thermal_cond.tex

\subsubsection{Eucken thermal conduction}

The Eucken thermal conduction is given by
\begin{equation}
\thermcond[s] = \vis[s]\left(\foreach \pre/\mod in {/vib,+/rot,+2.5/trans,+/elec} 
                {
                    \pre\specificHeat[\Vol,s]^{(\text{\mod})}
                }
                \right)
\label{thermal_cond:eucken:eq}
\end{equation}

\subsubsection{Pure species thermal conduction}

The thermal conduction of a species $s$ is given by
\begin{equation}
\begin{split}
\thermcond[s] & = \thermcond[s]^{(\text{rot})} + \thermcond[s]^{(\text{trans})} + \thermcond[s]^{(\text{vib})} \\
              & = \frac{\vis[s]}{\Mm[s]} \left(
                        \foreach \signe/\mode in {/vib,+/rot,+/trans}
                                {\signe \specificHeat[\Vol,s]^{(\text{\mode})}\thermcond[s]^{(\text{\mode})}}
                                          \right)
\end{split}
\label{transport:therm_cond}
\end{equation}
with
%%
\begin{equation}
\begin{split}
\thermcond[\Vol,s]^{(\text{vib})}   & = \mass[s] \frac{\diff[s s]}{\vis[s]} \\
\thermcond[\Vol,s]^{(\text{rot})}   & = \mass[s] \frac{\diff[s s]}{\vis[s]}\left(1 + \frac{2}{\pi}\frac{A}{B}\right) \\
\thermcond[\Vol,s]^{(\text{trans})} & = \frac{5}{2} \left(1 - \frac{2}{\pi}\frac{\specificHeat[\Vol,s]^{(\text{rot})}}{\specificHeat[\Vol,s]^{(\text{trans})}}\frac{A}{B}\right)
\end{split}
\label{thermal_cond:modes}
\end{equation}
%%
and
%%
\begin{equation}
\begin{split}
  A & = \frac{5}{2} - \mass[s] \frac{\diff[s s]}{\vis[s]} \\
  B & = \rotRelax + \frac{2}{\pi} \left(\frac{5}{3}\frac{\specificHeat[\Vol,s]^{(\text{rot})}}{\Rg} + \mass[s] \frac{\diff[s s]}{\vis[s]}\right)
\end{split}
\label{thermal_cond:A_B}
\end{equation}
%%
the rotational relaxation number is given by
\begin{equation}
\rotRelax[,s](\Temp) = \rotRelax[,s](298)\frac{F(298)}{F(\Temp)}
\label{thermal_cond:Zrot}
\end{equation}
with the function $F$ defined as
\begin{equation}
F(T) = 1 + \frac{\pi^{\frac{3}{2}}}{2}\sqrt{\left(\frac{\LJdepth[s]}{\Boltzmann\Temp}\right)}
         + \left(\frac{\pi^2}{4} + 2\right)\left(\frac{\LJdepth[s]}{\Boltzmann\Temp}\right)
         + \pi^{\frac{3}{2}}\left(\frac{\LJdepth[s]}{\Boltzmann\Temp}\right)^{\frac{3}{2}}
\label{therm_cond:F}
\end{equation}

The specific heat are given by the thermodynamic module (not documented here yet).