Relative diffusion coefficients calculation

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

The crossover concentration calculated for polystyrene with this expression is in the range of the reported values(12,13) Finally, the above discussion points out that c is not a sharp dividing line It is therefore not physically meaningful to scale the concentration axis in a plot of D/Dq versus concentration divided by the crossover concentration Equation 13 demonstrates that it is better to plot the relative diffusion coefficient vs the weight concentration of the polymer, as was done by Munch et al (12) ... 

FIGURE 5.15 Relative diffusion coefficient /)(/)/A> as a function of X = rm/rp for diffusion of spherical molecules of radius rm through a cylindrical capillary of radius rv calculated according to Eq. (5.26). 

FIGURE 13.5 Value of the particle relative diffusion coefficient 1)1) as a function of the relative interparticle distance u — hja) of two equal-sized spheres. Calculated according to Eq. (13.6). 

Anderson and Wennerstrom [33] calculated the geometrical obstruction factors of the self-diffusion of surfactant and solvent molecules in ordered bicontinuous microstructures, which serve as good approximations also for the disordered bicontinuous microemulsions and L3 (sponge) phases. The geometrical obstruction factor is defined as the relative diffusion coefficient DIDq, where D is the diffusion coefficient in the structured surfactant system and Z)q is the diffusion coefficient in the pure solvent. In a bicontinuous microemulsion the geometrical obstruction factor depends on the water/oil ratio. An expansion around the balanced (equal volumes of water and oil) state gives, to leading order. 

An assessment of the relative diffusion rates of ionic and molecular species in the PAN-based electrolyte may be made from the diffusion coefficients calculated for ferrocene from cyclic voltammograms. Some data are presented in Table 3.8. The ratio of diffusion coefficients of ferrocene in the PAN-based polymer electrolyte and PC/LiC104 liquid electrolyte at room temperature is the same as that obtained for the conductivity of LiC104 in these electrolytic media. It may be noted here that the ferro-cene/ferrocenium couple has been shown [36] to be useful for the overcharge protection of secondary Li batteries. 

Membranes of the polymers were evaluated as to their relative diffusion coefficients for glucose and oxygen. Relative diffusion coefficients were calculated using the formula... 

The free volume in IPNs has not been investigated extensively. The first attempt to estimate this value was done [65] using the data on vapor sorption based on the theory developed by Fujita [122]. The sequential IPNs based on styrene-DVB copolymer (network I) and cross-linked PU (network II) were studied. The PU content was up to 0.24 by mass. From the data on the sorption kinetics the interdiffusion coefficients Dy, solvent self-diffusion coefficient D, and relative diffusion coefficient D were found. The value of D was calculated from the relation Dy = I> (1 - Vg), where Vg is the volume fraction of a solvent in the IPN. According to Fujita, the change in self-diffusion coefficient in isothermal conditions is described by the equation ... 

Table I shows the results of calculating a soil diffusion coefficient and soil diffusion half-lives for the pesticides. The 10% moisture level specified means that the soil is relatively dry and that 40% of the soil volume is air available for diffusion. Complete calculations were not made for methoxychlor, lindane, and malathion because, based on Goring s criteria for the Henry s law constant, they are not volatile enough to diffuse significantly in the gas phase. This lack of volatility is reflected in their low values of X. These materials would move upward in the soil only if carried "by water that was moving upward to replace the water lost through evapotranspiration at the surface. Mirex has a very high Henry s law constant. On the basis of Goring s criteria, Mirex should diffuse in the soil air but, because of its strong adsorption, it has a very large a and consequently a very small soil air diffusion coefficient. The behavior of Mirex shows that Goring s criteria must be applied carefully.

Thus, the time that is necessary to attain a certain coverage, 6, or the time necessary to cover the surface completely (9 = 1) is inversely proportional to the square of the bulk concentration (cf. Fig. 4.10b). Assuming molecular diffusion only, 8 is of the order of 2 minutes for a concentration of 10 5 M adsorbate when the diffusion coefficient D is 10 5 cm2 s1 and rmax = 4 1010 mol cm 2 1). Considering that transport to the surface is usually by turbulent diffusion, such a calculation illustrates that the formation of an adsorption layer is relatively rapid at concentrations above 10 6 M. But it can become slow at concentrations lower than 10 6 M. 

The diffusion parameter calculated by the root time method is an average parameter, and is generally considered to be operative over the range of time from initial diffusion flux to near steady state flux conditions. The method is applicable for the situation where adsorption and desorption occur, and for various pH values of the influent. The closer (DE) is to (DB) in Fig. 5 d, the greater is the accuracy of the D coefficient. It is important to note that in the case of low pH values of the influent, desorption of cations from a clay soil could produce conditions where C2 > C1. Accordingly, the experimental values for relative change in concentration would then become negative. 

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

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