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Vibrational-Chemical Kinetics in Mars Entry Problems
Abstract
The paper deals with kinetic theory methods modelling of reacting gas flows near spacecrafts entering the Mars atmosphere. For mixtures containing CO2 molecules, the complete kinetic scheme including all vibrational energy transitions, dissociation, recombination and exchange chemical reactions is proposed. For the prediction of gas dynamic parameters and heat transfer to the surface of a spacecraft, a detailed approach taking into account state-to-state CO2 vibrational and chemical kinetics as well as multi-temperature approaches based on quasi-stationary vibrational distributions are used. A more accurate but complicated and time consuming state-to-state model is applied for the numerical simulation of a one-dimensional flow in a boundary layer near the entering body surface. More simple quasi- stationary three-temperature, two-temperature and one-temperature approaches are used for the numerical study of a 2-D viscous shock layer under entry conditions. The vibrational distributions near the surface are far from the local vibrational and chemical equilibrium and a noticeable difference is found between the values of CO2 vibrational-specific energies at the surface obtained by means of the state-to-state and quasi-stationary approaches. At the same time, for all considered approaches, the kinetic model for vibrational distributions and chemical reactions has a weak influence on the heat transfer to the non-catalytic vehicle surface.
... Kustova et al. [87] provide a straightforward procedure to obtain accurate rate coefficients for the V-V transfer between CO2 and CO. The rate coefficients of vibrational energy transitions between the lowest vibrational states are computed using experimental data from [88] and can be calculated using the expression: ...
... The value of the An constants can be found in table 1 of [87]. The remaining rate coefficients (for higher states) are calculated on the basis of the harmonic oscillator modified for polyatomic molecules. ...
... We compared the values obtained in this work following the procedure from Kustova et al. [87] for the process CO2(00 0 11) + CO(0) → CO2(00 0 01) + CO(1) with values determined experimentally by Rosser et al. [89] (linear dependence between 300 and 900K within experimental error), Starr et al. [90] and Blauer and Nickerson [68] and found a good agreement, as shown in Figure 4. ...
This work explores the effect of O 2 addition on CO 2 dissociation and on the vibrational kinetics of CO 2 and CO under various non-equilibrium plasma conditions. A self-consistent model, previously validated for pure CO 2 discharges, is further extended by adding the vibrational kinetics of CO, including electron impact excitation and de-excitation (e-V), vibration-to-translation relaxation (V-T) and vibration-to-vibration energy exchange (V-V) processes. The vibrational kinetics considered include levels up to v = 10 for CO and up to v 1 = 2 and v 2 = v 3 = 5, respectively for the symmetric stretch, bending and asymmetric stretch modes of CO 2 , and accounts for e-V, V-T in collisions between CO, CO 2 and O 2 molecules and O atoms and V-V processes involving all possible transfers involving CO 2 and CO molecules. The kinetic scheme is validated by comparing the model predictions with recent experimental data measured in a DC glow discharge, operating at pressures in the range 0.4 - 5 Torr (53.33 - 666.66 Pa). The experimental results show a lower vibrational temperature of the different modes of CO 2 and a decreased dissociation fraction of CO 2 when O 2 is added to the plasma but an increase of the vibrational temperature of CO. On the one hand, the simulations suggest that the former effect is the result of the stronger V-T energy-transfer collisions with O atoms which leads to an increase of the relaxation of the CO 2 vibrational modes; On the other hand, two main mechanisms contribute to the lower CO 2 dissociation fraction with increased O 2 content in the mixture: the back reaction, CO(a ³ Π r ) + O 2 → CO 2 + O and the recombinative detachment O ⁻ + CO → e + CO 2 .
... Kustova et al. [77] provide a straightforward procedure to obtain accurate rate coefficients for the V-V transfer between CO2 and CO. ...
... The value of the An constants can be found in table 1 of [77]. The remaining rate coefficients (for higher states) are calculated on the basis of the harmonic oscillator modified for polyatomic molecules. ...
... For VV1,2-CO, we use the scaling ( → + 1)( 1,2 + 1 → 1,2 ) = 0→1 * ( 1 + 1) * ( 2 + 1) * ( + 1). We compared the values obtained in this work following the procedure from Kustova et al. [77] for the process CO2(00 0 11) + CO(0) → CO2(00 0 01) + CO(1) with values determined experimentally by Rosser [79] et al. (linear dependence between 300 and 900K within experimental error), Starr et al. [80] and Blauer and Nickerson [63] and found a good agreement, as shown in Fig. 3. ...
This work explores the effect of O2 addition on CO2 dissociation and on the vibrational kinetics of CO2 and CO under various non-equilibrium plasma conditions. A self-consistent model, previously validated for pure CO2 discharges, is further extended by adding the vibrational kinetics of CO, including electron impact excitation and de-excitation (e-V), vibration-to-translation relaxation (V-T) and vibration-to-vibration energy exchange (V-V) processes. The vibrational kinetics considered include levels up to v = 10 for CO and up to v1=2 and v2=v3=5, respectively for the symmetric stretch, bending and asymmetric stretch modes of CO2, and accounts for e-V, V-T in collisions between CO, CO2 and O2 molecules and O atoms and V-V processes involving all possible transfers involving CO2 and CO molecules. The kinetic scheme is validated by comparing the model predictions with recent experimental data measured in a DC glow discharge, operating at pressures in the range 0.4 - 5 Torr (53.33 - 666.66 Pa). The experimental results show a lower vibrational temperature of the different modes of CO2 and a decreased dissociation fraction of CO2 when O2 is added to the plasma but an increase of the vibrational temperature of CO. On the one hand, the simulations suggest that the former effect is the result of the stronger V-T energy-transfer collisions with O atoms which leads to an increase of the relaxation of the CO2 vibrational modes; On the other hand, the back reactions with O2 contribute to the lower CO2 dissociation fraction with increased O2 content in the mixture.
... Simulation of nonequilibrium kinetics of carbon dioxide relates the most computationally complex problems of the modern physical mechanics. Such modeling has many applications in various fields, such as plasma chemistry [1], laser technology [2], and simulation of spacecraft entering the atmospheres of Venus and Mars [3,4]. In recent decades, the most rigorous approach based on the state-to-state description of vibrational and chemical relaxation was noticeably developed [5,4,6]. ...
