diffusion.tex

\subsubsection{Lewis diffusion}

Lewis diffusion coefficient\footnote{or diffusivity} is expressed at
the mixture level as
\begin{equation}
\diff[s] = \Le \frac{\thermcond[s]}{\mass[s] \specificHeat[\Press,s]}
\label{diff:Lewis}
\end{equation}


\subsubsection{Binary molecular diffusion}

The diffusion model used in \Antioch\ is at the species
level the molecular binary diffusion model as computed from kinetics
theory, assuming the ideal gas hypothesis.

The molecular binary diffusion coefficient (\unit{m^2\,s^{-1}}) is given by:
\begin{equation}
\diff[ij] = \frac{3}{16} \sqrt{\frac{2\Boltzmann^3\Temp^3}{\pi\Mass_{ij}}}\frac{1}{\Press\LJdia_{ij}^2\Stockmayer{1}}
\label{bimol_diff:eq}
\end{equation}
with $\Mass_{ij}$ the reduced mass (\unit{kg})
\begin{equation}
\Mass_{ij} = \left\{\begin{array}{l@{\qquad}l}
                \frac{\Mass_i\Mass_j}{\Mass_i + \Mass_j} & \text{if } i \neq j \\
                \Mass_i                                  & \text{else}
                    \end{array}
              \right.
\label{diffusion:bimol:red_mass}
\end{equation}
$\LJdia_{ij}$ the Lennard-Jones collision diameter (\unit{m})
between species $i$ and $j$:
\begin{equation}
\LJdia_{ij} = \frac{1}{2}\left(\LJdia_i + \LJdia_j\right) \xi^{-\frac{1}{6}}
\label{diffusion:bimol:LJdia}
\end{equation}
and \Stockmayer{1}\ the integrated collision interval, fitted from
the tables given in \citet{Monchick1961}, the parameter $\xi$
is given by \ref{reduced_parameter:xi}.

Given the ideal gaz assumption, noting $\Rg = \Boltzmann\Navo$, and
and $\Mass\Navo = \Mm$ with
\Navo\ the Avodagro number (\NavoEquation)
\begin{equation}
\diff[ij] = \frac{3}{16}\sqrt{\frac{2\Boltzmann}{\Navo\pi}} 
                \sqrt{\frac{\Temp}{\Mm[ij]}}\frac{1}{\left(\sum_s\conc_s\right)\LJdia[ij]^2\Stockmayer{1}}
\label{diffusion:bimol:simplified}
\end{equation}