Mechanisms of Coupled Vibrational Relaxation and Dissociation in

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Mechanisms of Coupled Vibrational Relaxation and Dissociation in Carbon Dioxide Iole Armenise, and Elena Kustova J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b03266 • Publication Date (Web): 21 May 2018 Downloaded from http://pubs.acs.org on May 21, 2018

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The Journal of Physical Chemistry

Mechanisms of Coupled Vibrational Relaxation and Dissociation in Carbon Dioxide 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 E-mail: [email protected]

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract A complete vibrational state-specific kinetic scheme describing dissociating carbon dioxide mixtures is proposed. CO2 symmetric, bending and asymmetric vibrations and dissociation-recombination are strongly coupled through inter-mode vibrational energy transfers. Comparative study of state-resolved rate coefficients is carried out; the effect of different transitions may vary considerably with temperature. A non-equilibrium 1-D boundary layer flow typical to hypersonic planetary entry is studied in the stateto-state approach. To assess the sensitivity of fluid-dynamic variables and heat transfer to various vibrational transitions and chemical reactions, corresponding processes are successively included to the kinetic scheme. It is shown that vibrational-translational (VT) transitions in the symmetric and asymmetric modes do not alter the flow and can be neglected whereas the VT2 exchange in the bending mode is the main channel of vibrational relaxation. Inter-mode vibrational exchanges affect the flow implicitly, through energy redistribution enhancing VT relaxation; the dominating role belongs to near-resonant transitions between symmetric and bending modes as well as between CO molecules and CO2 asymmetric mode. Strong coupling between VT2 relaxation and chemical reactions is emphasized. While vibrational distributions and average vibrational energy show strong dependence on the kinetic scheme, the heat flux is more sensitive to chemical reactions.

Introduction Carbon dioxide remains in the focus of extensive studies in physical chemistry and chemical physics due to its importance for many applications. In particular, CO2 vibrational-chemical kinetics is crucial for laser chemistry, 1–3 plasma assisted gas conversion, 4–8 Mars space exploration programs including hypersonic planetary entry. 9–13 The peculiarity of CO2 kinetics is the existence of multiple channels of vibrational relaxation due to transitions within and between vibrational modes; relations between the frequencies of various types of vibrations cause highly probable near-resonant inter-mode transitions which lead to strong coupling 2


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of vibrational modes; this coupling can significantly affect the vibrational distributions and flow parameters. Growing interest to CO2 kinetics is also confirmed by recent studies of cross sections for elementary vibrational energy transitions and dissociation from various vibrational states. 14–17 State-to-state simulations become a powerful tool for studying non-equilibrium flows since they are able to take into account different mechanisms of vibrational relaxation, chemical reactions and their mutual effect. First state-resolved kinetic schemes for CO2 molecules and CO2 -N2 mixtures were proposed for laser applications; 1 in these schemes only the lowest vibrational states were included as well as just several basic transitions between them. A general scheme taking into account all possible CO2 vibrational states was proposed later 18 and applied in the simplified form to hypersonic flow modeling. 10–13 Moreover, the kinetic-theory based algorithms for the calculation of state-specific transport coefficients and corresponding diffusive and heat fluxes developed in 19,20 were generalized for polyatomic molecules; 10,11,21 this allows evaluating contribution of various dissipative processes including diffusion of vibrational energy to the total energy transfer. The peculiarity of the model developed in the above studies is that the different vibrational modes of the CO2 molecule are assumed to be strongly coupled. Generally speaking, it is possible to include in the kinetic model all kinds of state-to-state vibrational transitions like vibrational-translational VT1 , VT2 and VT3 transitions in symmetric, bending and asymmetric modes, inter-mode energy exchanges within CO2 molecule VV1−2 , VV2−3 , VV1−2−3 , inter-mode energy exchanges between molecules of different chemical species, as well as state-specific dissociation/recombination and exchange reactions. Note that in the models applied previously 10,11 some important energy transitions were missed; in the present study we overcome this limitation. In the complete kinetic scheme describing dissociating CO2 flows with coupled vibrational modes, the total number of transitions is about tens of thousands. 10 One should confess that flow simulations based on such a detailed model, while accurate, are extremely time and

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resource consuming. In recent works, 4,5 following to the model proposed by Fridman, 22 a CO2 molecule is simulated using only a few symmetric stretching and bending vibrational levels but the whole asymmetric stretching ladder; the modes are assumed independent which reduces considerably the possible number of vibrational states; the kinetic scheme includes several selected vibrational transitions and, at the same time, detailed electron-heavy particle collision kinetics. Another reduced scheme taking into account about 70 coupled CO2 states and suitable for modeling the afterglow of a pulsed DC glow discharge is proposed in. 23 Using such simplified models is quite attractive from the computational point of view but requires further assessment in high-temperature flows. An alternative way to simplify the problem is to implement multi-temperature models for CO2 flow simulations 9,12,24,25 or use reduced-order representations formulated by grouping the vibrational states into a set of macroscopic bins 26–28 . In the frame of multi-temperature modeling, coupling of different vibrational modes can be achieved by introducing common temperatures of the modes related by near-resonant inter-mode vibrational transitions and also by using non-Boltzmann vibrational distributions accounting for anharmonicity of vibrations. 29 Although multi-temperature models are efficient computationally and can be used in aerospace applications, their level of detail is often not sufficient to catch fine effects important for plasma chemistry. The objectives of the present work are: 1) to develop a complete kinetic model of hightemperature carbon dioxide flows by introducing all relevant energy transfer processes and vibrational state-resolved chemical reactions into the kinetic scheme; 2) to analyze the rate coefficients of various transitions in a wide range of vibrational states; 3) to study the sensitivity of vibrational distributions, fluid-dynamic variables and heat transfer to the collisional processes included into the model and identify the main channels of vibrational and chemical relaxation. This will help to assess and improve simplified schemes and provide the basis for developing new, more realistic multi-temperature models. The paper is organized as follows. First, we describe the possible kinetic processes such as

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intra- and inter-mode transitions, review available data on their rate coefficients, and carry out a thorough analysis of the rate coefficients of all kinetic processes under a wide range of conditions. After that we solve numerically a set of 1-D Boundary Layer equations coupled to master equations for vibrational state populations adding successively different kinds of energy exchanges and chemical reactions into the source terms. Thanks to a state-to-state transport model 19,20 based on the modified Chapman–Enskog theory, the heat conductivity, viscosity, diffusion and thermal diffusion coefficients as well as heat and diffusion fluxes are calculated. Peculiarities of non-equilibrium vibrational distributions, mixture vibrational energy and the resulting heat flux are discussed; the surface heat flux is compared to that measured experimentally. 30,31