... Such modeling has many applications in various fields, such as plasma chemistry [1], laser technology [2], and simulation of spacecraft entering the atmospheres of Venus and Mars [3,4]. In recent decades, the most rigorous approach based on the state-to-state description of vibrational and chemical relaxation was noticeably developed [5,4,6]. ...
... In addition, intermediate calculations at an each step of the solver's computational scheme include the estimation of hundreds of thousands rate coefficients of vibrational energy transitions and chemical reactions. In some studies [1,4], "cut" systems of vibrational levels and simplified models for vibrational energy exchanges were used for state-to-state modeling of carbon dioxide relaxation. ...
Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a computationally complex problem requiring the solution of a huge system of stiff differential equations. In the present study, we use parallel numerical scheme, based on the extended backward differential formula with adaptive timestep strategy. Using this scheme, we study the contribution of various types of energy exchanges to the solution of kinetics equations. The comparison of the results obtained with the use of the ”cut” and full state-to-state approaches is also shown in the paper.
... Coming back to Eq. (13), we must emphasize that this equation gives only the maximum contribution of a pure vibrational mechanism in the absence of any VV and VT energy transfer processes. The situation may be largely improved by solving an appropriate vibrational master equation of the three modes of CO 2 [235,236,88]. A lot of problems arise in this description, the most important of them being the calculation of complete data set for VV and VT processes occurring in the system. ...
... Of course, QCT calculations can be a further improvement in the process provided the PES used for VV and VT rates by Lombardi et al. and allow to add reactive channels and further check by comparison with quantum classical results. Figure 21: The VT 2 probabilities are shown as a function of temperature and compared with the corresponding results from SSH theory [235]. Figure 22: The VV 12 and VV 3 probabilities, as a function of temperature, obtained from QCTs are compared with the corresponding results from SSH theory [235]. ...
... Figure 21: The VT 2 probabilities are shown as a function of temperature and compared with the corresponding results from SSH theory [235]. Figure 22: The VV 12 and VV 3 probabilities, as a function of temperature, obtained from QCTs are compared with the corresponding results from SSH theory [235]. Energy exchange probabilities obtained by QCT are shown in Figures 21 and 22 for VV and VT processes as a function of the temperature T and compared with the corresponding values from the SSH theory from Ref. [235]. Figure 21 shows the probability of VT 2 energy exchange (vibration-translation exchange involving a quantum of bending vibrational energy). ...
The modeling of atmospheric gas, interacting with the space vehicles in re-entry conditions in planetary exploration missions, requires a large set of scattering data for all those elementary processes occurring in the system. A fundamental aspect of re-entry problems is represented by the strong non-equilibrium conditions met in the atmospheric plasma close to the surface of the thermal shield, where numerous interconnected relaxation processes determine the evolution of the gaseous system towards equilibrium conditions. A central role is played by the vibrational exchanges of energy, so that collisional processes involving vibrationally excited molecules assume a particular importance. In the present paper, theoretical calculations of complete sets of vibrationally state-resolved cross sections and rate coefficients are reviewed, focusing on the relevant classes of collisional processes: resonant and non-resonant electron-impact excitation of molecules, atom-diatom and molecule-molecule collisions as well as gas-surface interaction. In particular, collisional processes involving atomic and molecular species, relevant to Earth (N2, O2, NO), Mars (CO2, CO, N2) and Jupiter (H2, He) atmospheres are considered.
... The QCT results are compared to those obtained by SSH scaling from [118]. In the VT processes (blue and green plots) the final excited symmetric stretching and bending states can be any one of (1,0,0), (1,1,0), (0,1,0), (0,2,0), (0,3,0) (a) (b) Fig. 4 a VT energy transfer in CO2+CO2 collisions, STS rate coefficients for the process in Eq. 10 as a function of temperature, comparison between QCT results and SSH rates from [176] and [118], respectively. b CO2+N2 collisions, rate coefficients for the process in Eq. 11, comparison between rigid (no effect of deformations) and flexible PES Furthermore, we simulated over a range of temperatures a second VT process, corresponding to the exchange (gain or loss) of a quantum of bending energy CO 2 (0, m, 0) + CO 2 → CO 2 (0, m ± 1, 0) + CO 2 (10) and make the comparison with the equivalent process from [176], where the corresponding thermal rates were obtained by SSH theory extended for polyatomic molecules (see Fig. 4a). ...
... In the VT processes (blue and green plots) the final excited symmetric stretching and bending states can be any one of (1,0,0), (1,1,0), (0,1,0), (0,2,0), (0,3,0) (a) (b) Fig. 4 a VT energy transfer in CO2+CO2 collisions, STS rate coefficients for the process in Eq. 10 as a function of temperature, comparison between QCT results and SSH rates from [176] and [118], respectively. b CO2+N2 collisions, rate coefficients for the process in Eq. 11, comparison between rigid (no effect of deformations) and flexible PES Furthermore, we simulated over a range of temperatures a second VT process, corresponding to the exchange (gain or loss) of a quantum of bending energy CO 2 (0, m, 0) + CO 2 → CO 2 (0, m ± 1, 0) + CO 2 (10) and make the comparison with the equivalent process from [176], where the corresponding thermal rates were obtained by SSH theory extended for polyatomic molecules (see Fig. 4a). Finally, as an example involving N 2 , we run QCT calculations for the following VT transfer process on a range of temperatures CO 2 (0, 0, 1) + N 2 (0) → CO 2 (1, 0, 1) + N 2 (0) (11) obtaining the thermal rate coefficients, in order to assess the effects of the molecular vibrations on the intermolecular interactions (see V inter in Eq. 7). ...
Numerous applications have required the study of CO2 plasmas since the 1960s, from CO2 lasers to spacecraft heat shields. However, in recent years, intense research activities on the subject have restarted because of environmental problems associated with CO2 emissions. The present review provides a synthesis of the current state of knowledge on the physical chemistry of cold CO2 plasmas. In particular, the different modeling approaches implemented to address specific aspects of CO2 plasmas are presented. Throughout the paper, the importance of conducting joint experimental, theoretical and modeling studies to elucidate the complex couplings at play in CO2 plasmas is emphasized. Therefore, the experimental data that are likely to bring relevant constraints to the different modeling approaches are first reviewed. Second, the calculation of some key elementary processes obtained with semi-empirical, classical and quantum methods is presented. In order to describe the electron kinetics, the latest coherent sets of cross section satisfying the constraints of “electron swarm” analyses are introduced, and the need for self-consistent calculations for determining accurate electron energy distribution function (EEDF) is evidenced. The main findings of the latest zero-dimensional (0D) global models about the complex chemistry of CO2 and its dissociation products in different plasma discharges are then given, and full state-to-state (STS) models of only the vibrational-dissociation kinetics developed for studies of spacecraft shields are described. Finally, two important points for all applications using CO2 containing plasma are discussed: the role of surfaces in contact with the plasma, and the need for 2D/3D models to capture the main features of complex reactor geometries including effects induced by fluid dynamics on the plasma properties. In addition to bringing together the latest advances in the description of CO2 non-equilibrium plasmas, the results presented here also highlight the fundamental data that are still missing and the possible routes that still need to be investigated.
