Diffusion coefficient mixture averaged

The coefficients are defined for infinitely dilute solution of solute in the solvent L. However, they are assumed to be valid even for concentrations of solute of 5 to 10 mol.%. The relationships are available for pure solvent, and could be used for mixture of solvents composed of molecules of close size and shape. They all refer to the solvent viscosity which can be estimated or measured. Pressure has a negligible influence on liquid viscosity, which decreases with temperature. As a consequence, pressure has a weak influence on liquid diffusion coefficient conversely, diffusivity increases significantly with temperature (Table 45.4). For mixtures of liquids, an averaged value for the viscosity should be employed. 

Similarity Relations for One-Dimensional, Constant-Area Channel Flow with Chemical Reactions. Similarity relations between stagnation temperature and mass fractions obtain during flow in a channel of constant cross section, provided a binary mixture approximation is used for the diffusion coefficient, the Lewis number is set equal to unity, the Prandtl number is set equal to 3/4, and a constant value is employed for the species and average isobaric specific heats. [The assumption that the species (cPii) and average (cp = 2YiCp,i) isobaric specific heats are... 

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

Mixture-Averaged Diffusion Coefficient The ordinary diffusion coefficient of a species k into a mixture may be evaluated as... 

For the purpose of illustration, take a simple level of theory representing diffusion coefficients as mixture-averaged values (Eq. 3.89). Accordingly the flux term in Eq. 3.124 depends on the mole-fraction gradient... 

This section describes approximate (mixture-averaged) methods to determine transport properties in a mixture from the pure species values and binary diffusion coefficients. These techniques can offer an attractive compromise between accuracy and computational expense. 

No reliable mixture-averaged theory is available for computing the thermal diffusion coefficient D[. When thermal diffusion is important, the rigorous multicomponent theory described next should be used to obtain D[. 

Species fluxes calculated by either the multicomponent (Section 12.7.2) or the mixture-averaged (discussed subsequently in Section 12.7.4) formulations are obtained from the diffusion velocities V, which in turn depend explicitly on the concentration gradients of the species (as well as temperature and pressure gradients). Solving for the fluxes requires calculating either all j-k pairs of multicomponent diffusion coefficients Dy, or for the mixture-averaged diffusion coefficient D m for every species k. 

In the case of the mixture-averaged formulation, we desire to calculate Dkm (i.e., a diffusion coefficient for diffusion of species k into a mixture of other gases). This can result in savings in computational cost. At the same time the results in this section are approximations, although in some cases good ones. 

See also Problem 12.12.) The mixture-averaged diffusion coefficient of Eq. 12.176 was derived for use in calculating the molar diffusion velocity J with respect to the molar average velocity V, as in Eq. 12.172. 

To calculate the mixture-averaged diffusion coefficient relating the mass flux with respect to the mass-average velocity V in terms of the mass fraction gradient, write an expression analogous to Eq. 12.158 ... 

Using the assumptions mentioned above, the mixture-averaged diffusion coefficient Dkm appropriate for use in Eq. 12.177 can be calculated from (see Problem 12.13)... 

By the same approach as outlined above, the derived mixture-averaged diffusion coefficient D km is [211]... 

Beginning with Eqs. 12.174 and 12.175, and making the approximation that the velocities of all species j k are equal, derive the mixture averaged approximation for the diffusion coefficient... 

Solve this two-point boundary-value problem by any of the methods mentioned in Section 12.8 or Problem 12.16. Calculate the mixture-averaged diffusion coefficients Dkm using Eq. 12.176. Report the molar fluxes Ni, N2 (mol/m2-s) for SO2 and H2O. 

Evaluate the four mixture-average diffusion coefficients D m, based on Eq. 12.178. 

Calculate the four species diffusion velocities V using these mixture-averaged diffusion coefficients and Eq. 12.177. 

Tn the critical region of mixtures of two or more components some physical properties such as light scattering, ultrasonic absorption, heat capacity, and viscosity show anomalous behavior. At the critical concentration of a binary system the sound absorption (13, 26), dissymmetry ratio of scattered light (2, 4-7, II, 12, 23), temperature coefficient of the viscosity (8,14,15,18), and the heat capacity (15) show a maximum at the critical temperature, whereas the diffusion coefficient (27, 28) tends to a minimum. Starting from the fluctuation theory and the basic considerations of Omstein and Zemike (25), Debye (3) made the assumption that near the critical point, the work which is necessary to establish a composition fluctuation depends not only on the average square of the amplitude but also on the average square of the local... 

Fig. 16. Translational diffusion coefficient distributions G(D) of a simulated polymer mixture at two scattering angles ( , 17° and O , 14°). The mixture contains two polystyrene standards of distinctly different weight average molar masses (3.0 x 105 and 5.9 x 106 g/mol) and a high mass polystyrene...

Note that the polymer affects only water molecules situated in the vicinity of the polymer chains. Thus, the estimated diffusion coefficient corresponds only to these water molecules and is not dependent on the polymer concentration. The averaged self-diffusion coefficient estimated for the entire polymer-water mixture should be different depending on the polymer concentration. 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Ferrimagnetic nanoparticles of magnetite (Fc304) in diamagnetic matrices have been studied. Nanoparticles have been obtained by alkaline precipitation of the mixture of Fe(II) and F(III) salts in a water medium [10]. Concentration of nanoparticles was 50 mg/ml (1 vol.%). The particles were stabilized by phosphate-citrate buffer (pH = 4.0) (method of electrostatic stabilization). Nanoparticle sizes have been determined by photon correlation spectrometry. Measurements were carried out at real time correlator (Photocor-SP). The viscosity of ferrofluids was 1.01 cP, and average diffusion coefficient of nanoparticles was 2.5 10 cm /s. The size distribution of nanoparticles was found to be log-normal with mean diameter of nanoparticles 17 nm and standard deviation 11 nm. 

The molecules in a gas mixture continually collide with each other, and the diffusion process is strongly influenced by this collision process. The collision of like molecules is of little consequence since both molecules are identical and it makes no difference which molecule crosses a certain plane. The collisions of unlike molecules, however, influence the rate of diffusion since unlike molecules may have different masses and thus different momeniums, and thus the diffusion process is dominated by the heavier molecules. The diffusion coefficients and thus diffusion rales of gases depend strongly on temperature since the temperature is a measure of the average velocity of gas molecules. Therefore, the diffusion rales are higher at higher temperatures. 

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