Thermodynamically Consistent Vibrational-Electron Heating: Generalized Model for Multi-Quantum Transitions

TL;DR

This paper extends a thermodynamically consistent vibrational-electron heating model to include multi-quantum transitions, improving accuracy in high-energy regimes by accounting for hot-band effects and ensuring proper thermal relaxation.

Contribution

The authors generalize a previous single-quantum model to incorporate multi-quantum overtone transitions, enhancing its validity for high-temperature plasma applications.

Findings

Previous models neglecting hot-band transitions have significant heating errors.

The generalized model accurately predicts electron temperature convergence to vibrational temperature.

The new formulation ensures thermodynamic consistency at equilibrium.

Abstract

Accurate prediction of electron temperature ($T_{\rm e}$) is critical for non-equilibrium plasma applications ranging from hypersonic flight to plasma-assisted combustion. We recently proposed a thermodynamically consistent model for vibrational-electron heating [Phys. Fluids 37, 096141 (2025)] that enforces the convergence of $T_{\rm e}$ to the vibrational temperature ($T_{\rm v}$) at equilibrium. However, the original derivation was restricted to single-quantum transitions, limiting its validity to low-temperature regimes ($T_{\rm e} \lesssim 1.5$ eV). In this Letter, we generalize the model to include multi-quantum overtone transitions, extending its applicability to high-energy regimes. We demonstrate that previous models neglecting hot-band transitions incur a systematic heating error of $\exp(-\theta_{\rm v}/T_{\rm v})$, where $\theta_{\rm v}$ is the characteristic vibrational temperature. This error exceeds 40% when $T_{\rm v}$ is greater than $\theta_{\rm v}$, effectively preventing thermal relaxation. To correct this, we derive a formulation where the total heating rate is a summation of channel-specific cooling rates $Q_{\rm e-v}^{(m)}$, each associated with a quantum jump $m$, scaled by a thermodynamic factor $\exp(m\theta_{\rm v}/T_{\rm e}-m\theta_{\rm v}/T_{\rm v})$. This generalized model preserves thermodynamic consistency by ensuring zero net energy transfer at equilibrium.