Applicability of the linearized theory of the Maxwell–Stefan equations

Abstract

The present article aims for a better understanding of the applicability of the linearized theory of the Maxwell-Stefan equations for multi-component diffusion. An analysis of the theory's accuracy is performed with respect to the classical two-bulb diffusion experiments by Duncan and Toor, from which the results are transferred to more general scenarios. It is shown that for an accurate linearized theory it is essential to have a quasi-stationary and quasi-one-dimensional flux, and also a so-called reference point. Two examples illustrating the theory's failure in case of unmet prerequisites are presented: a three-bulb configuration and a two-dimensional diffusion case. For the first setup the linearized theory results in negative concentrations, for the second it requires influxes at openings that are actually outlets.

Maxwell-Stefan-theory-based lattice Boltzmann model for diffusion in multicomponent mixtures

“…Considering the fact that the MS theory based continuum equations (15) are only a special case of nonlinear coupled diffusion equations, some available LB models for diffusion or convection-diffusion equations can be extended to solve the MS theory based continuum equations. In this work, we would propose a MRT-LB model for these continuum equations, and incorporate the cross collision terms in this model to reflect the coupling effects among different species which is similar to that in Ref.…”

“…(4b). In the following, we would adopt the MS based continuum equations (15) to study the diffusion in multicomponent mixtures.…”

“…Through a combination of Eqs. (53a) and (60), i.e., ε × (53a) + ε 2 × (60), we can correctly recover the MS theory based continuum equations for all species [see (15)]. Now let us focus on how to calculate the gradient term ∇ξ i (1 ≤ i ≤ n − 1), which can be used to determine the diffusion flux [see Eq.…”

“…Compared to the average mole fraction ξ1 , these curious results of the average mole fraction ξ2 are caused by the cross effects among different species, which can be confirmed by Eq. (15).…”

“…In this approach, usually the MS theory based continuum equations are first linearized where the effective diffusivities are assumed to be constants, then some mathematical methods are applied to derive the solutions of the linearized equations [1,9,10]. Although this approach can be used to reveal the complex diffusion mechanisms in multicomponent mixtures [11][12][13][14], it is usually limited to one-dimensional problems, and sometimes may also bring some undesirable errors due to the assumption of composite-independent effective diffusivities in the linearized equations [15]. The other one is numerical approach [16,17].…”