... The shock-layer flow was simulated with JAXA's in-house JONATHAN solver [49,50] using a simplistic approach based on the two-temperature chemistry model from Park et al. [51] and relaxation models for harmonic oscillators, whereas the adequacy of Park et al.'s [51] two-temperature model needs further assessment for CO 2 expanding flows [22,47,52]. Extension to selfconsistent models [53][54][55][56] was recently implemented to compute CO 2 expansion. In the meantime, the Park et al. [51] model was retained and a parametric analysis on the freestream conditions was undertaken. ...
... It is plausible that the amount of vibrational energy in the third mode remains unchanged and freezes as the gas flows over the spacecraft model, which results in emission of similar order of magnitude and would support the recent theoretical findings from [64]. Therefore, these results trigger the development of a new CO 2 vibrational dynamics model, which is based on the theoretical developments from [53][54][55][56], where the vibrational temperatures of CO 2 are decoupled from each other and from the other molecules. ...
This paper presents the first efforts in measuring carbon dioxide infrared radiation under nonequilibrium expansion, such as encountered in wakes, when carbon dioxide undergoes recombination. This work is motivated by the lack of radiation measurements in this flow and thermodynamic regimes, whereas carbon dioxide infrared radiation is deemed to contribute significantly to the afterbody heating withstood by a Martian spacecraft. The Hypervelocity Expansion Tube facility at Japan Aerospace Exploration Agency’s Chofu Aerospace Center is operated at the flight binary-scaled conditions of a mission currently considered at Japan Aerospace Exploration Agency. The test flow in front and after a scaled aeroshell is characterized with a combination of analytical formulations, numerical simulations, flow measurements, and schlieren diagnostics. Pressure and shock standoff measurements are presented. The thermodynamic state of the flow and its degree of nonequilibrium are discussed. Spatial infrared radiation measurements are obtained from the shock layer to the wake of the model. A noticeable increase of carbon dioxide infrared radiation is observed in the wake of the spacecraft: the reasons of which are discussed. Shock standoff distance measurements are complemented by carbon dioxide infrared spectra measurements. The latter outline the highly complicated vibrational energy distribution of carbon dioxide in the expansion tube and the subsequent radiation, as well as the possible shortcomings of the multitemperature models. Therefore, these first measurements of carbon dioxide nonequilibrium radiation in high-enthalpy facilities pose new challenges to the community and will enable the upgrade and validation of the multitemperature and state-to-state models, as well as radiation models currently in use.
... We consider the model of a simple harmonic oscillator. The processes of internal energy exchange in CO 2 which are important for hypersonic applications are discussed in [7][8][9]13]. Energy transitions taken into account in the present study are listed in Table 1. Here , , and are the vibrational quantum numbers of corresponding modes. ...
... For the direct VT 2 , VV 23 , and VV 123 processes the reaction rate coefficients are specified in [13] in the form , ...
The mechanism of internal energy exchange in CO2 including vibration-translation and different types of vibration-vibration exchange applicable for DSMC calculations is proposed. The microscopic probabilities of internal energy exchange processes transitions are obtained in the analytical form on the basis of known reaction rate coefficients using the inverse Laplace transform.
... We considered the temperature range from 200 to 2000 K and compared the trend of the VT 2 probabilities as a function of temperature with the corresponding results as obtained from the application of the SSH theory of energy transfer. These results have been extracted from Ref. 81. The SSH simplified FIG. ...
... 11. The VT 2 probabilities are shown as a function of temperature and compared with the corresponding results from SSH theory (from Ref. 81). The SSH probabilities are intended for transitions involving the gain or loss of a bending quantum of energy from a CO 2 molecule and for any accessible value of the initial bending quantum number. ...
Carbon dioxide molecules can store and release tens of kcal/mol upon collisions, and such an energy transfer strongly influences the energy disposal and the chemical processes in gases under the extreme conditions typical of plasmas and hypersonic flows. Moreover, the energy transfer involving CO2 characterizes the global dynamics of the Earth-atmosphere system and the energy balance of other planetary atmospheres. Contemporary developments in kinetic modeling of gaseous mixtures are connected to progress in the description of the energy transfer, and, in particular, the attempts to include non-equilibrium effects require to consider state-specific energy exchanges. A systematic study of the state-to-state vibrational energy transfer in CO2 + CO2 collisions is the focus of the present work, aided by a theoretical and computational tool based on quasiclassical trajectory simulations and an accurate full-dimension model of the intermolecular interactions. In this model, the accuracy of the description of the intermolecular forces (that determine the probability of energy transfer in molecular collisions) is enhanced by explicit account of the specific effects of the distortion of the CO2 structure due to vibrations. Results show that these effects are important for the energy transfer probabilities. Moreover, the role of rotational and vibrational degrees of freedom is found to be dominant in the energy exchange, while the average contribution of translations, under the temperature and energy conditions considered, is negligible. Remarkable is the fact that the intramolecular energy transfer only involves stretching and bending, unless one of the colliding molecules has an initial symmetric stretching quantum number greater than a threshold value estimated to be equal to 7.
... The CO2-CO V-V rate coefficient comes from our previous work [38,161] and is based on [162], while the rate coefficient for CO2-N2 V-V used in the present work is taken from [163]. We compare in Figure 1 the rate coefficients used in this work for the processes (6) and (7) [163]. ...
