Vibration-Dissociation Coupling
in Nonequilibrium CO^j'N% Mixtures
Alexander V. Eremin*, Ekaterina A. Nagnibeda^, Elena V.Kustova^, Valeria V. Shumova*
* Institute for High Energy Densities Izhorskaya si. 13/19 127412 Moscow, Russia
t Math, and Mech. Dept, St.Petersburg Uni.} Bibliotechaya pi. 2, 198904 St.Petersburg, Russia
Abstract. The vibrat ion- dis so elation kinetics of GO>2/ N>2 mixture is studied using three different models:
1) state-to-state approach, 2) quasi-stationary and 3)step-ladder approximations. Numerical simulations are
aimed to describe the results of experimental findings in mixtures 2000ppm GO2 + N2 and 2000ppm GO2 +
10%JV2 + Ar at Teg=2326-2855 K and Peg=0.75-2.59 bar. The yield of O-atoms from the dissociating CO2
is calculated in quasi-stationary and in step-ladder approximations and compared with the experimental one
measured by ARAS technique during ~100/us. The most interesting results are obtained at the early stage
of vibrational intermode exchange in GO2 by means of state-to-state approach. The vibrational exchange
of highly excited states (HES) of CO2 and N2 is studied on the base of step-ladder solution.
INTRODUCTION
Peculiarities of vibration-dissociation coupling in nonequilibrium mixtures oiCO^/N^ are of current interest
due to the importance of these processes in the computations of flight trajectories, as well as in laser chemistry.
The most complicated problem for numerical simulation of chemical reactions in high temperature gas flows
is the description of vibration-dissociation coupling of polyatomic molecules, especially under conditions of
considerable depletion of vibrational distributions, or under multi-mode nonequilibrium conditions. The kinetic
master equations (KME) must be solved in all these cases. Two different approaches to the solution of KME
exist: exact methods based on the state-to-state approach, and approximate ones using different kinds of
quasi-stationary smooth energy distributions.
State-to-state models of vibration-dissociation coupling of di-atomic molecules are developed by many authors, here we follow the model given in the paper [1]. State-to-state vibrational kinetics of pure CO^ behind
shock waves has been considered in [2] and compared with the solutions obtained on the basis of nonequilibrium quasi-stationary distributions for three-atomic molecules derived in [3]. To describe the vibrational
distributions of highly excited states (HES) for dissociating polyatomic molecules the different kinds of diffusion
assumptions are often used, such as the so-called "step-ladder" model, recently developed in [4].
The main goal of this study is to extend the advantages given by the state-to-state method to the description
of vibration-dissociation coupling of tri-atomic molecules like CO^^ and to compare the results of state-to-state
calculations with the step-ladder solution. The results of calculations are verified by the experimental data
of [5], where dissociation of CO2 under multi-mode vibrational nonequilibrium conditions in shock heated
mixtures CO^j' N^/Ar has been investigated.
In order to analyze the results of measurement of the CO^ dissociation rate coefficient under conditions
of vibrational nonequilibrium, the time-behavior of the distinct GO*} vibrational levels behind shock waves in
GO^I' N*}/Ar mixtures should be studied. This problem is considered in the present paper using two approaches.
I
STATE-TO-STATE MODEL
We consider a flow of a gas mixture containing CO^ and N% molecules behind a plane shock wave in the frame
of the Euler approximation. The flow is assumed to be one-dimensional and stationary. In the state-to-state
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
672
approach the gas dynamic conservation equations should be coupled with the master equations for vibrational
level populations and atomic number densities. The equations for populations ni±iii3 of vibrational levels ii,
«2, is of three CO^ modes and populations n^4 of N2 have a form:
d(VTlj
jl j )
v tlt3t3
_.—_
T7T7
_.— _
™T7
W1
V"V'
V"V'
* * *
^--^
«« 1«« 2«« 3
«« 1«« 2«« 3
* 1*
*2*
*3
*
** 1** 2** 3
W1
W'
j-
' = V Rvvf. + V RTVf. . + Rvv*r3 + R ^r 3 + R ?r3 + fl ^r 2 - 3 + R ?r* + E^-rec, (l)
1 **
* **
2*3
* **1 **2 * 3
* * *
'
V
'
= 0 , . . . , L f c , A = 1 , 2 , 3.
7
——
dx
1 A4
*
'
9 A4
*
'
(
4.
9j
*•
}
(2)
Here subscripts "1", "2", "3", "4" correspond to the CO^ symmetric, bending and asymmetric modes and to
N2 respectively, / is the additional quantum number, Lk is the number of excited levels considered in the k-ih
mode. Molecular vibrations are simulated by the anharmonic oscillator model.
