Maxwell-Stefan diffusion

A model for describing the diffusion in multicomponent systems is called Maxwell-Stefan diffusion (also Stefan-Maxwell diffusion ) . The equations that describe these transport processes were developed by James Clerk Maxwell for dilute gases and Josef Stefan for liquids in parallel and independently of one another. The Maxwell-Stefan equation is:

${\displaystyle {\frac {\nabla \mu _{i}}{R\,T}}=\nabla \ln a_{i}=\sum _{j=1 \atop j\neq i}^{n}{{\frac {\chi _{i}\chi _{j}}{{\mathfrak {D}}_{ij}}}({\vec {v}}_{j}-{\vec {v}}_{i})}=\sum _{j=1 \atop j\neq i}^{n}{{\frac {c_{i}c_{j}}{c^{2}{\mathfrak {D}}_{ij}}}\left({\frac {{\vec {J}}_{j}}{c_{j}}}-{\frac {{\vec {J}}_{i}}{c_{i}}}\right)}}$

• ∇: Nabla operator
• χ: Mole fraction
• μ: chemical potential
• a: activity
• i, j: indices for component i and j
• n: number of components
• ${\displaystyle {\mathfrak {D}}_{ij}}$: Maxwell-Stefan diffusion coefficient
• ${\displaystyle {\vec {v}}_{i}}$: Speed ​​of component i
• ${\displaystyle c_{i}}$: Molar concentration of the component i
• c: total substance amount concentration
• ${\displaystyle {\vec {J}}_{i}}$: Mass flow of component i

The equation assumes a homogeneous flow, i.e. H. the absence of velocity gradients, as is particularly the case with stationary media.

The basic assumption of the theory is that a deviation from the equilibrium between molecular friction and thermodynamic interactions leads to diffusion flow. The molecular friction between two components is proportional to their speed difference and the amount of substance. In the simplest case, the gradient of the chemical potential is the driving force of the diffusion. For more complex systems, such as electrolytic solutions , and other driving forces, such as pressure gradients, the equation must be expanded to include terms for additional 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 Fick's diffusion coefficients and are therefore not tabulated. Also, the diffusion coefficients can only be determined for the binary and ternary case with justifiable effort. For three-component systems there are a number of approximation formulas for predicting the Maxwell-Stefan diffusion coefficients.

A great advantage of the theory is that systems can be considered in which the “classical” Fickian diffusion theory fails. For example, negative diffusion coefficients are also not excluded in the Maxwell-Stefan theory.

It is possible to derive Fick's theory from the Maxwell-Stefan theory.

Individual evidence

1. ^ JC Maxwell: On the dynamical theory of gases , The Scientific Papers of JC Maxwell, 1965, 2 , 26-78.
2. ↑ J. Stefan: On balance and movement, especially the diffusion of mixtures , session reports of the Imperial Academy of Sciences, Vienna, 2nd section a, 1871, 63 , 63–124.
3. ↑ ^{a b} R. Taylor, R. Krishna: Multicomponent Mass Transfer . Wiley, 1993.
4. ^ RB Bird, WE Stewart, EN Lightfoot: Transport Phenomena . 2nd Edition. Wiley, 2007.
5. ^ EL Cussler: Diffusion - Mass Transfer in Fluid Systems . 2nd Edition. Cambridge University Press, 1997.
6. ↑ ^{a b} S. Rehfeldt, J. Stichlmair: Measurement and calculation of multicomponent diffusion coefficients in liquids , Fluid Phase Equilibria, 2007, 256 , 99-104.