Effect of Asymmetric Mode on CO2 State-to-State Vibrational

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Effect of Asymmetric Mode on CO2 State-to-State VibrationalChemical Kinetics Iole Armenise*,† and Elena Kustova‡ †

CNR NANOTEC_PLASMI Lab, Via Amendola 122/D, 70126, Bari, Italy Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

J. Phys. Chem. A Downloaded from pubs.acs.org by UNIV OF TEXAS AT EL PASO on 10/27/18. For personal use only.

‡

ABSTRACT: Coupled state-to-state vibrational-chemical kinetics, gas dynamics, and heat transfer in the five-component mixture of dissociated CO2 are studied using the complete three-mode kinetic model and the reduced scheme involving mainly the vibrational states of the asymmetric mode. The emphasis is on the effect of asymmetric vibrations on the rate of dissociation, fluid dynamic variables, and heat flux. It is shown that intermode vibrational energy transitions between CO and CO2 asymmetric mode may considerably decrease the rate of dissociation; the presence of CO in the mixture quickly depletes high vibrational states and thus inhibits dissociation at low temperatures. The reduced model overpredicts populations of highly located states of the asymmetric mode, especially when intermode VV transitions are neglected; therefore, using the simplified model in flows with dominating dissociation may yield overestimated dissociation rate. In the hypersonic flow along the stagnation line, the influence of asymmetric vibrations on the fluid dynamics and heat transfer is weak; the main role belongs to chemical reactions and VT transitions in the bending mode. In this case, the computationally efficient simplified model can be used to predict macroscopic variables and heat flux without significant loss of accuracy.

1. INTRODUCTION Development of reliable models for kinetics of carbon dioxide molecules is of great importance in many modern applied problems of physical chemistry and fluid dynamics. Design of space vehicles for Mars exploration programs requires simulations of high-temperature flows of dissociating CO2 and high-fidelity models for the transport properties.1−5 Addressing environmental problems needs modeling CO2 kinetics at the ambient temperature; thus conversion of greenhouse CO2 remains in the focus of attention of multiple scientific groups.6−13 For the greenhouse gas conversion, one of the most attractive techniques is based on low-temperature plasma-assisted or vibrational-assisted CO2 decomposition.6,7 Although mechanisms of vibrational-assisted dissociation are now actively studied, yet there is a lot of uncertainty in our knowledge of key processes allowing for CO2 conversion into useful chemical species without powerful energy deposition. In order to identify appropriate physical-chemical mechanisms, advanced models for coupled CO2 vibrational-chemical kinetics are needed. State-to-state modeling of polyatomic species10,11,14−18 is the best tool for in-depth studies of different energy exchanges coupled to chemical reactions. Development and assessment of full accurate kinetic models accounting for all possible collisional processes is crucial for the sensitivity analysis of flow characteristics to different processes included to the kinetic scheme and for identifying dominating relaxation mechanisms under various conditions. Subsequent reduction of detailed models is of great importance for engineering applications. Indeed, the full models are extremely time and © XXXX American Chemical Society

resource consuming since they involve thousands of states and millions of transitions. There are several possible ways for the model reduction: (1) uncoupling vibrational modes and taking into account just a few, the most important, states and transitions while keeping other advantages of the state-to-state modeling;7,10,11 (2) grouping the vibrational states into a set of macroscopic bins;5,19 (3) using reliable multitemperature models taking into account intermode energy exchanges;1,3,20−22 (4) methods based on the drift-diffusion Fokker−Planck equation.6,23 However, the full models remain the main source of information to assess the accuracy of reduced models in the absence of reliable experimental data. In our previous work,18 we have analyzed several dominating mechanisms of coupled CO2 vibrational relaxation and dissociation. It is shown that the VT2 exchange in the bending mode is the main channel of vibrational relaxation. Intra- and intermode VV exchanges, while accounted separately, do not affect the fluid dynamics. These processes are resonant or nearresonant and therefore there is no vibrational energy gain or loss during these collisions; vibrational energy remains isolated and does not affect flow parameters and heat flux. However, the intermode VV exchanges, when included together with the VT processes, enhance the VT2 relaxation because they cause vibrational energy redistribution between the CO2 modes. One of the important issues which is not addressed in18 is the role of the asymmetric CO2 vibrational mode in the Received: August 3, 2018 Revised: September 17, 2018 Published: October 15, 2018 A

DOI: 10.1021/acs.jpca.8b07523 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A vibrational-chemical kinetics, fluid dynamics and heat transfer. In the modern models of vibrationally assisted dissociation,7−10 the main idea is that dissociation proceeds through the high states of the asymmetric mode; thus pumping-up the vibrational energy in this mode yields significant enhancement of the dissociation rate under low-temperature conditions. Up to now, this assumption has not been thoroughly assessed using a complete kinetic model taking into account coupling of various CO2 vibrational modes and intermode vibrational energy exchanges; several important relaxation mechanisms have not been studied in the literature yet. The present study is aimed in filling this gap. The objectives of the present work are (1) to improve the state-to-state model of carbon dioxide vibrational-chemical kinetics and dynamics developed previously16 by introducing all energy transfer processes responsible for excitation or deactivation of the asymmetric mode, (2) to check the influence of intermode transitions on the populations of asymmetric mode and parameters of a nonequilibrium flow along the stagnation line, (3) to assess the assumption about preferential CO2 dissociation through excited asymmetric mode and to identify kinetic processes enhancing or inhibiting the dissociation, (4) to asses the reduced model based on the energy transitions involving mainly the excited vibrational states of the asymmetric mode and a few lowest states of the symmetric and bending modes. The paper is organized as follows. First, we describe the two kinetic models, the complete one18 and the simplified model limiting the number of vibrational states to those considered in refs 7 and 8. Then we analyze the rate coefficients of processes which can affect the population of asymmetric mode. Next, we assess the kinetic models solving numerically the set of 1-D boundary layer equations coupled to master equations for vibrational state populations and discuss the mechanisms of vibrationally assisted dissociation.

