Distribution of the Rotational Temperature of С₂ Molecules in High-Temperature Regions in a Supersonic Airflow under Injecting Ethylene, Propane, and Oxygen in the Discharge Area

Abstract

The distribution of the rotational temperature of C₂ molecules in the anode–cathode gap of electric-discharge modules placed in a supersonic flow, observed in the discharge regions under injections of e-thylene, propane, and oxygen, are studied with the use of emission spectroscopy. The rotational temperatures of the molecules are assumed to be close to gas-kinetic ones under experimental conditions. The analysis of the alternative way of comparing the gas-kinetic temperature with the rotational temperature of CN molecules shows the latter to be close to a higher vibrational temperature, apparently because CN molecules mainly originate in chemical reactions with the participation of strongly excited components arising under electron impacts. The correlation of the voltage across the discharge gap and the temperatures in the zones of energy release with the dynamic air pressure in the vicinity of the anode is derived.

APPENDIX

The multiparameter fitting (see below) is based on the analysis [12] of the calculations of electron-rotational-vibrational (ERV) C₂ spectra.

High-intensity ERV molecular bands which corresponded to the Swan C₂(d³П_g → a³П_u) system are considered in the analysis, taking into account the fact that the electric dipole transition between the triplet electronic states d³П_g and a³П_u of the C₂ molecule is determined by the selection rules, the bond type (interactions between electrons and particle nuclei), and the symmetry properties of these states [13, 14].

The power per unit volume ${{\varepsilon }_{{mn}}}\left( {{{{\tilde {\nu }}}_{{{\text{rad}}}}} - {{{\tilde {\nu }}}_{{mn}}}} \right)$ emitted by the C₂ molecules during the electric dipole transition $m \to n$ (m and n mean sets of quantum numbers) per unit wavenumber range was calculated within the quantum electrodynamics and physical kinetics [13–18]. The position of the C₂ ERV line in the emission spectrum, which corresponds to the $m \to n$ transition, is determined by the wavenumber ${{\tilde {\nu }}_{{mn}}}$. It was assumed that the wavelengths of the electromagnetic field are much larger than the size of the C₂ molecule. This condition is valid in the ultraviolet, visible, and infrared wavelength regions, which are of interest for practical spectroscopy. The electromagnetic field interacting with the C₂ molecule is weak. This interaction results in the absorption and emission of one photon. The photon energy is equal to the energy difference between the quantum states of the C₂ molecule, which are specified by the selection rules for the electric dipole transition of the C₂ molecule. The emission spectra of the C₂ molecule in the high-temperature zone between the electrodes are mainly due to the spontaneous emission of the C₂ molecule in the d³П_g excited state. The dependence of ${{\varepsilon }_{{mn}}}\left( {{{{\tilde {\nu }}}_{{{\text{rad}}}}} - {{{\tilde {\nu }}}_{{mn}}}} \right)$ on the particle concentration ${{N}_{m}}$ in the excited radiating state d³П_g is defined by the expressions [13–18]:

$$\begin{gathered} {{\varepsilon }_{{mn}}}\left( {{{{\tilde {\nu }}}_{{{\text{rad}}}}} - {{{\tilde {\nu }}}_{{mn}}}} \right) = {{S}_{{mn}}}\left( {{{{\tilde {\nu }}}_{{{\text{rad}}}}} - {{{\tilde {\nu }}}_{{mn}}}} \right)\varepsilon _{{mn}}^{{\operatorname{int} }}\left( {{{{\tilde {\nu }}}_{{mn}}}} \right), \\ \varepsilon _{{mn}}^{{\operatorname{int} }}\left( {{{{\tilde {\nu }}}_{{mn}}}} \right) = {{A}_{{mn}}}{{N}_{m}}hc{{{\tilde {\nu }}}_{{mn}}}. \\ \end{gathered} $$

(1)

Here, $\varepsilon _{{mn}}^{{\operatorname{int} }}\left( {{{{\tilde {\nu }}}_{{mn}}}} \right)$ is the power emitted by C₂ molecules per unit volume and ${{A}_{{mn}}}$ is the spontaneous emission probability. The function ${{S}_{{mn}}}\left( {{{{\tilde {\nu }}}_{{{\text{rad}}}}} - {{{\tilde {\nu }}}_{{mn}}}} \right)$ of the wavenumber ${{\tilde {\nu }}_{{{\text{rad}}}}}$ shows the real profile of the ERV line, which corresponds to the electric dipole transition of the C₂ molecule.