This work explores the effect of N 2 addition on CO 2 dissociation and on the vibrational kinetics of CO 2 and CO under various non-equilibrium plasma conditions. A self-consistent kinetic model, previously validated for pure CO 2 and CO 2 -O 2 discharges, is further extended by adding the kinetics of N 2 . The vibrational kinetics considered include levels up to v = 10 for CO, v = 59 for N 2 and up to v 1 = 2 and v 2 = v 3 = 5, respectively for the symmetric stretch, bending and asymmetric stretch modes of CO 2 , and account for electron-impact excitation and de-excitation (e-V), vibration-to-translation (V-T) and vibration-to-vibration energy exchange (V-V) processes. The kinetic scheme is validated by comparing the model predictions with recent experimental data measured in a DC glow discharge operating in pure CO 2 and in CO 2 -N 2 mixtures, at pressures in the range 0.6 - 4 Torr (80.00 - 533.33 Pa) and a current of 50 mA. The experimental results show a higher vibrational temperature of the different modes of CO 2 and CO and an increased dissociation fraction of CO 2 , that can reach values as high as 70 %, when N 2 is added to the plasma. On the one hand, the simulations suggest that the former effect is the result of the CO 2 -N 2 and CO-N 2 V-V transfers and the reduction of quenching due to the decrease of atomic oxygen concentration; on the other hand, the dilution of CO 2 and dissociation products, CO and O 2 , reduces the importance of back reactions and contributes to the higher CO 2 dissociation fraction with increased N 2 content in the mixture, while the N 2 (B ³ Π g ) electronically excited state further enhances the CO 2 dissociation.
... For complex problems, such as simulation of nonequilibrium kinetics of carbon dioxide [3,4] or biomodelling, classical numerical methods cannot be applied explicitly due to the huge computational costs or errors that accumulate in rigid numerical solutions of rigid systems. This leads to the need to develop separate numerical methods for each problem [2]. ...
The paper considers the possibility of using neural networks in solving differential equations and systems. The paper investigates the influence of various basic neural network topologies, various optimizers, activation functions, fast access links and some heuristic methods on the quality of neural network training and the accuracy of solving the differential equation.
... CO 2 (0n0) → CO 2 (000) + hν (5) which have been recognized 5758 to be the main contributor to the cooling of upper planetary atmospheres (Venus, Earth, Mars) trough the removal of translation energy from oxygen atoms by the bending mode of CO 2 (eq. 4), and the subsequent radiation of energy away from the atmospheric layer (either towards the ground or towards Space eq. 5). ...
A heavy-impact vibrational excitation and dissociation model for CO$_2$ is presented. This state-to-state model is based on the Forced Harmonic Oscillator (FHO) theory which is more accurate than current state of the art kinetic models of CO$_2$ based on First Order Perturbation Theory. The first excited triplet state $^{3}$B$_{2}$ of CO$_2$, including its vibrational structure, is considered in our model, and a more consistent approach to CO$_2$ dissociation is also proposed. The model is benchmarked against a few academic 0D cases and compared to decomposition time measurements in a shock tube. Our model is shown to have reasonable predictive capabilities, and the CO$_2$ $+$ O $\leftrightarrow$ CO $+$ O$_2$ is found to have a key influence on the dissociation dynamics of CO$_2$ shocked flows, warranting further theoretical studies. We conclude this study with a discussion on the theoretical improvements that are still required for a more consistent analysis of the vibrational dynamics of CO$_2$, discussing the concept of vibrational chaos and its possible application to CO$_2$. The necessity for further experimental works to calibrate such state-to-state models is also discussed, with a proposed roadmap for novel experiments in shocked flows.
... Current state of the art vibrationally specific kinetic models of CO 2 using the SSH model [4,5] are based on extrapolation of data and scaling of known experimental quantities. These lead to non-physical values of collision probabilities at high-temperatures which are pervasive in atmospheric entry flows. ...
... Further extension to triatom-triatom collisions would be very useful in the CO 2 -CO 2 collision dynamics, of large technological interest in the CO 2 destruction by plasma treatment (Fridman 2008;Lebouvier et al. 2013;Kozák and Bogaerts 2014;Pietanza et al. 2015). The vibrational kinetics of CO 2 is essentially based on data from SSH model (see for example (Kustova et al. 2014)) and QCT calculations (Lombardi et al. 2015). The two sets show discrepancies by orders of magnitude in the temperature range 200-2000 K (Lombardi et al. 2015). ...
Accurate modeling of low-temperature plasmas requires molecular collision input data, including the detail over the whole ladder of vibrational states. Obtaining this kind of detailed data is a big challenge, both theoretically and experimentally. These data can be calculated with the aid of simple models (for inelastic processes essentially based on forced harmonic oscillator mechanism), or with molecular dynamics at various levels, namely quasiclassical (QCT), semiclassical (SC), approximate and exact quantum mechanical methods, with computational cost rapidly increasing with accuracy. However, while accurate methods can become unfeasible when applied to the wide total energy ranges typically required in plasma modeling, more approximate semiclassical methods rapidly become efficient and accurate for increasing total energy, as shown in the literature. The best strategy is to study the limits of application of less accurate methods, to use them as a seamless continuation of accurate calculations on the total energy axis. In this sense, it is the current development about inelastic processes treated by QCT and SC methods. The aspect of special interest is the indication of a criterion for easily extracting the reliable QCT contribution to the inelastic process, treating the missing contributions by other SC methods in a restricted range. This procedure allows to optimize the use of different methods to maintain both a high level of accuracy and a high computational efficiency. As a consequence of the study about inelastic processes, a better comprehension and possible treatment of the dissociation mechanisms are obtained, with an indication of reliability of QCT results about this kind of process.
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... As discussed in many papers [25][26][27][28] and textbooks [29,30], CO 2 is a linear and symmetric molecule, with an axis of symmetry along the nuclei and a plane of symmetry perpendicular to its axis. This leads to three vibrational modes: one symmetric stretching mode, one asymmetric stretching mode, and one double degenerated bending mode. ...
A kinetic model describing the time evolution of ∼70 individual CO2(X1Σ⁺) vibrational levels during the afterglow of a pulsed DC glow discharge is developed in order to contribute to the understanding of vibrational energy transfer in CO2 plasmas. The results of the simulations are compared against in situ Fourier transform infrared spectroscopy data obtained in a pulsed DC glow discharge and its afterglow at pressures of a few Torr and discharge currents of around 50 mA. The very good agreement between the model predictions and the experimental results validates the kinetic scheme considered here and the corresponding vibration-vibration and vibration-translation rate coefficients. In this sense, it establishes a reaction mechanism for the vibrational kinetics of these CO2 energy levels and offers a firm basis to understand the vibrational relaxation in CO2 plasmas. It is shown that first-order perturbation theories, namely, the Schwartz-Slawsky-Herzfeld and Sharma-Brau methods, provide a good description of CO2 vibrations under low excitation regimes.