modes are strongly coupled and are not assumed independent. Therefore, to ensure that the vibrational energy does not exceed the dissociation threshold, we add a constraint Ev1,v2,v3 < D while calculating the vibrational energy Ev1,v2,v3. For the full model, the total number of possible CO2 vibrational states is 9018, which is rather difficult to implement for real flow simulations. It is shown in ref 16 that the reduced model taking into account only 1224 states provides a satisfactory accuracy and, at the same time, allows one to greatly decrease the computational costs. This model is applied further to construct the complete kinetic scheme. The following kinetic processes are taken into account for the complete scheme:16−18,21 vibrational−translational VTm transitions in the symmetric stretching, bending and asymmetric stretching mode in a collision with an arbitrary partner M = CO2, CO, O2, and O; intermode VV1−2, VV2−3, and VV1−2−3 exchanges within CO2 molecule; intermode VVm−k exchanges between molecules of different chemical species, VV3−CO, VV2−CO, and VV1−2−CO; state-specific dissociation−recombination reactions. Finally, to assess the mechanisms of vibrationally assisted CO2 dissociation through the asymmetric mode, we introduce the intramode VV3 exchanges. The main focus in the present study is on the coupling of various vibrational energy transitions in CO2 asymmetric mode and dissociation-recombination. The electronically excited states and complex CO kinetics including vibrational− electronic transitions, radiation, and Bouduard reactions24−26 are not considered to keep the computational costs reasonable. We are aware that these processes can affect the kinetics, and we plan to include them in our further studies as well as to remove the limit on the dissociation energy, thus taking into account all vibrational states; for anharmonic vibrations including highly excited states can influence the contribution of intramode VV transitions. The above kinetic processes are written in the general form in ref 18. In the present study, the emphasis is to the processes affecting the asymmetric vibrations, and particular attention is given to VT3 transitions

2. THEORY Carbon dioxide molecule in the ground electronic state has three vibrational modes: symmetric stretching with the wavenumber ω1 = 1345.04 cm−1, doubly degenerated bending (ω2 = 667.25 cm−1) and asymmetric stretching (ω3 = 2361.71 cm−1). We denote as v1, v2, v3 the quantum numbers of symmetric, bending and asymmetric modes, respectively. Since the molecule is linear, its rotational energy is introduced similarly to that of diatomic molecules. The wave numbers of vibrational modes are related by the expressions ω1 ≈ 2ω2, ω3 ≈ ω1 + ω2; the first one is almost exact whereas the latter one is approximate. These relations cause near-resonant intermode vibrational energy exchanges, namely VV1−2, VV2−3, VV1−2−3. In our study, we consider the CO2 molecule as well as the products of its dissociation, CO, O2, and O. It is worth mentioning that the wavenumber of asymmetric vibrations and that of CO molecule are close ω3 ≈ ωCO; therefore, the exchange VV3−CO is also near-resonant. 2.1. Complete Kinetic Scheme. In the complete kinetic scheme, we assume that three CO2 modes are coupled due to anharmonicity of vibrations. In this case, the vibrational state v of the molecule is characterized by the set of vibrational quantum numbers v = (v1,v2,v3); the numbers vk may vary in the range vk = 0, ..., Lk, where the last vibrational level Lk in each mode is determined by the dissociation threshold D. Since D = 8.83859 × 1019 J, we get L1 = 34, L2 = 67, and L3 = 20. An important feature of anharmonic vibrations is that the

CO2 (v1 , v2 , v3) + M F CO2 (v1 , v2 , v3 − 1) + M

(1)

VV2−3, VV1−2−3 intermode exchanges CO2 (v1 , v2 , v3) + M F CO2 (v1 , v2 − 3, v3 + 1) + M (2)

CO2 (v1 , v2 , v3) + M F CO2 (v1 − 1, v2 − 1, v3 + 1) + M (3)

VV3−CO intermolecular exchange CO2 (v1 , v2 , v3) + CO(w) F CO2 (v1 , v2 , v3 − 1) + CO(w + 1)

(4)

and VV3 transitions CO2 (v1 , v2 , v3) + CO2 (w1 , w2 , w3 + 1) F CO2 (v1 , v2 , v3 + 1) + CO2 (w1 , w2 , w3)

(5)

This latter process is included for the first time to the detailed state-to-state kinetic scheme. In what follows, we refer this detailed complete kinetic scheme as a “three-mode kinetics” (3-MK) model. The total number of considered transitions in the 3-MK model is more B


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The Journal of Physical Chemistry A

