User:Etheros/multicomponent transport draft

Multicomponent mass transfer refers to migration of more than one species under the influence of driving forces. This includes the binary diffusion modeled by Fick's law as well as migration of more than two species given usually by Maxwell-Stefan equations. The theories of multicomponent mass transport look at the interactions of species with each other and with the surroundings from which the motion of the species can be derived.

Background

The work on multicomponent mass transport was started by James Clerk Maxwell and Joseph Stefan. Stefan's work focused on diffusion of mixtures, while Maxwell's work was focused on kinetic theories of gases. The equations of multicomponent mass transport were developed independently and in parallel by Maxwell for dilute gases and Stefan for fluids. The idea of the Maxwell-Stefan model is to balance the driving forces in a system by friction forces in the system . First principle based Boltzmann type models can also be used to model multicomponnet mass transport, which use the Boltzamann equation for the species. These methods are usually employed with Lattice Boltzmann Methods which require rigorous computation computations.

General Multicomponent Mass Transport Model

The transport of the species in a mixture is driven by driving forces. The driving forces for mass transport can act alone or together on a species. The net driving force on a species $i$ is given as  : $\mathbf {d} _{i}={\dfrac {1}{c_{t}RT}}\left[\underbrace {c_{i}\nabla _{T,p}\mu _{i}} _{\text{Chemical Potential Gradient}}+\underbrace {(c_{i}{\overline {V}}_{i}-\omega _{i})\nabla p} _{\text{Net Pressure Gradient}}-\underbrace {\rho _{i}\left(\mathbf {F} _{i}-\sum _{j=1}^{n}\omega _{j}\mathbf {F} _{j}\right)} _{\text{External Force}}+\underbrace {\sum _{j=1 \atop j\neq i}^{n}\left({\dfrac {x_{i}x_{j}}{{D}_{ij}}}\right)\left({\dfrac {D_{i}^{T}}{\rho _{i}}}-{\dfrac {D_{j}^{T}}{\rho _{j}}}\right)\nabla \ln T} _{\text{Thermal Collision}}\right]$,

• $\nabla$: vector differential operator
• $\omega$: Mass fraction
• $x$: Mole fraction
• $\mu$: Chemical potential
• i, j: Indexes for component i and j
• n: Number of components
• ${D}_{ij}$: Maxwell–Stefan-diffusion coefficient
• $c_{i}$: Molar concentration of component i
• $c_{t}$: Total molar concentration
• ${\overline {V}}_{i}$: Partial molar volume
• R: Universal gas constant
• T: Temperature
• $\rho _{i}$: Density of species i
• p : Total pressure
• $\mathbf {F} _{i}$: External force on species i
• $D_{i}^{T}$: Thermal diffusion coefficient of species i

The external forces on the species can be gravity, electromagnetic forces or any other field forces. The governing force is countered by the friction forces present in the system. Usually the interspecies friction is an important term:

$Fr_{ij}=\sum _{j=1 \atop j\neq i}^{n}{\dfrac {x_{i}\mathbf {N} _{j}-x_{j}\mathbf {N} _{i}}{c_{t}{D}_{ij}}}$,

• $\mathbf {N} _{i}$: Molar flux of species i

Further, additional friction force can be attributed to the boundaries of the system. The interaction of the species with the boundaries (the walls).