Kinetic Scheme The linear CO2 molecule is characterized by three types of vibrations associated to symmetric stretching, doubly degenerated bending and asymmetric stretching vibrational modes. We conventionally assign numbers 1, 2, and 3 to symmetric, bending, and asymmetric vibrations respectively. Thus the wave numbers of vibrations are ω1 = 1345.04 cm−1 , ω2 = 667.25 cm−1 , ω3 = 2361.71 cm−1 , and v1 , v2 , v3 are the quantum numbers of corresponding modes. If the modes are assumed independent, the molecule can be considered as a set of three independent oscillators with corresponding frequencies and quantum numbers; for such a model, the numbers of vibrational states in each mode L1 , L2 , L3 are specified by the CO2 dissociation energy D. Starting from the value D = 8.83859 · 10−19 J we obtain L1 = 34, L2 = 67, L3 = 20, and the total number of vibrational states is about 120. When the vibrational modes are coupled, the vibrational state of the molecule is given by the tern v = (v1 , v2 , v3 ), the numbers vk take the values vk = 0, ..., Lk with the constraint that the total vibrational energy does not exceed the dissociation threshold, Ev1 ,v2 ,v3 < D. In this case, the total number of possible CO2 vibrational states is 9018; however, without considerable loss of accuracy, it can be reduced to 1224, see 10 for the discussion. 5



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In the complete kinetic scheme we consider the following processes: 10,11,24 • Vibrational-translational VTm transitions in the symmetric stretching, bending and asymmetric stretching mode, respectively:

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

(1)

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

(2)

CO2 (v1 , v2 , v3 ) + M * ) CO2 (v1 , v2 , v3 − 1) + M.

(3)

Here M stands for the collision partner, M=CO2 , CO, O2 , and O. • Inter-mode VV1−2 , VV2−3 , VV1−2−3 exchanges within CO2 molecule:

CO2 (v1 , v2 , v3 ) + M * ) CO2 (v1 − 1, v2 + 2, v3 ) + M,

(4)

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

(5)

CO2 (v1 , v2 , v3 ) + M * ) CO2 (v1 − 1, v2 − 1, v3 + 1) + M.

(6)

It is worth mentioning that the wave numbers of CO2 molecules are related by the expressions ω1 ≈ 2ω2 , ω3 ≈ ω1 + ω2 ; the first one is almost exact whereas the latter one is approximate. Therefore the energy gain or loss in the considered inter-mode VVm−k transitions is quite small, and the transitions are near-resonant. The energy variation in the VV1−2 transitions (4) is practically zero, and consequently these transitions are the most probable among the processes (4)–(6). • Inter-mode VVm−k exchanges between molecules of different chemical species, VV3−CO ,

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VV2−CO and VV1−2−CO :

CO2 (v1 , v2 , v3 ) + CO(w) * ) CO2 (v1 , v2 , v3 − 1) + CO(w + 1),

(7)

CO2 (v1 , v2 , v3 ) + CO(w) * ) CO2 (v1 , v2 − 3, v3 ) + CO(w + 1),

(8)

CO2 (v1 , v2 , v3 ) + CO(w) * ) CO2 (v1 − 1, v2 − 1, v3 ) + CO(w + 1).

(9)

Once again, the above collisions are near-resonant since the wave number of CO and ω3 are close to each other. • State-specific dissociation-recombination reactions, with M = CO2 , CO:

CO2 (v1 , v2 , v3 ) + M * ) CO(w) + O + M.

(10)

It is worth mentioning that electronically excited states may affect the CO2 molecule recombination. Thus the CO2 molecules that recombine tend to form a triplet CO2 , followed by a transition between triplet and singlet electronic states that ultimately results in the formation of singlet CO2 . 32 This effect is not included to the model yet, and will be taken into account in the future work. Moreover, since in the present work we give particular emphasis to the coupling of various vibrational energy transitions and dissociation-recombination, the exchange reaction CO2 +O=CO+O2 is not considered. We are aware that it can affect the kinetics and we plan to include it in further studies. Kinetic processes (2) and (5)–(6), (8)–(9), (10) were included to the model in our previous works, however their mutual effect was not systematically studied up to now except some preliminary results reported in. 33 In the present study, we include to the model successively all processes (1)–(10), investigate their interplay and identify the main channels of coupled vibrational relaxation and dissociation. Moreover, the kinetic scheme is considerably improved by including all possible collision partners in the vibrational energy transitions whereas in the earlier studies, only CO2 molecule was considered as a partner. Note that 7



VV3−CO transitions (7) are now studied for the first time in the frame of detailed state-tostate simulations.

Rate coefficients of vibrational transitions and reactions The rate coefficients of vibrational energy transitions (2) and (5)–(9) in our previous works 10–12 were calculated by adopting the SSH theory 34 modified for polyatomic molecules. 24 This means that while the rates of energy transitions between the lowest states are calculated from the experimental values reported by Achasov, 35 for the transitions between higher states, the rules prescribed by the SSH model are applied. For other transitions, if no experimental data are available, the rate coefficients were directly calculated with the SSH theory. 34,36 Simulations performed in 33 using various data sets (pure SSH model against experimental values combined with SSH model for higher states) showed a weak effect of these rate coefficients on the fluid-dynamic variables and heat flux, therefore in the present study we limit our consideration with only one data set (pure SSH model). Unfortunately the results of recent molecular dynamic simulations of the cross sections for vibrational energy transitions 14,15 cannot be applied in our study since the cross sections are obtained only for a few selected states, whereas we need a full data set for our modeling. The rate coefficients for dissociation from any vibrational state are calculated using the Treanor–Marrone model 37 generalized in 24 for polyatomic gases. The state-specific recombination rate coefficients are obtained using the detailed balance relations. In order to identify the main mechanisms of vibrational relaxation and chemical reactions, we have carried out a systematic assessment of the rate coefficients of various kinetic processes under different conditions. Low and high temperatures 700 and 4555 K are chosen since they correspond to the minimum and maximum temperatures in the test case considered in the next sections. First we compare the rate coefficients of VT transitions between the first and ground vibrational states in various vibrational modes, see Fig. 1. It is clearly seen that VT2 transitions in the bending mode dominate in the whole temperature range whereas 8


