Minimum Error Fickian Diffusion Coef®cients for Mass Diffusion in Multicomponent Gas Mixtures

Mass diffusion in multicomponent gas mixtures is governed by a coupled system of linear equations for the diffusive mass ̄uxes in terms of thermodynamic driving forces, known as the generalized Stefan±Maxwell equation. In computations of mass diffusion in multicomponent gas mixtures, this coupling between the different components results in considerable computational overhead. Consequently, simpli®ed diffusion models for the diffusive mass ̄uxes as explicit functions of the driving forces are an attractive alternative. These models can be interpreted as an approximate solution to the Stefan±Maxwell equation. Simpli®ed diffusion models require the speci®cation of `̀ effective'' diffusion coef®cients which are usually expressed as functions of the binary diffusion coef®cients of each species pair in the mixture. Current models for the effective diffusion coef®cients are incapable of providing a priori control over the error incurred in the approximate solution. In this paper a general form for diagonal approximations is derived, which accounts for the requirement imposed by the special structure of the Stefan±Maxwell equation that such approximations be constructed in a reduced-dimensional subspace. In addition, it is shown that current models can be expressed as particular cases of two general forms, but not all these models correspond to the general form for diagonal approximations. A new minimum error diagonal approximation (MEDA) model is proposed, based on the criterion that the diagonal approximation minimize the error in the species velocities. Analytic expressions are derived for the MEDA model's effective diffusion coef®cients based on this criterion. These effective diffusion coef®cients automatically give the correct solution in two important limiting cases: for that of a binary mixture, and for the case of arbitrary number of components with identical binary diffusivities. Although these minimum error effective diffusion coef®cients are more expensive to compute than existing ones, the approximation will still be cheaper than computing the exact Stefan±Maxwell solution, while at the same time being more accurate than any other diagonal approximation. Furthermore, while the minimum error effective diffusion coef®cients in this work are derived for bulk J. Non-Equilib. Thermodyn. 1999 Vol. 24 No. 1 # Copyright 1999 Walter de Gruyter Berlin New York diffusion in homogeneous media, the minimization procedure can in principle be used to derive similar coef®cients for diffusion problems in heterogeneous media which can be represented by similar forms of the Stefan±Maxwell equation. These problems include diffusion in macroand microporous catalysts, adsorbents, and membranes. 1. Introduction In this paper a rational approach to constructing diagonal approximations to the Stefan±Maxwell equation which governs mass-diffusion in multicomponent gaseous mixtures is presented. These approximations, like the Fickian diffusion assumption, decouple the full linear system by specifying effective binary diffusion coef®cients, thereby reducing computational expense. The necessary background for mass diffusion in multicomponent gas mixtures is established in Section 2. In order to familiarize the reader with some of the previous modeling efforts, three important simpli®ed diffusion models are described in Section 3. The modeling issues associated with constructing simpli®ed diffusion models are itemized in Section 4. In Section 5 the solution of the Stefan±Maxwell equation is detailed, which leads to the resolution of some of these modeling issues. This section also helps to establish a clear connection between the general form of existing approximations, which is given in Section 6, and the Stefan±Maxwell solution. The Stefan±Maxwell solution for two special limiting cases is explored in Section 7, which reveals the special structure of this equation. With the aid of this development a critical appraisal of existing models is given in Section 8. In Section 9 a new approach to constructing simpli®ed diffusion models is described, and the general form of diagonal approximations is given in Section 10. The particular form of these diagonal models that minimizes the error incurred in the approximation is derived in Section 11. Implications of this work are discussed in Section 12, and the salient conclusions are summarized in the ®nal section. The minimization procedure developed in this work can also be used to derive similar effective diffusion coef®cients for diffusion problems in heterogeneous media which can be cast in the Stefan±Maxwell equation framework. 2. Background Consider a multicomponent ideal gas mixture with N different chemical species. This system may be characterized by the mixture mass density , the mass-averaged velocity u, the internal energy e, and the species mass fractions Y ; ˆ 1; . . . ;N, which together constitute N‡ 5 unknowns. The mass fraction Y of species is de®ned as Y ˆ = ; ˆ 1; . . . ;N; …1† where is the mass density of the species (note that P ˆ , whence P Y ˆ 1). These N‡ 5 unknowns ( , u, e, Y ) are related by N‡ 5 conservation equations of mass, momentum, energy and species [1]. In particular, the species 2 S. Subramaniam J. Non-Equilib. Thermodyn. 