Multicomponent mass diffusion

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 applicability of the multicomponent mass diffusion models to chemical reactor engineering is assessed in the following section. Emphasis is placed on the first principles in the derivation of the governing flux equations, the physical interpretations of the terms in the resulting models, the consistency with Pick s first law for binary systems, the relationships between the molar and mass based fluxes, and the consistent use of these multicomponent models describing non-ideal gas and liquid systems. 

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 generalized multicomponent Maxwell-Stefan equation (2.298) represents the third multicomponent mass diffusion formulation considered in this book. The rigorous form of Maxwell-Stefan equation can be expressed as ... 

Lattice Boltzmann modeling of three-dimensional, multicomponent mass diffusion in a solid oxide fuel cell anode. J. Fuel Cell Sci. Technol., 7 (1), 011006-011008. 

In the second edition of the textbook by Bird et al. [9] from 2002, a third alternative definition of the Fickian multicomponent mass diffusion coefficients was proposed, deviating from relation (2.309) by the sign convention only. These diffusion coefficients were thus defined by ... 

Applications of Multicomponent Mass Diffusion Flux Models in Chemical Reactor Engineering... 

Rout KR, Solsvik J, Nayak AK, Jakobsen HA (2011) A numerictil study of multicomponent mass diffusion and convection in porous pellets for the sorption-enhanced steam methane reforming and desorption processes. Chem Eng Sci 66 4111 126... 

Smoluchowski M (1918) Versuch einer mathematischen Theotie der Koagulationskrnetik koUoider Losungen. Zeitschrift fiir PhysikaUsche Chemie, Leipzig, Band XCll pp 129-168 Solsvik J, Jakobsen HA (2011) Modeling of multicomponent mass diffusion in porous spherical pellets application to steam methane reforming and methanol synthesis. Chem Eng Sci 66 1986-2000... 

Solsvik J, Jakobsen HA (2012) A survey of multicomponent mass diffusion flux closures for... 

Solsvik J, Jakobsen HA (2013) Multicomponent mass diffusion in porous pellets effects... 

Multicomponent Mass Diffusion Flux Models C.9 The Mass Based Maxwell-Stefan Flux Model... 

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

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-... 

Hot-Wall Reactors. Because of the large mass diffusivities and nearly isothermal conditions (except for the entrance zone) in hot-wall, low-pressure reactors (50 Pa), multicomponent diffusion and chemical reactions are critical... 

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. 

Experimental Observations. Most of the experimental observations on multicomponent droplet vaporization use two-component droplets 49, 62, 63,-64), The observations on the temporal behavior of the droplet temperature (49,64), size (62,63), and composition (64) all indicate that the vaporization processes are controlled by the volatihty differentials rather than by hquid-phase mass diffusion. Since the fuels used in these experiments are quite nonviscous, the above results then indicate that internal circulation of suflBcient strength has been generated by the prevailing forced and/or natural convection. 

MULTICOMPONENT MASS TRANSFER DIFFUSION MODEL FOR THE ADSORPTION OF ACID DYES ON ACTIVATED CARBON... 

There is a large body of literature that deals with the proper definition of the diffusivity used in the intraparticle diffusion-reaction model, especially in multicomponent mixtures found in many practical reaction systems. The reader should consult references, e.g.. Bird, Stewart, and Lightfoot, Transport Phenomena, 2d ed., John Wiley Sons, New York, 2002 Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993 and Cussler, Diffusion Mass Transfer in Fluid Systems, Cambridge University Press, 1997. 

SOLUTION To predict multicomponent Fick diffusivities in the mass average velocity reference frame we combine Eq. 4.2.2 to predict the Fick diffusivities in the molar average velocity reference frame... 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

The ratio of driving forces Axi/Ax2 plays an important role in enhancing diffusional interaction effects in multicomponent mass transfer. Thus, a small cross-coefficient k 2 may be linked to a large Ax2, resulting in large interaction effects. The criteria presented above are a little different from those discussed in Section 5.2, where Fick diffusion coefficients and the mole fraction gradients were used. The physical significance is, however, the same. 

Equations 9.3.15 and 9.3.16 represent an exact analytical solution of the multicomponent penetration model. For two component systems, these results reduce to Eqs. 92.1. Unfortunately, the above results are of little practical use for computing the diffusion fluxes because they require an a priori knowledge of the composition profiles (cf. Section 8.3.5). Thus, a degree of trial and error over and above that normally encountered in multicomponent mass transfer calculations enters into their use. Indeed, Olivera-Fuentes and Pasquel-Guerra did not perform any numerical computations with this method and resorted to a numerical integration technique. 

The procedure for computing the multicomponent Fick diffusion coefficients in the mass average velocity reference frame was illustrated in Example 4.2.5 and the steps shown there have been repeated for this example with the result... 

A number of investigators used the wetted-wall column data of Modine to test multicomponent mass transfer models (Krishna, 1979, 1981 Furno et al., 1986 Bandrowski and Kubaczka, 1991). Krishna (1979b, 1981a) tested the Krishna-Standart (1976) multicomponent film model and also the linearized theory of Toor (1964) and Stewart and Prober (1964). Furno et al. (1986) used the same data to evaluate the turbulent eddy diffusion model of Chapter 10 (see Example 11.5.3) as well as the explicit methods of Section 8.5. Bandrowski and Kubaczka (1991) evaluated a more complicated method based on the development in Section 8.3.5. The results shown here are from Furno et al. (1986). 

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. 

Taylor, R. and Krishnamurthy, R., Film Models for Multicomponent Mass Transfer.—Diffusion in Physiological Gas Mixtures, Bull. Math. Biol., 44, 361-376 (1982). 

The 15 chapters fall into three parts. Part I (Chapters 1-6) deals with the basic equations of diffusion in multicomponent systems. Chapters 7-11 (Part II) describe various models of mass and energy transfer. Part III (Chapters 12-15) covers applications of multicomponent mass transfer models to process design. 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

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