AMK model uses the Treanor−Marrone dissociation model instead of the ladder-climbing model; (3) electronically excited states and exchange chemical reactions are neglected in the AMK model. 2.3. Rate Coefficients of Vibrational Transitions and Reactions. The vibrational-chemical kinetics and fluiddynamic variables depend on the rate coefficients of the involved processes. Rate coefficients of VTm, VVk−m vibrational energy transitions in CO2 molecules were discussed in details in our previous work.18 In this Section, we focus on the rates of processes involving mainly the CO2 asymmetric mode as well as on the CO2 dissociation-recombination rates. First, let us consider the state-resolved rate coefficients of dissociation and recombination reaction

than 700 000. It is clear that applying the complete scheme in the flow simulations is extremely time-consuming. Therefore, it is interesting to assess simplified schemes. 2.2. Simplified Scheme Based on Asymmetric Vibrations. For harmonic vibrations, the modes are uncoupled; this yields a significant simplification of the model. The molecule can be considered as a set of three independent oscillators with corresponding wave numbers ωk; the number of vibrational states in each mode corresponds to the dissociation threshold; as is stated above, L1 = 34, L2 = 67, and L3 = 20. The total number of vibrational states is obtained by straightforward summation of Lk and is equal to 124. Further simplification based on the Fridman model6 is used in the work of Kozak and Bogaerts,8 where the CO2 molecule is assumed to have only a few vibrational states: all the possible levels v3 of the vibrational asymmetric mode at fixed v1 = v2 = 0, the levels v2 = 0−4 of the bending mode at fixed v1 = v3 = 0, the levels v1 = 0−2 of the symmetric mode at fixed v2 = v3 = 0, and the levels v2 = 0−2 at fixed v1 = 1 and v3 = 0. Thus, the total number of vibrational states is equal to 29. Implementation of such a reduced model can greatly improve the efficiency of nonequilibrium flow simulations. One of the objective of the present paper is to evaluate the range of validity for the above simplified model. In our complete 3-MK model, we limit the CO2 vibrational energy by 3 eV in order to reduce computational costs.16 To be consistent with the 3-MK model, in the simplified model we set the same limit, and therefore the number of possible states in the asymmetric mode is 10, and the total number of vibrational states decreases to 18. Applying the reduced set of vibrational states to our original full model yields a really lower number of VTm, VVk−m, and dissociation−recombination processes. The kinetic processes included to the scheme are the same as for the complete 3-MK model, however the total number of transitions in the reduced model is less than 3500. In particular, in the simplified scheme, there are 4 VT1, 6 VT2, 9 VT3, 4 VV1−2, 1 VV2−3, and 1 VV1−2−3 processes for each CO2 collisional partner; 36 VV3 processes, 2 × LCO VV1−2−CO, 2 × LCO VV2−CO, and 9 × LCO VV3−CO processes, where LCO is the last CO vibrational level, 18 × (LCO + 1) dissociation-recombination processes (all the CO possible vibrational levels and all the possible combination of (v1, v2, v3) are taken into account). It is seen that the latter kinetic scheme is based mainly on the asymmetric mode vibrational states. In the following sections, this simplified model is referred to as ”asymmetric mode kinetics (AMK)”. It is worth mentioning that the AMK model has been developed for the discharge6,27 that presents different conditions with respect to high enthalpy flows. The discharge plasma is at room temperature, reducing the importance of the vibrational quenching, and second, CO is formed during the discharge, therefore its contribution in the discharge evolution becomes relevant at later times. However, as shown in,46 the AMK model predicts the heat flux in the boundary layer with a satisfactory accuracy; it is interesting to assess how it reproduces other flow parameters under high enthalpy conditions. The last note on the AMK model is that, on the one hand, it is similar to the original Fridman model since it utilizes the same vibrational ladder. On the other hand, there are some significant differences: (1) more channels of vibrational energy relaxation are included to the present AMK model (in particular, various kinds of intermode exchanges); (2) the

CO2 (v1 , v2 , v3) + CO2 F CO(w) + O + CO2

(6)

To calculate them, we use the Treanor−Marrone model28 generalized for the polyatomic molecules.21 This model represents the state-resolved dissociation rate coefficient kdiss, v1,v2,v3 as a product of thermal equilibrium dissociation rate coefficient keq diss given by the Arrhenius law and the nonequilibrium factor depending on the CO2 vibrational state: eq kdiss, v1, v2 , v3(T ) = kdiss (T )

CO2 Zvibr (T ) CO2 Zvibr (−U )

ij Ev , v , v i 1 1 yyz expjjj 1 2 3 jjj + zzzzzz U {{ k k kT

(7)

Here CO2 Zvibr (T ) =

ij Ev , v , v yz (v2 + 1) expjjj− 1 2 3 zzz kT { k v1, v2 , v3

∑

(8)

is the equilibrium vibrational partition function of CO2, v2 + 1 is the statistical weight related to the degeneracy of the bending mode, k is the Boltzmann constant, T is the temperature, and U is the parameter describing preferential character of dissociation from highly excited states. The parameters in the Arrhenius law are taken from29 the parameter U has the dimension of temperature and is set to the fraction of dissociation energy, U = D/(6k). The above formula gives no information about the vibrational level w of the formed CO molecule, therefore we assume that CO is formed with the Boltzmann vibrational distribution at the local temperature. The rate coefficient for recombination krec, v1,v2,v3 to the CO2 vibrational state (v1, v2, v3) is calculated using the detailed balance principle: ji mCO2 zyz k rec, v1, v2 , v3(T ) = kdiss, v1, v2 , v3jjj z j mCOmO zz k { CO2 Z ( T ) × h3(v2 + 1) rot CO Zrot

