conditions of absorption. Diffusion boundary layer approximation, Fluid Dynamics, Vol. 14, No. 6, pp. 862-866,1979. 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

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. 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

In Section 17.8.3 we discussed the catalytic combustion of methane within a single one of the tubes in a honeycomb catalyst, illustrated in Fig. 17.18. The high velocity, and thus the dominance of convective over diffusive transport, makes the boundary layer approximations valid for this system. We will model the catalytic combustion performance in one of the honeycomb channels in this problem. 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

Wilson (79, 80) pointed out that A is not the dimensionless thickness of the diffusion boundary layer scaled with D/Vg, as originally suggested by Burton et al. (74), except in the limit at which the velocity field in the layer is dominated by the bulk flow, that is, X >> 1. In this case, the analysis reduces to the one first presented by Levich (81), and the integral in equation 25 is approximated as follows ... 

Knowing an experimental value of k, it is possible to evaluate the diffusion coefficient of the atoms of a dissolving solid substance across the diffusion boundary layer at the solid-liquid interface into the bulk of the liquid phase using equations (5.6) and (5.7). Its calculation includes two steps. First, an approximate value of D is calculated from equation (5.6). Then, the Schmidt number, Sc, and the correction factor, /, is found (see Table 5.1). The final, precise value is evaluated from equation (5.7). In most cases, the results of these calculations do not differ by more than 10 %. Values of the diffusion coefficient of some transition metals in liquid aluminium are presented in Table 5.9.303... 

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