Diffusion normalized

The quantities (7.5) can be determined in some simple cases. In the simplest case, when no hydrodynamic interaction is assumed, one uses equation (3.8) with matrix (3.10) and, omitting the diffusive normal mode with the label 0, has... 

Why is diffusion normally such a slow process, despite the rapid movement of gas molecules Under what conditions does diffusion take place with the speed of molecular motion ... 

The figure shows strong anisotropy in the diffusion of zeolitic water, the dark band having moved further from the 201 than from the 001 face. is given in Table II the activation energies, , in the Arrhenius equation Da =Do exp — E/RT are 5400 and 9140 cal/mole, respectively, for diffusion normal to 201 and 001. 

Composition profiles are shown in Figure 8.13. Ditfusional interactions are quite limited in this system due to the similar nature of all three components. As a result, all three components diffuse normally (Fig. 8.13). ... 

A set of Fick s diffusional equations are applied to the case, linear diffusion normal to the electrode with simultaneous chemical reactions. 

In this case the differential equation corresponds to diffusion normal and radial (two-dimensional) to the electrode. They obtained the following equations describing the faradaic impedance in the case of a slow charge transfer when only Ox is initially present in the solution, its concentration being Cq ... 

We may note that the thermal boundary layer in this case is asymptotically thin relative to the boundary layer for a solid body. This is a consequence of the fact that the tangential velocity near the surface is larger, and hence convection is relatively more efficient. From a simplistic point of view, the larger velocity means that convection parallel to the surface is more efficient, and hence the time available for conduction (or diffusion) normal to the surface is reduced. Thus, the dimension of the fluid region that is heated (or within which solute resides) is also reduced. Indeed, if we define Pe by using a characteristic length scale lc and a characteristic velocity scale uc, heat can be conducted a distance... 

The implantation of MeV Si ions into a Si substrate can also suppress boron-enhanced diffusion normally associated with a high boron concentration layer (Shao et al. 2003). Junction depths of 20 nm were achieved in samples implanted with 0.5 keV B ion at a dose of 1015 cm-2 following a 1,000°C thermal anneal. 

In another approach, the interfacial diffusion of the nanoparticles was determined using two photobleaching methods fluorescence loss induced by photobleaching (FLIP) and fluorescence recovery after photobleaching (FRAP). It was found that the lateral diffusion of the nanoparticles at the interface as well as the diffusion normal to and from the interface deviated by about four orders of magnitude from the values obtained in free solution [46],... 

Within the ellterate layer, radicals would diffuse normally, undergoing their usual reactions. However. if the etlierale layer had the right thickness, most of the solvent attack could he by radicals that escaped Iron) it into the bulk solution. When one of these radicals diffused back to the interlace between liquid phases, both the viscosity and the polarity of the ellterate would act to inhibit its reentering the ellterate. If solvent attack were slowed hy solvent deuieration. radicals would be diverted to c, mostly, rather than r. because they would not diffuse to Mg/ (Figure 7.41). 

A molecule of R generated at the edge of the disk (r = r ) must diffuse normal to the disk to reach the ring, since is zero at y = 0. It is then swept in a radial direction and then moves by diffusion and convection in the y direction to reach the inner edge of the ring. This path can be described by some average trajectory and some time-dependent distance y from the electrode surface. Integration of (9.5.6) yields... 

Diffusion normal to the electrode is also taken into account by the procedures described in Section B.1.2. 

The penetration theory of Higbie replaces the stagnant fluid by intermittently static and moving eddies that arrive at the interface from the bulk stream, stay for a period of time in the interface (during which molecular diffusion normal to... 

Now, it is necessary to discuss the mass transfer coefficient for component j in the boundary layer on the vapor side of the gas-liquid interface, fc ,gas, with units of mol/(area-time). The final expression for gas is based on results from the steady-state film theory of interphase mass transfer across a flat interface. The only mass transfer mechanism accounted for in this extremely simple derivation is one-dimensional diffusion perpendicular to the gas-liquid interface. There is essentially no chemical reaction in the gas-phase boundary layer, and convection normal to the interface is neglected. This problem corresponds to a Sherwood number (i.e., Sh) of 1 or 2, depending on characteristic length scale that is used to define Sh. Remember that the Sherwood number is a dimensionless mass transfer coefficient for interphase transport. In other words, Sh is a ratio of the actual mass transfer coefficient divided by the simplest mass transfer coefficient when the only important mass transfer mechanism is one-dimensional diffusion normal to the interface. For each component j in the gas mixture. 

Obviously, the gas-liquid interface is curved, but MTBLTuquid/ /gas 1 at very high Schmidt numbers. Under these conditions, the effect of curvature is not important. The steady-state microscopic mass balance for benzene (B) in the liquid phase with one-dimensional diffusion normal to the interface and first-order irreversible chemical reaction is... 

Fig. 9 Probability distribution function (p r) (a) and effectiveness factor rj k) (b) corresponding to the tracer exchange curves in the limiting cases of dominating single-file diffusion, normal diffusion and surface barriers as a function of the quotient of r and Tintra- From [74] with permission...

Figure 9 displays the probability distribution function (p r) and the effectiveness factor r] k), which have been calculated via Eqs. 36 and 34 from the tracer exchange curves in the limiting cases of single-hle diffusion, normal diffusion and barrier confinement. The fact that in all cases the residence time distribution function is found to decrease monotonically may be easily rationalized as a quite general property. Due to the assumed stationarity of the residence time distribution function, the number of molecules with a residence time r is clearly the same at any instant of time. The number of molecules with a residence time r + At may therefore be considered as the number of molecules with a residence time r minus the number of molecules which will leave the system in the subsequent time interval At. Therefore, (p x) must quite generally be a monotonically decaying function. 

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