In the model for emission spectra calculation, the inhomogeneous Doppler and homogeneous ERV line broadening [19] caused by the thermal motion of C₂ molecules and the interaction between heavy particles (molecules and atoms), respectively, were considered. When homogeneous broadening is taken into account in the model, the line broadening due to collisions between heavy particles is assumed to be much more significant than the natural broadening, and the line shift due to particle collisions at the gas-kinetic temperature ${{T}_{g}}$ ≥ 2000 K is much smaller than its Doppler broadening. The homogeneous broadening of ERV lines in the spectrum of the C₂ molecule was estimated with the use of models of hard spheres (with different approximations of the dependence of the particle collisional cross section on the temperature ${{T}_{g}}$) and models where the interaction potential in particle collisions is approximated by the Lennard–Jones or Born–Meier potential [20]. Accounting for the homogeneous and inhomogeneous line broadening determined the dependence of the ${{\varepsilon }_{{mn}}}\left( {{{{\tilde {\nu }}}_{{{\text{rad}}}}} - {{{\tilde {\nu }}}_{{mn}}}} \right)$ value on the interaction potential parameters and the translational gas temperature ${{T}_{g}}$ in the high-temperature zone.

The use of the adiabatic approximation [13‒18] for the diatomic molecule C₂ admits the series of transformations of Eq. (1):

$$\begin{gathered} \varepsilon _{{mn}}^{{\operatorname{int} }}\left( {{{{\tilde {\nu }}}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}} \right) = \frac{{16{{\pi }^{3}}c}}{3}\tilde {\nu }_{{J{\kern 1pt} 'J{\kern 1pt} ''}}^{4}{{S}_{{e_{{st}}^{'}e_{{st}}^{''}}}}\left( {{{r}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}}} \right){{q}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}} \\ \times \,\,\frac{{{{S}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}}}{{\left( {2J{\kern 1pt} ' + 1} \right)K{\kern 1pt} '}}{{N}_{{J{\kern 1pt} 'v{\kern 1pt} 'e_{{st}}^{'}}}}. \\ \end{gathered} $$

Here, $m$ and $n$ are expressed in terms of the quantum numbers $v{\kern 1pt} ',J{\kern 1pt} '$ and $v{\kern 1pt} '',J{\kern 1pt} ''$, respectively. The superscript “'” designates the upper radiating state $e_{{st}}^{'}$ = d³П_g of the C₂ molecule, and the superscript ′′, the lower electronic state $e_{{st}}^{''}$ = a³П_u; J′ and J′′ are the quantum numbers of the total angular momentum of molecular rotation; ${{\tilde {\nu }}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}$ is the wavenumber which determines the position of the ERV line in the molecular spectrum; ${{S}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}$ is the strength of the ERV line (Hönl–London factor); ${{S}_{{e_{{st}}^{'}e_{{st}}^{''}}}}({{r}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}})$ is the strength of the $J{\kern 1pt} ',v{\kern 1pt} ',e_{{st}}^{'} \to J{\kern 1pt} '',v{\kern 1pt} '',e_{{st}}^{''}$ electronic transition; ${{N}_{{J{\kern 1pt} 'v'e_{{st}}^{'}}}}$ is the concentration of C₂ molecules in the excited ERV state $J{\kern 1pt} ',v{\kern 1pt} ',e_{{st}}^{'} = {{d}^{3}}{{\Pi }_{g}}$; $K{\kern 1pt} '$ is the normalizing coefficient of the Hönl–London factor [18]; ${{r}_{{v'v''}}}$ is the $r$-centroid; and ${{q}_{{v'v''}}}$ is the Franck–Condon factor. The values of ${{S}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}$ and the sets of quantum numbers $J{\kern 1pt} ',v{\kern 1pt} ',e_{{st}}^{'}$ and $J{\kern 1pt} '',v{\kern 1pt} '',e_{{st}}^{''}$ are determined by the intermediate coupling by Hund ([13], pp. 426–430), symmetry properties of the upper and lower ERV states of the C₂ molecule, and selection rules for electric dipole transitions $J{\kern 1pt} ',v{\kern 1pt} ',e_{{st}}^{'} \to J{\kern 1pt} '',v{\kern 1pt} '',e_{{st}}^{''}$ [13‒15].

