On the Maxwell-Stefan approach to diffusion: a general resolution in the transient regime for one-dimensional systems

The diffusion process in a multicomponent system can be formulated in a general form by the generalized Maxwell-Stefan equations. This formulation is able to describe the diffusion process in different systems, such as, for instance, bulk diffusion (in the gas, liquid, and solid phase) and diffusion in microporous materials (membranes, zeolites, nanotubes, etc.). The Maxwell-Stefan equations can be solved analytically (only in special cases) or by numerical approaches. Different numerical strategies have been previously presented, but the number of diffusing species is normally restricted, with only few exceptions, to three in bulk diffusion and to two in microporous systems, unless simplifications of the Maxwell-Stefan equations are considered. In the literature, a large effort has been devoted to the derivation of the analytic expression of the elements of the Fick-like diffusion matrix and therefore to the symbolic inversion of a square matrix with dimensions n x n (n being the number of independent components). This step, which can be easily performed for n = 2 and remains reasonable for n = 3, becomes rapidly very complex in problems with a large number of components. This paper addresses the problem of the numerical resolution of the Maxwell-Stefan equations in the transient regime for a one-dimensional system with a generic number of components, avoiding the definition of the analytic expression of the elements of the Fick-like diffusion matrix. To this aim, two approaches have been implemented in a computational code; the first is the simple finite difference second-order accurate in time Crank-Nicolson scheme for which the full mathematical derivation and the relevant final equations are reported. The second is based on the more accurate backward differentiation formulas, BDF, or Gear's method (Shampine, L. F. ; Gear, C. W. SIAM Rev. 1979, 21, 1.), as implemented in the Livermore solver for ordinary differential equations, LSODE (Hindmarsh, A. C. Serial Fortran Solvers for ODE Initial Value Problems, Technical Report; https://computation.llnl.gov/casc/odepack/odepack_ home.html (2006).). Both methods have been applied to a series of specific problems, such as bulk diffusion of acetone and methanol through stagnant air, uptake of two components on a microporous material in a model system, and permeation across a microporous membrane in model systems, both with the aim to validate the method and to add new information to the comprehension of the peculiar behavior of these systems. The approach is validated by comparison with different published results and with analytic expressions for the steady-state concentration profiles or fluxes in particular systems. The possibility to treat a generic number of components (the limitation being essentially the computational power) is also tested, and results are reported on the permeation of a five component mixture through a membrane in a model system. It is worth noticing that the algorithm here reported can be applied also to the Fick formulation of the diffusion problem with concentration-dependent diffusion coefficients.