Analytical Models for Electron-Vibration Coupling in Nitrogen Plasma Flows

Abstract

A model for the electron-vibration energy exchange in nitrogen based on a Landau-Teller-type rate equation is presented. Analytical expressions of relaxation times are derived for cutoff harmonic oscillator and Morse oscillator approximations, assuming that the relaxation proceeds by way of a continuous series of Boltzmann distributions over the vibrational states. Comparisons with the direct numerical integration of the master equation have shown that the use of a two-temperature relaxation time allows accurate modeling of the nitrogen vibrational energy relaxation rate for various conditions encountered in high-enthalpy ows with electron and vibrational temperatures up to 40,000 K. Nomenclature a = constant in the analytical expression of rate coef cients for v > 0 ! w > v transitions dE n v / dt = nondimensionalvibrational energy relaxation rate E v = vibrational energy per unit volume, J/m 3 E ¤ v = vibrational energy calculated with vibrational levels in Boltzmann equilibrium at the electron temperature T e , J/m 3 E v, init = vibrational energy at 0-s time, J/m 3 J = rotational quantum number k B = Boltzmann constant, J/K k v,w = electron-vibration(E-V) rate coef cient of the v ! w transition, m 3 /s N t = total N 2 number density, N v , m ¡ 3 N v = number density of the vibrational level v of N 2 , m ¡ 3 n e = electron number density, m ¡ 3 p e = electron partial pressure, n e k B T e , atm T e = electron temperature, K T v , T exci = vibrational temperature, K t = time, s t n = nondimensionaltime v m = total number of excited vibrational levels D ² = constant of the Morse oscillator, cm ¡ 1 ²(v) = energy of the vibrational level v of N 2 , J ² 1 = energy of the rst vibrational level of N 2 , J r v, w = E-V cross section of the v ! w transition, m 2 s e = E-V relaxation time, s Supercripts HO = truncated harmonic oscillator MO = Morse oscillator

Non-equilibrium vibrational and electron energy distributions functions in atmospheric nitrogen ns pulsed discharges and μs post-discharges: the role of electron molecule vibrational excitation scaling-laws

“…The first scaling-law we consider was reported in Ref. [36] and was obtained by fitting the data calculated by Huo et Figure 1. Some series of cross sections (on the left) and the corresponding rate coefficients (on the right) for resonant vibrational-excitation of N 2 by electron-impact calculated by Laporta et al [31,32].…”

“…Recently, a cross section data set due to Huo et al [35] was combined with extended experimental results to derive a new scaling-law [36]. It should be noted that a similar scaling-law was proposed by Gordiets et al limiting however its application to v = 10 [37].…”

Electron-vibration energy exchange models in nitrogen-containing plasma flows

“…8 and in particular in Fig. 9 they are compared with those of Lee 6 and Bourdon et al 9 for T v = 0. Present results lie between the other two, somewhat closer to the more recent results of Bourdon et al…”

Theoretical Performance of a Magnetohydrodynamic-Bypass Scramjet Engine with Nonequilibrium Ionization

“…It is well known that the interaction between electrons and the vibrational mode of nitrogen molecules is very fast. 10 At the electron temperature of 3000 K and total number density of 1:4 £ 10 24 m ¡3 , the vibrational temperature of N 2 and the electron temperature equalize within about 10 ¡7 s. The vibrational temperature of O 2 molecules will approach the vibrational temperature of N 2 because of the so-called vibration-to-vibration interaction,though at a slower rate. 6 By assumingthatthe vibrationaltemperaturesof N 2 and O 2 are the same as the electron temperature,one arrivesat the well-known twotemperature model.…”