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VT3 exchanges in the asymmetric mode have the lowest rate; the difference between the rate coefficients practically does not vary with the temperature. The important role of VT2 transitions is not surprising since the vibrational frequency of bending mode is much less compared to that of the stretching modes. Therefore we conclude that the preferential channel of VT relaxation in CO2 is associated with bending vibrations. 10-10

3 -1

-1

rate coefficients (cm s part )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

10-12 10-14

1

10-16 3 10-18 1 - VT k 1

10-20

f

2 - VT k 2

f

3 - VT k 3

10-22 1000

2000

3000

4000

f

5000

T

Figure 1: Rate coefficients of VTm transitions between the first and ground vibrational states as functions of temperature. More careful analysis of the VT2 rate coefficients is performed in Figs. 2–3. Fig. 2 shows the VT2 rate coefficients as functions of v2 for fixed T = 700, 4555 K and fixed quantum numbers of the remaining modes. Both forward (deactivation) and backward (excitation) processes are considered. For low temperature, the VT2 rate coefficients depend noticeably on the excitation of other modes. Thus excitation of the asymmetric mode leads to a considerable increase in the rate coefficient. The rates of excitation and deactivation processes differ significantly. For high temperature, both effects become weak: the vibrational state of stretching modes do not affect the rate of VT2 transitions; the rates of forward and backward processes become close to each other. The effect of the collision partner on the VT2 rate coefficients is shown in Fig. 3. At T = 700 K, the VT2 rate coefficients are almost the same if the collisional partner is a diatomic molecule, CO or O2 . On the other hand, the rates are up to half order of magnitude 9



10-11 T = 700 K

-1

VT rate coefficients (cm s part )

2 2' 10-13

1

1

3

f

2 - v =8, v =0, k 1

3

3'

3

1'

2 2'

10-9 1 - v =0, v =0, k

2

1 - v =0, v =0, k

1

1

4

8

12

16

20

24

3

28

3

f

2 - v =8, v =0, k

f

1

3 - v =0, v =5, k

a 0

10-8

3 -1

3'

2

3 -1

1'

10-12

10-14

b

T = 4555 K

1 3

-1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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3

f

3 - v =0, v =5, k

f

1

32

0

v

4

8

12

16

20

24

3

28

f

32

v

2

2

Figure 2: Forward and backward VT2 rate coefficients versus the bending mode quantum number, v2 , for different v1 , v3 ; T = 700, 4555 K. Curves 1,1’: v1 = v3 = 0; curves 2, 2’: v1 = 8, v3 = 0; curves 3, 3’: v1 = 0, v3 = 5. Curves 1–3 correspond to forward processes whereas curves 1’–3’ to the backward ones. lower when the collisional partner is another CO2 molecule and up to half order of magnitude higher if the partner is an oxygen atom O. Thus at low temperatures, atomic oxygen is the most efficient partner in VT2 relaxation. At T = 4555 K, the rates are higher and almost do not depend on the collisional partner: they are just a bit lower if the collisional partner is another CO2 molecule. Let us compare now rates of various inter-mode processes. In Fig. 4, the rate coefficients of VT2 , VV1−2 , VV2−3 , VV1−2−3 , and VV2−CO are plotted as functions of v2 . The modes which do not participate in the corresponding transitions are assumed to be in the ground state. One can notice a strong interplay between VT2 , VV1−2 and VV2−CO processes. While for low temperatures VV1−2 transitions are dominating, under high-temperature conditions the main role belongs to VT2 relaxation. Inter-mode exchanges VV2−3 and VV1−2−3 are less probable but yet are not negligible. Thus in a flow, when temperature changes sharply, the relaxation mechanisms may switch over; various VV transitions may both enhance or quench VT relaxation. Peculiarities of VV1−2 rate coefficients are shown in Fig. 5. For this kind of transitions, 10


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10-10

T = 700 K

v =0 1

1

-1

2 3

1

10-12

f

10-13

2 - k , CO

2

f

3-k ,O

3

4

a

10-9

1 - k , CO f

2 - k , CO

12

16 v

20

24

28

2

f

3-k ,O f

32

0

2

4-k ,O

b

f

8

3 1

4-k ,O 4

2

2

f

0

10-8

2

1 - k , CO

10-14

v =0

3 -1

10-11

3 -1

2


-1


3

4

T = 4555 K

v =0

v =0

f

4

8

12

16 v

2

20

24

28

32

2

Figure 3: VT2 rate coefficients (forward transitions) versus v2 for v1 = v3 = 0 for different collision partners; T = 700, 4555 K.

T = 700 K

T = 4555 K

v =0

v =0

1 3

10

1

10-13

-1

10-12

3 -1

-1 3 -1

1

2

v =0

-11


10 rate coefficients (cm s part )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

10-14 4 1 - VT k

10-15

2

2 - VV 10-16 10

3

3 - VV 4 - VV

-17

5 - VV 10-18 0

5

10 v

1-2 2-3

f

k k

1-2-3 2-CO

f

v =0

-9

1

3

10-10

2 5

10-11 10-12 4 10-13 3

10-14

1 - VT k

f

2

k

10

f

2 - VV

-15

k

3 - VV

f

10-16

15

0

1-2 2-3

5

k k

5 - VV

f

1-2-3 2-CO

k

f

k

f

f

10 v

2

4 - VV

f

15

2

Figure 4: Rate coefficients of VT2 and inter-mode VV transitions versus v2 . Quantum numbers of the modes not participating in the exchange are zero. T = 700, 4555 K.

11



the rates of forward and backward processes are very close; the reason is that the transitions are nearly resonant and therefore may easily occur in both directions. The effect of symmetric mode excitation (see curves with v1 = 8) is important and causes noticeable increase in the VV1−2 rate coefficients. On the contrary, the rate coefficients are practically insensitive to the asymmetric mode excitation. Rate coefficients of VV2−3 and VV1−2−3 transitions are more sensitive to excitation of other modes (see Fig. 6), and the ratio kf /kb of forward and backward process rate coefficients is greater. This ratio however decreases with temperature. The effect of collision partner on the rates of VV2−3 and VV1−2−3 transitions is rather weak. Concerning VV1−2 transitions, their rates are slightly lower when the molecule, instead of another CO2 molecule, collides with a diatomic species and significantly lower (in some cases at T = 4555 K more than one order of magnitude) when it collides with an oxygen atom. T = 700 K

T = 4555 K

10-8

1 -1

2'

3 -1

3 -1

1'


-1


2 10-11 3 10-12

3'

