State-to-State Theory of Vibrational Kinetics and
Dissociation in Three-Atomic Gases
E.V. Kustova, E.A. Nagnibeda
Department of Mathematics and Mechanics, Saint Petersburg University,
198904} Bibliotechnaya pL 2} Saint Petersburg, Russia
Abstract. The paper deals with the kinetic theory of gas mixtures containing polyatomic molecules with
several vibrational modes. The excitation of the rotational and vibrational degrees of freedom and dissociation are taken into account. The main attention is focused on the modeling of multi-level vibrational
kinetics and vibration-dissociation coupling of linear three-atomic molecules. The model is applied for the
study of shock heated gas flows and some new features of non-equilibrium kinetics of CO?, behind shock
waves are found.
I
INTRODUCTION
The kinetic theory of complex molecules with more than two vibrational modes is much less developed
compared to the theory for diatomic molecules. In particular, difficulties rise due to the existence of many
kinds of vibrational energy transitions within and between different modes. These transitions can produce nonBoltzmann distributions, which influence essentially the transport properties and dissociation rates [1~4]. Up to
now the mechanism of dissociation of vibrationally excited polyatomic molecules is not sufficiently understood.
There exist several non-equilibrium models of GQ<z kinetics based on the two-temperature or multitemperature quasi-stationary vibrational distributions [1,5,6]. However, under strong non-equilibrium conditions when the characteristic rates of all vibrational energy transitions become comparable with the dissociation rate, the quasi-stationary approximation is not valid and the state-to-state approach has to be used. In
the recent years the multi-level kinetic theory approach has been developed for reacting mixtures of diatomic
molecules, and applied for different gas flows. Some important features of this method are shown in [7]. Only
very few results have been obtained for CO^ molecules in the frame of the multi-level approximation [8,2].
The state-to-state vibrational and dissociation kinetics in CO^ flows behind shock waves has been investigated
in [2—4] where a comparison with experimental data and approximate models is given, and the importance of
the rigorous multi-level model in the beginning of the relaxation zone is shown.
In the present paper the multi-level kinetic theory developed previously for diatomic molecules [7] is extended
for mixtures with dissociating linear three-atomic molecules.
II
KINETIC EQUATIONS AND DISTRIBUTION FUNCTIONS
We consider the conditions when relaxation of translational and rotational energy proceeds much faster
compared to vibrational relaxation and chemical reactions:
Ttr < Trot <^ T~vibr < ^react ~ 0.
(1)
Here rt r , r ro t, Tvibr and rreact are the corresponding relaxation times, 9 is the mean time of changing of
macroscopic parameters. In this case the kinetic equations for the distribution functions /c,-j for every chemical
species c, rotational and vibrational energy levels j and i over velocity uc, time and space co-ordinates t and
r can be written in the form:
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
620
vjcij
cij
~~dT~
__ - / jtr
cij
,
' ~~d^~ ~~ ~e ^
rrot\ ,
cij
'
rvibr
cij
,
rreact
cij
(()\
'
^ *
The integral operators in the right hand side describe, correspondingly, the collisions leading to the translational, rotational and vibrational energy transitions and chemical reactions, e is the ratio of characteristic times
of rapid and slow processes (e ^C 1).
Under condition (1) the state-to-state approach has to be used. In this approach, the asymptotic solution of
Eq. (2) in the each approximation is expressed in terms of the vibrational level populations n c ^, macroscopic
velocity v and gas temperature T [7].
In the present paper we consider mixtures containing dissociating three- atomic molecules, diatomic molecules
and atoms. In the ground electronic state a linear three- atomic molecule has three vibrational modes: symmetric (z'i), twice degenerated bending (il2) and asymmetric (23) mode, quantum number / describes the projection
of the momentum of the bending vibrations onto the axis of the molecule. Therefore for three-atomic molecules
the vibrational quantum number i represents a set of quantum numbers corresponding to the different modes:
i = (*1,*2>*3)-
In the zero-order approximation of the generalized Chapman-Enskog method the distribution functions for
molecules take the form:
HciS
m
—
^c
£
~
f\
/ox
where sc-1 is the rotational statistical weight, ec? is the rotational energy, Z£r, Z™ot are the translational and
rotational partition functions, mc is the mass of a particle, cc =u c — v is the peculiar velocity, k is the Boltzmann
constant. For three-atomic molecules i = (ii, il2, is) , im = 0, 1, ..., Lm, m = 1, 2, 3, Lm is the number of excited
vibrational levels in the rath mode. For diatomic molecules i = 0,1,..., LCJ Lc is the number of excited
vibrational levels of the cth diatomic species.
