The classical description of diffusion, which is based on Fick's Law (the assumption. gradient), is justified only in mixtures in which one component

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1 Solution of the Maxwell-Stefan Equations Background The classical description of diffusion, which is based on Fick's Law (the assumption of a linear relationship between the flux of a species and its concentration gradient), is justified only in mixtures in which one component (usually referred to as the solvent) is present invery large excess. In this regard it is not well suited to the description of diffusion in multicomponent gaseous mixtures, which can assume the entire range of compositions. More specifically, analyses based on Fick's Law are fundamentally incapable of accounting for apparently unphysical phenomena that have been experimentally observed in multicomponent gaseous diffusion, such as a net flux of a species in the opposite direction to, or even in the absence of, a concentration gradient in that species. [See, for example, J.B. Duncan and H.L. Toor, An experimental study of three component gas diffusion,", AIChE Journal, 8(): 38-4 (96).] In contrast, counterintuitive phenomena such as these can be accounted for by the Maxwell-Stefan (M-S) equations, in which coupling between the fluxes N j of diffusing species is explicitly taken into account. For a gas of constant overall concentration c, the gradient rx i of the mole fraction of each species is related not only to its own flux, but to the fluxes of all the other species, according to rx i = X j6=i cd ij (x i N j x j N i ); where D ij = D ji is the binary diffusivity ofi in j (or j in i), which canin principle be estimated from the kinetic theory of gases. The reciprocal diffusivity that appears on the right-hand side of this equation has the physical significance of a drag coefficient, expressing the average force experienced by the two species as a result of their relative motion. The physical foundations and applications of the M-S equations are described thoroughly in a review by Krishna and Wesselingh [R. Krishna and J.A. Wesselingh, The Maxwell-Stefan approach to mass transfer,"chemical Engineering Science, 5(6): 86 9 (997)].

2 Dimensionless Form For numerical calculations, the equations can be conveniently be rewritten in terms of dimensionless variables. Thus, the length coordinates X; Y; Z can be replaced by thex = X=L; y = Y=L; z = Z=L, where L is a reference length, and the fluxes N i can be replaced with dimensionless fluxes n i defined as n i = L cd 0 N i ; where D 0 is a reference diffusivity, which can be assigned an arbitrary value. One plausible choice would be to set it equal to the smallest of the pairwise diffusivities obtained from the Chapman-Enskog formalism. Alternatively, one could simply set it equal to the diffusivity of the pure Nth component, corresponding to the dependent mole fraction x N = x ::: x N. In these terms, the transport equations rx i = X j6=i cd ij (x i N j x j N i ) in which the gradient is taken with respect to X; Y; Z, become L rx i = X j6=i cd ij cd 0 L (x in j x j n i ) in which the gradient is taken with respect to x; y; z, or rx i = X j6=i D ij (x i n j x j n i ); where D ij = D ij =D 0 is a dimensionless diffusivity. Application of the M-S equations to time-dependent phenomena requires introduction of further equations relating the time dependence of the mole fraction of each species to its rate of transport by diffusion and convection, and its consumption by homogeneous or heterogeneous reactions. In the absence of convection and reaction, accumulation or depletion of a species in an infinitesimal control volume arises from differences in the diffusive flux of that species across opposite faces of the volume. As the dimensions of the control volume are shrunk to zero, the conservation law = r N i:

3 is recovered. For a three-dimensional Cartesian coordinate system (X; Y; Z), the divergence (r ) term that appears on the right-hand side is r N i : When the dimensionless spatial coordinates and dimensionless fluxes are introduced, = L r (cd 0 L n i) and these equations become, after = r n i; where fi D 0 t=l is a dimensionless time. Nonlinearity The advantages of the more rigorous theory of diffusion expressed by the M-S equations are, unfortunately, offset to a large extent by the difficulties in the solution of these equations. The linear parabolic partial differential equation that results from Fick's Law is the subject of a vast literature, and many analytical solutions are available. In contrast, the nonlinear coupling of each concentration gradient to the fluxes of all the species has the practical effect of limiting analytical solutions to steady-state problems in which the fluxes can be independently specified, or most can be set equal to zero. The latter class of problems generally reduce to the solution of a transcendental equation for one or more of the nonvanishing fluxes, and includes several classical problems in chemical engineering, suchastheevaporation of a liquid through a gas film [see, for example, R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena," New York: Wiley (96), p. 539.] The theoretical analysis of the Duncan-Toor (96) experiments was based on an earlier method developed by Toor [H.L. Toor, Diffusion in three-component gas mixtures,", AIChE Journal, 3(): ((957))], in which the transport equations were reduced to a pair of nonlinear equations for the two independent fluxes, but this approach does not seem to be applicable to more complex cases. This paper contains references to other particular cases in which analytical solutions are possible. 3