The production terms in Eqs. (1), (2) describe the change of ni±iii3j n^ 4 as a result of numerous vibrational
energy transitions: terms R. .tk. (k = 1,2,3,4) describe VV exchanges of vibrational quanta within each
kth mode at the collision of two similar molecules, R^Yf. correspond to TV exchange of translational and
vibrational energy of the kih mode. Among both VV and TV processes only the single-quantum transitions
vv'_
vv'_ _
vv'_
4 3
are taken into account as the more probable ones. The terms R. .t rm, R. .j1."2"3, R^
~ describe non-resonant
VV1 exchange between different vibrational modes. Among VV1 processes we consider the near-resonant
two-quantum VV{_2 exchange between the symmetric and bending modes, three-quantum VV2_3 exchange
between the asymmetric and bending modes, VV{_2_3 exchange between all three vibrational modes, and
VVv_4 exchange between the asymmetric mode of CO^ and N2. The expressions for Rr?1?
.el.t2^.3 in terms of the
level populations and probabilities of vibrational energy transitions are given in [2].
The production terms Rls~rec describe the dissociation-recombination process of CO^ at the collision with
any partner M. Neglecting recombination due to its weak influence on the parameters of a shock heated gas
one can write
r>dis
J t - -j! •
_
\ ^
7 dis(M)
— —77j- j-i j- > n^fK, ,i , ,
/o\
^oj
M
where /?.*?,. is the rate coefficient of dissociation of a molecule CO^ at the vibrational state («i, i2, is) caused
by the collision with the partner M.
The equations for the number densities of O atoms and CO molecules have the the form:
d(vnco) _
The state-to-state rate coefficients of various vibrational energy transitions are calculated using the SSH
theory [7] generalized to take into account anharmonicity of vibrations. The Treanor-Marrone model [8] is
used for the state-to-state dissociation rate coefficients.
II
MODE-LADDER MODEL
In this section the master equations have been solved on the basis of the step-ladder approach, previously
used in [10].
The mode-ladder model is based on the following assumptions: 1) all modes are considered separately as
independent "ladders" with numerous processes of energy transfer between them; 2) the simplified KME,
describing only the dominant energy exchange channels, are solved for the system of lowest vibrational levels
in each mode; 3) higher levels are considered in the step-ladder approximation for each mode with the energy
density of states p ( i k ) analogously to [10].
673
For the simplicity the CO2 molecule is considered as a system with two types of vibrations: 1) symmetric
vibrations which include the symmetric (i/i) and bending (1/2) CO2 modes; 2) asymmetric vibrations corresponding to the CO^ asymmetric mode (^3). All CO2 vibrations are assumed to be harmonic oscillators,
while N<2 molecule is simulated as a Morse oscillator with vibrational frequency 1/4. The following vibrational
levels are considered separately: 3 lower levels of symmetric vibrations possessing the energy corresponding
to the energy of 3 levels of bending vibrations; the 1st level of asymmetric vibrations and the 1st level of N2
vibrations.
The complete description of energy exchange processes and chemical reactions included into the kinetic
scheme one can find in [5]. They are: 1) the conventional relaxation channels for low-lying vibrational levels
of different modes with rate constants /?20j ^22 > &23} &34? 2) step-ladder activation-deactivation of higher vibrational levels with the united step-size AE for each mode m(m = 1,2) and the rate coefficients kmm (m=l,2),
written for the actual numbers i, i — 1 of the corresponding physical vibrational levels in harmonic oscillator
approximation; 3) the energy exchange processes between HES with rates k^\E>E* ~ P' &233
k^\E<E* = 0;
&34 E&EQ ~ /3 - &34, where (3 and (3 are the fitting parameters, EQ is the dissociation threshold, E* w 3 eV
is the threshold of continuum spectrum evaluated in [5]; 4) the spontaneous decay of over-barrier HES is described by the Rice-Ramsperger-Kassel-Markus (RRKM) approximation, following [10]; 5) the decomposition
of under-barrier states is considered in the frame of the Treanor-Marrone assumption ( [8]), similarly to the
model discussed in section II.
