Mass transfer Stefan-Maxwell equations

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

Chapter 3 dealt with the problem of the reaction kinetics for different gas-solid reactions, while chapter 5 dealt with the mass and heat transfer problems for porous as well as non-porous catalyst pellets. In chapter 5 different degrees of complexities and rigor were used. In chapter 5, the analysis started with the simplest case of non-porous catalyst pellets where the only mass and heat transfer Coefficients are those at the external surface which depend mainly on the flow conditions around the catalyst pellet and the properties of the reaction mixture. It was shown clearly that j-factor correlations are adequate for the estimation of the external mass and heat transfer coefficients (k, h) associated with these resistances. For the porous catalyst pellets different models with different degrees of rigor have been used, starting from the simplest case of Fickian diffusion with constant diffusivity, to the rigorous dusty gas model based on the Stefan-Maxwell equations for multicomp>onent diffusion. 

Transport in OSN membranes occurs by mechanisms similar to those in membranes used for aqueous separations. Most theoretical analyses rely on either irreversible thermodynamics, the pore-flow model and the extended Nemst-Planck equation, or the solution-diffusion model [135]. To account for coupling between solute and solvent transport (i.e., convective mass transfer effects), the Stefan-Maxwell equations commonly are used. The solution-diffusion model appears to provide a better description of mixed-solvent transport and allow prediction of mixture transport rates from pure component measurements [136]. Experimental transport measurements may depend significantly on membrane preconditioning due to strong solvent-membrane interactions that lead to swelling or solvent phase separation in the membrane pore structure [137]. 

The exact formulations of the fluxes N ) depend on the particular model being used for mass transfer principally, the whole scope is feasible, from Pick s law to the complete set of Stefan-Maxwell equations. Since the only component of importance for the gas-liquid mass transfer is hydrogen, which has limited solubility in the liquid phase, the simple two-film model along with Pick s law was used, yielding the flux expression... 

In this paper a transfer model will be presented, which can predict mass and energy transport through a gas/vapour-liquid interface where a chemical reaction occurs simultaneously in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. On the basis of this model a numerical study will be made to investigate the consequences of using the Maxwell-Stefan equation for describing mass transfer in case of physical absorption and in case of absorption with chemical reaction. Despite the fact that the Maxwell-Stefan theory has received significant attention, the incorporation of chemical reactions with associated... 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

The modeling of RD processes is illustrated with the heterogenously catalyzed synthesis of methyl acetate and MTBE. The complex character of reactive distillation processes requires a detailed mathematical description of the interaction of mass transfer and chemical reaction and the dynamic column behavior. The most detailed model is based on a rigorous dynamic rate-based approach that takes into account diffusional interactions via the Maxwell-Stefan equations and overall reaction kinetics for the determination of the total conversion. All major influences of the column internals and the periphery can be considered by this approach. 

Assume that a simple film model exists for the mass transfer, equilibrium is established at the gas-liquid interface, and the diffusion occurs at isobaric and isothermal conditions. Also assume that neither helium nor argon is absorbed so that N2=N3 = 0. Then, the Maxwell-Stefan equations for the diffusion of argon and helium are... 

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... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the Maxwell-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair- directly based on the Maxwell-Stefan diffusivities. 

Kaczmarski et al. used a similar model for the calculation of the band profiles of the enantiomers of 1-indanol on a chiral phase in HPLC [29,57]. These authors ignored the external mass transfer and assumed that local equilibrium takes place for each component between the pore surface and the stagnant fluid phase in the macropores (infinite fast kinetics of adsorption-desorption). They also assumed that surface diffusion contribution is much faster than pore diffusion and neglected pore diffusion entirely. Instead of the single file Maxwell-Stefan diffusion, these authors used the generalized Maxwell-Stefan diffusion (see Chapter 5).The calculation (see below) requires first the selection of equations to calculate the surface molecular flux [29,57,58],... 

The competitive equilibrium isotherm model best fitting the FA experimental data for the R and S enantiomers of 1-phenyl-l-propanol on cellulose tiibenzoate was the Toth model. This model was used to calculate the elution profiles of samples of mixtures of the two enantiomers [29]. The General Rate model combined with the Generalized Maxwell-Stefan equation (GR-GMS) was used to model and describe surface diffusion (see Chapter 5). The mass transfer kinetics is slow and this model fits the experimental data well over a wide concentration range with one single set of numerical parameters to account for the band profiles in a wide range of concentrations, as shown in Figure 16.24. 

Algorithm 8.1 Algorithm for Calculation of Mass Transfer Rates from an Exact Solution of the Maxwell-Stefan Equations... 

