Boundary layer, diffusion developing

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

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... 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

However, the two-sink model as well as other existing adsorption (sink) models do not seem to be able to describe the strong asymmetry between the adsorption/desorption of VOCs on/from indoor surface materials (the desorption process is much slower than the adsorption process). Diffusion combined with internal adsorption is assumed to be capable of explaining the observed asymmetry. Diffusion mechanisms have been considered to play a role in interactions of VOCs with indoor sinks. Dunn and Chen (1993) proposed and tested three unified, diffusion-limited mathematical models to account for such interactions. The phrase unified relates to the ability of the model to predict both the ad/absorption and desorption phases. This is a very important aspect of modeling test chamber kinetics because in actual applications of chamber studies to indoor air quality (lAQ), we will never be able to predict when we will be in an accumulation or decay phase, so that the same model must apply to both. Development of such models is underway by different research groups. An excellent reference, in which the theoretical bases of most of the recently developed sorption models are reviewed, is the paper by Axley and Lorenzetti (1993). The authors proposed four generic families of models formulated as mass transport modules that can be combined with existing lAQ models. These models include processes such as equilibrium adsorption, boundary layer diffusion, porous adsorbent diffusion transport, and conveetion-diffusion transport. In their paper, the authors present applications of these models and propose criteria for selection of models that are based on the boundary layer/conduction heat transfer problem. 

Walsham and Edwards [20] have developed a model which satisfactorily describes the evaporation behaviour of solvent blends. They based their work on that of Sletmoe [21], who found that the solvent evaporation rate was limited by boundary layer diffusion into the vapour phase. On this basis the rate of evaporation should be proportional to the vapour pressure of the solvent. Deviations in evaporation behaviour from ideality were directly analogous to deviations from Raoult s law of vapour pressures and could be corrected using activity coefficients. Since for most solvent molecules activity coefficients are largely deter-... 

Consider the electrolyte hydrodynamic flow condition shown in Figure 7.17. This is amass transfer found in electrowinning and electrorefining cells, in which the electrolyte motion is upwards on the anode surface due to the generation of oxygen bubbles, which enhances the mass transfer. The reader can observe in this model that the electrolyte motion is more intense at the top than at the bottom of the electrode as indicated by the arrows. If the current flows, then the electrolyte motion increases and descends to the bottom of the cathode. This is the case in which a significant convective molar flux is superimposed on the Pick s diffusion molar flux. The concentration gradient in the fluid adjacent to the vertical electrode (plate) surface causes a variation in the fluid density and the boundary layer (tf ) develops upwards from laminar to turbulent conditions [2,7]. 

Diffusion-blading loss. This loss develops because of negative velocity gradients in the boundary layer. Deceleration of the flow increases the boundary layer and gives rise to separation of the flow. The adverse pressure gradient that a compressor normally works against increases the chances of separation and causes significant loss. 

Adhesion development depends on diffusion of the CPO component of the primer through the crystalline boundary layers followed by swelling and entanglement with the rubber rich layer [75]. 

Sweating, the other powerful heat loss mechanism actively regulated by the thermoregulatory center, is most developed in humans. With about 2,6 million sweat glands distributed over the skin and neurally controlled, sweat secretion can vary from 0 to 1 I7(h m ). The other, lesser, passive evaporative process of the skin is from the diffusion of water. The primary resistance to this flow is the stratum corneum or outermost 15 pm of the skin. The diffusion resistance of the skin is high in comparison to that of clothing and the boundary layer resistance and as a result makes water loss by diffusion fairly stable at about 500 grams/day. 

The explicit mathematical treatment for such stationary-state situations at certain ion-selective membranes was performed by Iljuschenko and Mirkin 106). As the publication is in Russian and in a not widely distributed journal, their work will be cited in the appendix. The authors obtain an equation (s. (34) on page 28) similar to the one developed by Eisenman et al. 6) for glass membranes using the three-segment potential approach. However, the mobilities used in the stationary-state treatment are those which describe the ion migration in an electric field through a diffusion layer at the phase boundary. A diffusion process through the entire membrane with constant ion mobilities does not have to be assumed. The non-Nernstian behavior of extremely thin layers (i.e., ISFET) can therefore also be described, as well as the role of an electron transfer at solid-state membranes. 

