Coefficient of binary diffusion

The coefficient D12 is called the coefficient of binary diffusion, though it is frequently designated simply as D. In gases, Du is practically independent of composition, it increases with temperature, and is inversely proportional to pressure. 

Here the expression in parentheses is the dimensionless driving force of ordinary diffusion for j-th component, Dij are coefficients of binary diffusion. 

Recall that u is the mobility, D is the coefficient of binary diffusion, related to with mobility by the equation D = ATo . Equations (5.85) and (5.86) are known as the Nernst-Planck equations. 

The coefficient of sell-diffusion does not appear to have an anomaly near the critical point. For the engineer, however, the mutual dift usion coefficient is the more important property. The binary dilfusion coefficient approaches zero at the mixture critical point ("critical slowing-down"). In dilute mixtures, however, the decrease of the binary dilfusion coefficient is not seen until the critical line is approached very closely. For many practical purposes, such as supercritical extraction and chromatography, the mixture is dilute, and it can be assumed that the coefficient of binary diffusion is intermediate between that in the vapor and that in the liquid. Since the diffusion coefficient decreases roughly inversely proportional to the density, dilfusion in supercritical solvents is much more rapid than in liquid solvents, thus increasing the speed of diffusion-controlled chemical processes. 

States Hard-Sphere Model for the Diffusion-Coefficients of Binary Dense-Plasma Mixtures. 

To expedite the evaluation of transport properties, one could fit the temperature dependent parts of the pure species viscosities, thermal conductivities, and pairs of binary diffusion coefficients. Then, rather than using the complex expressions for the properties, only comparatively simpler polynomials would be evaluated. The fitting procedure must be carried out for the particular system of gases that is present in a given problem. Therefore the fitting cannot be done once and for all but must be done once at the beginning of each new problem. 

Such fits may also be done for each pair of binary diffusion coefficients in the system,... 

The local density augmentation caused by the large isothermal compressibility of the fluid may conceivably influence k i or ka. We assume that the lifetime of the clusters is extremely short and thus there is no effect on kd, based on the molecular dynamics study of Petsche and Debenedetti (29) and experimental measurements of binary diffusion coefficients near the critical point. It seems more likely that a higher local density would affect k i due to an increase in the number of... 

All the proposed expressions require the knowledge of the diffusion coefficients. The binary diffusion coefficient determination is non trivial and it is usually made experimentally. 

The method of Blanc [16] permits calculation of the gas-phase effective multicomponent diffusion coefficients based on binary diffusion coefficients. A conversion of binary diffusivities into effective diffusion coefficients can be also performed with the equation of Wilke [54]. The latter equation is frequently used in spite of the fact that it has been deduced only for the special case of an inert component. Furthermore, it is possible to estimate the effective diffusion coefficient of a multicomponent solution using a method of Burghardt and Krupiczka [55]. The Vignes approach [56] can be used in order to recalculate the binary diffusion coefficients at infinite dilution into the Maxwell-Stefan diffusion coefficients. An alternative method is suggested by Koijman and Taylor [57]. 

In this case, it is well known that the process occurs in steady state. To understand this process, one must consider it as a special case of binary diffusion, where the diffusivity of the Pd atoms is zero. Consequently, the frame of reference is the fixed coordinates of the solid Pd thin film. The interdiffusion or chemical diffusion coefficient is the diffusivity of the mobile species [20], that is, hydrogen. Then, the hydrogen flux in the Pd thin film is given by... 

With the help of Equation 5.107, as was previously done with Equation 5.86, we obtain a transport or chemical diffusion coefficient that is a result of Fick s laws. We now interpret the meaning of this coefficient if we consider diffusion in a microporous solid, as a special case of binary diffusion, where A is the mobile species and the diffusivity of the microporous framework atoms is zero, then, the frame of reference are the fixed coordinates of the porous solid consequently, we have a particular case of interdiffusion where the diffusion coefficient is simply the diffusivity of the mobile species [12,20],... 

The Maxwell-Stefan diffusion coefficients represent binary diffusivities for ideal and many nonideal mixtures, they are independent of the concentration of the species in the multicomponent mixtures. 

The behavior of the Fick diffusion coefficient in nonideal systems may be complicated, while the Maxwell-Stefan diffusion coefficients behave quite well, and are always positive for binary systems. In nonideal binary systems, the Fick diffusivity varies with concentration. As seen in Figure 6.1, water-acetone and water-ethanol systems exhibit a minimum diffusivity at intermediate concentrations. Table 6.1 displays the dependency of binary diffusivity coefficients on concentration for selected alkenes in chloroform at 30°C and 1 atm. As the nonideality increases, mixture may split into two liquid phases at certain composition and temperature. 