... Different approaches can be also used to model CO 2 vibrational ladder. Instead of taking into account only the first low lying symmetric levels and all the CO 2 (00v) asymmetric levels, the following possibilities could be investigated, considering: (1) a complete mixing of the three modes as in the attempts of Kustova and Armenise [51,52] (2) the first few levels as in in the CO 2 lasers [53] adding a continuum of levels from a given level on (3) the use of an uncoupled discrete model for the three vibrational modes of CO 2 [54]. Options 1-2 need of a complete re-examination of the cross sections, especially concerning the electron molecule ones, while the uncoupled model can use simplified data set of cross sections for the three modes [55]. ...
A self-consistent time dependent model, based on the coupling between the Boltzmann equation for free electrons, the non equilibrium vibrational kinetics for the asymmetric mode of CO2 and simplified global models for the dissociation and ionization plasma chemistry, has been applied to conditions which can be met under pulsed microwave (MW), dielectric barrier discharge (DBD) and nanosecond pulsed discharges (NPD). Under MW discharge type conditions, the selected pulse duration generates large concentration of vibrational excited states, which affects the electron energy distribution function (eedf) through the superelastic vibrational collisions. Moreover, in discharge conditions, plateaux appear in the vibrational distribution function (vdf) through the vibrational-vibrational up pumping mechanism, persisting also in the post discharge. In post discharge conditions, also the eedf is characterized by plateaux due to the superelastic collisions between cold electrons and the CO2 electronic state at 10.5 eV. The plateau in vdf increases the dissociation of pure vibrational mechanism (PVM), which can become competitive with the dissociation mechanism induced by electron molecule collisions. The PVM rates increase with the decrease of gas temperature, generating a non-Arrhenius behaviour. The situation completely changes under DBD and NPD type conditions characterized by shorter pulse duration and higher applied E/N values. Under discharge conditions, both vdf and eedf plateaux disappear, reappering in the afterglow.
... The number of levels is therefore about 10000. Same authors have developed a state-to-state model for CO 2 including all the levels separately[48,49,50,51]to investigate the relaxation in the boundary layer of hypersonic flow. The rate coefficients have been obtained in the framework of SSH theory adapted for polyatomic molecules. ...
The paper will present a brief overview of the applications of state-to-state kinetics in modeling fluid dynamics. The research activities ranges from hypersonic entry (boundary layer, shock wave) to ground test facilities, from MHD
interaction to DBD flow control. The state-to-state model in fluid dynamics in the last years is rapidly diffusing, promising new interesting developments in the next future.
... To understand better the results, one should construct databases for both coupled and decoupled oscillators, taking into account the work made in these years by Bogaerts 11,12 and Kustova and Armenise. 52,53 Particular attention should be paid to the direct electron impact dissociation process, including the influence of electron molecule vibrational and dissociation excitation cross sections of CO 2 on the electron energy distribution function and dissociation mechanisms in cold pure CO 2 plasmas, including its dependence on the vibrational quantum number(s). The assumption of a simple vibrational threshold shift of the dissociation cross section 11,12,[16][17][18]45 should be eliminated. ...
A new set of e-V processes linking the first ten vibrational levels of the symmetric mode of CO2 is derived by using a decoupled vibrational model and inserted in the Boltzmann equation for the electron energy distribution function (eedf). The new eedf and dissociation rates are in satisfactory agreement with the corresponding ones obtained by using the e-V cross sections reported in the data base of Hake and Phelps (H-P). Large differences are, on the contrary, found when the experimental dissociation cross sections of Cosby and Helm (C-H) is inserted in the Boltzman equation. Comparison of the corresponding rates with those obtained by using the low energy threshold energy, reported in the H-P data base, shows differences up to orders of magnitude, which decrease with the increasing of the reduced electric field. In all cases, we show the importance of superelastic vibrational collisions in affecting eedf and dissociation rates either in the direct electron impact mechanism (DEM) or in the pure vibrational mechanism (PVM).
This work discusses the potential of combining non-thermal plasmas and conducting membranes for in situ resource utilization (ISRU) on Mars. By converting different molecules directly from the Martian atmosphere, plasmas can create the necessary feed-stock and base chemicals for processing fuels, breathing oxygen, building materials, and fertilizers. Different plasma sources operate according to different principles and are associated with distinct dominant physicochemical mechanisms. This diversity allows exploring different energy transfer pathways leading to CO[Formula: see text] dissociation, including direct electron-impact processes, plasma chemistry mediated by vibrationally and electronically excited states, and thermally driven dissociation. The coupling of plasmas with membranes is still a technology under development, but a synergistic effect between plasma decomposition and oxygen permeation across conducting membranes is anticipated. The emerging technology is versatile, scalable, and has the potential to deliver high rates of production of molecules per kilogram of instrumentation sent to space. Therefore, it will likely play a very relevant role in future ISRU strategies.
В статье исследуются поперечные колебания неоднородной круглой тонкой пластины. С помощью метода возмущений получены асимптотические формулы для частот свободных колебаний пластины, толщина и модуль Юнга которой линейно зависят от радиуса. Проанализировано влияние условий закрепления края пластины на частоты и поведение частот при фиксированной массе пластины. Для низших частот пластины асимптотические результаты сравниваются с результатами конечно-элементного анализа.
A heavy particle impact vibrational excitation and dissociation model for CO2 is presented. This state-to-state model is based on the forced harmonic oscillator (FHO) theory, which is more accurate than current state-of-the-art kinetic models of CO2 based on first-order perturbation theory. The first excited triplet state 3B2 of CO2, including its vibrational structure, is considered in our model, and a more consistent approach to CO2 dissociation is also proposed. The model is benchmarked against a few academic zero-dimensional (0D) cases and compared to decomposition time measurements in a shock tube. Our model is shown to have reasonable predictive capabilities, and the CO2 + O ↔ CO + O2 reaction is found to have a key influence on the dissociation dynamics of CO2 shocked flows, warranting further theoretical studies. We conclude this study with a discussion on the theoretical improvements that are still required for a more consistent analysis of the vibrational/dissociation dynamics of CO2.