3/2

(2π kT)−3/2

ij Ev , v , v − EwCO + ECO − ECO − EO yz j zz 2 × expjjj− 1 2 3 zz j z kT k {

(9) CO Zrot 2,

h is the Planck constant, mc is the mass of species c, ZCO rot are the rotational partition functions for CO2 and CO molecules, ECO is the CO molecule vibrational energy for w the state w, and Ec is the formation energy for c species. In Table 1, the dissociation rate coefficients calculated at the last vibrational state of each mode for the ground states of the C


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The Journal of Physical Chemistry A Table 1. Dissociation Rate Coefficients, kdiss, v1,v2,v3(T), cm3 s−1 part−1 v1

v2

v3

T = 700 K

T = 4555 K

17 0 0

0 33 0

0 0 9

1.79 × 10−18 5.87 × 10−19 4.23 × 10−20

5.04 × 10−13 4.01 × 10−13 2.33 × 10−13

also worth noting that the Treanor−Marrone model is more realistic than the ladder-climbing mechanism used in the Fridman model; the ladder-climbing model yields underpredicted dissociation rates since it assumes dissociation only from the last excited vibrational state. In Figure 3, the dissociation and recombination rate coefficients (forward and backwardreaction 6) are plotted. Although from this graph it is not possible to associate a particular rate to the corresponding tern of vibrational modes, this figure still gives some interesting information, representing an overview of all the possible rates. The first observation is straightforward and follows from eq 7 and eq 9; nevertheless, the following point deserves to be commented on: while there is a biunique correspondence between the CO2 vibrational energy and the dissociation rate coefficients, there is a spread of the recombination rate coefficients around their average values; the recombination rate coefficients do not form a regular curve but a set of points in the graph. The second observation that comes out from the Figure 3 is that, for each temperature, both the dissociation and the recombination rate coefficients increase with the CO2 vibrational energy, however, the rate of increase is different under various conditions. For low temperature and low vibrational energy, recombination is dominating (although the rates of both reactions are low); for high temperature, recombination rate is negligible in the whole range of energy states, which is indeed not surprising. Let us discuss now the rates of different vibrational energy transitions involving the asymmetric mode, like the intramode VV3 , and the intermode VV3−CO , VV2−3 , and VV1−2−3 transitions, and compare them with the dissociation and recombination rates. Data on the rate coefficients of multiple vibrational energy exchanges in CO2 are limited to several experimental studies of the transitions between the lowest states30−34 and a few molecular dynamic simulations of the cross sections for selected vibrational energy transitions.35,36 These results are not applicable in our study since we need a full data set for the whole range of vibrational energies. In the present study, the rate coefficients of vibrational energy transitions are calculated using the SSH theory.37,38 Although the SSH theory is rather approximate, it provides a satisfactory agreement with experimental data, for instance in the afterglow of a pulsed DC glow discharge.11

remaining modes are given for different temperatures. For the low temperature, the rate of dissociation from the asymmetric mode is considerably lower than that from the other modes. For higher temperatures, the rate coefficients are of the same order. Thus, under low-temperature conditions, pumping the symmetric and bending modes can provide more efficient dissociation than excitation of asymmetric vibrations. Although the dissociation energy is limited to 3 eV, this formal conclusion is also valid for the full model, since the dependence of the rate coefficients on the vibrational states is practically linear, see Figures 1 and 2. However, in reality, dissociation mechanism through the symmetric mode can be considered only if vibrational modes are uncoupled. When intermode vibrational energy exchanges are taken into account, pumping the symmetric mode is hardly feasible: resonant VV1−2 transitions and subsequent fast VT2 relaxation prevent population of high states in the first mode.18 On the other hand, since VT3 transitions are practically frozen at low temperatures (see discussion below and in ref 18), the asymmetric mode may provide an efficient channel for vibrationally assisted dissociation. Nevertheless there are some limitations in this mechanism, and they are discussed hereafter. In Figure 1, the dissociation rate coefficients as functions of the asymmetric mode quantum number, v3, at different bending mode quantum numbers, v2, are plotted when T = 700 K. From this figure it is clear that the dissociation rates increase with increasing the vibrational level of each mode, at fixed values of the other two modes. The dissociation rates increase by many order of magnitude when the CO2 molecule passes from the ground to the excited states. For high temperature, the trend is the same, but the values are several order of magnitude higher, see Figure 2. Thus, we see that coupling of vibrational modes may enhance dissociation from the asymmetric one if other modes are sufficiently excited. It is

Figure 1. Dissociation rate coefficients versus v3, at different v2: v1 = 0 (a), v1 = 10 (b). T = 700 K. D


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Figure 2. Dissociation rate coefficients versus v3: at different v2 and v1 = 0 (a); at different v1 and v2 = 0 (b). T = 4555 K.

Figure 3. Dissociation and recombination rate coefficients as functions of CO2 vibrational energy: T = 700 K (a); T = 4555 K (b).

