Cantera/Diffusion

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

Taken from information 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 u_{k,diff} $

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

$ u_{k,diff} = u_{k,mass diff} + u_{k,therm diff} $

These are defined (assuming the one dimension is denoted $ x $) as:

$ u_{k,multicomponent mass diff} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \dfrac{ d X_j }{dx} $

or more generally,

$ u_{k,multicomponent multi diff} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \nabla X_j $

and the thermal diffusion term is:

$ u_{k,multicomponent therm diff} - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \dfrac{dT}{dx} $

or,

$ u_{k,multicomponent therm diff} = - \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:

$ u_{k,mixavg mass diff} = - \dfrac{1}{X_k} D_{km} \dfrac{d X_k}{dx} $

$ u_{k,mixavg mass diff} = - \dfrac{1}{X_k} D_{km} \nabla X_k $

and the thermal diffusion term is:

$ u_{k,mixavg therm diff} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \dfrac{dT}{dx} $

$ u_{k,mixavg therm diff} = - \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} $