transport_mixture.tex

From species to mixture consists in reducing the
dimension of the three quantities discussed above.
Viscosity and thermal conduction passed from
per species to per mixture, diffusion either
(Lewis model) stays a per species characterization
or (binary molecular diffusion) passes from
per couple species to per species. Tab.~\ref{spec_to_mix}
summarizes this.
%
\begin{table}
\centering
\begin{tabular}{l@{\hspace{3pt}}c@{\hspace{3pt}}c}\toprule
Model  & Species level                                    & Mixture level \\\midrule
\multicolumn{3}{l}{\hspace{\parindent}Viscosity}\\\cmidrule(lr){1-2}
Blottner     & ($\{\vis[1],\vis[2],\dots,\vis[\nspecies]\}$) & \vis[\text{mixture}] \\
Sutherland   & ($\{\vis[1],\vis[2],\dots,\vis[\nspecies]\}$) & \vis[\text{mixture}] \\
Pure species & ($\{\vis[1],\vis[2],\dots,\vis[\nspecies]\}$) & \vis[\text{mixture}] \\\\
\multicolumn{3}{l}{\hspace{\parindent}Thermal conduction}\\\cmidrule(rl){1-2}
Eucken       & ($\{\thermcond[1],\thermcond[2],\dots,\thermcond[\nspecies]\}$) & \thermcond[\text{mixture}] \\
Pure species & ($\{\thermcond[1],\thermcond[2],\dots,\thermcond[\nspecies]\}$) & \thermcond[\text{mixture}] \\\\
\multicolumn{3}{l}{\hspace{\parindent}Diffusion}\\\cmidrule(lr){1-2}
Lewis                 &  ($\{\diff[1],\diff[2],\dots,\diff[\nspecies]\}$) & $\{\diff[1],\diff[2],\dots,\diff[\nspecies]\}$ \\
Bimolecular diffusion & $\left(\begin{array}{cccc}
                          \diff[1,1]         & \diff[1,2]         & \dots  & \diff[1,\nspecies]\\
                          \diff[2,1]         & \diff[2,2]         & \dots  & \diff[2,\nspecies]\\
                          \vdots             &  \vdots            & \vdots & \vdots \\
                          \diff[\nspecies,1] & \diff[\nspecies,2] & \dots  & \diff[\nspecies,\nspecies]\\
                         \end{array}\right)
                          $ & $\{\diff[1],\diff[2],\dots,\diff[\nspecies]\}$ \\
\bottomrule
\end{tabular}
\caption{\label{spec_to_mix}Species to mixture values of viscosity, thermal conduction and diffusion
                with respect to the different models available in \Antioch.}
\end{table}


\subsubsection{The Wilke model}
Or mixture-averaged model.

\paragraph{Diffusion}
In case of Lewis model, nothing need to be done.

For the bimolecular diffusion model, we use:
\begin{equation}
\diff[s] = \frac{1-\massfrac[s]}{\sum_{j\ne s}^{\nspecies}\frac{\molarfrac[s]}{\diff[js]}}
         = \frac{\sum_{j\neq s}^{\nspecies}\molarfrac[j]\Mm[j]}{\Mm[\text{mixture}]\sum_{j\ne s}^{\nspecies}\frac{\molarfrac[s]}{\diff[js]}}
\label{Wilke:diff}
\end{equation}

\paragraph{Viscosity} The Wilke formul\ae\ for viscosity, independant of
the viscosity model, is
\begin{equation}
\vis[\text{mixture}] = \sum_{s=1}^{\nspecies} \frac{\molarfrac[s]\vis[s]}{\sum_{j=1}^{\nspecies}\molarfrac[j]\Phi_{sj}}
\label{Wilke:viscosity}
\end{equation}
with
\begin{equation}
\Phi_{sj} = \frac{
                \left[
                     1 + \sqrt{\frac{\vis[s]}{\vis[j]}\sqrt{\frac{\Mm[j]}{\Mm[s]}}}
                \right]^2
                }{\sqrt{8\left(1 + \frac{\Mm[s]}{\Mm[j]}\right)}}
\label{Wilke:viscosity:Phi}
\end{equation}

\paragraph{Thermal conduction}
Thermal conduction mixture value is using also the Wilke rule:
\begin{equation}
\thermcond[\text{mixture}] = \sum_{s=1}^{\nspecies} \frac{\molarfrac[s]\thermcond[s]}{\sum_{j=1}^{\nspecies}\molarfrac[j]\Phi_{sj}}
\label{Wilke:therm_cond}
\end{equation}

Note that \ChemKin\ transport is using an approximation of
the Wilke rule:
\begin{equation}
\thermcond[\text{mixture}] = \frac{1}{2} \left[
                                                \sum_{s=1}^{\nspecies} \molarfrac[s]\thermcond[s] + 
                                                \frac{1}{\sum_{s=1}^{\nspecies} \frac{\molarfrac[s]}{\thermcond[s]}}
                                        \right]
\label{ChemKin:therm_cond}
\end{equation}