Mooney et al. [70] investigated the effect of pH on the solubility and dissolution of ionizable drugs based on a film model with total component material balances for reactive species, proposed by Olander. McNamara and Amidon [71] developed a convective diffusion model that included the effects of ionization at the solid-liquid surface and irreversible reaction of the dissolved species in the hydrodynamic boundary layer. Jinno et al. [72], and Kasim et al. [73] investigated the combined effects of pH and surfactants on the dissolution of the ionizable, poorly water-soluble BCS Class II weak acid NSAIDs piroxicam and ketoprofen, respectively. 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

The convective diffusion theory was developed by V.G. Levich to solve specific problems in electrochemistry encountered with the rotating disc electrode. Later, he applied the classical concept of the boundary layer to a variety of practical tasks and challenges, such as particle-liquid hydrodynamics and liquid-gas interfacial problems. The conceptual transfer of the hydrodynamic boundary layer is applicable to the hydrodynamics of dissolving particles if the Peclet number (Pe) is greater than unity (Pe > 1) (9). The dimensionless Peclet number describes the relationship between convection and diffusion-driven mass transfer ... 

COSILAB Combustion Simulation Software is a set of commercial software tools for simulating a variety of laminar flames including unstrained, premixed freely propagating flames, unstrained, premixed burner-stabilized flames, strained premixed flames, strained diffusion flames, strained partially premixed flames cylindrical and spherical symmetrical flames. The code can simulate transient spherically expanding and converging flames, droplets and streams of droplets in flames, sprays, tubular flames, combustion and/or evaporation of single spherical drops of liquid fuel, reactions in plug flow and perfectly stirred reactors, and problems of reactive boundary layers, such as open or enclosed jet flames, or flames in a wall boundary layer. The codes were developed from RUN-1DL, described below, and are now maintained and distributed by SoftPredict. Refer to the website http //www.softpredict.com/cms/ softpredict-home.html for more information. 

Although Rs values of high Ks compounds derived from Eq. 3.68 may have been partly influenced by particle sampling, it is unlikely that the equation can accurately predict the summed vapor plus particulate phase concentrations, because transport rates through the boundary layer and through the membrane are different for the vapor-phase fraction and the particle-bound fraction, due to differences in effective diffusion coefficients between molecules and small particles. In addition, it will be difficult to define universally applicable calibration curves for the sampling rate of total (particle -I- vapor) atmospheric contaminants. At this stage of development, results obtained with SPMDs for particle-associated compounds provides valuable information on source identification and temporal... 

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. 

Equation (4) states that the linear deposition rate vj is a diffusion controlled boundary layer effect. The quantity Ac is the difference in foulant concentration between the film and that in the bulk flow and c is an appropriate average concentration across the diffusion layer. The last term approximately characterizes the "concentration polarization" effect for a developing concentration boundary layer in either a laminar or turbulent pipe or channel flow. Here, Vq is the permeate flux through the unfouled membrane, 6 the foulant concentration boundary layer thickness and D the diffusion coefficient. 

For developing concentration diffusion layers in channel or pipe flow the concentration boundary layer thickness is expressible by the proportionality... 

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

These features become even more accentuated if the dimensionless diffusion coefficient D is made smaller. In the limit D - 0, as may be expected, the profiles tend to be virtually uniform across most of the region, with ass and j3ss given by eqn (9.33). At the edges of the reaction zone the system develops thin boundary layers, with thickness of order D 12. 

As long as there is no phase change involved, the influence of the transition zone on mass transfer is negligible. The position of the boundary layer is slightly shifted, but the exchange flux is scarcely affected. This is no longer true if the boundary separates two different media, for instance, the water of a lake from the sediments. In this case the drop of diffusivity D(x) and the increase of the partition ratio KAIB (Eq. 19-27) do not coincide (Fig. 19.10). Let us first develop the necessary mathematical tools to describe this new situation and then discuss an example for which the influence of the boundary layer may be relevant. 

In the seventies, the growing interest in global geochemical cycles and in the fate of man-made pollutants in the environment triggered numerous studies of air-water exchange in natural systems, especially between the ocean and the atmosphere. In micrometeorology the study of heat and momentum transfer at water surfaces led to the development of detailed models of the structure of turbulence and momentum transfer close to the interface. The best-known outcome of these efforts, Deacon s (1977) boundary layer model, is similar to Whitman s film model. Yet, Deacon replaced the step-like drop in diffusivity (see Fig. 19.8a) by a continuous profile as shown in Fig. 19.8 b. As a result the transfer velocity loses the simple form of Eq. 19-4. Since the turbulence structure close to the interface also depends on the viscosity of the fluid, the model becomes more complex but also more powerful (see below). 

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