For atoms one has:
\
"
Distribution functions (3) represent the Maxwell-Boltzmann distributions over velocities and rotational energies. The vibrational level populations are found from the equations of detailed vibration-dissociation kinetics
coupled with the macroscopic equations.
Ill
MACROSCOPIC EQUATIONS
In the each approximation of the generalized Chapman-Enskog method the closed set of equations for the
populations ni±iii3 of various vibrational states of three-atomic molecules, for the vibrational level populations
nci of diatomic species, atomic number densities nc, macroscopic velocity and gas temperature follows from
the kinetic equations (2).
For the mixture containing dissociating GO*} molecules (with products of dissociation CO and O), nondissociating N<2 molecules and inert gases these equations in the Euler approximation of non- viscous nonconductive flow take the form:
(5)
lit
dv
?——- -f gradp = 0,
dt
(6)
•>— + pdivv = 0,
(7)
dt
i•
A
l
^
:
CX + Rty^
5
*
,,
ryvibr
?
4
• _ n
i
5
*
*
{
™ = °' ->£ m,
r
*
5
T
:
m = 1, 2, 3,
(8)
(Q\
5
\
~
621
*&•
/
(1Q)
Here p is the density, p is the pressure, U is the total energy per unit mass which in the state-to-state approach
depends on the gas temperature, vibrational level populations of molecular species and atomic number densities.
Subscript "4" corresponds to the vibrational level of N% molecule. The right hand sides of master equations for
level populations (8) and (9) describe the change of population of each vibrational state as a result of different
kinds of vibrational energy exchanges and dissociation. It should be noted that CO molecules appearing as the
result of CO^ dissociation are supposed to be in thermal equilibrium. Also we neglect in this study dissociation
of N%, recombination and the simultaneous transitions of translational, rotational and vibrational energy.
The expressions for R^^i ,-3 and Ri4 contain the state-to-state rate coefficients of vibrational energy transitions
and dissociation. The vibrational energy transitions include TV (VT) transitions between translational and
vibrational degrees of freedom, VV exchange of vibrational quanta within the same mode of colliding molecules,
VV exchanges of vibrational energy between different modes of the same molecule and between different
molecules. It should be noted that the amount of energy exchanges contributing to the production terms of
Eqs. (8), (9) is very big, and it is interesting to estimate the influence of diverse energy transitions on the
macroscopic parameters and dissociation rate.
The numerical solution of Eqs. (5)-(10) has been obtained in the relaxation zone behind a plane shock
wave for the mixtures 2000ppm CO^ + W%N2 + Ar and 2000ppm CO% + N%. The level populations n{liiia
and 7i,-4, number densities of CO and O, gas temperature and velocity have been found in dependence on the
distance from the shock front. It is clear that system (5)-(10) contains a large number of equations for level
populations of different vibrational levels of three CO% modes and of N%. The numerical solution of the stateto-state master equations consumes a lot of computational time, therefore it is important to find a possibility
of reducing of the master equations to the more simple relaxation equations under non-equilibrium conditions.
The reduced models based on the quasi-stationary distributions are considered in the next section.
Another problem appearing at the realization of the state-to-state model is the lack of data on the rate coefficients of different kinds of vibrational energy transitions within and between modes. The diverse experimental
results describe only several vibrational transitions. In the present paper the rate coefficients of vibrational
energy transitions between the lowest vibrational states are computed using the recent data from [9,10], the
remaining rate coefficients are calculated on the basis of the SSH theory [11] modified for polyatomic molecules
and taking into account the anharmonic effects. The rate coefficients of dissociation from each CO% vibrational
level are found using the Treanor-Marrone model [12] extended for three-atomic molecules. Thus the dissociation state-to-state rate coefficient Ar.*?, . ^ can be connected with the thermal equilibrium dissociation rate
coefficient k^s * by the expression:
idiss(M) __ %vibr(T)
(6iiil2i3 /_!_
I li
6XP
' 2«'3 " Zvibr(-U)
V * \T
J_\ \ ieq(M)
diss
UJJ
where Si^i3 is the vibrational energy of CO% molecule (the vibrational spectra are simulated using the anharmonic oscillator model), Zv^r is the equilibrium vibrational partition function, U is the parameter of the
model.
