Cantera/Diffusion

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
• $\mathbb {D} _{jk}$ - 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 $x$) as:

$V_{km}={\dfrac {1}{X_{k}{\overline {W}}}}\sum _{j=1}^{K}W_{j}D_{kj}{\dfrac {dX_{j}}{dx}}$

or more generally,

$V_{km}={\dfrac {1}{X_{k}{\overline {W}}}}\sum _{j=1}^{K}W_{j}D_{kj}\nabla X_{j}$

and the thermal diffusion term is:

$V_{kT}-{\dfrac {D_{k}^{T}}{\rho Y_{k}}}{\dfrac {1}{T}}{\dfrac {dT}{dx}}$

or,

$V_{kT}=-{\dfrac {D_{k}^{T}}{\rho Y_{k}}}{\dfrac {1}{T}}\nabla T$

Mixture-Averaged

The mixture-averaged diffusive flux model assumes the diffusive flux for species k, $J_{k}$, depends only on the gradient of species k:

$V_{km}=-{\dfrac {1}{X_{k}}}D_{km}{\dfrac {dX_{k}}{dx}}$

$V_{km}=-{\dfrac {1}{X_{k}}}D_{km}\nabla X_{k}$

and the thermal diffusion term is:

$V_{kT}=-{\dfrac {D_{k}^{T}}{\rho Y_{k}}}{\dfrac {1}{T}}{\dfrac {dT}{dx}}$

$V_{kT}=-{\dfrac {D_{k}^{T}}{\rho Y_{k}}}{\dfrac {1}{T}}\nabla T$

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

$D_{km}={\dfrac {1-Y_{k}}{\sum _{j\neq k}{K}{\dfrac {X_{j}}{\mathbb {D} _{jk}}}}}$