The databases have been created for the calculation of the $\varepsilon _{{mn}}^{{\operatorname{int} }}\left( {{{{\tilde {\nu }}}_{{mn}}}} \right)$ value based on a number of works (for example, [13, 18, 21–26]); they include the spectroscopic constants of electronically excited states of the C₂ molecule; and values of ${{S}_{{e_{{st}}^{'}e_{{st}}^{''}}}}({{r}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}})$, ${{q}_{{v'v''}}}$, wavelengths, and the probability of radiative transitions of the C₂ molecule. The ${{S}_{{e_{{st}}^{'}e_{{st}}^{''}}}}({{r}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}})$ values are calculated by the semiempirical expression [18]:

$${{S}_{{e_{{st}}^{'}e_{{st}}^{''}}}}\left( {{{r}_{{v'v''}}}} \right) = 33{{\left( {1 - 0.52{{r}_{{v'v''}}}} \right)}^{2}}.$$

The positions of rotational lines ${{\tilde {\nu }}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}$ in the spectrum are determined with allowance for the intermediate type of coupling by Hund, the selection rules, and the symmetry properties of the ERV states by the relationship [13–15, 18]:

$${{\tilde {\nu }}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}} = {{\tilde {\nu }}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}} + F\left( {J{\kern 1pt} ',v{\kern 1pt} '} \right) - F\left( {J{\kern 1pt} '',v{\kern 1pt} ''} \right).$$

To calculate the spectral rotational terms $F\left( {J{\kern 1pt} ',v{\kern 1pt} '} \right)$ and $F\left( {J{\kern 1pt} '',v{\kern 1pt} ''} \right)$ in the vibrational states $v{\kern 1pt} '$ and $v{\kern 1pt} ''$, respectively, the expressions from [13–18] are used, which take into account rotational-vibrational, centrifugal, and spin-orbital interactions. The ensemble of the ERV lines observed in the C₂ molecule emission spectrum is caused by the transitions between different rotational levels $J{\kern 1pt} '$ and $J{\kern 1pt} ''$ at fixed values of the vibrational states $v{\kern 1pt} '$ and $v{\kern 1pt} ''$. The value of ${{\tilde {\nu }}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}}$ is constant in the calculation of ${{\tilde {\nu }}_{{J{\kern 1pt} 'J{\kern 1pt} ''}}}$ for the specific electronic-vibrational transition $\nu {\kern 1pt} ',e_{{st}}^{'}$ = d³П_g → $\nu {\kern 1pt} '',e_{{st}}^{''}$ = a³П_u [14, 15]. It determines the beginning of the electronic-vibrational band (zero line) and is calculated in the model by the expression [13–18]:

$${{\tilde {\nu }}_{{v{\kern 1pt} 'v{\kern 1pt} ''}}} = {{\tilde {\nu }}_{{e_{{st}}^{'}e_{{st}}^{''}}}} + G\left( {v{\kern 1pt} '} \right) - G\left( {v{\kern 1pt} ''} \right).$$

The value of ${{\tilde {\nu }}_{{e_{{st}}^{'}e_{{st}}^{''}}}}$ is defined as the difference between the values of the upper ${{T}_{{e_{{st}}^{'}}}}$ and lower ${{T}_{{e_{{st}}^{''}}}}$ states of the spectral electronic terms which correspond to the $e_{{st}}^{'}$ = d³П_g and $e_{{st}}^{''}$ = a³П_u states:

$${{\tilde {\nu }}_{{e_{{st}}^{'}e_{{st}}^{''}}}} = {{T}_{{e_{{st}}^{'}}}} - {{T}_{{e_{{st}}^{''}}}}.$$

This difference is constant during the electronic-vibrational transition $\nu {\kern 1pt} ',e_{{st}}^{'}$ = d³П_g → $\nu {\kern 1pt} '',e_{{st}}^{''}$ = a³П_u. The values of $G\left( {v{\kern 1pt} '} \right)$ and $G\left( {v{\kern 1pt} ''} \right)$ are determined by the spectral vibrational terms which correspond to the upper $e_{{st}}^{'}$ = d³П_g and lower $e_{{st}}^{''}$ = a³П_u electronic states of the C₂ molecule. The values of $G\left( v \right)$ are calculated in the model based on the relation from the works [13‒18].