4 A highly interesting paper by Baker [Daniel R. Baker, Reducing nonlinear systems of transport equations to Laplace's equation," SIAM Journal of Applied Mathematics, 53(): (993)] demonstrates how the nonlinearity can be removed by expressing the fluxes in terms of a scalar potential, Q viz., n i = ο i rq; where ο i is a constant, and the potential function satisfies Laplace's equation: r Q =0: A direct consequence of this representation is that the divergence of the fluxes vanishes identically: r n i =0: If the constants ο i are chosen so that ο = and ο i = M i =M,whereM i and M are the molar masses of species i and, respectively, the mass-average velocity of the fluid is ρv = M N + :::+ M m N m = 0; this situation is usually referred to as equimolar counterdiffusion. This transformation has the effect of reducing the transport equations to a system of linear differential equations for the mole fractions with independent variable Q, and the condition that all the mole fractions sum to unity yields a nonlinear boundary condition for Q. Nonlinear boundary value problems for Laplace's equation can be solved by using the appropriate Green function to set up a nonlinear integral equation, which can be solved by substitutioniteration. Subsequent literature references to this important work have been concerned with applications to the solution of the Nernst-Planck equations for combined ionic diffusion and migration in electrolyte solutions. No further work has been published on the application of this method to the M-S equations, or to unsteady conditions. Since the conservation law equates the rate of change of the mole fraction to the negative divergence of the species flux, the representation of the species fluxes in terms of a harmonic potential is fundamentally unsuited to analysis of transient diffusion phenomena. Furthermore, the flux and its divergence can be expected to vary with position as well as with time. These two observations suggest that it might be possible to generalize Baker's approach to unsteady problems by dividing the space into elements within which the 4

5 fluxes are again represented as n i = ο i rq, but the potential in the kth such element is instead a solution of the Poisson equation r Q = fl k ; where the constant fl k is not necessarily zero. The time-dependence of the each mole fraction is then expressed = r n i = ο i fl k ; in view of the definition of the constants ο i, these expressions clearly sum to zero, which indicates consistency with the equimolal diffusion assumption as well as the addition of the mole fractions to unity. When this scalar potential representation of the fluxes is substituted into the M-S equations, and the multi-variable chain rule is applied in the form rx i = dx i dq rq; the gradient ofq cancels, resulting in a system of linear differential equations for the vector x of mole fractions as a function of Q: dx X i dq = (x j ο i x i ο j ): D j6=i ij Collecting terms in each mole fraction, this can be written in the form d dq 6 4 x. x m = 6 4 P j6= οj D j ο ο D P j6=. οm Dm οm Dm D οj D j ο D m ο D m.... P j6=m οj Dmj x. x m : For example, for a three-component system, the matrix is A = 6 4 ο cd ο 3 cd 3 ο cd ο 3 cd 3 ο cd ο cd ο 3 cd 3 ο cd 3 ο 3 cd 3 ο 3 cd 3 ο cd ο 3 cd : 5