SOLUTION In preparation for the estimation of the mass transfer coefficients in the liquid phase we must first compute the Maxwell-Stefan diffusion coefficients. Equation 4.2.18 is used for this task as illustrated below. 

In any event, we hope it is now well understood that mass transfer in multicomponent systems is described better by the full set of Maxwell-Stefan or generalized Fick s law equations than by a pseudobinary method. A pseudobinary method cannot be capable of superior predictions of efficiency. For a simpler method to provide consistently better predictions of efficiency than a more rigorous method could mean that an inappropriate model of point or tray efficiency is being employed. In addition, uncertainties in the estimation of the necessary transport and thermodynamic properties could easily mask more subtle diffusional interaction effects in the estimation of multicomponent tray efficiencies. It should also be borne in mind that a pseudobinary approach to the prediction of efficiency requires the a priori selection of the pair of components that are representative of the... 

In this textbook we have eoneentrated our attention on mass transfer in mixtures with three or more species. The rationale for doing this should be apparent to the reader by now multicomponent mixtures have characteristics fundamentally dijferent from those of two component mixtures. In fact, a binary system is peculiar in that it has none of the features of a general multicomponent mixture. We strongly believe that treatments of even binary mass transfer are best developed using the Maxwell-Stefan equations. We hope that this text will have the effect of persuading instructors to use the Maxwell-Stefan approach to mass transfer even at the undergraduate level. 

The explicit method of Taylor and Smith (1982) for mass transfer in ideal gas mixtures is an exact solution of the Maxwell-Stefan equations for two component systems where all matrices are of order 1. Does the generalized explicit method derived in Exercise 8.40 reduce to the expressions given in Section 8.2 for a film model of mass transfer in binary systems ... 

Equation A.6.1 arises when the Maxwell-Stefan equations are solved for the case of steady-state, one-dimensional mass transfer, as discussed in Chapter 8. The matrices [A ] and [O] are as defined in Chapter 8, is the molar density of the mixture and a scalar, and (Ax) is a column matrix of mole fraction differences. All matrices in Eq. A.6.1 are of order n — 1 where n is the number of components in the mixture. For the purposes of this discussion we shall assume that the matrices [/ ] and [O] have already been calculated. The matrix function [0][exp[] - [7]] denoted by [2], can be computed using Sylvester s expansion formula (see, however, below) so the immediate problem is the calculation of the column matrix (7) from... 

Brocker, S. and Schulze, W., A New Method of Calculating Ternary Mass Transfer with a Non-Transferring Species Based on Gilliland s Parametric Solution of the Maxwell-Stefan Equations for the Film Model, Chem. Eng. Commun., 107, 163-172 (1991). 

Taylor, R. and Webb, D. R., Film Models for Multicomponent Mass Transfer Computational Methods I—the Exact Solution of the Maxwell-Stefan Equations, Comput. Chem. Eng., 5, 61-73 (1981). 

We also feel that portions of the material in this book ought to be taught at the undergraduate level. We are thinking, in particular, of the materials in Section 2.1 (the Maxwell-Stefan relations for ideal gases). Section 2.2 (the Maxwell-Stefan equations for nonideal systems). Section 3.2 (the generalized Fick s law). Section 4.2 (estimation of multicomponent diffusion coefficients). Section 5.2 (multicomponent interaction effects), and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of mass transfer in binary mixtures that is normally included in undergraduate courses. 

Write and solve the Maxwell-Stefan equations for multicomponent mixtures of ideal gases in terms of the corresponding binary mass-transfer coefficients. 

The simplest approach is to calculate binary mass-transfer coefficients F.. from the corresponding empirical correlation, substituting the MS diffusivity D. for the Fick diffusivity in the Sc and Sh numbers. The Maxwell-Stefan equations are, then, written in terms of the binary mass-transfer coefficients. For ideal gas multicomponent mixtures and one-dimensional fluxes, they become... 

When mass-transfer rates are moderate to high, an additional correction term is needed in equations (6-101) and (6-102) to correct for distortion of the composition profiles. This correction, which can have a serious effect on the results, is discussed in detail by Taylor and Krishna (1993). An alternative approach would be to numerically solve the Maxwell-Stefan equations, as illustrated in Examples 1.17 and 1.18. The calculation of the low mass-transfer fluxes according to equations (6-94) to (6-104) is illustrated in the following example. 

The most fundamentally sound way to model mass transfer in multiMaxwell-Stefan theory [5, 8]. The Maxwell-Stefan equations for mass transfer in the vapor and liquid phases respectively are given by... 

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