Speaker
Description
\section{Introduction}
%\label{sec:intro}
In the last years, state-to-state (StS) chemical kinetics have been used to model high-enthalpy flows in 2D configurations in dissociating air~\cite{bonelli2024finite,guo2024investigation,wang2023high}. The recent interest in the exploration of ice giant planets~\cite{blanc2021science} needs the construction of kinetic schemes for Hydrogen/Helium mixture plus some impurities (CH$_{4}$), which give a relevant contribution to the radiation emission in the shock layer.
A pure H$_{2}$/He StS kinetic scheme, originally developed to describe volume sources of negative ion production~\cite{celiberto2023advanced,colonna2017vibrational}, has been used to model EAST shock tube experiment in conditions of Saturn entry~\cite{cruden2017shock}, reproducing the ionisation profile~\cite{colonna2020ionization}. It has been shown that ionisation is more efficiently initiated by the reaction $2\text{H}_{2}\rightleftharpoons \text{H}_{3}^{+}+\text{H}^{-}$ than by ionising recombination ($2\text{H}\rightleftharpoons \text{H}_{2}^{+}+e^{-}$), the latter limited by the dissociation kinetics.
A StS model accounting for ionisation should also describe the free electron kinetics, through the solution of the Boltzmann equation~\cite{colonna2022two}, and, together with the vibrational levels, the evolution of electronically excited states of atoms and molecules~\cite{colonna2001influence}.
A multi-temperature (mT) model rigorously derived from the StS is here presented. The state-specific cross section database has been updated so as to include the most accurate data available for the electron-impact induced inelastic and dissociative processes of H$_2$~\cite{scarlett2021complete} and the mT model still follows the kinetics of selected electronically excited states of molecular hydrogen. The equation for vibrational energy relaxation includes also the contribution of chemical processes.
\section{From State-to-State to multi-Temperature}
\label{sec:StS-to-mT}
A mT model assumes that the internal distributions follow the Boltzmann function with independent internal temperatures $T_{v}$. Given a process like $\text{H}_{2}(v)+X \rightleftharpoons \cdots$
with $k_{v}^{d}$ and $k_{v}^{r}$ direct and reverse rate respectively, the kinetic equation contains the term proportional to $k_{v}^{d}(T)f_{v}(T_{v})$ and to $k_{v}^{r}(T)$, where $f_{v}$ is the vibrational distribution. The global rates are given by
\begin{equation}
\begin{array}{ll}
K^{d}(T,T_{v})&=\sum_{v}k_{v}^{d}(T)f_{v}(T_{v})\
K^{r}(T)&=\sum_{v}k_{v}^{r}(T)
\end{array}
\label{e:colonna:globalRateGen}
\end{equation}
and the contribution to the internal energy by
\begin{equation}
\begin{array}{ll}
\mathcal{L}(T,T_{v})&=\sum_{v}\varepsilon_{v}k_{v}^{d}(T)f_{v}(T_{v})\
\mathcal{G}(T)&=\sum_{v}\varepsilon_{v}k_{v}^{r}(T)
\end{array}
\label{e:colonna:globalGainLoss}
\end{equation}
where the symbols $\mathcal{L}$ and $\mathcal{G}$ indicate the rate of energy loss and gain respectively.
It should be noted that only $K^{d}$ and $\mathcal{L}$ depend on $T_{v}$, while $K^{r}$ and $\mathcal{G}$ are only a function of $T$. The reverse rates are not independent and can be calculated from the direct ones using the detailed balance principle
\begin{equation}
K^{r}(T)=K^{d}(T,T)/K_{eq}(T).
\label{e:colonna:detBal}
\end{equation}
A similar procedure is followed to calculate global rates for processes induced by electron collision,
$e^{-}+\text{H}_{2}(v)$.
The first step consists in calculating the StS rates as a function of the electron temperature $T_{e}$
\begin{equation}
k^{d}{v}(T{e})=\int_{\varepsilon^{\star}{v}}^{\infty}f^{M}{e}(\varepsilon,T_{e})u(\varepsilon)\sigma_{v}(\varepsilon)\varepsilon
\label{e:colonna:electronRateTe}
\end{equation}
where $f^{M}_{e}$ is a Maxwell electron energy distribution function, $\varepsilon$ the electron energy, $u$ the electron velocity and $\sigma_{v}$ the cross section of the process. The quantity $\varepsilon^{\star}_{v}$ is the threshold corresponding to the amount of energy lost by electrons in the process. Then the global rates as a function of $T_{e}$ and $T_{v}$ can be calculated using Eqs.~\ref{e:colonna:globalRateGen},\ref{e:colonna:globalGainLoss}. In the absence of an applied electromagnetic field, it is reasonable to assume that free electrons are in equilibrium with the gas temperature~\cite{colonna2020ionization}. For the sake of completeness, the contribution of these processes to the electron energy equation are given by
\begin{equation}
\begin{array}{ll}
\mathcal{L}(T,T_{v})&=\sum_{v}\varepsilon_{v}^{\star}k_{v}^{d}(T)f_{v}(T_{v})\
\mathcal{G}(T)&=\sum_{v}\varepsilon_{v}^{\star}k_{v}^{r}(T)
\end{array}
\label{e:colonna:elEnGainLoss}
\end{equation}
that should be added to the Landau-Teller terms due to elastic collisions with heavy particles.
The numerical values of the $K^{d}$ have been fitted as a function of $T$ and $T_{v}$ with a 2D best fitting procedure. Quantum effects and anharmonicity of the vibrational levels make the Arrhenius expression not accurate enough to fit the data in the long range. In order to preserve continuity full-range expressions have been used, accurate in the temperature interval $100\div 10^{5}$. The fitting expression is generally given as a function $T$
\begin{equation}
K^{d}(T,T_{v})=f(T;{c_{i}(T_{v})})
\label{e:colonna:FittingFunc}
\end{equation}
whose coefficients $c_{i}$ are functions of $T_{v}$. The functions used are generally the sum of Arrhenius or of sigmoids
\begin{equation}
\sigma_{f}(x)=a\frac{e^{(x-c)/\ell}}{e^{(x-c)/\ell}+e^{-(x-c)/\ell}}.
\label{e:colonna:FittingFuncBis}
\end{equation}
\section{Conclusions}
\label{sec:conclusion}
A novel mT model has been derived from accurate StS dynamical data. This is the first step toward a mT model of the entry in Ice Giants, including impurities already present in the atmosphere or evaporated from the vehicle surface. As in the traditional mT approaches, this model cannot take into account the departure from the Boltzmann distributions. However, corrections are possible, by adding an equation for the tail of the distributions~\cite{colonna2008recombination} to correct the rates with the contribution of highly-excited vibrational levels.
\subsection*{\sc acknowledgements}
This research has been funded under ESA Contract No. 4000139351/22/NL/MG "Development of a state-to-state CFD code for the characterization of the aerothermal environment of Ice Giants planets entry capsules".
%\bibliographystyle{IEEEbib}
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\end{thebibliography}
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% -------------------------------------------------------------------------
Summary
A hybrid multi-temperature model for ionising H$_{2}$/He mixture has been constructed from state-specific data. The model considers a vibrational temperature $T_{v}$ for the hydrogen molecule, adding some electronic states of H$_{2}$ and H. Data have been fitted by analytical functions in a wide range of $T$ and $T_{v}$.
