Flux expressions Stefan-Maxwell equation

The Stefan-Maxwell equations are normally written in terms of fluxes, Nj. Since Ni = c,y, Equation (C.1.9) can be expressed in terms of fluxes as ... 

The n-1 equations given by eq. (8.2-34) and the physical constraint of the form (8.2-43) will form the necessary n equations for solving for the fluxes N. What we shall show in this section that the Stefan-Maxwell equations can be inverted to obtain the useful flux expression written in terms of concentration gradients, instead of concentration gradient in terms of fluxes. 

By using the Grahams law equation (8.6-22) into the Stefan-Maxwell equation (8.6-20a), we obtain the following equation expressing the flux in terms of concentration gradient for the component 1 ... 

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... 

The Chapman-Enskog kinetic theory cf gases (Hirschfelder et al., 1964) is used to describe the multicomponent diffusion flux of species i in a mixture of n gas species and expressed as the Stefan-Maxwell equation (Bird et al, 2002). The diffusion flux of species i is given as... 

A third approach is suggested by Hugo s formulation of material balances at the limit of bulk diffusion control, described in Section 11.3. Hugo found expressions for the fluxes by combining the stoichiometric conditions and the Stefan-Maxvell relations, and this led to no inconsistencies since there are only n - 1 independent Stefan-Maxwell relations for the n fluxes. An analogous procedure can be followed when the diffusion is of intermediate type, using the dusty gas model equations in the form (5.10) and (5.11). Equations (5.11), which have the following scalar form ... 

We consider a nonreacting mixture such as iso-propanol (1) and water (2), evaporating into ambient air at constant temperature. Assuming the physical equilibrium condition (VLE) at the vapor-liquid interface and applying the Stefan-Maxwell-flux equations to the liquid phase and linear flux equations to the air-diluted gas phase, we derive the following expression for the relative flux XT... 

For multicomponent gas mixtures the generalised Maxwell-Stefan (GMS) equations should be used. Krishna [87b] derived an expression for the flux of specimen / ... 

The introductory Section 3.1.2.5 in Chapter 3 identifies the negative chemical potential gradient as the driver of targeted separation, and the relevant species flux expression is developed in Section 3.1.3.2 (see Example 3.1.9 also). Section 3.1.4 introduces molecular diffusion and convection and basic mass-transfer coefficient based flux expressions essential to studies of distillation and other phase equilibrium based separation processes. Section 3.1-5.1 introduces the Maxwell-Stefan equations forming the basis of the rate based approach of analyzing distillation column operation. After these fundamental transport considerations (which are also valid for other phase equilibrium based separation processes), we encounter Section 3.3.1, where the equality of chemical potential of a species in all phases at equilibrium is illustrated as the thermodynamic basis for phase equilibrium (Le. = /z ). Direct treatment of distillation then begins in Section 3.3.7.1, where Raouit s law is introduced. It is followed by Section 3.4.1.1, where individual phase based mass-transfer coefficients are reiated to an overall mass-transfer coefficient based on either the vapor or liquid phase. 

Equations 2.3.17 are the generalized Maxwell-Stefan (GMS) relations and the are the Maxwell-Stefan diffusion coefficients we encountered earlier. These equations are more useful when expressed in terms of the molar fluxes... 

In Section 8.3 we presented a derivation of an exact matrix solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures. Although the final expression for the composition profiles (Eq. 8.3.12), is valid whatever relationship exists between the fluxes (i.e., bootstrap condition), the derivation given in Section... 

Slattery (1981) presents a solution of the Maxwell-Stefan equations for the special case when two molar fluxes are zero, N- = N2 = 0. Write down expressions for 4>, [P], (y), and show that, for this special case, the eigenvalue solutions are equivalent to the expressions given by Slattery. 

For dilute gas mixtures we may employ the linearity postulate in irreversible thermodynamics to obtain the transport fluxes for heat and mass. The fundamental theory is examined in chap 2 and we simply refer to the expressions (2.456) and (2.457). Moreover, a particular form of the generalized Maxwell-Stefan equations, i.e., deduced from (2.298) in chap 2, is given by ... 

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. 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [77] or the average species velocity is used [96]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. 

However, if convective transport of heat and species mass in porous catalyst pellets have to be taken into account simulating catal3dic reactor processes, either the Maxwell-Stefan mass flux equations (2.394) or dusty gas model for the mass fluxes (2.427) have to be used with a variable pressure driving force expressed in terms of mass fractions (2.426). The reason for this demand is that any viscous flow in the catalyst pores is driven by a pressure gradient induced by the potential non-uniform spatial species composition and temperature evolution created by the chemical reactions. The pressure gradient in porous media is usually related to the consistent viscous gas velocity through a correlation inspired by the Darcy s law [21] (see e.g., [5] [49] [89], p 197) ... 

Gibbs-Duhem restriction on the chemical potential (eq. 8.5-3). Eq.(8.5-5) is the generalized Maxwell-Stefan constitutive relation. However, such form is not useful to engineers for analysis purposes. To achieve this, we need to express the chemical potential in terms of mole fractions. This is done by using eq. (8.5-2) into the constitutive flux equation (8.5-5). 

The Ki, values for each species i and the enthalpies used in the energy balance equations for any stage ra are obtained from conventional approaches used in multistage distillation analysis. However, the species flux is expressed in terms of the sum of a convective component and a diffusive component. The diffusive component is modeled using the Maxwell-Stefan approach (Section 3.1.5.1) for this complex multicomponent system in a matrix framework. For an illustrative introduction, see Sender and Henley (1998). 

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