Figure 4. Rate coefficients of VV3 transitions as functions of v3 for fixed vibrational states of other modes: T = 700 K (a); T = 4555 K (b).

same order of magnitude and take really close values. Moreover in some cases the forward rates are higher than the backward ones, in other cases the opposite happens, therefore, in the average, there is a kind of balance among the VV3 transitions involving all CO2 molecules in the different vibrational states. In the following discussion, we will see that this behavior of the rates causes an almost negligible

In Figure 4, the forward and backward rate coefficients of the VV3 vibrational energy transitions 5 are given as functions of the asymmetric stretching mode quantum number v3 at fixed quantum numbers v1 = v2 = 0 of two other modes, and for the fixed vibrational states (w1 = 0, w2 = 0, w3 = 0) and (w1 = 0, w2 = 16, w3 = 0) of the collision-partner CO2 molecule. One can see that the forward and backward rate coefficients are of the E


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Figure 5. Rate coefficients of VV3−CO transitions versus CO2 vibrational energy for the fixed vibrational state of the CO molecule: w = 0 (a); w = 66 (b). T = 700 K.

Figure 6. Rate coefficients of different VVm−k transitions and dissociation versus v3 at v1 = v2 = 0: T = 700 K (a); T = 4555 K (b).

contribution of these processes to the fluid dynamics. Nevertheless, the rates of these transitions are high enough to affect the upper vibrational states of the CO2 asymmetric mode, which is the basic assumption in the model of vibrational assisted dissociation proposed in.6−8 In Figure 5, the forward and backward rate coefficients of the intermode VV3−CO process as functions of the CO2 vibrational energy for the fixed vibrational state of the CO molecule are presented for T = 700 K. For the case of the ground CO state, w = 0, the forward and backward reaction rate coefficients are comparable. At high w, the rates of forward transitions leading to deactivation of the CO2 asymmetric mode excited states are considerably higher than the backward ones. A similar trend is observed for higher temperatures. Comparing the rate coefficients of dissociation and VV3−CO transitions, we can emphasis that for low temperatures, the rates of intermode vibrational transitions are significantly higher than those of dissociation, even for the high CO2 vibrational energy. This means that the presence of CO in the mixture may prevent the mechanism of vibrationally assisted dissociation described in.7,8 Pumping the high vibrational states of the CO2 asymmetric mode will result in the energy redistribution between CO2 and CO rather than in CO2 dissociation. Similar dissociation inhibition at low temperatures may be caused by intermode VV2−3, VV1−2−3 transitions,

although to a lesser extent. The rates of VV2−3, VV1−2−3 transitions are lower than the rate of VV3−CO exchange18 but still higher than the dissociation rate under the same conditions. This is clearly shown in Figure 6 where the rate coefficients of the cited processes are plotted as functions of the asymmetric mode quantum number, v3, both at T = 700 and 4555 K. At T = 700 K and v3 < 10, all intermode vibrational transitions are more efficient than dissociation from the asymmetric mode. For T = 4555 K, coupling of dissociation and vibrational energy transitions becomes more important, VV3−CO exchange remains the dominating channel for the asymmetric mode deactivation. It is worth noting that if we consider the full ladder with L3 = 20, then, taking into account the slope, the rate of dissociation may become greater than those of VVk−m transitions. Since implementation of the full ladder to the real flow simulations is still unfeasible due to the computational costs, in our future work we plan to study this effect for the case of 0-D spatially homogeneous relaxation. Finally, the rate coefficients of VT3 transitions are very low,18 even compared to the rate of dissociation from the upper states; thus, the VT3 transitions do not contribute to the kinetics and fluid dynamics. 2.4. Test Case Description. To assess various kinetic schemes in a real nonequilibrium reacting flow, one has to solve fluid-dynamic equations coupled with the equations of F


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The Journal of Physical Chemistry A detailed vibrational-chemical kinetics. In the Euler approximation, the equations do not include the mass and energy fluxes, which simplifies numerical simulations but does not allow evaluating the effect of nonequilibrium kinetics on the heat transfer. On the other hand, the heat flux is the observable which can be extracted from experiments and thus provide the validation of the kinetic model. On the basis of this reasoning, we have decided to apply our model to study a hypersonic boundary layer problem under conditions specified in the available experiments. We consider a 1-D flow of the CO2/CO/O2/O/C mixture along the stagnation line. The set of governing equations written in self-similar coordinates along the normal to the surface, η, has the following form:16 ∂ 2cv ∂η

2

+ f Sc

∂cv = Sv , v = 1, ..., N ∂η

∂ 2θ ∂θ + f Pr = ST 2 ∂η ∂η

Figure 7. Test case sketch.

In the experiments, the model was made with Macor, which is a ceramic with a low catalytic efficiency. Therefore, in the present study, the surface is assumed to be noncatalytic with zero mass fraction derivatives.

(10)

3. RESULTS AND DISCUSSION In this section, we analyze, by adopting the 3-MK model, the role of different energy exchange processes involving the CO2 asymmetric mode in the hypersonic boundary layer flow. The emphasis is to the intramode VV3, the intermode VV3−CO, VV2−3, and VV1−2−3, and the vibrational−translational VT3 transitions with respect to other kinetic processes. The reason to focus on the above processes is to assess the role of asymmetric vibrations in the kinetics and fluid dynamics. Moreover, to investigate CO2 behavior in different physical problems, the full 3-MK model is both time and memory consuming from the numerical point of view, and the simplified AMK model is rather attractive. In the last part of this section, we compare the results obtained with the 3-MK and AMK models. 3.1. Effect of Vibrational Transitions in the Asymmetric Mode. Let us discuss first the vibrational distributions obtained using different kinetic schemes. In our previous study,18 it was shown that the rate of VT3 transitions is considerably lower compared to other vibrational exchanges. In order to assess the role of this vibrational−translational energy exchange in the boundary layer flow, we ran a test case where only VT3 transitions are included to the kinetic scheme. In Figure 8, the vibrational distributions in CO2 molecules are given as functions of the vibrational energy for η = 0 (on the