-13

1

3

1-2

1 - v =0, v =0, k

10-14

f

VV

10

1-2

VV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2 - v =8, v =0, k 1

0

f

1

10-9

1'

2 10

-10

2' 3

10-11

3' 10-12 1 - v =0, v =0, k 1

10-13

1

4

8

12

16

20

24

3

f

28

32

3

f

2 - v =8, v =0, k 1

3 - v =0, v =5, k

a

10-15

3

b

3

f

3 - v =0, v =5, k 1

10-14 0

v

4

8

12

16

20

24

3

28

f

32

v

2

2

Figure 5: Forward and backward VV1−2 rate coefficients versus v2 ; T = 700, 4555 K. Curves 1,1’: v1 = v3 = 0; curves 2, 2’: v1 = 8, v3 = 0; curves 3, 3’: v1 = 0, v3 = 5. Curves 1–3 correspond to forward processes, 1’–3’ to the backward ones. Let us now discuss the VV transitions involving CO molecules. In Fig. 7, the rate coefficients of VV2−CO , VV1−2−CO , and VV3−CO exchanges are given as functions of v2 or v3 . For low temperature and low vibrational states, VV3−CO process is dominating. At T = 4555 K, the VV3−CO rate coefficient becomes comparable with that of VV2−CO transition 12


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T = 700 K

T = 700 K

2 - v =8, v =0, k

f

2

1

-1

1

2

f

3 - v =0, v =16, k 1

2

f

3 -1

-13

3 10

1 - v =0, v =0, k

2 - v =8, v =0, k


10

1 - v =0, v =0, k 1

3'

-14

2'

10-15

2

1

-17

3' 10

-13

1

2

f

2

f

3 - v =0, v =16, k

3

1

2

f

2' 2 10-14 1' 1

10-15

VV

10

1-2-3

1'

10-16

2-3

3 -1

-1


a

VV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10-18

b

10-16 0

2

4

6

8

0

v

2

4

6

8

v

3

3

Figure 6: Forward and backward VV2−3 (a) and VV1−2−3 (b) rate coefficients versus v3 ; T = 700 K. Curves 1,1’: v1 = v3 = 0; curves 2, 2’: v1 = 8, v3 = 0; curves 3, 3’: v1 = 0, v3 = 5. Curves 1–3 correspond to forward processes, 1’–3’ to the backward ones. around v2 =v3 =5 and is lower than the VV2−CO rate coefficient at higher v3 . With rising T and v2 , the contribution of VV2−CO transitions to CO2 relaxation is of importance. As is shown below, this process affects the rate of VT2 relaxation removing some energy from the bending mode. The effect of excited states of different CO2 modes on the rates of VV2−CO , VV1−2−CO , and VV3−CO transitions is shown in Figs. 8 and 9. While excitation of both stretching vibrations weakly affects the rate of VV2−CO exchange, excitation of the symmetric mode yields considerable increase in the rate coefficients of VV1−2−CO transitions. At 700 K, both the VV2−CO and the VV1−2−CO backward transition rates are higher than the forward ones when the colliding CO molecule is in its ground vibrational level, however for increasing CO vibrational energy their behavior is the opposite (see Fig. 9.b). At T = 700 K, the VV3−CO forward rates are generally higher than the backward ones, except the case of low CO vibrational levels and excited symmetric stretching and bending modes. This feature holds at T = 4555 K too, even if in this case the difference between the forward and the backward rates is small.

13



v

v

3

-1

10-12 1 10-13 1 - VV

k

1-2-CO

10-14

k

3 - VV

k

3-CO

15 v

20

25

10-9

3

10-10

1

1 - VV

f

2 - VV

2-CO

10

2

3 -1

2

5

w=0

10-8


3

0

3

T = 4555 K

10-11

3 -1

-1


10-7

w=0

T = 700 K

10-11

f

2 - VV 3 - VV

f

30

0

5

10

15 v

2

20

k

1-2-CO 2-CO 3-CO

f

k

f

k

f

25

30

2

Figure 7: Rate coefficients of VV transitions involving CO molecules versus v2 and v3 (for VV3−CO ). Quantum numbers of the modes not participating in the exchange are zero. T = 700, 4555 K.

a -1


1' 10-11

10-12

10

-13

10

-14

3 -1

1 2' 3 2

3'

2-CO

1

3

1-2-CO

1 - v =0, v =0, k

f

2 - v =8, v =0, k 1

f

3 - v =0, v =5, k

w=0

0

3

1

5

10

15

20

3

25

2'

10-12

2

1' 3' 3 1

10-13

1 - v =0, v =0, k

10-14

1 1

f

0

v

3

f f

3 - v =0, v =5, k

w=0

30

3

2 - v =8, v =0, k

VV

3 -1

b

T = 700 K

-1


T = 700 K

VV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 36

1

5

10

15

20

3

25

f

30

v

2

2

Figure 8: Forward and backward rate coefficients of VV2−CO (a) and VV1−2−CO (b) transitions versus v2 , w = 0; T = 700 K. Curves 1,1’: v1 = v3 = 0; curves 2, 2’: v1 = 8, v3 = 0; curves 3, 3’: v1 = 0, v3 = 5. Curves 1–3 correspond to forward processes, 1’–3’ to the backward ones.

14


Page 15 of 36

10-11

2

-1

1

2' 3

1'

3-CO

10-12

1

2

f

10-12

f

v =0 1

1

2

4

6

2

8

v =0

1-k

v =0

1' - k

2

3 - v =0, v =16, k

w=0

0

2

1'

VV

2 - v =8, v =0, k 1

10

-11

3-CO

1 - v =0, v =0, k

10-13

1

3 -1

3 -1

3'

b

T = 700 K


-1


10-10

a

T = 700 K

VV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


f

b

3

10-13

10

0

v

10

f

20

30

40

50

60

70

w

3

Figure 9: Forward and backward VV3−CO rate coefficients (a) versus the asymmetric stretching mode quantum number, v3 , at the ground vibrational level, w = 0, of CO molecule; curves with prime correspond to backward processes. (b) versus the CO vibrational level, w. T = 700 K. Finally, the state-resolved dissociation and recombination rate coefficients are given in Fig. 10 for low and high temperatures. Under low temperature conditions, the dissociation rate coefficients increase sharply with the vibrational state of each CO2 vibrational mode. It is worth mentioning that in this case there is a range of triplets (v1 , v2 , v3 ) for which the dissociation rates are lower then the corresponding recombination ones. For high temperature, as expected, the recombination rates are always many orders of magnitude lower than the dissociation ones. Another observation is that the dissociation and recombination rate coefficients are up to one order of magnitude lower when the collisional partner is a CO molecule (compared to the case of CO2 molecule as a collision partner). To summarize this section, we can emphasize that the most probable processes in the CO2 vibrational kinetics are VT2 , VV1−2 , VV3−CO , VV2−CO . One can expect that they give the main contribution to the variation of macroscopic flow variables in non-equilibrium flows. However, before we can recommend to neglect other transitions in real gas flow simulations, the following effects have to be assessed. First, the rates of different vibrational energy transitions and dissociation are strongly coupled; excitation of the modes not directly

