Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species.

More information and context is available at http://public.ca.sandia.gov/chemkin/docs/oppdif.pdf

Diffusitivites

Notation for some diffusivities should be defined before describing models for multicomponent or mixture-averaged diffusive fluxes. Three diffusivities are defined:

$D_{kj}$ - multicomponent species diffusivity
$D_{km}$ - mixture-averaged diffusivity
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{D}_{jk}} - binary diffusivity

The diffusive flux $\mathbf {J}$ is modeled as being proportional to species gradients. The diffusivities are the proportionality constants of that relationship.

Multicomponent

The multicomponent diffusive flux model assumes the diffusive flux $\mathbf {J}$ is proportional to every species gradient. This means that each species has a distinct contribution to the diffusive flux from the gradient of each species in the mixture:

(Assuming domain/diffusion is one-dimensional)

$\mathbf {J} _{k}=J_{k}=\rho V_{k}$

where $V_{k}$ is the diffusive flux velocity for species k.

There are two contributions to the diffusion velocity $V_{k}$ : the multicomponent diffusion velocity, and a contribution to diffusion velocity via thermal gradients:

$V_{k}=V_{km}+V_{kT}$

These are defined (assuming the one dimension is denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} ) as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{km} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \dfrac{ d X_j }{dx} }

or more generally,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{km} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \nabla X_j }

and the thermal diffusion term is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{kT} - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \dfrac{dT}{dx} }

or,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{kT} = - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \nabla T }

Mixture-Averaged

The mixture-averaged diffusive flux model assumes the diffusive flux for species k, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle J_k} , depends only on the gradient of species k:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{km} = - \dfrac{1}{X_k} D_{km} \dfrac{d X_k}{dx} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{km} = - \dfrac{1}{X_k} D_{km} \nabla X_k }

and the thermal diffusion term is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{kT} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \dfrac{dT}{dx} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_{kT} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \nabla T }

The mixture-averaged diffusivities are computed from the binary diffusivities as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_{km} = \dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk} } } }

Cantera/Diffusion

From charlesreid1

Diffusitivites

Multicomponent

Mixture-Averaged