The intensity distribution in certain regions of the ERV emission spectrum of molecules during electric discharges can be characterized by different values of the rotational temperature (“cold” and “hot” regions of the molecular emission spectra) [13]. The appearance of cold and hot sections in the spectra is due to the physicochemical processes with the participation of molecules in electronically excited states. In contrast to the models for calculating emission spectra ([27–29] and others), the computational codes we have created in this study allow us to determine the set of T_rot($e_{{st}}^{'}$) values corresponding to different rotational terms and selected groups of quantum numbers.

Multiparameter Fitting of the Calculated to the Experimental Spectra

Before the fitting procedure, the calculated and experimental spectra are converted into one format convenient for the comparison. Alternation of the intensity $I_{c}^{{}}\left( \lambda \right)$ versus the wavelength λ in the spectra is normalized to the intensity maximum.

The procedure of multiparameter fitting of the calculated to measured spectra consists of the following steps.

To speed up the processing of the experimental data, a data set is created: during the first stage of the fitting, a library of intensity dependences $I_{c}^{{}}\left( \lambda \right)$ is calculated for the bands belonging to sequences ($\Delta v = 0, \pm 1$) of the Swan system (d³П_g → a³П_u) of the C₂ molecule at the rotational temperatures T_r (d³П_g) = 1000–8000 K. During this stage, the rotational temperature T_r (d³П_g) is assumed to be equal to ${{T}_{g}}$ and the vibrational temperature T_V (d³П_g), which corresponds to the Boltzmann distribution function of C₂ molecules over the electronic-vibrational levels $v{\kern 1pt} '$ of the state $e_{{st}}^{'}$ = d³П_g. The T_r (d³П_g) maxima and minima are determined from the comparison with the experimental spectra, at which

$$I_{c}^{{\min }}\left( \lambda \right) \leqslant {{I}_{{\exp }}}\left( \lambda \right) \leqslant I_{c}^{{\max }}\left( \lambda \right).$$

(2)

Here, $I_{c}^{{\min }}\left( \lambda \right)$ and $I_{c}^{{\max }}\left( \lambda \right)$ are the intensities calculated under the condition of minimal differences $I_{{\exp }}^{{}}\left( \lambda \right) - I_{c}^{{\min }}\left( \lambda \right)$ and $I_{c}^{{\max }}\left( \lambda \right) - I_{{\exp }}^{{}}\left( \lambda \right)$. Figure 11 shows the calculated intensity distributions as functions of the wavelength in the band sequence ($\Delta v = 0, \pm 1$) of the Swan system of the C₂ molecule at different values of T_r (d³П_g) and equilibrium between the translational and internal degrees of freedom of the molecule. According to the calculations, the spectral region which corresponds to short wavelengths strongly changes with T_r (d³П_g).

Fig. 11.

Intensity I_c(λ) distribution in the Swan sequence bands of the C₂ molecule in the translational temperature range ${{T}_{g}}$ = 1000–8000 K at equilibrium between the translational and internal degrees of freedom of the molecule: (a) ΔV = –1, (b) ΔV = 0, (c) ΔV = +1.

During the second stage of fitting, five ${{{{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}} \mathord{\left/ {\vphantom {{{{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}} {{{N}_{{e_{{st}}^{'}v{\kern 1pt} ' = 0}}}}}} \right. \kern-0em} {{{N}_{{e_{{st}}^{'}v{\kern 1pt} ' = 0}}}}}$ values ($v{\kern 1pt} '$ = 1–5) found during the first stage at the temperature T_r (d³П_g) are changed to satisfy condition (2); ${{T}_{g}}$ = T_r (d³П_g).

During the third stage, the T_r (d³П_g) value is changed at the ${{{{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}} \mathord{\left/ {\vphantom {{{{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}} {{{N}_{{e_{{st}}^{'}v{\kern 1pt} ' = 0}}}}}} \right. \kern-0em} {{{N}_{{e_{{st}}^{'}v{\kern 1pt} ' = 0}}}}}$ values found during the second stage to satisfy condition (2). As in the previous stages, ${{T}_{g}}$ = T_r (d³П_g).

