Diffusion corrected diffusivity

MaxweU-Stefan surface diffusivity (corrected diffusivity) Damkohler number... 

Fuller s equation, applied for the estimation of the coefficient of diffusion of a binary gas mixture, at a pressure greater than 10 bar, predicts values that are too high. As a first approximation, the value of the coefficient of diffusion can be corrected by multiplying it by the compressibility of the gas /... 

The derivation of the flux relations given here is a simplified treatment which yields the correct algebraic form but does not provide estimates of the various diffusion parameters. A more complete discussion is given by Mason et al. [21]. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

Corrections to the MP4/6-311G(d,b) Energy. Higher-level basis functions, if they are prudently chosen, should be better than lower-level functions. Thus the energy of, for example, a diffuse function, [MP2/6-311 - - G(d,p)] should be lower (more negative) than the same function in which diffuse electron density is not taken into account [MP2/6-31 lG(d,p)], provided that the levels of elecUon... 

The rate of chemical diffusion in a nonfiowing medium can be predicted. This is usually done with an equation, derived from the diffusion equation, that incorporates an empirical correction parameter. These correction factors are often based on molar volume. Molecular dynamics simulations can also be used. 

Density functional theory calculations have shown promise in recent studies. Gradient-corrected or hybrid functionals must be used. Usually, it is necessary to employ a moderately large basis set with polarization and diffuse functions along with these functionals. 

As with the diffusion coefficient, sedimentation coefficients are frequently corrected for concentration dependence and reduced to standard conditions ... 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

X 10 m /s. Diffusion coefficients may be corrected for other conditions by assuming them proportional to Schmidt numbers depend only weaMy on temperature (113). 

A rapid increase in diffusivity in the saturation region is therefore to be expected, as illustrated in Figure 7 (17). Although the corrected diffusivity (Dq) is, in principle, concentration dependent, the concentration dependence of this quantity is generally much weaker than that of the thermodynamic correction factor d ap d a q). The assumption of a constant corrected diffusivity is therefore an acceptable approximation for many systems. More detailed analysis shows that the corrected diffusivity is closely related to the self-diffusivity or tracer diffusivity, and at low sorbate concentrations these quantities become identical. 

Fig. 7. Variation of (a) intracrystaUine diffusivity and (b) corrected diffusivity (Dq) with sorbate concentration for / -heptane in a commercial sample of 5A 2eolite crystals Q 409 K A, v> 439 K (ads, des) x, 462 K +, 491 K. Reproduced by permission of National Research Council of Canada from ref 17.

The temperature dependence of the corrected diffusivity follows the usual Eyring expression... 

If the gas has the correct composition, the carbon content at the surface increases to the saturation value, ie, the solubiUty limit of carbon in austenite (Fig. 2), which is a function of temperature. Continued addition of carbon to the surface increases the carbon content curve. The surface content is maintained at this saturation value (9) (Fig. 5). The gas carburizing process is controlled by three factors (/) the thermodynamics of the gas reactions which determine the equiUbrium carbon content at the surface (2) the kinetics of the chemical reactions which deposit the carbon and (J) the diffusion of carbon into the austenite. 

The Displacement Distance theory suggests that since the stmcture of the flame is only quantitatively correct, the flame height can be obtained through the use of the displacement length or "displacement distance" (35,36) (eq. 12), where h = flame height, m V = volumetric flow rate, m /s and D = diffusion coefficient. 

Class B direct dyes have poor leveling power and exhaustion must be brought about by controlled salt addition. If these dyes are not taken up uniformly in the initial stages it is extremely difficult to correct the urdevelness. They are dyes that have medium—high affinity and poor diffusion. In their apphcation the cellulose is entered into a dyebath containing ordy dye. The salt is added gradually and portionwise as the temperature is increased and possibly the final additions made after the dyebath has come to the bod. 

The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combinea to add just enough diffusion to ehminate the oscillations without smoothing the solution too much. The algorithm is comphcated and lengthy but well worth the effort (Refs. 37, 107, and 270). 

Catchpole-Kinp examined binaiy diffusion data of near-critical fluids in the reduced density range of 1 to 2.5 and found that their data correlated with average deviations of 10 percent and a maximum deviation of 60 percent. They observed two classes of behavior. For the first, no correction fac tor was required R = 1). That class was comprised of alcohols as solvents with aromatic or ahphatic solutes, or carbon dioxide as a solvent with ahphatics except ketones as solutes, or... 

Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restiicied to binary diffusion at infinite dilution D°s of lo self-diffusivity D -. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits neghgible composition effects and deviations from thermodynamic ideahty. Conversely, liquid-phase diffusion almost always involves volumetiic and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

Pinto-Graham Pinto and Graham studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M. 

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