(11)

Here cv = ρv/ρ are the mass fractions of each species including all vibrational states (N comprises the total number of vibrational states in molecules and two atomic species), θ = T/Te is the dimensionless temperature related to the temperature Te at the boundary layer external edge. Sc and Pr are the Schmidt and Prandtl numbers, respectively; f is the stream function. The production term Sv in eqs 10 describes the rate of species formation due to chemical reactions and vibrational energy transitions included to the kinetic scheme; the term ST in eq 11 is responsible for the energy production caused by vibrational-chemical kinetics. The expressions for the source terms are given in ref 16. To evaluate the heat flux in a flow, we use the technique proposed earlier in.16 In the state-to-state kinetic models, the heat flux is specified by the heat conductivity of translational and rotational degrees of freedom, mass diffusion of chemical species, thermal diffusion and diffusion of vibrational energy carried by excited molecules.39−41 The accurate algorithm for the calculation of corresponding state-resolved transport coefficients is developed in ref 40. It includes solving the transport linear systems in each cell of the computational grid; the transport coefficients are functions of temperature and molar fractions of all vibrational states and atomic species. Keeping in mind that the transport systems are of the order of N, and the number of diffusion coefficients is about N2, implementation of this transport model to the fluid-dynamic solver is hardly feasible. In order to calculate the heat flux, we use the postprocessing technique proposed in.42,43 First we extract the fluid-dynamic variables obtained along the stagnation line while solving eqs 10−11; then, we calculate the state-resolved transport coefficients; and finally, using the gradients of T and cv, we calculate the heat flux. The boundary conditions used in our study correspond to the experiment of Hollis2,44,45 conducted in the HYPULSE facility under the following free stream conditions: velocity is 4788 m/s, density is 5.738 × 10−3 kg/m3, pressure is 1182 Pa, and temperature is 1090 K. We set the surface temperature Tw = 700 K, external edge temperature Te = 4555 K, the edge pressure, pe = 1.269 × 105 Pa, the inverse of the residence time in the boundary layer, β = 1.283 × 105 s−1, and the mixture composition at the edge is 22.6662% of CO2, 41.5502% of CO, 5.6685% of O2, 30.115% of O, and 10−4% of C (see Figure 7).

Figure 8. Vibrational distributions in CO2 molecules as functions of the vibrational energy for η = 0 (surface) and η = 6.4 (external edge). Only VT3 transitions are included in the kinetic scheme. G


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Figure 9. Surface CO2 vibrational distributions as functions of v3 at different values of v1 and v2: (a) only VV3 transitions are included; (b) full 3MK scheme.

surface) and η = 6.4 (at the external edge). It is seen that, if only VT3 transitions are taken into account, the initial Boltzmann distribution calculated at Te = 4555 K does not change through the boundary layer and remains exactly the same at the cold surface. Therefore, VT3 transitions are completely frozen in the flow and contribute neither to kinetics nor to fluid dynamics. Since the fluid dynamic variables calculated with and without VT3 processes coincide in the whole range of η, corresponding figures are not included to the discussion.

asymmetric mode level populations calculated taking into account different kinds of intermode and intramode VV transitions. For all cases, the distributions are quite similar. Thus, we conclude that higher population of the asymmetric mode can be attributed mainly to the recombination process and not to the vibrational energy transitions. It is interesting to assess how the coupling of vibrational modes affects the VV3 exchange process. For this purpose, we included to the kinetic scheme either VV3 transitions from arbitrary states of the remaining modes or VV3* transitions involving only the ground states of the symmetric and bending modes, v1 = v2 = 0. The vibrational distributions obtained with VV*3 exchanges are exactly the same as shown in Figure 8 although the rate of VV*3 exchange is much higher than that of frozen VT3 processes. The absence of any effect of uncoupled VV3* transitions on the flow is explained by the fact that the vibrational energy in this case remains practically isolated. For the case of VV3 processes taking into account excitation of other modes, the surface distributions are slightly different from those at the boundary layer edge. However, the overall effect of VV3 transitions on the fluid dynamic variables is rather weak. In Figure 11, the surface vibrational distributions are shown as functions of vibrational energy for the kinetic schemes taking into account and neglecting VV 3 and VV 3−CO transitions. First the test case neglecting both VV3 and VV3−CO exchanges is considered (blue signs). One can see that adding the VV3 exchange (Figure 11a) yields more condensed vibrational level populations in the high energy domain (red signs); on the average, the distributions are slightly lower when VV3 processes are taken into account. Adding the VV3−CO transitions leads to a significant decrease in the vibrational state populations on the surface and spread distributions at intermediate and high energies (Figure 11b). A similar effect is obtained while including VV2−3 and VV1−2−3 transitions, but to a lesser extent. Thus, intermode VV transitions can be a kind of ”bottleneck” in the mechanism of vibrationally assisted dissociation through asymmetric vibrations. The rate of pumping of the high states in the asymmetric mode by VV3 and other processes can be lower than the rate of energy loss in this mode due to the energy redistribution between the CO2 and CO molecules. This can inhibit CO2 dissociation from the asymmetric mode.