The next stages of the fitting repeat the second and third stages until the best agreement between the measured and calculated intensity dependences $I_{c}^{{}}\left( \lambda \right)$ in the bands of different sequences ($\Delta v = 0, \pm 1$) of the Swan system of the C₂ molecule. If the T_r (d³П_g) values, which correspond to the bands from different sequences, coincide, then we analyze the conditions under which the T_r (d³П_g) value found can be identified with the gas-kinetic temperature ${{T}_{g}}$ in the high-temperature zone. They correspond to [13].

The relaxation time of the energy $\tau _{{R - T}}^{{}}$ stored in the ERV degrees of freedom of the C₂ molecule should be longer than the translational relaxation time $\tau _{{T - T}}^{{}}$of the molecule and shorter than the spontaneous radiative decay time ${{\tau }_{{{{{e'}}_{{st}}}}}}$, which corresponds to the d³П_g → a³П_u transition of the C₂ molecule: ${{\tau }_{{T - T}}} < {{\tau }_{{R - T}}} \ll {{\tau }_{{e_{{st}}^{'}}}}$. To estimate $\tau _{{R - T}}^{{}}$, the characteristic time is assumed to be comparable with the translational relaxation time of the C₂ molecule in order of magnitude. The $\tau _{{T - T}}^{{}}$ value is on the order of magnitude of the average time of collisions of C₂ molecules with particles in the high-temperature zone, $\tau _{{T - T}}^{{}} \approx$ (2–5) × 10^–9 s in the tempe-r-ature range from 1000 to 5000 K; and ${{\tau }_{{e_{{st}}^{'}}}}$ = (1–8) × 10^–7 s [18]. The condition ${{\tau }_{{R - T}}} \ll {{\tau }_{{e_{{st}}^{'}}}}$ is satisfied in the high temperature zone.

The characteristic times of direct and reverse inelastic processes and chemical reactions with the participation of C₂ molecules in the excited d³П_g state are assumed to be much longer than the times $\tau _{{R - T}}^{{}}$ and ${{\tau }_{{e_{{st}}^{'}}}}$.

The residence time of C₂ molecules in the excited d³П_g state in the high-temperature zone should be longer than the times $\tau _{{R - T}}^{{}}$ and ${{\tau }_{{e_{{st}}^{'}}}}$. This condition is satisfied since $\tau _{L}^{{}} \approx$ (3–7) × 10^–5 s [30]. As is ascertained in this study, T_r (d³П_g) increases with the energy input in the discharge zone. This also indirectly points to the fact that T_r (d³П_g) = T_g. Thus, the gas-kinetic temperature ${{T}_{g}}$ coincides with the rotational temperature ${{T}_{r}}\left( {{{d}^{3}}{{\Pi }_{g}}} \right)$ in the high-temperature zone.

The results of calculation of the rotational temperature T_r(d³П_g) via multiparameter fitting are shown in Fig. 12. The symbols correspond to the intensities measured under the conditions of an electric longitudinal discharge discussed in detail in [8–10]. The solid curves show the intensity calculated. The distribution function over the ERV levels of the C₂ molecule in the d³П_g triplet state is satisfactorily described by the Boltzmann distribution; the rotational temperatures corresponding to different sequences coincide within the error (±300 K) and lie in the T_r (d³П_g) 2000–2300 K range. Propane C₃H₈ with the mass flow rate ${{G}_{{{{{\text{C}}}_{{\text{3}}}}{{{\text{H}}}_{{\text{8}}}}}}}$ = 1 g/s is introduced into the interelectrode region. The emission spectra were recorded in a cross section of the discharge channel located at the distance X = 21 mm from the leading edge of the cathode [8–10].

Fig. 12.

Distributions of the measured (symbols) and calculated (solid lines) intensity I_c(λ) in the bands of the Swan C₂ sequence under imbalance between the rotational and vibrational degrees of freedom of the molecule in the ${{d}^{3}}{{\Pi }_{g}}$ state: (a) Δ$v$ = –1, T_r (d³П_g) = 2230 ± 300 K, and T_V (d³П_g) = 5800 ± 900 K for $v$ = 0–2; (b) Δ$v$ = 0, 2300 ± 300 K, and 6600 ± 900 K; (c) Δ$v$ = +1, 2000 ± 300 K, and 6000 ± 900 K.