15



T = 700 K T = 4555 K

a

10-19

dissociation (cm3s-1part-1) and recombination (cm6s-1part-2) rate coefficients

dissociation (cm3s-1part-1) and recombination (cm6s-1part-2) rate coefficients

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 36

10-21 10-23 1

10-25

3 2

10-27 10-29

3'

1'

2'

10-31 10-33 1 - v =0, v =0, k

10-35

1

3

d

2 - v =8, v =0, k 1

10-37

3

d

3 - v =0, v =5, k 1

10-39 0

4

8

12

16

20

24

3

28

d

10-13

3 2

10-17

1

10-21 1 - v =0, v =0, k 1

10-25

1 1

3 3

d d

-33

10-37 10-41

1'

2'

3'

b

10-45

32

d

3 - v =0, v =5, k

10-29 10

3

2 - v =8, v =0, k

0

4

8

v

12

16

20

24

28

32

v

2

2

Figure 10: Dissociation and recombination rate coefficients versus v2 for different v1 , v3 ; T = 700, 4555 K. Curves 1,1’: v1 = v3 = 0; curves 2, 2’: v1 = 8, v3 = 0; curves 3, 3’: v1 = 0, v3 = 5. Curves 1–3 correspond to dissociation processes, 1’–3’ to the recombination ones. involved to the transitions may considerably affect the rate coefficients. Moreover, different relaxation channels are dominating under low temperature and high temperature conditions. All these effects are evaluated in the next sections for the non-equilibrium boundary layer flow where the gas temperature varies from the hot external edge towards the cold surface.

Boundary layer equations To investigate vibrational-chemical kinetics and heat transfer in the CO2 /CO/O2 /O/C mixture, we have chosen to study a hypersonic boundary layer problem. The set of governing equations is written in self similar coordinates, along the normal to the surface, η: 10 ∂cv ∂ 2 cv + f Sc = Sv , 2 ∂η ∂η

v = 1, ..., N

∂ 2θ ∂θ + f Pr = ST . ∂η 2 ∂η

16


(11) (12)

Page 17 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The unknowns are the mass fractions of each species, cv = ρv /ρ, and the normalized temperature, θ = T /Te . The N species are each vibrational state of the considered molecules and each atom. Sc and Pr are the Schmidt and Prandtl numbers, respectively; f is the stream function. The selected boundary conditions are the surface and edge temperatures, Tw = 700 K and Te = 4555 K, the pressure, pe = 1.269 · 105 Pa, the inverse of the residence time in the boundary layer, β = 1.283 · 105 s−1 , the edge composition CO2 % = 22.6662, CO% = 41.5502, O2 % = 5.6685, O% = 30.115, C% = 10−4 . This test case (except Tw ) corresponds to the experiment of Hollis 30,31,38 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. The surface is assumed non-catalytic according to the conditions of experiment. It is worth noting that studying a partially catalytic surface is of particular interest for aerospace applications. However for this purpose we need information on the vibrational state-resolved surface recombination coefficients. To the best of our knowledge such data for the CO2 surface recombination are not available yet. The source terms Sv , ST in the right hand size of the equations contain the rates of chemical reactions and vibrational energy transitions listed in the previous sections. More details on the equations and source terms are provided in. 10 For the heat transfer prediction, we apply the technique proposed in. 10 In the framework of the modified Chapman–Enskog theory, 19,20,39 from the molecular vibrational distributions and the atomic densities, the transport properties and in particular the heat flux are calculated: q = −λ∇T − p

X cv

DTcv dcv +

X

ncv Vcv

cv

5 cv c kT + rot +εv + εc , 2

(13)

λ is the partial thermal conductivity coefficient specified by the translational and rotational degrees of freedom, DT cv , dcv are the thermal diffusion coefficient and diffusive driving force of chemical species c (c = CO2 , O2 , CO, C, O) at the vibrational state v, ncv , εcv are the number density and vibrational energy of species c in the corresponding vibrational state v, k is the Boltzmann constant, rot is the rotational energy averaged with the 17



Page 18 of 36

Boltzmann equilibrium distribution, εc is the formation energy of species c, Vcv are the diffusion velocities for each vibrational state. The state-resolved transport coefficients are calculated using the algorithm developed in 20 on the basis of the fluid-dynamic variables obtained along the stagnation line while solving Eqs. (11)–(12). The heat flux is calculated using the post-processing technique proposed in. 40,41 In the heat flux formula (13) four different terms can be distinguished, 39 i.e. the Fourier flux due to heat conduction, qF , the flux due to thermal diffusion, qT D , the flux due to mass diffusion, qM D , and the flux due to diffusion of vibrational energy transferred by excited molecules, qDV E : q = qF + qM D + qT D + qDV E .

(14)

These contributions are studied in the next section for various models.

Results and discussion Equations (11)–(12) were solved numerically using an iterative finite difference method (see 10 for the details). More than 20 test cases were studied by including to the kinetic scheme various collisional processes, one by one and in different combinations. The results are discussed below.