An advanced model for the calculation of electron energy distribution functions (eedfs), vibrational distributions, and electronic excited state densities of reacting CO2 in microwave (MW) discharges has been developed for clarifying: (1) the role of electronic states of the relevant neutral species in affecting the eedf and (2) the contribution to the CO2 dissociation of the electron impact and heavy particle dissociation mechanisms. To model the discharge, the power density typical of MW discharges is used as a parameter. Different case studies including optically thick and thin plasmas and the dependence of the CO2 dissociation rates on the gas temperature are investigated. The results show that at a low gas temperature, i.e., 300 K, the heavy-particle dissociation mechanism, also called the pure vibrational mechanism, prevails on the electron impact dissociation one, while at a high gas temperature, i.e., 2000 K, the two mechanisms become competitive and the global behavior strongly depends on the choice of electron impact dissociation cross sections. Large differences appear in the eedf, especially in the post-discharge regime, when considering thick and thin plasmas. In the thick case, a well-structured eedf appears as a result of superelastic collisions mainly involving the electronic states of the relevant neutral species. In the thin plasma, many peaks disappear because the concentration of the excited states strongly decreases. Finally, our model gives the results of conversion and energy efficiency as well as vibrational distributions in satisfactory agreement with the corresponding results calculated by the Antwerp group.
The aim of this paper is to develop a line by line model for CO2 vibrational nonequilibrium radiation and to investigate nonequilibrium effects in the case of some applications involving high temperature expanding flows. A vibrational specific collisional relaxation model is developed and is incorporated in a multi-temperature thermodynamic description of the gas mixture, in order to compute vibrational level populations along the flow. The HITEMP-2010 spectroscopic database is employed with a model for level energy splitting to provide line by line absorption and emission total and per-vibrational mode specific spectra. The specific spectra allow us to derive the radiative source terms to be used in the multi-temperature model if a coupled approach is required. The model is applied first to a simple conical nozzle flow and then to the plume of a high altitude solid propellant rocket engine. It is shown that, for the considered applications, the partial freeze of vibrational excitation in the expanding flow increases significantly the radiative intensity escaping from the mixture.
The state-to-state vibrational kinetics of a CO2/O2/CO/C/O/e⁻ mixture in a hypersonic boundary layer under conditions compatible with the Mars re-entry is studied. The model adopted treats three CO2 modes (the two degenerated bending modes are approximated as a unique one) as not independent ones. Vibrational-translational transitions in the bending mode, inter-mode exchanges within CO2 molecule and between molecules of different chemical species as well as dissociation-recombination reactions are considered. Attention is paid to the electron-CO2 collisions that cause transitions from the ground vibrational state, CO2(0,0,0), to the first excited ones, CO2(1,0,0), CO2(0,1,0) and CO2(0,0,1). The corresponding processes rate coefficients are obtained starting from the electron energy distribution function, calculated either as an equilibrium Boltzmann distribution at the local temperature or by solving the Boltzmann equation.
Results obtained either neglecting or including in the kinetic scheme the electron-CO2 collisions are compared and explained by analysing the rate coefficients of the electron-CO2 collisions.
The present work aims to derive a set of thermomechanical relaxation rate parameters and chemical reaction rate coefficients relevant to future interplanetary missions. It also attempts to assess the impact of thermochemical nonequilibrium phenomena on radiative heating rates for the stagnation point of the Martian entry vehicle.
In this paper the non-equilibrium vibration-chemical kinetics, diffusion and heat transfer in polyatomic gas mixtures containing CO 2 molecules are studied using state-to-state and quasi-stationary multi-temperature kinetic theory approaches. The influence of mixture composition on the gas dynamic parameters and heat flux is discussed, the role of different internal modes in the heat transfer is estimated.
In this paper, the influence of non-equilibrium CO2 kinetics
on gas dynamic parameters and transport properties in a viscous shock
layer near blunt bodies is studied. This problem is important for
accurate prediction of macroscopic parameters and heat transfer near a
space vehicle re-entering the Mars atmosphere. The hypersonic flow of
reacting CO2/CO/O2/C/O mixture is studied
numerically using the accurate kinetic theory models for transport
properties and chemical reaction rates proposed in our previous papers
on the basis of three-temperature vibrational distributions. The kinetic
theory algorithms were implemented directly to the computational fluid
dynamics solvers. The transport properties and gas dynamic parameters in
2-D viscous shock layer near blunt bodies of different shape are studied
for various test cases under Mars re-entry conditions. The influence of
non-equilibrium effects on gas flow parameters and heat transfer is
discussed; a quite different structure of a shock layer is found for
various body shapes. The comparison with the results obtained using more
simple models for CO2 kinetics, transport coefficients, and
reaction rates is shown.
The vibrational relaxation rates of binary mixtures of CO with the additives CO2, N2O and COS have been measured in incident shock waves over a temperature range 1200–2000°K. The fundamental vibration of CO was found to be closely coupled to the asymmetric stretching vibration in each additive and so the rate determining step was not V‐V energy exchange between CO and the additive, but intramolecular energy transfer within the polyatomic species. For CO∕CO2 this step is where the forward rate constant is given by For the other additives the rate constants for the analogous reactions are given by These results are in accord with those from recent laser fluorescence measurements and can be reconciled with shock measurements on other systems.
State-to-state vibrational kinetics and transport models of a mixture containing triatomic CO2 molecules are developed. The models are implemented into a hypersonic boundary layer solver specially upgraded for this purpose. Although at the moment only vibrational–translational transitions in the bending mode (VT2), inter-mode exchanges within CO2 molecule (VV1-2-3), and inter-mode exchanges between molecules of different chemical species (VV1-2-CO) are taken into account, the approach can be generalized to include more complete kinetics.In order to overcome problems caused by the computational load of the state-to-state vibrational kinetics of a triatomic molecule, a Reduced Model is proposed and compared with the Full one.
Rate constants have been measured for the vibrational relaxation of CO2 and CH3F in the temperature range 300-150 K by the collision partners 4He and 3He. These results are compared with those calculated with a vibrational close-coupling, rotational infinite-order sudden approximation.
The paper deals with the kinetic theory of gas mixtures containing polyatomic molecules with several vibrational modes. The excitation of the rotational and vibrational degrees of freedom and dissociation are taken into account. The main attention is focused on the modeling of multi-level vibrational kinetics and vibration-dissociation coupling of linear three-atomic molecules. The model is applied for the study of shock heated gas flows and some new features of non-equilibrium kinetics of
CO
2
behind shock waves are found.
