Diffusive mass flux multicomponent

In the foregoing discussion the diffusive mass fluxes are written in terms of the diffusion velocities, which in turn are determined from gradients of the concentration, temperature, and pressure fields. Such explicit evaluation of the diffusion velocities requires the evaluation of the multicomponent diffusion coefficients from the binary diffusion coefficients. 

Evaluate the four multicomponent diffusion velocities k using Eq. 12.166. Verify that the sum of the diffusive mass fluxes is zero,... 

In general, the diffusive mass flux is composed of diffusion due to concentration gradients (chemical potential gradients), diffusion due to thermal effects (Soret diffusion) and diffusion due to pressure and external forces. It is possible to include the full multicomponent model for concentration gradient driven diffusion (Taylor and Krishna, 1993 Bird, 1998). In most cases, in the absence of external forces, it is... 

Show that in a multicomponent system the summation of the diffusive mass flux of all the components is zero ... 

All the experimental data in Table 6.1 refer to pure gases. Separation experiments, in which surface diffusion is the separation mechanism, are scarcely reported. Feng and Stewart (1973) and Feng, Kostrov and Stewart (1974) report multicomponent diffusion experiments for the system He-Nj-CH in a y-alumina pellet over a wide range of pressures (1-70 bar), temperatures (300-390 K) and composition gradients. A small contribution of surface diffusion (5% of total flow) to total transport could be detected, although it is not clear, which of the gases exhibits surface difiusion. The data could be fitted with the mass-flux model of Mason, Malinauskas and Evans (1967), extended to include surface diffusion. 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

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... 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

For diffusion in isothermal multicomponent systems the generalized driving force was written as a linear function of the relative velocities (m/ — My). In the general case, we must allow for coupling between the processes of heat and mass transfer and write constitutive relations for and q in terms of the (m — My) and V(l/r). With this allowance, the complete expression for the conductive heat flux is... 

Cotrone, A. and De Giorgi, C., A Rigorous Method for Evaluating Concentration Profiles and Mass Fluxes in Multicomponent Isothermal Gaseous Mixtures Diffusing Through Films of Given Thickness, Ing. Chim. Ital., 7(6), 84-97 (1971). 

It is possible to justify several alternative definitions of the multicomponent diffusivities. Even the multicomponent mass flux vectors themselves are expressed in either of two mathematical forms or frameworks referred to as the generalized Fick- and Maxwell-Stefan equations. 

An alternative to the complete Maxwell-Stefan model is the Wilke approximate formulation [103]. In this model the diffusion of species s in a multicomponent mixture is written in the form of Tick s law with an effective diffusion coefficient instead of the conventional binary molecular diffusion coefficient. Following the ideas of Wilke [103] we postulate that an equation for the combined mass flux of species s in a multicomponent mixture can be written as ... 

In reactive flow analysis the Pick s law for binary systems (2.285) is frequently used as an extremely simple attempt to approximate the multicomponent molecular mass fluxes. This method is based on the hypothesis that the pseudo-binary mass flux approximations are fairly accurate for solute gas species in the particular cases when one of the species in the gas is in excess and acts as a solvent. However, this approach is generally not recommend-able for chemical reactor analysis because reactive mixtures are normally not sufficiently dilute. Nevertheless, many industrial reactor systems can be characterized as convection dominated reactive flows thus the Pickian diffusion model predictions might still look acceptable at first, but this interpretation is usually false because in reality the diffusive fluxes are then neglectable compared to the convective fluxes. 

The design of a complete set of governing equations for the description of reactive flows requires that the combined fluxes are treated in a convenient way. In principle, several combined flux definitions are available. However, since the mass fluxes with respect to the mass average velocity are preferred when the equation of motion is included in the problem formulation, we apply the species mass balance equations to a (/-component gas system with q — independent mass fractions Wg and an equal number of independent diffusion fluxes js. However, any of the formulations derived for the multicomponent mass diffusion flux can be substituted into the species mass balance (1.39), hence a closure selection optimization is required considering the specified restrictions for each constitutive model and the computational efforts needed to solve the resulting set of model equations for the particular problem in question. 