Under conditions of experiments performed, GO*} dissociation is assumed to proceed only through the lowest
spin-forbidden channel CO2(X1Y<) -> CO2(3B2) ->• CO + O(3P) according to the results of [10]. The
fast reactions of O-atoms with N2 and secondary bimolecular reactions in the system of O-C-N-atoms are
considered in the conventional form. The activation energy of the reaction O + N2 —> N + NO is reduced by
the mean vibration energy of nitrogen molecules (mode 1/4). The mode-ladder model has been validated first
under vibrationally equilibrium conditions, i.e. k^is in Ar and in N2 has been calculated for the case when
vibrational relaxation of N2 is completed.
Previous studies of the CO2 decay performed in [10] show that this process can be described in the frame of
a "weak collisions" assumption with the mean portion of energy transfered per collision < AE1 >?^ 0.7-rO.S kT.
The best fit with experimental value of Ea/k given in [11] and [12], is attained at < AE >= 0.8 kT. This
value has been used below.
Ill
RESULTS
The peculiarities of nonequilibrium CO2 dissociation have been studied first by means of the state-to-state
approach. Then the mode-ladder approximation has been used. The comparison of results is given below.
First, let us discuss the results obtained in the state-to-state approximation. The nonequilibrium stateto-state distributions of CO2 and number density of O behind a shock wave have been found as a result
of numerical solution of Eqs.(l - 2), (4) coupled with the conservation equations. Two different mixtures:
2000ppm CO2 + N2 and 2000ppm CO2 + 10%^ -\-Ar have been studied at the same free stream conditions:
MQ = 2, TO = 1258K, po = 0.11 Satin. The vibrational distributions in the free stream are assumed to be
equilibrium Boltzmann distributions with the gas temperature TQ. The distributions remain the same just
behind the shock (t = 0) because the vibrational energy transitions in the shock front are frozen. Thermal
equilibrium dissociation rate coefficients have been computed using the results of [13]:
kedfsAr} = 3.8 • 1024T-2'4(1 - exp(-3380/T)) exp(-65930/T), keJ(sNs} = 2keJ(sAr}, *$COa) = Q.5k^Ar\
the parameter U of the Treanor-Marrone model is supposed to be equal D/Qk. Collisions between CO2 and
CO molecules or O atoms causing the CO2 dissociation are neglected because of very low concentrations of
these species.
The number density of O atoms is given in Fig. 1 as a function of t. Solid and dashed curves correspond to
the mixtures CO2/N2/Ar and CO2/N2 respectively. One can notice the lower yield of oxygen as well as the
lower rate of dissociation in the latter case. It can be explained by two reasons. First, the temperature just
behind a shock and the equilibrium temperature Teq are higher in the mixture CO2/N2/Ar. Second, VV^_^
vibrational energy exchange of the CO2 asymmetric mode with cold N2 in the mixture CO2/N2 results in a
significant depopulation of the highly excited states of the third mode and therefore leads to a decrease of CO2
dissociation rate and O atoms yield. This effect is confirmed by the next figures.
674
1.E+00
1.E-01
1-E-021.E-03
1.E-041.E-05 1.E-06
FIGURE 1. Oxygen atom concentration as a function of time. Solid curve - CO2/N2/Ar, dashed curve - CO2/'N2.
Mo = 2, To = 1258K, po = O.llSatm. State-to-state approach.
FIGURE 2. Reduced population of selected levels of CO2 asymmetric mode as a function of time. Solid curves CO2/N2/Ar, dashed curves - CO2/N2. Curves 1,1' - i3 = 1, 2,2' - z 3 = 2, 3,3' - i3 = 3, 4,4' - z 3 = 4, 5,5' - i3 = 5.
Mo = 2, To = 1258K, po = O.llSatm. State-to-state approach.
The reduced populations of selected levels of each CO^ mode are presented in Figs. 2 - 4 as functions of time.
One can observe a non-monotonous behaviour of the populations of asymmetric mode in the very beginning of
the relaxation zone, which can be explained by the exchange of vibrational energy between the z/s mode of CO^
and vibrationally cold N%. The strong influence of the mixture composition on the populations of asymmetric
mode is also seen. The main reason for it is that VV^_ 4 vibrational transitions deplete the highly located
vibrational levels of the third mode. The effect of this exchange on the populations of the other modes is
much less, their behaviour remains monotonous. One can see also a fast establishment of the quasi-stationary
vibrational distributions in the bending mode, this fact is confirmed by experiments.
The vibrational distributions in the asymmetric mode at different time values are given in Fig. 5 versus 33.
A different shape of the vibrational distributions also shows a depletion of HES, which leads to a decrease
of the dissociation rate in the mixture CO^/N^ compared to CO^/N^/Ar. It is interesting to emphasize the
non-Boltzmann behaviour of the vibrational distributions in the third mode, which proves the importance of
the state-to-state solutions within the relaxation zone.