A quasi-stationary vibrational distribution in strongly excited multiatomic molecules due to vibrational exchange (distribution with plateau) is considered. Anharmonic coupling of vibrational modes makes it impossible to consider them in isolation, so that it is question of a distribution in the space of vibrational mode numbers. A formula is derived for the vibrational quantum flux, describing relaxation of the distribution with plateau. CO2 and NO2 molecules are considered as examples.
A state-to-state vibrational kinetics for air components including recombination-dissociation processes as well as the formation of NO through the reaction between vibrationally excited nitrogen molecules and atomic oxygen has been inserted in a monodimensional fluid dynamic code, describing the boundary layer surrounding a body under re-entry conditions. The results show that the formation of NO is strongly enhanced by the nonequilibrium vibrational distribution of N-2 formed during the recombination process. This kind of distribution is responsible for the non-Arrhenius behavior of dissociation constants of N-2 and O-2 as well as the NO formation rate as a function of instantaneous temperature.
By means of an analysis of currently available experimental data and theoretical models, expressions for the dissociation
rate constantk
0 are obtained for thermal equilibrium conditions. These expressions are necessary for describing the molecular dissociation
process under both thermal equilibrium and non-equilibrium conditions in the gas.k
0 values are presented for the O2, N2, NO, CO, CN, and C2 molecules at temperatures from 300 to 40,000 K, and their errors are estimated.
Vibrational relaxation data are surveyed in order to provide the rates of vibrational energy transfer for processes important in the CO2-N2 laser. A kinetic model is assumed for the vibrational energy transfer into and within the various vibrational modes of the molecules that make up a CO2-N2 laser, including the species H2O, O2, He, and H2. Experimental data are assembled and interpreted for the rate constants and the probabilities per collision for the various kinetic processes of the assumed mechanism as a function of temperature. For certain processes, the experimental data are reinterpreted in terms of more recent knowledge of vibrational energy transfer. The data are compared with theoretical calculations and various anomalies in those comparisons are discussed. The significance of the various vibrational energy transfer processes for understanding the operation of the CO2-N2 laser are contrasted with the state of knowledge of the rate information.
In this paper, simple analytical state-to-state rate coefficients for the dissociation–recombination and chemical exchange reactions are presented on the basis of kinetic theory in nonequilibrium excited diatomic gases. They take into account the excited vibrational and electronic states of the chemical species and are expressed according to the preferential character of the chemical reactions. Evolution of these rate coefficients varying according to the translational temperature, bringing into play molecules CO and C2, are discussed.
In this paper the kinetic description of a viscous reacting flow of CO2/CO/O2/C/O mixture is proposed on the basis of the kinetic theory methods. Vibrational excitation of CO2, CO and O2 molecules as well as non-equilibrium dissociation, recombination and exchange reactions are taken into account. A five-temperature model for a non-equilibrium flow is derived from the kinetic theory and used for evaluation of transport properties of the considered mixture at different temperatures and mixture composition. The paper presents the closed set of equations for macroscopic parameters, expressions for transport and source terms suitable for numerical simulations and results of calculations. The influence of non-equilibrium kinetics and mixture composition on transport properties in a flow is discussed.
Transport properties of carbon dioxide with strongly excited asymmetric vibrational mode are studied on the basis of kinetic theory treatment. The kinetic model takes into account anharmonism of molecular vibrations and different rates of various energy exchanges. The method of transport coefficients derivation is proposed. The calculation of nonequilibrium specific heats is carried out using a non-Boltzmann distribution of the molecules over vibrational energy levels. The effect of strong vibrational excitation and anharmonicity on thermal conductivity, shear and bulk viscosity coefficients is estimated.
The vibrational relaxation of the CO2 molecules in a mixture with Ar in supersonic expansion is studied. The kinetic equations considered describe the changes in the population of all the vibrational levels contained within the first six multiplets of CO2. The energy exchange and relaxation rate constants are calculated using a modified SSH theory. Existence and creation of a Treanor-Likal'ter distribution is examined. A checking procedure was developed to show to what extent the real distribution could be approximated by the TL distribution. The particular role of the VV energy transfer in the relaxation of the polyatomic CO2 molecules is examined. Unexpected new population inversions have been found for some intramultiplet transitions.
A comprehensive survey of relaxation rate data important in the modeling of nonequilibrium infrared radiation from exhaust plumes at altitude is presented. A model is developed which considers energy transfer among the first fourteen vibrational states of CO2, the first five of H2O, and up to the first three states of each of the diatomic molecules considered, i.e., of N2, O2, CO, OH, and H2. Available data are first assembled for those processes that have been subjected to experimental analysis. Where possible, statistical and/or quantum considerations are used to relate unmeasured rate constants to those whose values are known. Finally, resort is made to available theory to obtain estimates for the rate constants for the balance of the interactions important to the overall model.
Vibrational relaxation times in gases are calculated with the method of Zener using an exponential repulsion in a one‐dimensional model. The constants of the interaction potential are determined by fitting it to the data of Hirschfelder, et al. The great effect which some impurities have is accounted for either by their low mass and resultant high velocity or by ``near resonance'' transfers in which the vibrational quantum of the substratum is used partly to excite the vibration of the impurity, only the difference being transferred to translation. However, there are other impurities, the action of which cannot be explained in this manner. The theoretical values for the relaxation times are 10 to 30 times shorter than the experimental ones, which difference may be accounted for by the use of the one‐dimensional model. Macroscopic equations governing the more complex relaxation processes in polyatomic gases and gas mixtures are developed.
The paper deals with the numerical simulation of a supersonic viscous flow containing CO2 molecules near a space body entering the Mars atmosphere. The gas dynamic equations in a shock layer are coupled to the equations of vibrational and chemical kinetics in the mixture CO2/CO/O2/C/O using three theoretical models for CO2 vibrational excitation. Three‐temperature and two‐temperature non‐equilibrium approaches as well as the one‐temperature thermal equilibrium model have been applied. A comparison of gas flow parameters and heat transfer calculated on the basis of different approximations is presented, and the effect of CO2 vibrational non‐equilibrium is discussed. Transport coefficients in a flow are computed using rigorous kinetic theory algorithms which have been incorporated directly to the numerical schemes. The effect of bulk viscosity in a shock layer is studied.