Figure 10. Surface CO2 vibrational distributions as functions of v3 for v1 = v2 = 0. Different VVk−m and VV3 are taken into account.

In Figures 9−11, the effect of different VV transitions on the surface vibrational distributions is shown. In Figure 9, we compare the surface distributions in the asymmetric mode as functions of v3 calculated for various v1, v2 using the full 3-MK model and taking into account only VV3 transitions. Populations obtained with the full model are considerably higher, especially when other modes are in the ground state. Note that the energies corresponding to the excited states of symmetric v1 = 8 and bending v2 = 16 modes are approximately the same; we see that excitation of symmetric vibrations results in lower populations of the asymmetric mode. In order to understand which processes are responsible for higher distributions in the full model, in Figure 10, we compare the H


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Figure 11. Surface CO2 vibrational distributions as functions of vibrational energy: (a) comparison of the 3-MK model neglecting both VV3 and VV3−CO transitions and neglecting only VV3−CO transitions; (b) comparison of the 3-MK model neglecting both VV3 and VV3−CO transitions and the full 3-MK model.

Although the VV3 and VV3−CO transitions are important for the formation of nonequilibrium vibrational distributions on the surface, their impact on the fluid dynamics in the boundary layer is weak. The temperature and mixture composition along the stagnation line are insensitive to these transitions; the behavior of macroscopic variables is similar to that discussed in.18 As an illustration, the heat flux calculated including and neglecting VV3−CO exchange is plotted in Figure 13. The discrepancy on the surface is about 2−3%, which is quite negligible. Thus, the main processes affecting fluid dynamics are VT2 relaxation and recombination.

Let us discuss now the CO2 average vibrational energy obtained using the 3-MK model including different processes in the CO2 asymmetric mode. This observable is directly related to the CO2 vibrational distributions.18 It is calculated starting from the CO2 vibrational states mass fractions and the corresponding vibrational energies, as ρv v v CO2 Evibr = ∑ 1 2 3 Ev1v2v3 ρ v ,v ,v (12) 1 2 3

therefore it depends on the kinetic processes that are active in the boundary layer. In Figure 12, the average vibrational energy along the stagnation line is shown for different kinetic schemes. One can

Figure 13. Heat flux as a function of η calculated using the full 3-MK model and neglecting VV3−CO transitions. Figure 12. CO2 average vibrational energy as a function of η calculated for different kinetic models.

3.2. Comparison of 3-MK and AMK Models. In this subsection, a comparison of the results obtained using the full 3-MK and simplified AMK models is presented. In Figure 14, the surface vibrational distributions in the asymmetric mode are given as functions of v3. Populations of the asymmetric mode levels with v3 > 1 calculated on the basis of the AMK model are considerably higher with respect to those given by the full model. It is interesting to note that including VV3 and VV3−CO transitions yields different shape of the distribution. If we include VV 3 and neglect VV 3−CO exchanges, the distribution is close to the Boltzmann one. If both VV3 and VV3−CO transitions are neglected, the populations of

notice that both VV3 and VV3−CO exchanges practically do not affect the vibrational energy if accounted separately. Nevertheless, if VV3−CO exchange is neglected in the full kinetic scheme, it leads to overpredicted vibrational energy on the surface; the discrepancy is about 10−12%. This observation is similar to that discussed in our previous paper:18 near-resonant intermode transitions, if considered alone, do not contribute to the average vibrational energy. However, while included to the full model, they may enhance or slow down other relaxation processes. I


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2 values of ECO vibr , the difference is about 20−25%. On the other hand, at the edge of the boundary layer, the CO2 vibrational energy strongly depends on the applied model: the AMK model causes its significant underestimation. By definition, the CO2 vibrational energy (see eq 12) is determined by the CO2 vibrational distribution; the latter one at the external edge is the Boltzmann distribution16

ij Ev v v yz (v2 + 1) expjjj− 1 2 3 zzz Zvibr (13) k kT { where the partition function Zvibr is defined by eq 8. The Boltzmann distributions at the external edge temperature Te calculated for the two models are reported in Figure 16. At the medium and high vibrational energies, in the 3-MK nv1, v2 , v3 = nCO2

Figure 14. CO2 surface vibrational distributions at v1 = v2 = 0 as functions of v3 obtained using the 3-MK and AMK models. Comparison of the full models and models neglecting VV3, VV3−CO exchanges.

intermediate and high states of the asymmetric modes are essentially non-Boltzmann and show a kind of inversion. Keeping in mind preferential dissociation from highly excited states (see Figure 1), one can expect significantly overpredicted rate of CO2 dissociation when VV3 and VV3−CO transitions are excluded from the kinetic scheme. In Figure 15, the average vibrational energy and CO2 mass fractions obtained along the normal to the body surface using the 3-MK and AMK models either including or neglecting the VV3 and VV3−CO transitions are compared. Going from the external edge toward the surface, the VV3 and VV3−CO exchanges practically do not affect the CO2 vibrational energy up to η ≈ 3 for both models, and then VV3−CO transitions enhance its deactivation whereas VV3 do not contribute to the energy variation (see also Figure 12). This is in line with the CO2 vibrational distribution on the surface (see Figure 14), that is decreased by these processes. Following our previous discussion on the VV3 and VV3−CO relaxation channels, we can state that the main role in the asymmetric mode vibrational energy deactivation is played by the VV3−CO transitions. Another observation about the CO2 vibrational energy is that on the surface, the 3-MK and AMK models yield close

