Diffusion boundary-layer thickness

Fig. 5.5 Oxygen gradient measured in situ by a benthic lander in Skagerrak at the transition between the Baltic Sea and the North Sea at 700 m water depth. Due to the high depth resolution of the microelectrode measurements it was possible to analyze the microgradient within the 0.5 mm thick diffusive boundary layer. The framed part in the upper graph is blown up in the lower graph. (Data from Gundersen et al. 1995).

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 free-convection mass transfer at electrodes, as well as in forced convection, the concentration (diffusion) boundary layer (5d extends only over a very small part of the hydrodynamic boundary layer <5h. In laminar free convection, the ratio of the thicknesses is... 

The velocity of liquid flow around suspended solid particles is reduced by frictional resistance and results in a region characterized by a velocity gradient between the surface of the solid particle and the bulk fluid. This region is termed the hydrodynamic boundary layer and the stagnant layer within it that is diffusion-controlled is often known as the effective diffusion boundary layer. The thickness of this stagnant layer has been suggested to be about 10 times smaller than the thickness of the hydrodynamic boundary layer [13]. 

The diffusion boundary layer thickness depends on D, and consequently the viscosity of the medium, r, and the geometry of the microorganism. In the absence of flow, the diffusion boundary layer of large or planar surfaces (n> > (5) can be defined by [40,43] ... 

Fluid motion acts to decrease the diffusion boundary layer thickness. Strategies of the microorganism to increase solute flux by decreasing its size or surface concentrations of the solute, c°, will be examined in Section 6. In this section, the solute concentration at the surface of the organism, c°, is assumed to be zero, i.e. the cell is a perfect absorber (sink), since this will provide an upper limit for the importance of fluid motion. It is clear that if fluid motion has no effect for a perfect absorber, it will have no effect for an imperfect one. 

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 ... 

Linear diffusion0 of a 500 nm colloid through a diffusion boundary layer thickness of 50 pm 3 x 104... 

Curves corresponding to various values of St have been drawn in Figure 8.17. A standard chemical (or diffusion) boundary layer thickness 5 can be defined as the... 

A convenient estimate of the anomalous layer thickness (chemical boundary layer) is given by 8 = 3>/v. Indeed, the excess or deficit M of diffusing substance is equal to the area limited by the concentration profile and the initial distribution C0 in the liquid... 

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. 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

In Example 8.5, we saw how the diffusive boundary layer could grow. The boundary never achieves 1 m in thickness or even 5 cm in thickness, because of the interaction of turbulence and the boundary layer thickness. The diffusive boundary layer is continually trying to grow, just as the boundary layer of Example 8.5. However, turbulent eddies periodically sweep down and mix a portion of the diffusive boundary layer with the remainder of the fluid. It is this unsteady character of the turbulence... 

The eddy diffusion coefficients that we introduced in Chapter 5 were steady quantities, using mean turbulence quantities (e.g., the temporal mean of u C). This temporal mean character of eddy diffusion coefficients can be misleading in determining the thickness of a diffusive boundary layer because of the importance of unsteady characteristics. We will review some conceptual theories of mass transfer that have been put forward to describe the interaction of the diffusive boundary layer and turbulence. 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

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.

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... 

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... 

Equations for the thickness 8 of the diffusion boundary layer can be obtained by solving Eq. (372). Equations (369a) and (369b) are compatible with Eq. (373) only if... 

Diffusion-layer thickness Thermal-boundary layer... 

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 ... 

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