Cantera/Diffusion
From charlesreid1
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{ 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} } } $