1999 Vol 24 No. 1 conservation equation is: @Y @t ‡ u rY ˆ S ÿr ‰ Y… †V… †Š; ˆ 1; . . . ;N; …2† which states that following the mass-average motion of the ̄uid, the species mass fraction Y can change only by chemical reaction S , or by diffusion. The quantity Y… †V… † is termed the diffusive mass ̄ux of species . The summation convention in this paper is that repeated indices are summed over unless they are bracketed like in the diffusive mass ̄ux term of equation (2). If u denotes the average velocity of molecules of species , then the mass-averaged velocity of the mixture u is determined by the relation u ˆ Y u : …3† The mass diffusion velocity of species , denoted V , is de®ned as the difference between the mean molecular velocity of species and the mass-averaged velocity of the mixture, V u ÿ u: …4† Note that equations (3) and (4) together imply that the mass diffusion velocity must satisfy the constraint Y V ˆ 0: …5† The diffusive mass ̄ux of species relative to the mass-averaged velocity of the mixture u, is de®ned as J … †‰u… † ÿ uŠ ˆ Y… †V… †: …6† The N species conservation equations (eq. (2)) together with the 5 conservation equations for , u, and e, form a closed set in terms of the N‡ 5 unknowns, provided quantities such as the reaction rates S and the diffusion ̄uxes Y… †V… † can be related to the variables ( , u, e, Y ), and their gradients. In this paper we are concerned with the closure of the mass diffusion terms. The closure equation for the mass diffusion ̄uxes in terms of the mass-fraction gradients as given by the complete kinetic theory [1], is the Stefan±Maxwell equation. 2.1. The generalized Stefan±Maxwell equation Following Ramshaw [2], the generalized Stefan±Maxwell equation for the mass diffusion velocities V is written as a linear system of the form X X… †X… † D ‰V… † ÿ V… †Š ˆ G ; …7† Minimum error Fickian diffusion coef®cients 3 J. Non-Equilib. Thermodyn. 1999 Vol 24 No. 1 where X is the mole fraction of species , D is the binary diffusivity for the species pair ( , ) and the driving forces G are given by G ˆ rX ‡ …X ÿ Y †r In p‡ K r ln T ÿ 1 p ‰ … †F… † ÿ Y F Š; …8† where p is the pressure, T is the temperature, and F is the body force per unit mass acting on species . The coef®cients K are related to the thermal diffusion coef®cients DT ; [1], [2] by the relation K ˆ X X… †X… † D DT ; Y… † ÿ DT ; Y… † : …9† When the only non-zero contribution to the driving forces G is due to concentration gradients rX , equation (7) is commonly referred to as the Stefan±Maxwell equation [1]. When the driving forces are generalized to include the contributions due to pressure gradients, temperature gradients and body forces [2], equation (7) is referred to as the generalized Stefan±Maxwell equation. The sum of the driving forces G over all species is zero: X G ˆ 0: …10† Since the species conservation equation (eq. (2)) is in terms of mass fractions, it is convenient to express the mole-fraction gradients G (for simplicity, and without loss of generality, hereinafter it is assumed that the only non-zero contribution to the driving forces arises from concentration gradients) in equation (8) as a linear combination of mass fraction gradients H rY : G ˆ rX ˆ @X @Y rY ˆ T H ; …11† where the transformation matrix T is given by T… †… † ˆ X… † Y… † ‰1ÿ X… †Š …12† T ˆ X X… † Y… † ; 6ˆ : …13† With the relation between G and H , it is easy to see that equation (7) represents a closure of the diffusion ̄ux in terms of mass fraction gradients. The sum of the driving forces H over all species is also zero: X H ˆ 0: …14† 4 S. Subramaniam J. Non-Equilib. Thermodyn. 1999 Vol 24 No. 1 It is well known that the Stefan±Maxwell equation (eq. (7)) alone does not determine the mass diffusion velocities V uniquely [2]. The additional constraint on the massdiffusion velocity Y V ˆ 0 is needed to uniquely determine V . Since equation (7) is only a statement concerning velocity differences, its solution is indeterminate to within a constant vector. This indeterminacy implies that equation (7) may also be written in terms of the species velocity with u ˆ V ‡ u in place of V as X X… †X… † D ‰u… † ÿ u… †Š ˆ G : …15† The additional constraint is now Y u ˆ u, where u is determined by the momentum equation, and may be regarded as known. Some approximations to the Stefan±Maxwell equation also use the molar diffusion velocity V0 , which is de®ned as the difference between the mean molecular velocity of species and the molar-averaged velocity of the mixture: V0 u ÿ u0; …16† where the molar-averaged velocity of the mixture u0 is given by the relation u0 ˆ X u : …17† Note that equations (17) and (16) together imply that the molar diffusion velocity must satisfy the constraint