A Chapman Enskog method for the determination of transport terms
appearing in the Navier-Stokes equations governing reactive gaseous
flows is presented. Solutions for the particular case of vibrational
relaxation are developed and specific forms of the heat flux and stress
tensor are given. Due to the expansion of the distribution function, the
method fails in near-equilibrium regions. A modification of the method
is then presented which allows the matching between strong
nonequilibrium and equilibrium situations. Corresponding transport terms
are calculated taking into account some simplifying hypotheses. Two
examples of nonequilibrium boundary-layers are computed: the end-wall
boundary layer behind a reflected shock and the nozzle wall
boundary-layer in which vibration-vibration exchanges are preponderant.
The thermal conductivity of O2, N2 and CO2 has been calculated as a sum of translational, rotational, and vibrational components. The rotational component is a fixed fraction of the translational component of the thermal conductivity. The vibrational component is calculated explicitly and estimated or experimentally determined values of relaxation times and diffusion coefficients are not required. The results of the calculations compare favourably with experiment over a temperature range 400-1500K.
The paper presents results of a numerical simulation of a supersonic two-dimensional (2D) viscous flow containing CO2 molecules near a spacecraft entering the Mars atmosphere. The gas–dynamic equations in the shock layer are coupled to the
equations of non-equilibrium vibrational and chemical kinetics in the five-component mixture CO2/CO/O2/C/O. Transport and relaxation processes in the flow are studied on the basis of the rigorous kinetic theory methods; the
developed transport algorithms are incorporated in the numerical scheme. The influence of the vibrational excitation of CO2 and chemical reactions on the gas flow parameters and heat transfer is analyzed. The obtained results are compared with those
found using two simplified models based on the two-temperature and one-temperature vibrational distributions in CO2. The accuracy of the simplified models and the limits of their validity within the shock layer are evaluated. The effect
of bulk viscosity in a flow near a re-entry body is discussed. The role of different diffusion processes, chemical reactions,
and surface catalytic properties in a flow of the considered mixture in the shock layer is estimated.
KeywordsNon-equilibrium CO2 flows–Multi-temperature vibrational-chemical kinetics–Transport properties–Shock layer
A recently introduced bond-bond formulation of the intermolecular interaction has been extended to six-atom systems to the end of assembling a new potential energy surface (PES) and has been incorporated into a grid empowered simulator able to handle the modeling of the CO(2) + CO(2) processes. The proposed PES is full dimensional and accounts for the dependence of the intermolecular interaction on some basic physical properties of the colliding partners, including modulations induced by the monomer deformation. The used analytical formulation of the interaction involves a limited number of parameters, each having a clear physical meaning. Guess values for these parameters can also be obtained from analytical correlation formulae. Such estimates can then be fine tuned by exploiting experimental and theoretical information. The resulting PES well describes stretched and bent asymptotic CO(2) monomers as well as the CO(2)-CO(2) interaction in the most and less stable configurations. On this potential massive quasiclassical elastic and inelastic detailed scattering trajectories have been integrated, by exploiting the innovative computational technologies of the grid. The efficiency of the approach used and the reliability of the estimates of the dynamical properties obtained in this way is such that we can now plan a systematic evaluation of the state specific rate coefficient matrix elements needed for space craft reentry modeling. Here, we present probabilities and cross sections useful to rationalize some typical mechanisms characterizing the vibrational transitions of the CO(2) + CO(2) system on the flexible monomer proposed PES. On such PES, the key dynamical outcomes are: (a) there is a strong energy interchange between symmetric stretching of the reactants and bending of the products (and viceversa) while asymmetric stretching is strongly adiabatic (b) reactant energy is more efficiently allocated (with respect to the rigid monomers PES) as product vibration when reactant stretching modes are excited while the contrary is true when the reactant bending mode is excited.
In the present paper, a strongly non-equilibrium reacting CO2 flow is studied on the basis of the kinetic theory methods. An accurate description of kinetics, gas dynamics and transport properties taking into account complex internal structure of carbon dioxide molecules as well as different rates of vibrational energy exchanges and dissociation is proposed. The expressions for all transport coefficients are derived in the final form. The model, while taking into account multiple CO2 vibrational modes and main features of intra- and inter-mode energy transitions, is sufficiently simple and suitable for practical realization.
In this paper the influence of different vibrational distributions on heat transfer and diffusion in expanding nozzle flows is studied. Non-equilibrium flows of N2/N and O2/O mixtures with dissociation, recombination and excitation of vibrational levels are considered. Vibrational distributions, gas dynamic parameters as well as the transport coefficients and total energy flux are computed along the nozzle axis using four approaches of the transport kinetic theory: a rigorous state-to-state approximation, quasi-stationary two-temperature models for harmonic and anharmonic oscillators and a thermal equilibrium one-temperature model. A comparison of vibrational distributions and transport properties obtained in different approaches is presented.
Transport properties of multi-component reacting gas mixtures are studied on the basis of the kinetic theory in the case of strong vibrational and chemical nonequilibrium. Considered are the conditions when quasi-stationary distributions of the molecules over vibrational levels do not exist and level kinetic approach is developed. The formulas for the viscosity, diffusion and thermal conductivity coefficients in terms of the nonequilibrium level populations, gas temperature and elastic collision integrals are derived. The practical algorithm for the calculation of these coefficients is given and applied for the investigation of the heat transfer behind a plane shock wave.
Ab initio calculations of rate coefficients are reported for the vibrational relaxation of CO2 molecules in collision with helium and neon atoms. Self consistent-field computations have been performed to parameterise simple three-dimensional potential energy functions which have been used in vibrational close-coupling, rotational infinite-order-sudden calculations of rate coefficients. Excellent agreement is obtained between the calculated and experimental rate coefficients for the deactivation of the (0110) vibrational level in the He + CO2 system at temperatures of 300 K and above. The ab initio predictions of rate coefficients for relaxation of CO2 vibrational levels such as (1000) and (0200) should be useful in computer simulations of CO2 lasers.
A semiclassical model recently developed has been used to calculate probabilites for intra- and inter-molecular energy transfer in the NeCO2 systems. In this paper we are especially concerned with the inclusion of the anharmonic coupling terms.
The rate of molecular dissociation behind strong shock waves is calculated with the assumption that dissociation can occur preferentially from the higher vibrational levels. An exponential probability of dissociation from the various vibrational levels is employed using an anharmonic oscillator model. Results for the dissociation of oxygen in an argon diluent are presented. Vibrational non-equilibrium introduces a T−3 temperature dependence into the oxygen dissociation rate constant in the range 4000°–8000°K. A dissociation lag-time of the order of the extrapolated vibrational relax ation time is predicted immediately behind the shock front. The computed results are shown to be in agreement with available experimental results.
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