Vibrational distributions and vibrational energy An important macroscopic flow characteristic is its average vibrational energy. For CO2 , the CO2 average vibrational energy Evibr is calculated using the formula

CO2 Evibr =

X ρv v v 1 2 3 Ev1 v2 v3 , ρ v ,v ,v 1

2

3

18


(15)

Page 19 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where ρv1 v2 v3 /ρ is the mass fraction of CO2 molecules at the vibrational state (v1 , v2 , v3 ), Ev1 v2 v3 is the energy of the corresponding state. It is clear that the vibrational energy is specified by non-equilibrium vibrational distributions which are formed as a result of collisional processes. Let us consider first the vibrational distributions in the bending mode at different fixed quantum numbers of the remaining modes, v1 and v3 . In Figs. 11, 12, corresponding distributions are given at η = 0 (on the surface). One can notice that inter-mode vibrational energy exchanges weakly affect the surface distributions, see Fig. 11(a). Contribution of other processes is seen in Fig. 11(b). For v1 = v3 = 0, dissociation and recombination do not modify populations of low levels in the bending mode (with respect to the pure inter-mode VVk−m transitions); however chemical reactions lead to some increase in the high state populations. This is due to the dominant role of recombination near the surface. While VT2 transitions provide the main channel for the formation of surface distributions in the absence of chemical reactions, various inter-mode exchanges may enhance or inhibit the relaxation process; there is a competition between different transitions (see discussion below). When different kinds of transitions (VT2 and VVk−m ) and dissociation-recombination (D/R) are simultaneously included to the kinetic scheme, the distribution shape is far from the Boltzmann one, and has a pronounced plateau part in the middle and high levels. When the symmetric and asymmetric modes are excited (Fig. 12), the inter-mode transitions still weakly affect the distribution. On the other hand, when other processes are included, the distributions become completely different. Chemical reactions, if considered alone, strongly pump the vibrational state populations. When all kinds of vibrational transitions are included in the absence of dissociation-recombination, the populations are much lower compared to other cases. However, a competition between chemical reactions and vibrational energy transitions results in intermediate values of the bending mode populations, similar to the case when only inter-mode exchanges are taken into account. The role of dissociation and recombination in the formation of strongly non-equilibrium

19



1 - VV

a

2 - VV 3 - VV 4 - VV 5 - VV

1-2-3 2-CO 3-CO

-3

3 2 5

4 1

3

10-3

2

10-4 1 10

4

-5

1

2

2 - VV

10 0

5

10

15

20

25

30

4 - VT , VV

except VV

5 - VT , VV

except VV

2

k-m

2

-8

35

1-2

3 - D/R

10-7

3

5

1 - VT

10-6

v =v =0

10-4

3

10-2

2

10

v =v =0 1

2-3

2

wall CO vibrational distribution

5

10-1 1-2


10-2

0

5

k-m

10

3-CO 3-CO

15

v

20

, D/R

b

25

30

35

v

2

2

Figure 11: Surface vibrational distributions in the bending mode at v1 = v3 = 0. (a) All VVm−k exchanges; (b) Different schemes.

a

1

4 5

2

1

2

1 - VV 2 - VV 3 - VV 4 - VV 5 - VV 10-6 1

2

3

4

5

v =v =5 1

3

10-4 1 10

3

2

-5

5 4

10-6

2

-5

0

b

3

1 3 10

10-3

v =v =5


10-4


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 36

1-2 2-3 1-2-3

1 - VT

2

2 - VV

10-7

2-CO 3-CO

4 - VT , VV

except VV

5 - VT , VV

except VV

10-8 6

1-2

3 - D/R

0

v

1

2

k-m

2

k-m

2

3-CO 3-CO

3

4

, D/R 5

6

v

2

2

Figure 12: Surface vibrational distributions in the bending mode at v1 = v3 = 10. (a) All VVm−k exchanges; (b) Different schemes.

20


Page 21 of 36

distributions is seen in Figs. 13, 14 where the CO2 surface distributions are plotted as functions of vibrational energy. In triatomic molecules, the vibrational distribution as a function of vibrational energy takes the form of a “set of points ” and not a simple curve, as it happens to diatomic molecules; this feature is due to the non-biunique correspondence between the vibrational states of the molecule and the vibrational energies; indeed comparable energies can correspond to completely different vibrational quantum numbers terns (v1 ,v2 ,v3 ). 10 When only VT transitions are included to the kinetic scheme, the distribution shape is close to that of the Boltzmann distribution with highly populated low states and monotonic decrease in populations towards the high states. Dissociation and recombination results in considerable increase in the high state populations and, in the case of coupled VT2 transitions and D/R, to depletion of the low states. From this analysis we can conclude that the main role in the formation of surface distributions belongs to the recombination process whereas the dissociation noticeably contributes to the kinetics only close to the high-temperature boundary layer edge. It is worth noting that the VT3 transitions and recombination are practically uncoupled (see Fig. 14(b)). Contrarily, the VT2 exchange is strongly coupled to D/R processes; as a result of this interplay, complex vibrational distributions with a clearly marked plateau part are established at the surface. 10-1

2

VT + D/R

-2

2

10-3

10-4

10-5

10-6 a

10-7 0

VT + D/R

2

2

2

10

VT


10-1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

D/R

-2

10-3

10-4

10-5

10-6 b

10-7 500

1000

1500

2000

2500

3000

0

CO2 vibrational energy (meV)

500

1000

1500

2000

2500

3000


Figure 13: Surface vibrational distributions as functions of energy. (a) VT2 , VT2 +D/R; (b) D/R, D/R+VT2 . 21



10-1

3

VT + D/R

10-2

3

10-3

10-4

10-5

10-6 a

10-7 0

VT + D/R

3

2

2

VT


10-1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

D/R

10-2

10-3

10-4

10-5

10-6 b

10-7 500

1000

1500

2000

Page 22 of 36

2500

3000

0


500

1000

1500

2000

2500

3000


Figure 14: Surface vibrational distributions as functions of energy. (a) VT3 , VT3 +D/R; (b) D/R, D/R+VT3 . The average vibrational energy for various kinetic schemes is presented in Figs. 15–16. There are some interesting features in the dependence of vibrational energy on the kinetic scheme. VV exchanges (both intra- and inter-mode), while accounted separately, practically do not affect the average vibrational energy. These processes are resonant or near-resonant and therefore there is no vibrational energy gain or loss during these collisions; vibrational energy remains isolated and does not change. VT2 exchange is the main channel of vibrational relaxation. Deactivation (forward) process is more efficient than excitation (backward) at low temperatures (see Fig. 2). For low temperatures kf > kb ; for high temperatures close to the boundary layer external edge, the rates of forward and backward transitions are comparable. Due to the efficient deactivation near the surface, the majority of CO2 molecules occupy low energy states (see Fig. 13); this leads to a decrease in the total CO2 vibrational energy, see Fig. 15(a). When inter-mode exchanges are included, this causes vibrational energy redistribution between CO2 modes and, in general, enhancement of VT2 relaxation. Since VT1 and VT3 processes are slow, vibrational energy is transferred to the translational one mainly through the bending mode deactivation (Fig. 15(b)). The most efficient VV transition enhancing VT2 relaxation is the near resonant VV1−2 exchange since it has the