Figure 16. External edge vibrational distributions obtained using the full 3-MK and AMK models.

model there is a huge amount of vibrational states contributing to the CO2 vibrational energy; these states are omitted in the AMK model. This is the reason why, at the edge of the boundary layer, the AMK model underestimates the CO2 vibrational energy. The CO2 mass fraction (see Figure 15b) is slightly higher in the case of the AMK model, the difference on the surface is about 7%, and it is affected to a minimum extent by the VV3 and VV3−CO transitions. This can be explained by the fact that,

Figure 15. CO2 average vibrational energy (a) and mass fraction (b) obtained using the 3-MK and AMK models. Comparison of the full models and models neglecting VV3, VV3−CO exchanges. J


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between symmetric and bending modes with subsequent fast VT2 deactivation.18 Since VT3 exchange is practically frozen for all considered cases, pumping the high states of the asymmetric mode can be quite efficient to enhance dissociation. Nevertheless, it is important to account for intermode vibrational transitions which may significantly inhibit vibrationally assisted dissociation; these processes are commonly neglected in the kinetic schemes. The main process responsible for the decrease in the dissociation rate is the VV3−CO exchange resulting in fast depopulation of the high asymmetric mode states due to the energy transfer to CO molecules. In the presence of CO (as a product of CO2 dissociation), it is highly likely that the excited CO2 molecule will lose its energy in the VV3−CO transition rather than dissociate. In the absence of CO (at the early stage of dissociation), VV2−3 and VV1−2−3 transitions play a similar role, although to a lesser extent. These inhibition mechanisms have to be taken into account while designing the facilities for plasma assisted and vibrationally assisted CO2 conversion. In the hypersonic flow along the stagnation line, when the dominating chemical mechanism is recombination, the effect of asymmetric vibrations on the fluid dynamics and heat transfer is rather weak; the main role belongs to chemical reactions and VT2 transitions. In this case, to predict macroscopic variables and heat flux, the simplified AMK model can be used instead of the full 3-MK model without significant loss of accuracy; the AMK model is much more computationally efficient. However, in nonequilibrium flows governed by dissociation (shock heated flow regimes), the simplified model may cause considerable errors in the predicted values of fluid-dynamic variables and heat flux, especially if mode coupling and intermode exchanges are neglected. For the future work, we plan further improvement of the full model which, in the present form, still has a number of limitations. The contribution of CO kinetics, exchange reactions, excited electronic states has to be assessed; the accuracy of the rates, especially for highly excited states, has to be verified. It should be noted that the results obtained strongly depend on the calculation conditions and on the choice of the dependence of the rate coefficients on the relevant parameters. Development of a 0-D code implementing the improved full model, simulation of nonequilibrium kinetics under different conditions, and sensitivity analysis are considered as next steps.

in the boundary layer, the main chemical mechanism is recombination. It can greatly influence vibrational distributions, but recombination itself does not depend on the CO2 vibrational distributions and therefore on different kinds of VV exchange. Under conditions when dissociation is the dominating chemical channel, we expect a significant effect of the model on the mixture composition. The total heat flux along the stagnation line calculated using the 3-MK and AMK models is given in Figure 17. The heat flux

Figure 17. Total heat flux obtained using the full 3-MK and AMK models.

on the surface is slightly higher for the 3-MK model, the discrepancy is 12%. At η > 0.17, the AMK model yields a higher total heat flux. In any case, the difference obtained by using the two models is within 20%. It is worth noting that both models provide an acceptable agreement of the surface Fourier flux with the stagnation point heating of 8MW/m2 measured in refs 44 and 45. While the 3-MK model yields somewhat overpredicted heating, the AMK model slightly underestimates the flux; both models provide the surface heating within the uncertainty range of experimental measurements; see ref 46. Thus, for the heat transfer prediction in the boundary layer, the reduced AMK model can be used. However, for the dissociation-dominated regimes, we expect that the AMK model will overpredict the rate of CO2 dissociation, especially when intermode VV transitions involving asymmetric vibrations are neglected; this may result in a significant inaccuracy in the predicted rate of heat and mass transfer.

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4. CONCLUSIONS Different vibrational energy transitions in the CO2 asymmetric mode together with state-resolved dissociation-recombination are studied using the full 3-MK kinetic model as well as the simplified AMK scheme including a few vibrational states. Thorough analysis of the rate coefficients of VT3, VV3, VV3−CO, VV2−3, and VV1−2−3 as well as dissociation−recombination is performed taking into account coupling of CO2 vibrational modes. It is shown that, at low temperatures, the rate of dissociation from the high states of the first and second modes is greater than that from the asymmetric mode; the highest rate is found for the symmetric mode. On the other hand, considerable excitation of symmetric vibrations for the purpose of low-temperature CO2 decomposition is hardly attainable because of the resonant VV1−2 energy redistribution

AUTHOR INFORMATION

Corresponding Author

*(I.A.) E-mail: [email protected]. ORCID

Iole Armenise: 0000-0002-9440-0026 Elena Kustova: 0000-0001-5192-0390 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS E.K. received funding from Saint Petersburg University, Project 6.37.206.2016, and the Russian Foundation for Basic Research, Grant 18-08-00707. K


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Effect of Asymmetric Mode on CO2 State-to-State Vibrational

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