Let us present now the results obtained using the mode-ladder model. This model is based on a number of
additional assumptions and requires less computational time. The implementation of mode-ladder approach
permits to consider a greater energy range of excited states and thus gives an opportunity to get insight
into the mechanism of interaction also between HES. In Fig. 6 the results of mode-ladder calculations of the
675
6.E-04 -,
5.E-04 4.E-04 2
3.E-04 - Lr
2.E-04 -
k
^^1: ''««M..— .
1.E-04- r*x^!--.
O.E+000
9*v
A-t>
,
1
2
3
4
FIGURE 3. Reduced population of selected levels of CO2 bending mode as a function of time. Solid curves CO2/N2/Ar, dashed curves - CO2/N2. Curves 1,1' - i2 = 1, 2,2' - « 2 = 2, 3,3' - i2 = 3, 4,4' - i2 = 4, 5,5' - i2 = 5.
Mo = 2, To = 1258K, PQ = O.llSatm. State-to-state approach.
1.4E-04 -,
1.2E-04 1.0E-048.0E-05 |
6.0E-05 4.0E-05 2.0E-05 -
2
^^ —— ...2L...._ .................
f
n npj.nn
C)
2
4
6
8
10
FIGURE 4. Reduced population of selected levels of CO2 symmetric mode as a function of time. Solid curves CO2/N2/Ar, dashed curves - CO2/N2, Curves 1,1' - ii = 1, 2,2' - ii = 2, 3,3' - ii = 3 . M0 = 2, T0 = 1258K,
po = O.llSatm. State-to-state approach.
dissociation rate coefficient under different assumptions are adduced. One can see that in the case when
dissociation is supposed to proceed from the states of symmetric mode, k^is is higher than the experimental
value given, for instance, in [12,11], by the factor of 100. In the opposite case, if dissociation is supposed to
proceed from the states of asymmetric mode, kdis is by the factor of 10 lower than the experimental one, and
it is impossible to fit it through the variation of the rate coefficients of VV exchange within the second and
the third mode at low levels. This inference proves that the interaction of HES of symmetric and asymmetric
modes is very essential, and this fact could be explained by the high density of Fermi-resonance at energies
E* < E < EQ. This interaction between v<i — vs HES simulates also the roto-vibrational energy transfer at
high energies, which promotes the ladder climbing (that means, activation). The fitting parameter /? in the
rate constant of this process &23* has been varied in the range 1.0 < (3 < 10.0 and the best fit with experiments
is attained at /?=6.0. In Fig. 6 the k^ s value theoretically predicted by the weak collision unimolecular rate
theory according to data given in [14] is also represented.
Under nonequilibrium conditions in CO^/N^ mixtures, when the vibrational temperature of N% is essentially
lower than the translational one, the interaction between vibrations of CO^ and N% has a strong influence on the
populations of CO^ levels, and consequently, on the CO^ dissociation rate. Calculations have shown that the
VV1 exchange between the lowest CO^ and ^2 levels with conventional values of corresponding rate constants
676
1.E-03
i3
FIGURE 5. Reduced populations of asymmetric mode at different time values. Solid curves - CO2/N2/Ar, dashed
curves - CO2/N2. Curves 1,1' - t = 1.04/us, 2,2' - t = 5.03/us, 3,3' - t = 10/us. M0 = 2, T0 = 1258K, p0 = O.llSatm.
State-to-state approach.
3
.-1 -1
mol s
7
10 10 -
5
10 -
10
1B4/T,K4
3:
3.0
3.5
4.0
4,5
5.0
FIGURE 6. Rate constant of CO2 dissociation in Ar given in [11] (line marked by x) and in [12] (line marked by
*) in comparison with the different mode-ladder model approximations: 1 - decomposition from all HES of each CO2
modes is assumed, 2 - the energy exchange between HES is neglected, 3 - complete "mode-ladder" mechanism. Open
points - present data on kdis in vibrationally nonequilibrium conditions related to equilibrium temperature (D - mixture
2000ppm GO2+N>2] A - mixture 2000ppm CO2+10% JV 2 +Ar). Black point - theoretical value of conventional weak
collision unimolecular rate theory given in [14] (kdis = 5.33 - I0~6cm3mol~1s~1 at T=3000K).