If Eqs. (5-200) and (5-201) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-214)]. For gases, the values Dy of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AlChE J., 10,448 and 460 (1964) Stewart and Prober, Ind. Eng. Chem. Fundam., 3,224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor [Chem. Eng. J., 25, 47 (1982)], and Taylor and Webb [Comput Chem. Eng., 5, 61 (1981)]. 

The parameter a in Eq. 6 is the ratio of the effective binary coefficient of ordinary diffusion for the reactive species and the mixture of other void gases to the coefficient of ordinary self diffusion for the reactive species. As shown in the Appendix, this parameter depends only on the composition of the gas mixture and is independent of both the pressure and temperature for ideal gases. Thus for fixed gas composition, the parameter a is constant. 

The parameter a is the ratio of the effective binary coefficient of ordinary diffusion for the reactive, species and the mixture of other furnace gases to the coefficient of ordinary self... 

M. Gordon, References to Experimental Data on Diffusion Coefficients of Binary Gas Mixtures, National Engineering Laboratory Kept. No. 641, Department of Industry, Glasgow, Scotland (1977). 

Diffusion coefficients in binary liquid mixtures are of the order 10 m /s. Unlike the diffusion coefficients in ideal gas mixtures, those for liquid mixtures can be strong functions of concentration. We defer illustration of this fact until Chapter 4 where we also consider models for the correlation and prediction of binary diffusion coefficients in gases and liquids. 

For process engineering calculations it is almost inevitable that experimental values of D or f), even if available in the literature, will not cover the entire range of temperature, pressure, and concentration that is of interest in any particular application. It is, therefore, important that we be able to predict these coefficients from fundamental physical and chemical data, such as molecular weights, critical properties, and so on. Estimation of gaseous diffusion coefficients at low pressures is the subject of Section 4.1.1, the correlation and prediction of binary diffusion coefficients in liquid mixtures is covered in Sections 4.1.3-4.1.5. We do not intend to provide a comprehensive review of prediction methods since such are available elsewhere (Reid et al., 1987 Ertl et al., 1974 Danner and Daubert, 1983) rather, it is our purpose to present a selection of methods that may be useful in engineering calculations. 

The diffusion of species s in a multicomponent mixture can then be written in the form of Pick s law of binary diffusion, using the effective diffusion coefficient Dsrn instead of the binary diffusivity. The Wilke equation is defined by ... 

Vignes30 correlated the composition dependence of binary diffusion coefficients in terms of their infinite-dilution values and this thermodyxamic correction factor. 

FIGURE 2.3-4 Composition dependence of binary diffusion coefficients in liquids, (in units of cm2/ ) (a) acetone (A)-CC 4, (b) methanol (A)-HjO (c) ethanol (A)-H20. (d) acetone (A)-CH3CI. Reprinted with permission from A. Vigtics. lnd, Eng. Chem. Fundam,, 5, 198 (1966). Copyright 1966 Amarican Chemical Secisty,... 

Here, the coefficient D b of binary diffusion for distinguishable but otherwise equivalent particles will depend on the total concentration that is, on the degree of saturation... 

The theory for diffusion coefficients in liquids is not very well developed. Since the general theory for the calculation of binary diffusion coefficients in the the liquid phase is missing, semi-empirical equations are often used. These equations describe the diffusion of a dissolved... 

Consider first a diffusion process in a binary mixture, for example, the diffusion of a colored dye in water. If the dye is injected at any point in the liquid, it spreads throughout the liquid. The dye flows from the area in which it has high concentration to areas in which its concentration is lower. Simultaneously, one can observe the transfer of molecules of liquid in the opposite direction. In time, the process reaches equilibrium, diffusion ceases, and the solution becomes homogeneous. The cited case is a typical example of binary diffusion moreover, the coefficient of diffusion, Du, of the dye relative to water is equal to the coefficient of diffusion, D21, of water relative to the dye. 

The device illustrated in Figure 7.13 is used to measure the diffusion coefficient of binary gas pairs. Two glass tubes with equal cross-sections are joined together at one end with a flexible tube that is pinched in the middle. Each tube is fllled with a different gas and the clip is detached to initiate the experiment (t = 0). 

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