6 Thus, the system of differential equations for x can be written compactly as dx dq = Ax where the elements of the matrix A are A ij =( ffi ij ) ο j X ο j ffi ij ; cd ij cd ij and ffi ij is the Kronecker delta ( if i = j and 0 otherwise.) Integrating the differential equations for x from one boundary of the region at which the concentrations are fixed at x 0, and Q is zero, the solution is x = x 0 exp[aq]: To complete the solution, this expression is substituted into the boundary conditions to be satisfied by the mole fractions and/or the species fluxes. This typically results in a nonlinear boundary condition for the Laplace equation defining the potential. Some remarks on the practical implementation of this result might be helpful. The right-hand side of this result contains the exponential of a matrix times a scalar argument Q. Although this is to be interpreted as: j6=i exp[aq] =I + QA + Q! A + ::: where I is the identity matrix, this power series representation is never used in practice. Rather, the exponential is represented in the form exp[aq]= k= E k e kq ; where the scalars k are the eigenvalues of A. For symmetric matrices, the m m coefficient matrices E k can be constructed from the dyadic products of the corresponding eigenvectors. For unsymmetric matrices such asa defined above, the E k can be obtained from the first m powers of A, according to the procedure suggested by Harris et al. (00) [William A. Harris, Jr., Jay P. Fillmore, and Donald R. Smith, Matrix exponentials - another approach," SIAM Review, 43(4): (00)], which is based on the Cayley-Hamilton Theorem from linear algebra. 6

7 Construction of the Potential Function As shown by Baker, steady-state solutions of the M-S equations can be expressed in terms of a single solution of Laplace's equation subject to a nonlinear boundary condition, and the above analysis showed that a single solution of the Poisson equation can likewise be used to describe situations in which the flux divergences are constant. For unsteady problems the flux divergence can be expected to vary with position, the indicated solution is to divide the region of interest into elements that are small enough that the flux divergence is approximately constant. Since within each element, the potential is defined by a different function, additional conditions of continuity of the potential function and its derivative must be satisfied at the boundaries of adjacent elements. The piecewise definition of the potential function subject to continuity conditions is closely related to the problem of interpolation by polynomial spline functions. The process of constructing the potential function corresponding to the concentration profile existing at some instant of time is perhaps best illustrated by considering the simplest case of unsteady diffusion in a onedimensional region from z = 0 to z = L. The appropriate analytical form for the potential function within each element follows by integration of the Green function for the one-dimensional Poisson equation times a constant source density. For the equation d Q dz = f(z) over the interval [0,L], at the end points of which homogeneous Dirichlet conditions are satisfied, the Green function satisfies d g = ffi(z ) dz where is the coordinate of the source point, and is given by g(zj ) = 8 < : (L )z 0 L (L z) L Suppose that the source density is given by» z<» z<l f( ) =F; [z ;z ] 7

8 and is zero elsewhere in the interval. Then the potential is Q(z) = Z L 0 g(zj )f( )d ; and the case of interest here is when z [z ;z ]. Then Z z Z z Q = F (L )zd + F (L z) d L z L z = F ρ z[l(z z ) L (z z)] + (L z) (z z )] = Lz z(lz + z z )+ Lz : This shows that the required potential is a quadratic function of the coordinates. For the satisfation of the continuity and flux-matching conditions that apply at the boundaries between the adjoining elements, it proves more convenient to assume a more general form Taking the gradient of this function Q = ff k + fi k (z z k )+ fl k(z z k ) : dq dz = fi k + fl k (z z k ) shows that the fluxes vary linearly with z in each element, and that the flux divergences, which are proportional to d Q dz = fl k are constant. Continuity of the potential at the boundary between elements k and k + is expressed by ff k+ = ff k + fi k (z k+ z k )+ fl k(z k+ z k ) ; and the continuity of the derivative at the same point requires that fi k+ = fi k + fl k (z k+ z k ): 8 ff

9 If the elements are assumed to be of identical width, so that z k+ z k = h, k = ;:::;n, these equations can be solved recursively, as follows. Thus, for k =, the continuity and flux-matching conditions at the boundary are and for k =3, ff = ff + fi h + fl h fi = fi + fl h; ff 3 = ff + fi h + fl h fi 3 = fi + fl h: After performing the indicated substitutions, the latter two relations become ff 3 = ff +fi h +» 3 fl + fl h fi 3 = fi +(fl + fl )h: Proceeding similarly for higher values of k and in general, ff 4 = ff +3fi h + ff 5 = ff +3fi h +» 5 fl + 3 fl + fl 3 h fi 4 = fi +(fl + fl + fl 3 )h;» 7 fl + 5 fl + 3 fl 3 + fl 4 h fi 5 = fi +(fl + fl + fl 3 + fl 4 )h; ff k+ = ff + kfi h + h fi k+ = fi + h kx l= k l + fl k In this way, it is seen that for n such elements, the piecewise-defined potential function requires a total of n + coefficients: ff, fi,andfl fl n. In the special case when all fl = fl n =0,thePoisson equation reduces to Laplace's equation, for which the appropriate solution is a linear function of the single 9 kx l= fl k :

