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The '''Maxwell–Stefan diffusion''' (or '''Stefan–Maxwell diffusion''') is a [[Mathematical model|model]] for describing [[diffusion]] in multicomponent systems. The equations that describe these transport processes have been developed independently and in parallel by [[James Clerk Maxwell]]<ref name="Maxwell 1">J. C. Maxwell: ''On the dynamical theory of gases'', The Scientific Papers of J. C. Maxwell, 1965, '''2''', 26–78.</ref> for dilute gases and [[Josef Stefan]]<ref name="Stefan 1">J. Stefan: ''Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen'', Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, '''63''', 63-124.</ref> for fluids. The Maxwell–Stefan equation is <ref name="BSL">{{cite book|title=[[Transport Phenomena (book)|Transport Phenomena]]|edition=2|first1=R.B.|last1=Bird|first2=W.E.|last2=Stewart|first3=E.N.|last3=Lightfoot|publisher=Wiley|year=2007}}</ref> | |||
:<ref name="TaylorKrishna">{{cite book|title=Multicomponent Mass Transfer|first1=R.|last1=Taylor|first2=R.|last2=Krishna|publisher=Wiley|year=1993}}</ref><ref name="Cussler">{{cite book|title= Diffusion - Mass Transfer in Fluid Systems|edition=2|first1=E.L.|last1=Cussler|publisher=Cambridge University Press|year=1997}}</ref> | |||
:<math> | |||
\frac{\nabla \mu_i}{R\,T} | |||
= \nabla \ln a_i | |||
==\sum_{j==1\atop j\neq i}^{n}{\frac{\chi_i \chi_j}{\mathfrak{D}_{ij}}(\vec v_j-\vec v_i)} | |||
==\sum_{j==1\atop j\neq i}^{n}{\frac{c_ic_j}{c^2\mathfrak{D}_{ij}}\left(\frac{\vec J_j}{c_j}-\frac{\vec J_i}{c_i}\right)} | |||
</math> | |||
* ∇: [[del|vector differential operator]] | |||
* χ: [[Mole fraction]] | |||
* μ: [[Chemical potential]] | |||
* a: [[Activity (chemistry)|Activity]] | |||
* i, j: Indexes for component i and j | |||
* n: Number of components | |||
* <math>\mathfrak{D}_{ij}</math>: Maxwell–Stefan-diffusion coefficient | |||
* <math>\vec v_i</math>: Diffusion velocity of component i | |||
* <math>c_i</math>: [[Molar concentration]] of component i | |||
* c: Total molar concentration | |||
* <math>\vec J_i</math>: [[Flux]] of component i | |||
The equation assumes [[steady state]], that is the absence of velocity gradients. | |||
The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux.<ref name="Rehfeldt">S. Rehfeldt, J. Stichlmair: ''Measurement and calculation of multicomponent diffusion coefficients in liquids'', Fluid Phase Equilibria, 2007, '''256''', 99–104</ref> The molecular friction between two components is proportional to their difference in speed and their mole fractions. In the simplest case, the [[gradient]] of chemical potential is the driving force of diffusion. For complex systems, such as [[electrolytic]] solutions, and other drivers, such as a pressure gradient, the equation must be expanded to include additional terms for interactions. | |||
A major disadvantage of the Maxwell–Stefan theory is that the [[diffusion coefficient]]s, with the exception of the diffusion of dilute gases, do not correspond to the [[Fick's laws of diffusion|Fick's diffusion coefficients]] and are therefore not tabulated. Only the diffusion coefficients for the binary and ternary case can be determined with reasonable effort. In a multicomponent system, a set of approximate formulas exist to predict the Maxwell–Stefan-diffusion coefficient.<ref name="Rehfeldt"/> | |||
The Maxwell–Stefan theory is more comprehensive than the "classical" Fick's diffusion theory, as the former does not exclude the possibility of negative diffusion coefficients. It is possible to derive Fick's theory from the Maxwell–Stefan theory.<ref name="TaylorKrishna"/> | |||
==See also== | |||
*[[Pervaporation]] | |||
== References == | |||
<references /> | |||
{{DEFAULTSORT:Maxwell-Stefan diffusion}} | |||
[[Category:Diffusion]] | |||
Revision as of 16:46, 25 January 2014
The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. The equations that describe these transport processes have been developed independently and in parallel by James Clerk Maxwell[1] for dilute gases and Josef Stefan[2] for fluids. The Maxwell–Stefan equation is [3]
- ∇: vector differential operator
- χ: Mole fraction
- μ: Chemical potential
- a: Activity
- i, j: Indexes for component i and j
- n: Number of components
- : Maxwell–Stefan-diffusion coefficient
- : Diffusion velocity of component i
- : Molar concentration of component i
- c: Total molar concentration
- : Flux of component i
The equation assumes steady state, that is the absence of velocity gradients.
The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux.[6] The molecular friction between two components is proportional to their difference in speed and their mole fractions. In the simplest case, the gradient of chemical potential is the driving force of diffusion. For complex systems, such as electrolytic solutions, and other drivers, such as a pressure gradient, the equation must be expanded to include additional terms for interactions.
A major disadvantage of the Maxwell–Stefan theory is that the diffusion coefficients, with the exception of the diffusion of dilute gases, do not correspond to the Fick's diffusion coefficients and are therefore not tabulated. Only the diffusion coefficients for the binary and ternary case can be determined with reasonable effort. In a multicomponent system, a set of approximate formulas exist to predict the Maxwell–Stefan-diffusion coefficient.[6]
The Maxwell–Stefan theory is more comprehensive than the "classical" Fick's diffusion theory, as the former does not exclude the possibility of negative diffusion coefficients. It is possible to derive Fick's theory from the Maxwell–Stefan theory.[4]
See also
References
- ↑ J. C. Maxwell: On the dynamical theory of gases, The Scientific Papers of J. C. Maxwell, 1965, 2, 26–78.
- ↑ J. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 4.0 4.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 6.0 6.1 S. Rehfeldt, J. Stichlmair: Measurement and calculation of multicomponent diffusion coefficients in liquids, Fluid Phase Equilibria, 2007, 256, 99–104