The rigorous Fickian multicomponent mass diffusion flux formulation is derived from kinetic theory of dilute gases adopting the Enskog solution of the Boltzmann equation (e.g., [17] [18] [19] [89] [5]). This mass flux is defined by the relation given in the last line of (2.281) ... 

The multicomponent generalization of Pick s first law of binary diffusion is the second mass flux formulation on the Fickian form considered in this book. The generalized Pick s first law is defined b [72] [22] [62] [20] [96] ... 

It is further noted that the use of interfacial mass flux weighted transfer terms is generally not convenient treating multicomponent reactive systems, because the phase change processes are normally not modeled explicitly but deduced from the species composition dependent joint diffusive and convective interfacial transfer models. Moreover, the rigorous reaction kinetics and thermodynamic models of mixtures are always formulated on a molar basis. 

From the Maxwell-Stefan theory for multicomponent diffusion, Taylor and Krishna (1993) developed the following scheme to estimate the matrices of zero-flux multicomponent mass-transfer coefficients from binary-pair mass-transfer coefficients, Fy. These are obtained from correlations of experimental data, with the Chan and Fair (1984) correlations being the most widely used. For an ideal gas solution ... 

If we consider the system as a binary one with a surface-active material and bulk liquid, it is physically instructive to write the individual material balance relation for the surface excess concentration F (mol m ). The procedure for this is exactly as was carried out for the bulk binary system treated in Section 3.3. No chemical reaction at the interface is assumed, the system is considered to be dilute, the multicomponent mass flux is assumed to follow Pick s law, and the diffusion coefficients are taken to be constant. The expression for the surface concentration then becomes... 

The diffusional molar flux of component A is expressed via Pick s law in terms of the concentration gradient of A, only. Coupling between the diffusional mass flux of one species and all the independent mass fractions in the liquid phase is avoided by modeling this multicomponent diffusion problem as if it were a pseudobinary mixture. 

For dilute gases, the generalized multicomponent Fickian diffusion coefficients are strongly composition dependent. It follows that these diffusion coefficients do not correspond to the approximately concentration independent binary difffisivities, Dsr, which are available from binary diffusion experiments or kinetic theory since Dgr Dsr. In response to this Fickian model limitation it has been proposed to transform the Fickian diffusion flux model, in which the mass-flux vector, jj, is expressed in terms of the driving force, dj, into the corresponding Maxwell-Stefan form [95, 97, 142, 143] where d is given as a linear function of jj. The key idea is to rewrite the Fickian diffusion flux in terms of an alternative set of difffisivities (i.e., preferably the known binary difffisivities) which are less concentration dependent than the Fickian difffisivities. 

Problem Solving Methods Most, if not aU, problems or applications that involve mass transfer can be approached by a systematic-course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., iTmlticomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are apphed to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it apphes to many mass-transfer problems. 

The full treatment of multicomponent diffusion requires a diffusion matrix because the diffusive flux of one component is affected by the concentration gradient of all other components. For an N-component system, there are N-1 independent components (because the concentrations of all components add up to 100% if mass fraction or molar fraction is used). Choose the Nth component as the dependent component and let n = N 1. The diffusive flux of the components can hence be written as (De Groot and Mazur, 1962)... 

Apart from the diffusion step in the particle, when the uptake process occurs from a binary or multicomponent fluid mixture, there maybe an additional resistance to mass transfer associated with the transport of solutes through the fluid layer surrounding the particle. The driving force in this case is the concentration difference across the boundary layer, and the flux at the particle surface is... 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

The optimal Reynolds number defines the operating conditions at which the cylindrical system performs a required heat and mass transport, and generates the minimum entropy. These expressions offer a thermodynamically optimum design. Some expressions for the entropy production in a multicomponent fluid take into account the coupling effects between heat and mass transfers. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations including the effects of ordinary, forced, pressure, and thermal diffusion. 

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