Diffusion diffusional mixing

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

For very dilute solid solutions of B in A, the basic physics of diffusional mixing is the same as for (A, A ). An encounter between VA and BA is necessary to render the B atoms mobile. But B will alter the jump frequencies of V in its surroundings and therefore numerical values of the correlation factor and cross coefficient are different from those of tracer A diffusion. Since the jump frequency changes also involve solvent A atoms, in addition to fB, the numerical value of fA must be reconsidered (see next section). 

In this case, the electrogenerated R will encounter Z as it diffuses away from the electrode and will react to form P. This diffusional mixing of two reactants (R and Z in this example) is the basis for chronoabsorptometry (and other electrochemical techniques) as a kinetic method. The homogeneous chemical reaction of R will perturb its accumulation by a magnitude that is proportional both to k and to the concentration of Z. The influence of this perturbation is apparent from the concentration-distance profiles for R, which are illustrated for a rate constant of 107 L/(mol s) in Figure 3.9. 

Mixing of chemical differences in fluids through diffusion has been dealt with in some detail previously, for example, England et al. (1987) for oil and gas, and Smalley et al. (1995) for water. A general equation that gives an order of magnitude estimate of diffusional mixing times is... 

Mixing due to diffusion occurs when particles roll over a sloping powder surface. The random movement of particles on the free surface results in a redistribution of the particles. As was discussed in section 1.2.2.I, the movement of particles on a free surface can also give rise to segregation due to the percolation of fine particles if particles of different size are present in the mixture. Only when the particles within the mixture have identical properties can diffusional mixing be a truly randomizing process. 

Another modification of the diffusion method is the integral diffusion method developed by Yu.M. Gershenzon and B.F. Monin. In this method the cut of the nozzle is located at the inlet to the cavity of an ESR spectrometer. The reaction is studied directly in the zone of diffusional mixing of reactants. The ratio of the ESR signals in the presence and absence of reactant B is measured. Since the ESR signal is proportional to the total number of active qiecies in the diffusion cloud, the method is integral. 

Diffusional mixing is usually the result of a series of uncorrelated atomic movements of the type described above. Because the movements are not related to one another, any individual atom may move in any direction. However, the net movement of atoms depends upon the concentration gradient for a given species. The result is that the flux of atoms F across a given plane perpendieular to the direction of diffusion, x, may be described by Pick s first law ... 

Ordinary diffusion involves molecular mixing caused by the random motion of molecules. It is much more pronounced in gases and Hquids than in soHds. The effects of diffusion in fluids are also greatly affected by convection or turbulence. These phenomena are involved in mass-transfer processes, and therefore in separation processes (see Mass transfer Separation systems synthesis). In chemical engineering, the term diffusional unit operations normally refers to the separation processes in which mass is transferred from one phase to another, often across a fluid interface, and in which diffusion is considered to be the rate-controlling mechanism. Thus, the standard unit operations such as distillation (qv), drying (qv), and the sorption processes, as well as the less conventional separation processes, are usually classified under this heading (see Absorption Adsorption Adsorption, gas separation Adsorption, liquid separation). 

Confined flows typically exhibit laminar-flow regimes, i.e. rely on a diffusion mixing mechanism, and consequently are only slowly mixed when the diffusion distance is set too large. For this reason, in view of the potential of microfabrication, many authors pointed to the enhancement of mass transfer that can be achieved on further decreasing the diffusional length scales. By simple correlations based on Fick s law, it is evident that short liquid mixing times in the order of milliseconds should result on decreasing the diffusion distance to a few micrometers. 

With the carrier stream unsegmented by air bubbles, dispersion results from two processes, convective transport and diffusional transport. The former leads to the formation of a parabolic velocity profile in the direction of the flow. In the latter, radial diffusion is most significant which provides for mixing in directions perpendicular to the flow. The extent of dispersion is characterized by the dispersion coefficient/). 

In a detailed rotating-disk electrode study of the characteristic currents were found to be under mixed control, showing kinetic as well as diffusional limitations [Ha3]. While for low HF concentrations (<1 M) kinetic limitations dominate, the regime of high HF concentrations (> 1 M) the currents become mainly diffusion controlled. However, none of the relevant currents (J1 to J4) obeys the Levich equation for any values of cF and pH studied [Etl, Ha3]. According to the Levich equation the electrochemical current at a rotating disk electrode is proportional to the square root of the rotation speed [Le6], Only for HF concentrations below 1 mol 1 1 and a fixed anodic potential of 2.2 V versus SCE the traditional Levich behavior has been reported [Cal 3]. 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

---