22


Page 23 of 36

highest rate among inter-mode transitions. This effect can be seen from Fig. 15(a) where the vibrational energy is given for the kinetic schemes including successively VT2 , VV1−2 and both VT2 and VV1−2 transitions. 6 102

1.2 103

a

2

1 2

2

5 102 4.5 102 1

4 102 3.5 102

3 1 - VT

1 103

2 - VV

5

2

0

1

2

3

4

5

6

3

4 - D/R 5 - VT + D/R

7

1

6 - VT + D/R

8 102

2

7 - VT + D/R 3

6 6 102

3 1

4 10

1-2

3 - VT , VV 2

3 - VT

4

2

3 102 2.5 10

1 - VT 2 - VT


5.5 10 CO2 vibrational energy (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

2

b

1-2

7

0

η

1

2

3

4

5

6

7

η

Figure 15: Average vibrational energy as a function of η. (a) VT2 , VV1−2 , VT2 +VV1−2 ; (b) VTm , D/R processes are taken into account. When other inter-mode exchanges are included to the kinetic scheme, one can notice a competitive effect of various VV transitions (Fig. 16(a)). By adding, one by one, various inter-mode exchanges to the V T2 transitions, it was verified that while VV1−2 , VV2−3 , VV1−2−3 , VV3−CO exchanges enhance VT2 relaxation, VV2−CO , VV1−2−CO transitions reduce its rate and thus cause a slight increase in the vibrational energy. The reason for the latter effect is that the backward process (excitation of CO2 and deactivation of CO molecules) is more efficient under considered conditions, which can be seen from Figs. 8, 9; this increases population of CO2 middle and high energy states and, consequently, the vibrational energy. The overall effect of VV2−CO , VV1−2−CO collisions is however weak, and the main coupling effect is due to VV1−2 transitions. Including chemical reactions to the kinetic scheme yields considerable increase in the vibrational energy at the surface (see Fig. 15(b), Fig. 16(a)) due to formation of CO2 molecules as a result of recombination. There is strong coupling between chemical reactions and VT 23



6 102

1 - VT

b

2

1 103

3 - VV

4

2-CO


2 - VV CO2 vibrational energy (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 36

3-CO

4 - D/R

5 - VT , VV 2

k-m

except VV

3-CO

6 - full model

6 10

2

2 3 1 6 5

2 10

0

1

2

3

4

5

6

3

5 102

1

3' 2

1' 4.5 102

4' 4

4 102

2 3.5 10

2

3 10

2

2' 4

4'

1 - VT

1

2 - VT

2

2'

3 - VT

3

4 - full scheme except VV

a

2

5.5 102

7

3-CO

0

η

1

2

3

4

5

6

7

η

Figure 16: Average vibrational energy as a function of η. (a) Different schemes. (b) Effect of collision partners. Curves 1–4 correspond to CO2 as a collisional partner, curves 1’–4’ to all possible partners. transitions. Once again, the most important role belongs to VT2 exchanges; the effect of VT1 relaxation is rather weak whereas the contribution of VT3 transitions is negligible, indeed curve 4, corresponding to Dissociation/Recombination reactions, and curve 7, corresponding to the combined effect of VT3 processes and chemical reactions, coincide. Near the wall, the main chemical process is recombination; it tends to increase the vibrational energy. On the other side, VT relaxation tends to decrease the vibrational energy, and thus a competition between these processes occurs. Taking into account different collisional partners in VTm and VVk−m transitions (1)–(6) improves the accuracy of the predicted vibrational energy values (see Fig. 16(b)), however, the corrections are small. Thus the error in the CO2 vibrational energy calculation is about 5.4% for the VT1 process, less then 4.6% for VT2 and about 2.4% for VT3 . When the full kinetic scheme, except VV3−CO exchanges, is considered, the error in the CO2 vibrational energy is about 14%. This variable is however the most sensitive to including different partners to the model. For the same test case, the CO2 mass fraction and the heat flux are almost insensitive to the considered collisional partners. Finally, let us discuss briefly the effect of including the VV3−CO transitions to the kinetic 24


Page 25 of 36

scheme. This process has never been studied before in the frame of the hypersonic boundary layer problem. In Fig. 17, the surface vibrational distribution and the average vibrational energy are plotted for the full kinetic scheme and neglecting VV3−CO transitions. The influence of this energy exchange on the vibrational distribution at the middle and high states is clearly seen; neglecting it results in some spreading of vibrational populations whereas including VV3−CO transitions yields highly condensed distributions on the middle states. Moreover, taking into account this exchange leads to the considerable decrease in the average vibrational energy at the surface. Thus the role of VV3−CO transitions in the vibrational kinetics is important. However, as discussed later, its impact on the fluid dynamics and heat transfer is weak. 6 102 10

3-CO


2

b


-1

2 - full scheme


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10-2 10-3 10-4 10-5

5.5 102

5 102

4.5 102

4 102

1

3.5 102

10-6


2 10

3-CO

a

-7

0

500

1000

1500

2000

2500

3 10

3000

2 - full scheme

2

0


1

2

3

4

5

6

7

η

Figure 17: Surface vibrational distributions as functions of energy (a); Average vibrational energy as a function of η (b). Full scheme and the scheme neglecting VV3−CO exchange.

Fluid-dynamic variables and heat flux In this section, we discuss the effect of different kinetic processes included to the kinetic scheme on the fluid dynamics and heat fluxes. The gas temperature calculated along the stagnation line for various schemes is given in Fig. 18. It can be noticed that near the hot boundary layer edge, the temperature distribution is governed mainly by dissociation 25



reaction; including other processes practically does not affect the temperature. Neglecting chemical reactions yields lower boundary layer thickness, at η ≈ 5, the temperature is equal to Te . Near the surface, there is some difference in the temperature calculated neglecting inter-mode vibrational energy transitions, however the overall effect is weak. 5 103

4 103

1

2 3

T (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 36

3 103

2 103

2 3 1 - VT

1 103

2

2 - D/R 3 - full model

0

1

2

3

4

5

6

7

η

Figure 18: Gas temperature as a function of η. The mass fraction of CO2 molecules along the stagnation line is presented in Fig. 19. The mutual effect of vibrational relaxation and dissociation-recombination reactions appears, of course, in the mixture chemical composition, too. We would stress, as an example, that coupling of VT2 relaxation to dissociation-recombination reactions results in decreasing the CO2 mass fraction close to the surface of about 5%, whereas coupling of only VV12 exchange to D/R does not change the CO2 mass fraction near the wall. Coupling of all vibrational exchanges to dissociation-recombination reactions results in the overall decrease in the surface CO2 mass fraction up to 10%; this effect is associated with enhancing the VT relaxation by VV transitions. Including all possible collisional partners in vibrational energy transitions (1)–(6) practically does not affect the mixture chemical composition. The heat flux calculated using expression (13) as well as different contributions to the heat flux (14) are presented in Fig. 20. First of all, we emphasize that the surface Fourier flux is in a good agreement with the stagnation point heating of 8MW/m2 measured in 30,31 26


