Boundary thickness, diffuse

For diffusion in a biphasic system, there is the additional complication of the phase boundary. Therefore, diffusion in each phase will be described by Equation 2.11, but in the region of the phase boundary different rules apply to take into account the mass transfer of the reactant from one phase to the other. Where the solubility of the solute is the same in both phases, the rate of diffusion across the phase boundary J for a solute moving from the higher concentration [A]i to the lower concentration [A]2 through a film of thickness l is given by Equation 2.12, which also describes an exponential decrease in concentration, but... 

Formulation. Consider two unity thick diffusion layers of a mixture of 1, 1-, and 1, z-valent3 electrolytes with a common anion, adjacent to a planar ideally permselective cation-exchange membrane. Direct the axis x normally to the membrane and let x = 0 coincide with the outer boundary of the diffusion layer. The diffusion layers will thus be located at 0 < x < 1 and 1 + Aelectro-diffusional transfer of ions across the membrane and the diffusion layers is... 

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

Figure 10.4 Depiction of electrode roughness compared to diffusion layer thickness, vDt. Dotted line indicates approximate boundary of diffusion layer, with (A) diffusion layer thickness greater than surface roughness, resulting in an observed area equal to the projected area, and (B) diffusion layer thickness on the order of surface roughness, resulting in a larger apparent electrode area.

Koberstein J. T., Morra B. and Stein R. S., The determination of diffuse-boundary thickness of polymers by small angle X-ray scattering. J. Appl. Cryst. 13(1980) pp. 34-45. 

The mass-, heat- and momentum-transfer equations and their corresponding boundary conditions discussed so far are obviously very complex and their solutions are not trivial to obtain. Moreover, the thickness, diffusivities and conductivity of each layer in the membrane element are difficult to measure. It is, therefore, convenient and reasonable to consider the permselective membrane layer and the support layer(s) as an integral region with effective thickness, diffusivities and conductivity for the composite region. And it is also desirable to search for simpler models which are capable of providing the... 

FIGURE 2 Noyes-Whitney dissolution model where Ci is the drug solubility at the same conditions as the particle surface, Cb is the concentration in the bulk dissolution medium and A is the thickness of the boundary or diffusion layer. 

The concentration gradients in an asymmetric membrane are complex because the driving force for diffusion in the skin layer is the concentration gradient of gas dissolved in the dense polymer, and the driving force in the porous support layer is a concentration or pressure gradient in the gas-filled pore. When the porous layer is thick, diffusion does not contribute very much to the flux, and gas flows by laminar flow in the tortuous pores. For high-flux membranes, there may also be significant mass-transfer resistances in the fluid boundary layers on both sides. 

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).

Experimental studies of micellar systems were carried out using scattering methods [Selb et al., 1983 Rigby and Roe, 1984, 1986 Kinning et al., 1990, 1991]. Theoretical simulations of the scattering curves have been based on the assumptions that either an infinitely sharp boundary thickness or a diffuse interfacial thickness is equal to Al. In spite of these seemingly diverse principles the simulations were reasonably correct. 

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