Diffusion coefficients discontinuity

For instance, of great interest are through or continuous execution schemes available for solving the diffusion equation with discontinuous diffusion coefficients by means of the same formulae (software). No selection of points or lines of discontinuities of the coefficients applies here. This means that the scheme remains unchanged in a neighborhood of discontinuities and the computations at all grid nodes can be carried out by the same formulae without concern of discontinuity or continuity of the diffusion coefficient. 

Homogeneous through execution schemes are quite applicable in the cases where the diffusion coefficient is found as an approximate solution of other equations. For instance, such schemes are aimed at solving the equations of gas dynamics in a heat conducting gas when the diffusion coefficient depends on the density and has discontinuities on the shock waves. 

In nonporous membranes, diffusion occurs as it would in any other nonporous solid. However, the molecular species must first dissolve into the membrane material. This step can oftentimes be slower than the diffusion, such that it is the rate-limiting step in the process. As a result, membranes are not characterized solely in terms of diffusion coefficients, but in terms of how effective they are in promoting or limiting both solubilization and diffusion of certain molecular species or solutes. When the solute dissolves in the membrane material, there is usually a concentration discontinuity at the interface between the membrane and the surrounding medium (see Figure 4.55). The equilibrium ratio of the solute concentration in one medium, c, to the solute concentration in the surrounding medium, C2, is called the partition coefficient, K12, and can be expressed in terms of either side of the membrane. For the water-membrane-water example illustrated in Figure 4.55,... 

Colburn and Welsh (3) have discussed the application of individual transfer resistances in countercurrent liquid-liquid extraction and have considered in a preliminary way the effects of viscosity, density, and diffusion coefficients on individual H.T.U. They obtained correlations which indicated that, other variables being equal, the phase with the largest H.T.U. should be made discontinuous. Additional work is required to provide fundamental data for individual film resistances and individual H.T.U. which could be used to interpret existing empirical data and permit predictions from physical properties. 

Consider sensory hairs 2 pm in diameter that are 20 pm apart. Inserting these values into equation (21.22) and assuming that D is 2.5 x 10 6 m2/s (the diffusion coefficient for bombykol, the main component of the commercial silkmoth sex pheromone Adam and Delbriick, 1968) results in the prediction that these hairs are likely to interfere with each other s odorant interception when the air speed between the hairs is below 0.0125 m/s. This is not a discontinuous function - the sensory hairs will interfere with each other more at slower speeds and less at faster speeds. Another way of appreciating what this means quantitatively is to recognize that the root mean square displacement of a molecule (considering movement in one dimension) is... 

Dsd is the surface diffusion coefficient of Meads and f dt denotes the charge transfer resistance at line discontinuities at = 0. 

The distribution of these impurities or minor alloy constituents near lattice discontinuities is known to affect the chemical and mechanical properties of the contaminated materials for example the presence of sulfur on a metal surface can promote ) or retard - o) corrosion, modify the surface energy ) or cause considerable increase in the surface self-diffusion coefficient ). Sulfur accumulation along grain boundaries may induce intergranular weakness and render otherwise ductile materials brittle ), either by formation of precipitates " ) or by enhancement of hydrogen adsorption >227)... 

Based on these qualitative arguments, we may predict that for 8 < 1, in which the diffusion coefficient of the autocatalyst is greater than that of the reactant, the overall tendency will be a stabilization of the planar front. For 8 > 1, however, we anticipate that the planar front will lose stability as the destabilizing influence of the reactant diffusion becomes dominant. To test this prediction, Eqs. [91] were numerically integrated with 8 = 1 and 8 = 5. The initial conditions correspond to a discontinuity in the concentrations of A and B a = 0, p = 1 for X s Xq and a = 1, p = 0 for x > Xg at some convenient Xq for all y. To perturb the planar front, the middle third in the y direction was displaced sli tly forward. For 8 = 1, the perturbation eventually completely decays away to yield a planar front. For 8 = 5, the perturbation evolves to produce a distinctly nonplanar front in which four spatial oscillations in the y direction are displayed. The concentration profiles of a and P for the front are... 

Figure 3 A plot of diffusion coefficient vs. the reciprocal of temperature at constant density for naphthalene in carbon dioxide. The line denoted 7c separates the supercritical region on the left from the subcritical (liquid) region on the right Note that there is no discontinuity when the fluid changes from supercritical to subcritical. The solid lines represent constant densities, from top to bottom of 0.60, 0.70, 0.80, and 0.90gcm , even though temperature is changing continuously from left to right The dashed line indicates a constant pressure line.

---