Stagnant boundary layer

Ammonia gas is diffusing at a constant rate through a layer of stagnant air 1 mm thick. Conditions are such that the gas contains 50 per cent by volume ammonia at one boundary of the stagnant layer. The ammonia diffusing to the other boundary is quickly absorbed and the concentration is negligible at that plane. The temperature is 295 K and the pressure atmospheric, and under these conditions the diffusivity of ammonia in air is 1.8 x 10 5 m2/s. Estimate the rate of diffusion of ammonia through the layer. 

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 stagnant layer analysis offers a pedagogical framework for presenting the essence of diffusive burning. For the most part the one-dimensional stagnant layer approximated a two-dimensional boundary layer in which 6 = <5(x), with x the flow direction. For a convective boundary layer, the heat transfer coefficient, hc, is defined as... 

The term in the bracket can be regarded as an equivalent heat of combustion for the more complete problem. If this effect is followed through in the stagnant layer solution of the ordinary differential equations with the more complete boundary condition given by... 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

The velocity of the gases is high but it is always laminar flow. Over the susceptor there will be a boundary, or stagnant, layer where the velocity gradient decreases to zero. As the gases are heated, the silane and hydrocarbon will decompose and the species will diffuse through the boundary layer to grow on the reactor walls or on the substrate. A comprehensive study of this may be found in a paper by M. Leys and H. Veenvliet [41]. 

Though there is fluid flow in the bulk of the electrolyte, it is found that there is a layer adjacent to the electrode in which the electrolyte is stationary, or stagnant. Thus the electron acceptors travel by convection from the bulk up to the stagnant layer and then cross the remaining boundary layer by diffusion. This transport by a convection-with-diffusion mechanism has not been taken into account so far. The equations for the time and space variation of concentration [i.e., Eq. (7.178)], for the transition time [Eq. (7.190)], and for the time variation of potential [Eq. (7.192)] have been derived for convection-free conditions, and they break down when convection becomes significant. The first approximation theory given above, therefore, deviates from experiment if the constant current is applied sufficiently long (times on the order of seconds) for convection to be important. 

Another approximate limiting solution for a vertical enclosure (i.e., < 90°) is obtained, as mentioned before, by assuming that the flow consists of boundary layers on the hot and cold walls with an effectively stagnant layer between them and that the presence of these end walls has a negligible effect on the boundary layer flows. The assumed flow is therefore as shown in Fig. 8.31. 

An approximate model of the flow in a vertical porous medium-filled enclosure assumes that the flow consists of boundary layers on the hot and cold w alls with a stagnant layer between the two boundary layers, this layer being at a temperature that is the average of the hot and cold wall temperatures. Use this model to find an expression for the heat transfer rate across the enclosure and discuss the conditions under which this model is likely to be applicable. 

The aqueous boundary layer or the unstirred water layer (UWL) is a more or less stagnant layer, about 30-100 pm in thickness, composed of water, mucus, and glycocalyx adjacent to the intestinal wall that is created by incomplete mixing of the lumenal contents near the intestinal mucosal surface. The glycocalyx is made up of sulfated mucopolysaccharides, whereas mucus is composed of glycoproteins (mucin), enzymes. 

Figure 1. Schematic representation of the electrochemical cell. First insert concentration profile of the reactant in the stagnant layer in the vicinity of the electrode. Second insert mass transfer black box at the boundary between the stagnant layer and the bulk solution (x = (5).

The problem is then to relate the concentration gradient (dC/dx) at the stagnant layer boundary to that (9C/9x)x=o the electrode surface (compare Fig. 1). This is done via the integration of the differential equations, such as Eq. (175) in Chapter 1, which control the concentration profiles within the stagnant layer. In the most simple situation, the species whose bulk concentration is given in Eq. (1) undergoes no chemical reaction within the stagnant layer. Then from Eq. (175) (Chapter 1), its concentration flux is constant witliin the layer, and thus (dC/dx) = (9C/9x)x=o = (C — C =o)/ > where Cx=o... 

Stagnant boundary layer Assumes stagnant layer at the 1... 

