Thickness of the diffusion boundary layer

If the thickness of the diffusion boundary layer is smaller than b — a (and also smaller than a), one may consider that the diffusion takes place from the sphere to an infinite liquid. It should be emphasized here that the thickness of the diffusion boundary layer is usually about 10 % of the thickness of the hydrodynamic boundary layer (L3). Hence this condition imposes no contradiction to the requirements of the free surface model and Eq. (195). ... 

In natural waters, unattached microorganisms move with the bulk fluid [55], so that no flux enhancement will occur due to fluid motion for the uptake of typical (small) solutes by small, freely suspended microorganisms [25,27,35,41,56,57], On the other hand, swimming and sedimentation have been postulated to alleviate diffusive transport limitation for larger organisms. Indeed, in the planar case (large r0), the diffusion boundary layer, 8, has been shown to depend on advection and will vary with D according to a power function of Da (the value of a is between 0.3 and 0.7 [43,46,58]). For example, in Chapter 3, it was demonstrated that in the presence of a laminar flow parallel to a planar surface, the thickness of the diffusion boundary layer could be estimated by ... 

The basic assumption is that the rotating filter creates a laminar flow field that can be completely described mathematically. The thickness of the diffusion boundary layer (5) is calculated as a function of the rotational speed (to), viscosity, density, and diffusion coefficient (D). The thickness is expressed by the Levich equation, originally derived for electrochemical reactions occurring at a rotating disk electrode ... 

Lionbashevski et al. (2007) proposed a quantitative model that accounts for the magnetic held effect on electrochemical reactions at planar electrode surfaces, with the uniform or nonuniform held being perpendicular to the surface. The model couples the thickness of the diffusion boundary layer, resulting from the electrochemical process, with the convective hydrodynamic flow of the solution at the electrode interface induced by the magnetic held as a result of the magnetic force action. The model can serve as a background for future development of the problem. 

Figure 3.6. Schlieren photographs showing the changes in thickness of the diffusion boundary layer and the behavior of buoyancy-driven convection shown in relation to bulk supersaturation [1], [2]. The figure shows the (111) faceofaBa(N03)2 crystal from an aqueous solution. In region I, only the thickness of the diffusion boundary layer increases in region II, we see unstable lateral convection (HA) and intermittently rising plumes (IIB) and in region III we see steady buoyancy-driven convection.

Since the diffusion coefficient of the dust particles is very small, the thickness of the diffusion boundary layer is small compared to the radius R of the collector. Therefore, the concentration distribution and the rate of deposition can be calculated by substituting for the velocity and electric fields the expressions valid for y/R 1. In that region, Eqs. (182) and (183) become... 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

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 ... 

It is clear that under conditions of decomposition control the rate of dissolution of a solid in a liquid is independent of the thickness of the diffusion boundary layer and hence of the intensity of agitation of the liquid. By contrast, in the case of diffusion control the intensity of agitation of a liquid has a strong effect on the thickness of the diffusion boundary layer, thus influencing the value of the dissolution-rate constant, k. 

According to V.G. Levich299 (see also Refs 300, 302, 304), the thickness of the diffusion boundary layer at the rotating disc surface is determined by the equation... 

Compared to small molecules the description of convective diffusion of particles of finite size in a fluid near a solid boundary has to account for both the interaction forces between particles and collector (such as van der Waals and double-layer forces) and for the hydrodynamic interactions between particles and fluid. The effect of the London-van der Waals forces and doublelayer attractive forces is important if the range over which they act is comparable to the thickness over which the convective diffusion affects the transport of the particles. If, however, because of the competition between the double-layer repulsive forces and London attractive forces, a potential barrier is generated, then the effect of the interaction forces is important even when they act over distances much shorter than the thickness of the diffusion boundary layer. For... 

The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. 

To obtain a physical interpretation of these results, it is convenient to introduce the dimensionless thickness of the diffusion boundary layer as follows ... 

By comparing formulas (3.4.9)—(3.4.11) with (3.4.27)-(3.4.29), we see that for x 1 the thickness of the diffusion boundary layer near the free surface of the film, So Pe /2, is considerably less than that of the boundary layer near the solid surface, <5soi Pe-1 3. Accordingly, the diffusion flux to the free surface is larger than that to the solid surface. Moreover, the diffusion flux decreases more rapidly on the free surface than on the solid boundary with the increase of the distance from the input cross-section. These effects are due to the fact that the fluid moves much more rapidly near the free surface than near the solid boundary, where the no-slip condition is satisfied. 

While the thickness of the hydrodynamic boundary layer can be estimated by L / Re", a similar estimate L / Pe" is obtained for the thickness of the diffusion boundary layer 8p, with n, and nj having values less than unity. Therefore, a rough estimate of the ratio 8p / is Pr " at n, = nj. The kinematic viscosity of mobile water-like liquids is of the order of V 10" cm%, the diffusion coefficients of molecules and ions in aqueous solutions are of the order of D 10 cm%, those of macromolecules D 10 cm%. Thus, in water and in similar... 

Figure 6.7 Model concentration profile at the membrane-solution interface during electrodialysis (cation exchange membrane). F is the Faraday constant, Clf C2, C3 and C4, concentrations of desalting side solution, of the membrane—solution interface desalting side, at the membrane-solution interface (the concentrated side) and of the concentrated solution, respectively <5m, is the thickness of ion exchange membrane, <5i and d2 the thickness of the diffusion boundary layer at the desalting side and concentrated side, respectively t+ and t+ are the transport numbers of the cation in the solution and in the membrane, I is the current density, D and Dm are diffusion coefficients of electrolyte in the solution and in the membrane.

In connection with these studies, the thickness of the diffusion boundary layer can be directly observed by optical methods such as the Schlieren-diagonal method,7 linear laser interferometry8 and by the change in color of an indicator such as methyl red at the membrane-solution interface.9 Further, the concentration polarization at membrane-solution interfaces in electrodialysis has been experimentally and theoretically analyzed in detail.10... 

The correlating equation [67] established here can be used to evaluate the mass transfer coefficient and the thickness of the diffusion boundary layer, S(= d/sh). The thickness of this layer calculated for an organic solvent and aqueous solution were 10 -10 and 10 -10 cm, respectively, for the four types of quaternary salts studied. For a solute crossing a mass transfer resistance film, the transfer time can be approximately estimated by the following equation [131] ... 

There are various ways of using Equations 8 and 9 to obtain information about the solidification process. The simplest one is to do an order of magnitude analyses, OMA, of these equations. This yields immediately that on a relative basis the first, second and third terms are of order 1/t, U/6 and DL/62, where 6 is the approximate thickness of the diffusion boundary layer. Equating the first and last terms gives... 

Consider the diffusion process at an infinite fiat wall [3, 4], Estimate the thickness of the diffusion boundary layer. For simplicity, assume the process to be stationary. It is known from the theory of a viscous boundary layer that v/u S /L, therefore u8C/8x vdC/dy. Since Sd Su, the structure of velocity in diffusion layer is equal to the velocity in the immediate vicinity of the wall... 

We may then derive an expression for the diffusion flux, which turns out to be directed not from the wall, but toward it, and the thickness of the diffusion boundary layer. It is obvious that these characteristics will coincide with (6.46) and (6.47), but in these equations, Cjat should be replaced with Cq. 

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