The model used for the calculation of the emission spectra is distinctive in that it does not require any assumptions about the behavior of the distribution functions of C₂ molecules over the vibrational levels $v{\kern 1pt} '$. The comparison between the measured and calculated emission spectra in the high-temperature zone allows deriving the ${{N}_{{e_{{st}}^{'}v{\kern 1pt} }}}{\text{/}}{{N}_{{e_{{st}}^{'}{{v}_{{{\kern 1pt} '}}} = 0}}}$ values from the dependence

$$\ln ({{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}{\text{/}}{{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}_{{ = 0}}) = f\left( {hcG\left( {v{\kern 1pt} '} \right)} \right),$$

which is the desired distribution function of C₂ molecules over the electronic-vibrational levels in the state $e_{{st}}^{'}$ = d³П_g. In the case of its linear dependence on $hcG\left( {v{\kern 1pt} '} \right)$, the distribution function of C₂ molecules over electronic-vibrational levels $v{\kern 1pt} '$ is Boltzmann. The concept of the vibrational excitation temperature T_v (d³П_g) for the given vibrational levels $v{\kern 1pt} '$ of the state $e_{{st}}^{'}$ = d³П_g is introduced into the model. Its value is determined by the slope of the straight line constructed by the least squares method for the dependence of $\ln ({{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}{\text{/}}{{N}_{{e_{{st}}^{'}v{\kern 1pt} '}}}_{{ = 0}})$ on $hcG\left( {v{\kern 1pt} '} \right)$. Figure 13 shows the distribution function over the electronic-vibrational levels $v{\kern 1pt} '$ of the C₂ molecule in the ${{d}^{3}}{{\Pi }_{g}}$ excited state.

Fig. 13.

Distribution function over vibrational levels v of the C₂ molecule in the ${{d}^{3}}{{\Pi }_{g}}$ excited state (symbols) derived from processing of the emission spectra of the Swan sequences Δν = 1 (1) and 0 (2): calculation by the Boltzmann formula at the vibrational temperature T_ν (d³П_g) = 6000 K (line); ${{N}_{\nu }}$ is the concentration of C₂ mol-ecules excited to the vibrational level V; the conditions are the same as in Fig. 12.

It has been ascertained that the state of an ionized gaseous medium in the interelectrode zone is nonequilibrium [8–10]. For all sequences, the distribution function over six electronic vibrational levels ($v{\kern 1pt} '$ = 0–5) is not described by the Boltzmann distribution. The distribution function over three electronic vibrational levels ($v{\kern 1pt} '$ = 0–2) is Boltzmann with the temperature 5800–6600 K, which exceeds the rotational temperature (2000–2300 K). The temperature hierarchy corresponding to different degrees of freedom of the C₂ molecule has the form

$${{T}_{v}}({{d}^{3}}{{\Pi }_{g}}) > {{T}_{r}}({{d}^{3}}{{\Pi }_{g}}) \approx {{T}_{g}}.$$

The database of the measured and calculated spectra is used to validate and develop the model. Figure 14 shows the emission spectra of the C₂ molecule calculated within the model developed in this study and with models [27–29]. A satisfactory agreement is observed between the calculation results and rotational and vibrational temperatures determined with the use of different models.

Fig. 14.

Radiation intensity I_c(λ) in the bands of the Swan system of the C₂ molecule calculated by models from [27–29] (symbols) and this work (curve) for the sequence $\Delta \nu = 0$: (a and b) T_r (d³П_g) = 6000 [27] and 4000 K [28], respectively; (c) T_r (d³П_g) = 4000 K and T_ν (d³П_g) = 5000 K for ν = 0–1 [29].

Processing the emission spectra consists of identification of the spectrum, background subtraction, and determination of the spectral response and the transfer function of the optical system. It is assumed that the intensity of the observed emission spectra of C₂ molecules in the high-temperature zone is not distorted due to reabsorption, refraction, rereflection from elements of the optical system, etc.

During identification of the spectra from the high-temperature zone, tabular data [21] and the spectra comparison method [13] are used. The identified spectra from different published sources [13, 22–32] are used in the spectra comparison method. To determine the background intensity, spectral regions free of atomic lines and molecular bands are considered. The calibration characteristics of the optical system make it possible to take into account the distortion of the spectral intensity distribution due to the interaction of the electromagnetic radiation with the elements of the working part of the experimental setup and the optical system.