Diffusive Boundary Layers

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). ... 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Again, the form of the concentration profile in the diffusion boundary layer depends on the conditions which are assumed to exist at the surface and in the fluid stream. For the conditions corresponding to those used in consideration of the thermal boundary layer, that is constant concentrations both in the stream outside the boundary layer and at the surface, the concentration profile is of similar form to that given by equation 11.70 ... 

Note that Eqs. (4) and (5) implicitly consider the transfer across the interface as the rate-determining step in the ion transfer processes [51], and neglect other steps involved in the process such as the ion transport across the diffusion boundary layers [55] and across the diffuse electrical double layer [50]. 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

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]. 

Reduction of particle size increases the total specific surface area exposed to the solvent, allowing a greater number of particles to dissolve more rapidly. Furthermore, smaller particles have a small diffusion boundary layer, allowing faster transport of dissolved material from the particle surface [58]. These effects become extremely important when dealing with poorly water-soluble drugs, where dissolution is the rate-limiting step in absorption. There are numerous examples where reduction of particle size in such drugs leads to a faster dissolution rate [59-61], In some cases, these in vitro results have been shown to correlate with improved absorption in vivo [62-64]. 

Ploug, H., Stolte, W. and Jorgensen, B. B. (1999). Diffusive boundary layers of the colony-forming plankton alga Phaeocystis sp - implications for nutrient uptake and cellular growth, Limnol. Oceanogr., 44, 1959-1967. 

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 ... 

Four strategies are generally employed to demonstrate mass transfer limitation in aquatic systems. Most commonly, measured uptake rates are simply compared with calculated maximal mass transfer rates (equation (17)) (e.g. [48,49]). Uptake rates can also be compared under different flow conditions (e.g. [52,55,56,84]), or by varying the biomass under identical flow conditions (e.g. [85]). Finally, several recent, innovative experiments have demonstrated diffusion boundary layers using microsensors [50,51]. Of the documented examples of diffusion limitation, three major cases have been identified ... 

It is perhaps wise to begin by questioning the conceptual simplicity of the uptake process as described by equation (35) and the assumptions given in Section 6.1.2. As discussed above, the Michaelis constant, Km, is determined by steady-state methods and represents a complex function of many rate constants [114,186,281]. For example, in the presence of a diffusion boundary layer, the apparent Michaelis-Menten constant will be too large, due to the depletion of metal near the reactive surface [9,282,283], In this case, a modified flux equation, taking into account a diffusion boundary layer and a first-order carrier-mediated uptake can be taken into account by the Best equation [9] (see Chapter 4 for a discussion of the limitations) or by other similar derivations [282] ... 

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

Koch, E. W. (1994). Hydrodynamics, diffusion-boundary layers and photosynthesis of the seagrasses, Thalassia testudinum and Cymodocea nodosa, Mar. Biol., 118, 767-776. 

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... 

Liu Z, Dreybrodt W. Dissolution kinetics of calcium carbonate minerals in H20-C02 solutions in turbulent flow—the role of the diffusion boundary layer and the slow reaction H20 + CO2 <-> H+ + HC03. Geochim Cosmochim Acta 1997 61(14) 2879-2889. 

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. 

In addition to the interphase potential difference V there exists another potential difference of fundamental importance in the theory of the electrical properties of colloids namely the electro-kinetic potential, of Freundlich. As we shall note in subsequent sections the electrokinetic potential is a calculated value based upon certain assumptions for the potential difference between the aqueous bulk phase and some apparently immobile part of the boundary layer at the interface. Thus represents a part of V but there is no method yet available for determining how far we must penetrate into the boundary layer before the potential has risen to the value of the electrokinetic potential whether in fact f represents part of, all or more than the diffuse boundary layer. It is clear from the above diagram that bears no relation to V, the former may be in fact either of the same or opposite sign, a conclusion experimentally verified by Freundlich and Rona. 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the 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... 

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