summarized in [15] has no sufficient influence on the populations of HES of CO^ compared to the very intensive
VV exchange rates within the CO^ molecule itself. That means that the vibrational exchange between 1^2-^3
HES is very important. Therefore the interaction of higher CO^v^) and N% levels has been taken into account
with the rate constant £34*^*3, ^4), containing the fitting parameter /? , varied in the range 0.01</? <1.0. The
best fit with experiments is attained with /? =0.1. The value of/? reflects that the most effective process of VVexchange between HES of z/3 CO^ and 1/4 N2 is the process (702 (si, «2,
In Fig. 7 the example of simulation of O-atom yield during CO^ dissociation in vibrationally nonequilibrium
N<z is shown in comparison with the experimental data. Besides the complete kinetic scheme of the modeladder model, the results of calculations without VV and/or VV1 exchange of CC^l^s) decomposing HES
are presented. One can see that only the complete mode-ladder kinetic mechanism fits the experimental data
adequately.
677
0,0
100.0
200,0
300,0
400,0
500,0
FIGURE 7. Mode-ladder numerical calculations of O-atom concentration time-dependence in comparison with experimental data. Curve 1 - VV exchange between HES of CCb^) and CCbfVs) is neglected, curve 2 - VV' exchange
between CCb^s) and N2(V) is neglected. Curve 3 - complete kinetic scheme. Mixture 2000ppm CCb + N2, T e g=2754
K, p=2.09 bar.
[0].10"12cm-3
10
10
1Q10- 5 -
10
FIGURE 8. Oxygen atom concentration as a function of time calculated in state-to-state (solid curves) and
mode-ladder (dashed curves) approaches in the mixtures 2000ppm CO^ + 10%NZ + Ar (curves a) and 2000ppm
CO2 + N2 (curves b). M0 = 2, T0 = 1258K, p0 = O.llSatm.
IV
DISCUSSION
A comparison of the results obtained using the state-to-state and mode-ladder models is given in this section.
The O-atoms yield (n0) calculated in the both approaches is presented in Fig. 8. Solid curves correspond to the
state-to-state calculations, dashed curves represent the results obtained using the mode-ladder model (curves
a correspond to CO^/N^/Ar mixture, 6 to CO^/N^ mixture). One can notice the different behaviour of no for
the two models. Thus, in the mixture 2000ppmCfO2+10%Ar2+^ the mode-ladder solution gives more rapid
increase of O concentration compared to the state-to-state solution. This result is similar to the behaviour of
the quasi-stationary and state-to-state solution reported in [9]. In the mixture 2000ppmCfO2+^2 the modeladder solution gives a lower concentration of O. This difference may be explained by two reasons: 1) secondary
reactions of O, especially with N%, neglected in the state-to state solution and 2) the longer relaxation times
for higher levels given by the mode-ladder calculation. The role of secondary reactions in the early stage
of dissociation seems to be less important than the influence of vibrational distributions because of the low
concentration of O atoms.
One of the most significant conclusion of above analysis is that the apparent dissociation rate of carbon
678
dioxide diluted in nitrogen depends considerably on the mixture composition and alters during the course of
the process. Calculations show, that at sufficiently low temperatures the presence of nitrogen can reduce the
rate of CO^ dissociation up to the factor of 100 relatively to vibrationally equilibrium conditions.
CONCLUSIONS
The investigation of CO^ dissociation under conditions of vibrational nonequilibrium based on both state-tostate and mode-ladder approaches gives the insight into the mechanism of vibrational energy transfer between
different vibrational modes and into the process of vibration-dissociation coupling in the mixtures CO^ + N%
+ Ar.
The results obtained using both state-to-state and mode-ladder methods show:
1) Decomposition of GO*} molecules occurs from HES of the asymmetric mode.
2) For the correct prediction of CO^ dissociation rate it is important to take into account the nonequilibrium
non-quasi-stationary vibrational distributions.
3) Vibrational distributions within the 3rd (asymmetric) mode depend strongly on the mixture composition,
while distributions in the symmetric modes are practically independent of it. The efficient energy exchange
between HES of the symmetric and asymmetric modes takes place.
4) The strong energy exchange between HES of the asymmetric mode of CO^ and N% leads to a decrease of
the CO*i dissociation rate in presence of vibrationally cold N% •
It should be kept in mind that the accuracy of both implemented models is limited by the accuracy of the
Treanor-Marrone model for dissociation probability and the models for the rate coefficients for vibrational
transitions.
Finally one can conclude that the state-to-state approach gives more precise vibrational distributions and
dissociation rates but requires more computational time. Therefore one can choose the simulation method
according to the aim of investigation.
ACKNOWLEDGMENTS
Authors are grateful to INTAS (99-00464, 99-00701) for the financial support of this work.
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