Page 27 of 36

6.5 10-1

1 - VT

2

6 10

2 - D/R 3 - full model

-1

2

5 10-1

3

CO2

/ρ

5.5 10-1

ρ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.5 10

-1

4 10-1 1

3.5 10-1 3 10-1 0

1

2

3

4

5

6

7

η

Figure 19: CO2 mass fraction as a function of η. (see the black point with its error bar in Fig. 20 (b)). If no chemical reactions are included to the kinetic scheme, the maximum heat flux is achieved when all vibrational energy transitions are taken into account. The heat flux is not sensitive to VV transitions if they are considered separately (except some effect on qDV E , but it is small in the present case). On the other hand, VT2 relaxation gives a noticeable contribution to the heat transfer, and its effect is considerably enhanced by intramode VV transitions. Mass diffusion is absent on the surface in the present case since the wall is assumed non-catalytic. Thermal diffusion and heat conduction give almost equal contributions to the total energy flux. Therefore, for a non-catalytic surface, neglecting thermal diffusion may result in a significantly under-predicted heat flux. This result is in line with reported earlier for air, nitrogen and CO2 flows. 11,40,41 Thus, if no chemistry is included, the main effect on the heat flux is due to fluid dynamics causing the temperature gradient in a flow, and not due to kinetics. Including chemical reactions yields increase in the surface heat flux, even for the noncatalytic wall. Coupling VT2 relaxation and chemistry leads to a further increase in the surface heat flux (η = 0) but decrease in the heat flux in the flow at 0.3 < η < 1.5. The contribution of thermal diffusion decreases but still remains important (qF and qT D are of 27



1.4 107 1.3 107

2

1 107

2 2

1 107

3

9 106

3

8 106

1

4

8 106 0

0.2

0.4

0.6

0.8

1

F

4

6

1

9 106

6 106

7 106

4 106

2

1 3 4

2 106

2

2

1

4 106 0

0.1

0.2

0.3

0.4

0.5

4 - full model

b

a 0 0

1

2

3

4

5

6

7

0

1

2

3

η 6 106

3

6 106 5 106

2

2 - D/R 3 - VT , D/R

2 106

4

8 106

1 - VT

4 106

0

4 - full model Hollis' experimental value

1.2 107 1.1 107

6 10

2

8 106

3

q

-1

2 - D/R 3 - VT , D/R

4

-2

1 10

7

2

1.5 10

4

-1

1.2 10

7

1 - VT

7

[J s m ]

1.4 10

7

1 107

1.6 107

-2

Heat Flux [J s m ]

1.6 107

5 106

4

4

5

6

7

η

6 106

c

1 - VT

d

5 106

2

2 - D/R 3 - VT , D/R

4

4 106

1.3 106

2

4

4 - full model

6

4

4 106

-2

9.7 105

3

-1

-1

3

1 106

3 106

0 0.2

0.4

0.6

0.8

1

DVE

0

3

6.4 105

q

TD

[J s m ]

2

1

2 106

1

2 106

-2

[J s m ]

3 10

q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 36

1 - VT

2

2

2 - D/R 3 - VT , D/R

4

1 106

3.2 105

1

2

4 - full model

2 0

0 0

1

2

3

4

5

6

7

0

1

2

η

3

4 η

Figure 20: Heat fluxes as functions of η.

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5

6

7

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the same order of magnitude). The heat flux due to the mass diffusion is zero on the wall due to the non-catalytic boundary conditions; however in a flow qM D is of the same order as qF and has the maximum at η ≈ 0.1. The heat flux connected to the diffusion of vibrational energy is still small. Intra- and inter-mode vibrational energy exchanges do not affect the heat flux (except qDV E which is small).

Conclusions Coupling of vibrational relaxation and dissociation in CO2 is studied in a hypersonic boundary layer using the state-to-state approach. Both analysis of the rate coefficients and fluiddynamic problem simulations show the negligible role of VT1 and VT3 transitions in stretching vibrational modes in the kinetics, dynamics and heat transfer. The rates of VT2 and inter-mode exchanges are of the same order, and their contribution to fluid dynamics depends on the temperature. The main relaxation channel is the VT2 exchange in the bending mode. The inter-mode VVk−m exchanges, if considered alone, do not affect vibrational distributions and fluid-dynamic variables. However, if coupled to VT2 transitions, they may play an important role in the kinetics, enhancing or inhibiting VT relaxation. Competitive effects of various processes are also emphasized. Under considered conditions, the interplay of VT, VVk−m transfers and recombination is crucial for the formation of non-equilibrium surface distributions. The key processes are VT2 , VV1−2 , VV3−CO , and D/R. The contribution of other VV transitions is rather weak although non-zero. The vibrational distributions in the bending mode are strongly affected by excitation of stretching (symmetric and asymmetric) vibrations; this confirms once again strong coupling of various kinds of vibrations in CO2 . The average vibrational energy depends essentially on the kinetic scheme, especially on the surface; this variable can be recommended for accurate sensitivity analysis of fluid dynamics to the kinetic scheme. The temperature and heat flux on the non-catalytic surface are less sensitive to vibrational kinetics and are mainly governed by fluid dynamics and

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chemical reactions. However we expect much more important effects for a partially catalytic surface. In the future work we plan to assess simplified models of vibrational kinetics assuming uncoupled vibrational modes; to understand the mechanisms of state-resolved recombination under different conditions, and to develop new, more realistic multi-temperature models.

Acknowledgement This study is supported by Saint Petersburg University, Project 6.37.206.2016 and Russian Foundation for Basic Research, grant 18-08-00707.

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Heat Transfer in O2 /O Mixtures Near Catalytic Surfaces. J. Thermophys. Heat Transfer 2002, 16, 238–244.

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Graphical TOC Entry state-to-state vibrational kinetics hypersonic boundary layer

some processes rate coefficients

s urfac e, 700 K

36

edge, 4555 K


Page 36 of 36

Mechanisms of Coupled Vibrational Relaxation and Dissociation in

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