Maxwell diffusion coefficient

In the case of transport in the gas phase it is often convenient to use the Maxwell diffusion coefficient (M4), which is related to the Fick diffusion coefficient in the following way (04) for a phase that may be treated as an ideal solution (L4) ... 

In Equation 7.67 DAf (Ps) and Dm (Ps) are the Maxwell diffusion coefficients of A in a binary mixture respectively of P and Dt at surface pressure. Neither D, (Ps) nor DAn (P,) depend on the gas composition or pressure. DM is the Knudsen diffusion coefficient of A in the catalyst pores, which is also independent of the gas composition and the pressure. The term is the mole fraction of Dj that would have been obtained... 

DBk is the Knudsen diffusion coefficient of B inside the catalyst pores, Dgp(Ps) is the Maxwell diffusion coefficient of B in a binary mixture with P, the other diffusion coefficients are defined in a similar way. All other symbols are as defined before. Notice the following ... 

In these equations JA is the mole flux of A in moles of A per second per square metre flowing through surface area of the catalyst pores. This is not the same as the mole flux in moles of A per second per square metre flowing through surface area of the catalyst pellet. This is elaborated in Appendix E. The term DAP is the Maxwell diffusion coefficient of A in a binary mixture with P and DM is the Knudsen diffusion coefficient of A inside the catalyst pores. ka is the mole fraction of A. J, Dpdp Dn, kp etc. are similarly defined. VA is a factor that accounts for viscous flow inside the pores. If VA is much smaller than one, viscous flow can be neglected. We will neglect viscous flow for all components and substitute... 

Since the Maxwell diffusion coefficients are inversely proportional to the pressure, this can be written as... 

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)]. 

In the late 1800s, the development of the kinetic theory of gases led to a method for calculating mmticomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simnlta-neonsly by Stefan and Maxwell. The problem is to determine the diffusion coefficient D, . The Stefan-Maxwell equations are simpler in principle since they employ binary diffnsivities ... 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

Graham-Uranoff They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion. 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

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. 

As the pore size decreases, molecules collide more often with the pore walls than with each other. This movement, intermediated by these molecule—pore-wall interactions, is known as Knudsen diffusion. Some models have begun to take this form of diffusion into account. In this type of diffusion, the diffusion coefficient is a direct function of the pore radius. In the models, Knudsen diffusion and Stefan—Maxwell diffusion are treated as mass-transport resistances in seriesand are combined to yield... 

Note that the Stefan-Maxwell equations involve the binary diffusion coefficients, and not the ordinary multicomponent diffusion coefficients. 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

These constraints must be satisfied in the solution of the Stefan-Maxwell equations. At a point within a chemically reacting flow simulation, the usual situation is that the diffusion velocities must be evaluated in terms of the diffusion coefficients and the local concentration, temperature, and pressure fields. One straightforward approach is to solve only K — 1 of Eqs. 3.105, with the X4h equation being replaced with a statement of the constraint. For... 

Note that the Stefan-Maxwell equations involve the binary diffusion coefficients T>kj, not the ordinary multicomponent diffusion coefficients Dy. In this context, the are also sometimes refered to as the multicomponent Stefan-Maxwell diffusivities. 

Axial, film, and macropore Maxwell and Knudsen diffusion coefficients are estimated based on relatively standard formulations ( 6, 7, 8, 9), using estimates of physical properties shown in Table I to compute the diffusivity values for the systems studied. The effect of errors in the estimated values of these properties will be discussed later. 

The first hypothesis seems unlikely to be true in view of the rather wide variation in the ratio of carbon dioxide s kinetic diameter to the diameter of the intracrystalline pores (about 0.87, 0.77 and 0.39 for 4A, 5A and 13X, respectively (1J2)). The alternative hypothesis, however, (additional dif-fusional modes through the macropore spaces) could be interpreted in terms of transport along the crystal surfaces comprising the "walls" of the macropore spaces. This surface diffusion would act in an additive manner to the effective Maxwell-Knudsen diffusion coefficient, thus reducing the overall resistance to mass transfer within the macropores. 

Fick s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell-Stefan expression is used, the diffusion of... 

The mass transfer within a rigid droplet is determined by the Maxwell-Stefan diffusion. The appropriate diffusion coefficients experimentally determined... 

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