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 gas, liquid, and solid phase) and diffusion in microporous material (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 all of them suffer from limitations. In particular 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. These methods require the knowledge of the analytic expression of the elements of the Fick-like diffusion matrix and therefore a 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 and practically is the limiting step for the applicability of such methods to problems with a large number of components. This work addresses the problem of the numerical resolution of these 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. The method is applied on 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. 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.

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

ANGELI, Celestino
2009

Abstract

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 gas, liquid, and solid phase) and diffusion in microporous material (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 all of them suffer from limitations. In particular 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. These methods require the knowledge of the analytic expression of the elements of the Fick-like diffusion matrix and therefore a 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 and practically is the limiting step for the applicability of such methods to problems with a large number of components. This work addresses the problem of the numerical resolution of these 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. The method is applied on 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. 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.
2009
Microporous materials; microporous diffusion; Maxwell-Stefan; membranes; zeolites
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11392/1379888
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