10 space coordinate z. The above result is clearly consistent with this, since in that limit fi k+ = fi for all k and ff k+ = ff + kfi h -thetwo parameters ff, fi are then determined by the values of the potential at each end. Thus, the potential function appropriate for an unsteady problem also includes the steady-state function. The above discussion demonstrates the crucial importance of the parameters fl k, but makes no mention of how these are derived. As discussed in detail in the next section, it is at this point that the nonlinearities enter the analysis. Determination of the Flux Divergences Of the total of n + constants included in the potential function for the one-dimensional diffusion problem presented in the preceding section, ff can in fact be set equal to zero, and fi can be determined to represent the steady state concentration profile. The remaining n parameters fl :::fl n, the values of which define the divergences of the species fluxes and the rate of change of the mole fractions within each element, need to be selected so that the vectors mole fractions at adjacent nodesk, k + are consistent with the analytical solution. This is somewhat less straightforward than other collocation problems, in that there are m components of each vector that must be equated, but only one adjustable parameter, fl k. The best that can be done under such conditions is to apply a least-squares criterion to the quantity i= jx k+ exp[aq(z k+ )]x k j = i= jx k+ exp[aq(z k + h)]x k j ; which should have a minimum value of zero. Now if the matrix exponential is represented in the form exp[aq(z k + h)] = l= E l exp[ l Q(z k + h)]; where the matrices E l are obtained as described by Harris et al. (00), the mole fractions at x k+ are x k+ = " l= E l exp[ l Q(z k + h)] # 0 x k = l= exp[ l Q(z k + h)]e l x k :

11 Since E l is m m and x k is m, each of the l products E l x k is seen to be a column vector with ith component so that x k+;i = (E l x k ) i = l= j= E ij;l x k;j ; exp[ l Q(z k + h)] j= E ij;l x k;j : The requirement that the mole fractions at successive nodesk, k + are consistent is therefore expressed by the system of nonlinear equations 0 3 X 4 xk+;i exp[ l Q(z k + m E ij;l x k;j A5 =0; i= l= which are to be solved for fl :::fl n. The computational overhead associated with the solution of a system of n nonlinear equations at each iteration in fact turns out to be less severe than might first be supposed. This is because once ff and fi have been determined from the steady state concentration profile, the appearance of only fl in the potential for the first element[z ;z ], only fl, fl in the second [z ;z 3 ], etc., allows fl :::fl n to be determined by a forward recursion. Each step in this process can be achieved by means of one-dimensional optimization algorithms, which are known to be inherently more robust than their multivariate counterparts. Time-Stepping Algorithm As mentioned earlier, the parameters fl :::fl n define the rate of change of each mole fraction with respect to time. Although in principle, the j= = r n i = ο i fl k leads directly to the Euler approximation scheme x k;i (fi + ffifi) =x k;i (fi)+ο i fl k ffifi; in practice, this method is very seldom used, since unreasonably small step sizes ffifi are required to achieve results of satisfactory accuracy. But if the

12 entire calculation described above were incorporated into a subroutine that returned fl :::fl n as the values of the time derivatives of a function, the time-stepping algorithm could be performed by a wide variety ofmuch more accurate subroutines for the numerical integration of systems of ordinary differential equations. One possible choice is the fourth-order Runge-Kutta method; since this estimates the variable increments as the average of four function values, each time step would require four `passes' through the recursion for the flux divergences. The computational cost associated with multiple passes is possibly significant for adjustable-order predictor-corrector schemes, which potentially require even more function evaluations for each step. Interestingly, this proposed scheme seems to allow considerably more flexibility than the well-known numerical methods for the integration of the heat equation, in which the temporal and spatial integrations are more intimately linked.