On the formal derivation of the reactive Maxwell-Stefan equations from the kinetic theory

Abstract

We study multicomponent diffusion in a gaseous mixture undergoing a reversible chemical reaction. Under isothermal and non-isothermal conditions, we extend the Maxwell-Stefan (MS) theory to the reactive system, by deriving the reactive MS equations containing additional diffusion terms resulting from the chemical reaction. We start from the simple reacting spheres (SRS) kinetic model and consider a diffusive regime, for which both mechanical collisions and chemical reactions have comparable relaxation times, which are much smaller than the characteristic times of the flow. In the corresponding hydrodynamic limit, we formally derive the reactive MS equations from the momentum equation of the species and deduce explicit expressions for the diffusion coefficients.

From the Boltzmann Description for Mixtures to the Maxwell–Stefan Diffusion Equations

“…Their result allows to prove that, in the limit ε → 0, the Maxwell-Stefan equation are rigorously derived. The authors work in the three-dimensional torus T 3 .…”

“…In [3], the same authors modify the scaling, by supposing that both mechanical collisions and chemical reactions have comparable relaxation times, much smaller than the characteristic time of the flow. The reactive Maxwell-Stefan equations under this scaling have the form:…”

“…After describing these formal results, we summarize the result by Bondesan and Briant [8], which provides a rigorous description of the limiting procedure. The next chapter is devoted to the reactive case, and is essentially based on the recent articles [2], [3] and [4]. We conclude this review by quoting the most recent literature on some strictly related subjects.…”

Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model

“…Since our interest is on inert mixtures, we shall give the non-isothermal Maxwell-Stefan diffusion equations here for the completeness. They can be obtained from the moment equations ( 7) by using the scaled distribution function (15) obtained by MEP. The mass/concentrantion and momentum balance laws for the species are obtained taking ψ i (v) = 1 and ψ i (v) = v, respectively.…”

“…From the macroscopic point of view, Maxwell-Stefan diffusion model consists of mass balance equations for the constituents, and a kind of momentum balance relations adjoined with a closure relation due to their linear dependence. It can be derived either in the continuum framework [11,12], or starting from the Boltzmann equations (BEs) for mixtures [13,14,15,16]. In the latter approach, crucial step in derivation of Maxwell-Stefan equations is the asymptotic limit enabled through appropriate scaling of the governing equations.…”