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

# 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:

$V_{km} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \dfrac{ d X_j }{dx}$

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:

$V_{km} = - \dfrac{1}{X_k} D_{km} \dfrac{d X_k}{dx}$

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

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:

$D_{km} = \dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk} } }$