From Blelly and Schunk (1993), the 16-moment equation set gives:

$\begin{eqnarray*} \frac{\delta q_s^p}{\delta t} &=& \sum_{t \ne i} \frac{3}{2\pi}\nu_{st} \left[ k_b n_s m_s \sigma_{st}^t R_{st}^{(2)}(u_t-u_s) + \frac{\sigma_{st}^t}{\sigma_{st}^p}\left(-R_{st}^{(4)}q_s^t + \frac{n_s m_s}{n_t m_t}R_{st}^{(6)}q_t^t-R_{st}^{(8)}q_s^p + \frac{n_s m_s}{n_t m_t}R_{st}^{(10)}q_t^p\right) \right] -3n_s k_b T_s^p\frac{\delta U_s}{\delta t} \end{eqnarray*}$

...

$\begin{eqnarray*} \frac{\delta q_s^{\parallel}}{\delta t} &=& \sum_t \nu_{st}(U_t-U_s)T_sA^{*0p}_{st}k_bn_s + \nu_{st}\left(D^{*1pt}_{st}q_s^{\perp}+\frac{n_s}{n_t}D^{*4pt}_{st}q_t^{\perp}+D^{*1pp}_{st}q_s^{\parallel}+\frac{n_s}{n_t}D^{*4pp}_{st}q_t^{\parallel}\right) - 3n_sk_bT_s^{\parallel}\frac{\delta U_s}{\delta t}\\ &=& T_sk_bn_s\sum_t \nu_{st}A^{*0p}_{st}(U_t-U_s)\\ &+& n_s\sum_t \nu_{st}\left(D^{*1pt}_{st}\frac{q_s^{\perp}}{n_s}+D^{*4pt}_{st}\frac{q_t^{\perp}}{n_t}+D^{*1pp}_{st}\frac{q_s^{\parallel}}{n_s}+D^{*4pp}_{st}\frac{q_t^{\parallel}}{n_t}\right)\\ &-& 3n_sk_bT_s^{\parallel}\frac{\delta U_s}{\delta t}\\ \end{eqnarray*}$

We use the normalisation $\gamma_s^{\parallel} = \frac{q_s^{\parallel}}{n_s}\sqrt{\frac{m_s}{(k_bT_s^{\parallel})^3}}$ and $\gamma_s^{\perp} = \frac{q_s^{\perp}}{n_s}\sqrt{\frac{m_s}{(k_bT_s^{\perp})^3}}$ :

$\begin{eqnarray*} \frac{\delta q_s^{\parallel}}{\delta t} &=& T_sk_bn_s\sum_t \nu_{st}A^{*0p}_{st}(U_t-U_s)\\ &+& n_s\sum_t \nu_{st}\left( D^{*1pt}_{st}\frac{k_b^{\frac{3}{2}}T_s^{\perp \frac{3}{2}}}{m_s^{\frac{1}{2}}}\gamma_s^{\perp }+ D^{*4pt}_{st}\frac{k_b^{\frac{3}{2}}T_t^{\perp \frac{3}{2}}}{m_t^{\frac{1}{2}}}\gamma_t^{\perp }+ D^{*1pp}_{st}\frac{k_b^{\frac{3}{2}}T_s^{\parallel \frac{3}{2}}}{m_s^{\frac{1}{2}}}\gamma_s^{\parallel }+ D^{*4pp}_{st}\frac{k_b^{\frac{3}{2}}T_t^{\parallel \frac{3}{2}}}{m_t^{\frac{1}{2}}}\gamma_t^{\parallel }\right)\\ &-& 3n_sk_bT_s^{\parallel}\frac{\delta U_s}{\delta t}\\ \end{eqnarray*}$

$t_s = \frac{T_s}{T_0}$ is the normalised temperature. $u_s = \frac{U_s}{C_s}$ is the normalised velocity for specie $s$ with $C_s = \sqrt{\frac{k_b T_0}{m_s}}$ .

$\begin{eqnarray*} \frac{\delta \gamma_s^{\parallel}}{\delta t} &=& \frac{1}{n_s}\frac{m_s^{\frac{1}{2}}}{k_b^{\frac{3}{2}}T_s^{\parallel\frac{3}{2}}}\frac{\delta q_s^{\parallel}}{\delta t}\\ &=& \frac{1}{n_s k_b}\frac{1}{C_s}\frac{1}{t_s^{\parallel\frac{3}{2}}} \left[ t_s k_b n_s \sum_t \nu_{st}A^{*0p}_{st}(C_t u_t - C_s u_s) + \right.\\ &+& n_s k_b \sum_t \nu_{st}\left( D^{*1pt}_{st}C_s t_s^{\perp \frac{3}{2}}\gamma_s^{\perp }+ D^{*4pt}_{st}C_t t_t^{\perp \frac{3}{2}}\gamma_t^{\perp }+ D^{*1pp}_{st}C_s t_s^{\parallel \frac{3}{2}}\gamma_s^{\parallel }+ D^{*4pp}_{st}C_t t_t^{\parallel \frac{3}{2}}\gamma_t^{\parallel }\right)\\ &-& \left.3n_sk_bt_s^{\parallel}C_s\frac{\delta u_s}{\delta t}\right]\\ &=& t_s \sum_t \nu_{st}A^{*0p}_{st}(\frac{C_t}{C_s} u_t - u_s)\frac{1}{t_s^{\parallel\frac{3}{2}}}\\ &+& \sum_t \nu_{st}\left( D^{*1pt}_{st} \frac{t_s^{\perp \frac{3}{2}}}{t_s^{\parallel \frac{3}{2}}} \gamma_s^{\perp }+ D^{*4pt}_{st}\frac{C_t}{C_s} \frac{t_t^{\perp \frac{3}{2}}}{t_s^{\parallel \frac{3}{2}}} \gamma_t^{\perp }+ D^{*1pp}_{st} \gamma_s^{\parallel }+ D^{*4pp}_{st}\frac{C_t}{C_s} \frac{t_t^{\parallel \frac{3}{2}}}{t_s^{\parallel \frac{3}{2}}} \gamma_t^{\parallel }\right)\\ &-& \left.3t_s^{\parallel}\frac{\delta u_s}{\delta t}\frac{1}{t_s^{\parallel\frac{3}{2}}}\right]\\ \end{eqnarray*}$

Collision terms for ionic heat flux

Collision terms for neutral heat flux