The mechanism of dissolution was proposed by Nernst (1904) using a film-model theory. Under the influence of non-reactive chemical forces, a solid particle immersed in a liquid experiences two consecutive processes. The first of these is solvation of the solid at the solid-liquid interface, which causes the formation of a thin stagnant layer of saturated solution around the particle. The second step in the dissolution process consists of diffusion of dissolved molecules from this boundary layer into the bulk fluid. In principle, one may control the dissolution through manipulation of the saturated solution at the surface. For example, one might generate a thin layer of saturated solution at the solid surface by a surface reaction with a high energy barrier (Mooney et al., 1981), but this application is not commonly employed in pharmaceutical applications. 

Diffusion away from the surface. The rate of evaporation is determined in large part by the rate of diffusion through the thin stagnant layer, usually referred to as the boundary layer, of air at the surface. The thickness of this layer is determined by the mrbulence while diffusion rate depends on the diffusion coefficient in air and the concentration gradient. 

In considering the volatilization of contaminants, the factors that must be considered are (a) escape from the interface, (b) diffusion through the surface boundary layer, and (c) turbulent diffusion in the atmosphere. - The escape from the surface depends mainly on the vapor pressure of the contaminant at a given temperature, the molecular weight, and Henry s coefficient. After the contaminant has escaped from the surface, it must diffuse outward in the stagnant boundary layer that is normally present. Then, the contaminant will be transported away from the stagnant layer by advection and turbulent diffusion... 

Taking into account that Sand s equation is valid only as long as the change of concentration occurs within a stagnant layer undisturbed by convection and introducing the Nemst diffusion layer boundary 8 and hydrodynamic layer boundary Ah, the minimum current density that must be applied in the first pulse for electrodeposition of the second layer to take place is given by... 

When a (solid) surface moves in a liquid, or vice versa, there is always a layer of liquid adjacent to the surface that moves with the same velocity as the surface. The distance from the surface over which this stagnant liquid layer extends or, in other words, the location of the boundary between the mobile and the stationary phases, the so-called plane of shear or slip plane, is not exactly known. For smooth surfaces, the plane of shear is within a few liquid (water) molecules from the surface (see Figure 9.4), that is, well within the electrical double layer. The stagnant layer is probably somewhat thicker than the Stern layer, so that the plane of shear is located in the diffuse part of the electrical double layer. It follows that the potential at the plane of shear, that is, the electrokinetic potential or the zeta potential is somewhat lower than the Stern potential /j. Because the largest part of the potential drop in the... 

This latter equation is the basic equation of the boundary layer resistance model [17-19]. The boundary layer can be considered as a concentrated solution through which solvent molecules permeate, with the permeability of this stagnant layer depending very much on the concentration and the molecular weight of the solute. The resistance exerted by thislayer is far much greater for macromolecular solutes (ultrafiltration) relative to for low molecular weight. solutes (reverse osmosis). Because there is a concentration profile in the boundary layer, the permeability P of the solvent may be written as a function of the distance coordinate x w ith the boundaries x = 0 and x = 6. 

The external diffusion limitation (EDL) indicates that substrate transport through diffusion (stagnant) layer is a rate-limiting process [10]. At internal diffusion limitation (IDL), the substrate diffusion through the external diffusion layer is fast, and process is limited by the diffusion inside an enzyme membrane. The disadvantage of these approximate solutions is an error at the boundaries between the different approximate treatments. It is helpful to illustrate this approach by reference to a trivial problem of the substrate conversion in the biocatalytical membrane of the biosensor and at the concentration of the substrate less than the Michaelis-Menten constant Km)- The calculated profile of substrate concentration at steady-state or stationary conditions is shown in Fig. 1. 

On the other hand, at the steady state, a substrate flux through the boundary of stagnant layer/bulk solution is equal to the flux through the boundary of biocatalytical membrane/stagnant layer ... 

For a series of CVs with fixed scan rate, but varied electrode surface temperature, shape of voltammograms changes from classical peak form (without heating) till complete sigmoidal shape (where the increased electrolysis current has driven the concentration profile thickness very fast to the boundary of the stagnant layer, this way generating a diffusion layer of constant thickness). 

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