Mass transfer Maxwell-Stefan model

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

An important application of multicomponent mass transfer theory that we have not considered in any detail in this text is diffusion in porous media with or without heterogeneous reaction. Such applications can be handled with the dusty gas (Maxwell-Stefan) model in which the porous matrix is taken to be the n + 1th component in the mixture. Readers are referred to monographs by Jackson (1977), Cunningham and Williams (1980), and Mason and Malinauskas (1983) and a review by Burghardt (1986) for further study. Krishna (1993a) has shown the considerable gains that accrue from the use of the Maxwell-Stefan formulation for the description of surface diffusion within porous media. 

Section 15.6 describes the deficiencies in the Fickian model and points out why an alternative model (the fourth) is needed for some situations. The alternative Maxwell-Stefan model of mass transfer and diffusivity is explored in Section 15.7. The Maxwell-Stefan model has advantages for nonideal systems and multicomponent mass transfer but is more difficult to couple to the mass balances when designing separators. The fifth model of mass transfer, the irreversible thermodynamics model fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2QQQ Haase. 1990T is useful in regions where phases are unstable and can split into two phases, but it is beyond the scope of this introductory treatment. The... 

Maxwell-Stefan Model of Diffusion and Mass Transfer... 

Maxwell-Stefan model. The Maxwell-Stefan model is generally agreed to be a better model than the Fickian model for nonideal binary and all ternary systems. However, it is not as widely understood by chemical engineers, data collected in terms of Fickian diffusivities need to be converted to Maxwell-Stefan values, and the model can be more difficult to use. Use this model, coupled with a mass-transfer model, when the Fickian model fails or requires an excessive amount of data. 

MODELLING OF SIMULTANEOUS MASS AND HEAT TRANSFER WITH CHEMICAL REACTION USING THE MAXWELL-STEFAN THEORY—I. MODEL DEVELOPMENT AND ISOTHERMAL STUDY... 

Many industrial processes involve mass transfer processes between a gas/vapour and a liquid. Usually, these transfer processes are described on the basis of Pick s law, but the Maxwell-Stefan theory finds increasing application. Especially for reactive distillation it can be anticipated that the Maxwell-Stefan theory should be used for describing the mass transfer processes. Moreover, with reactive distillation there is a need to take heat transfer and chemical reaction into account. The model developed in this study will be formulated on a generalized basis and as a consequence it can be used for many other gas-liquid and vapour-liquid transfer processes. However, reactive distillation has recently received considerable attention in literature. With reactive distillation reaction and separation are carried out simultaneously in one apparatus, usually a distillation column. This kind of processing can be advantageous for equilibrium reactions. By removing one of the products from the reactive zone by evaporation, the equilibrium is shifted to the product side and consequently higher conversions can be obtained. Commercial applications of reactive distillation are the production of methyl-... 

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... 

Thus the gas/vapor/liquid-liquid mass transfer is modeled as a combination of the two-film model and the Maxwell-Stefan diffusion description. In this stage model, the equilibrium state exists only at the interface. 

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... 

Strictly speaking, Eqs. (13-69) and (13-70) are valid only for describing mass transfer in binary systems under conditions where the rates of mass transfer are low. Most industrial distillation and absorption processes, however, involve more than two different chemical species. The most fundamentally sound way to model mass transfer in multi-component systems is to use the Maxwell-Stefan (MS) approach (Taylor and Krishna, op. cit.). 

Frank, M. J. W., Kuipers, J. A. M., Krishna, R., and van Swaaij, W. P. M., Modelling of simultaneous mass and heat transfer with chemical reactions using the Maxwell-Stefan theory— U. Non-isothermal study. Chem. Eng. Sci. 50(10), 1661 (1995b). 

A theory of gas diffusion and permeation has recently been proposed [56] for the interpretation of experimental data concerning molecular-sieve porous glass membranes. Other researchers [57,58], on the basis of experimental evidences, pointed out that a Stefan-Maxwell approach has to be preferred over a simple Pick one for the modeling of mass transfer through zeolite membranes. 

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. 

The K j may be estimated using an empirical correlation or alternative physical model (e.g., surface renewal theory) with the Maxwell-Stefan diffusivity of the appropriate i-j pair D-j replacing the binary Fick D. Since most published correlations were developed with data obtained with nearly ideal or dilute systems where F is approximately unity, we expect this separation of diffusive and thermodynamic contributions to k to work quite well. We may formally define the Maxwell-Stefan mass transfer coefficient k - as (Krishna, 1979a)... 

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... 

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 ... 

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). 

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