IV
MULTI-TEMPERATURE MODELS
The master equations for vibrational level populations (8), (9) can be simplified and reduced to the less
number of macroscopic parameters using multi-temperature approximations.
1. 5-temperature approach. The 5-temperature approach takes place under the condition of rapid
VV exchange of vibrational quanta within the same vibrational mode (of CO*} and N%):
r
vv < rfyT ~ r$T < rdiss ~0,
&, m = 1,...,4.
(12)
Here TyV, TyyT1 are the characteristic times of vibrational energy exchanges within the same k-th mode
and between different k-th and m-th modes, TyT is the mean time of vibration-translation exchange in the
k-ih mode. In the 5-temperature approach the level populations are expressed in terms of the vibrational
temperatures of the first level in each mode T& (k = 1,2,3,4) and the gas temperature T. The vibrational
distribution n^ in N% has the form of the Treanor distribution:
622
s^2 is the energy of the first vibrational level of N 2 , Z^r{T^T^) is the non-equilibrium vibrational partition
function:
N2
»4
N2
\
" /
The expression for the level populations of CO2 ni±iii3 has been obtained in [1] in the next form:
** C* O 2
rn m
j
m
m \
ATi
* 1*2*3
--
j.«
...
^T2
— «j_
«
« ^ « j .
/ -i p" \
LT1
^ I
^
'
kT3
2
non-equilibrium vibrational partition function Zvib (T, TI, T2, TS) is given by:
^T2
kT3
si±iii3 is the vibrational statistical weight. Distribution (15) represents the generalized Treanor distribution.
A peculiarity of this distribution is that the vibrational population in each mode depends on the vibrational
state of the remaining modes.
Equations (8), (9) for the level populations in the 5-temperature case are reduced to four relaxation equations
for the vibrational temperatures T& (k = 1, 2, 3, 4).
2.
Four-temperature approach. The 4-temperature approach takes into account the rapid
VV ( 1-2 ) exchange between the symmetric and bending modes of CO^ caused by the Fermi-resonance:
TVV ^ Tv~v> < 4w3'4 ~ TVV~B < TVT < rdiss ~0,
k = 1, ..., 4.
(17)
1-2
In the 4-temperature approach the rapid VV ( ) energy exchange gives the equation connecting TI and T2,
and CO*i level populations are expressed in terms of Ti2 = T2, TS and T [5,1]:
&T1
&T1
&T1
(18)
non-equilibrium partition function also depends on three temperatures:
r-rCOK^m
rp x
^t/ifrr (T' T12^3) =
g
V^
I
«'i«' 2 »3 ~ V 2 ^ 1 + ^ J g Q l ^ ~ »3gQ0 0 l
(2g'i + » 2 ) g Q l i Q
«3^000l \
/-Qx
2^
^1^*3 6XP ———————————————PF————————————— ~ ————^————— ~
L^
' V19)
£
. ~^. /
\
*^12
^^3 /
Macroscopic equations (5)—(7) for /?, v, T in the 4-temperature approach are coupled with the equations for
3. Three-temperature approach. The 3-temperature model is valid in the case of rapid
exchange and also VV ( 3 ~ 4 ) vibrational quanta exchange between the asymmetric mode of CO2 and
-k
_l-2
^3-4 ~ _l,2-3
_l,2-4 . fc .
/j
VV ^ VV1 ^ VV1 ^- VV1 ^ VV1
VT ^ 'diss ^ " ^
Z. _ 1
/I
^ —±,...,4:.
In this case the temperatures TS and T4 are connected due to the rapid VV ( 3 ~ 4 ) exchange and, therefore, 7V2
level populations are defined by T% and T. The system of gas dynamic equations contains equations (5)—(7)
and the equations for Ti2 = T2 and Ts4 = TS .
This paper presents a comparison of the results obtained on the basis of the state-to-state and 4-temperature
models behind a shock wave.
623
V
RESULTS AND DISCUSSION
The system of macroscopic equations (5)—(10) has been solved numerically for the flow behind a plane shock
wave using both state-to-state and 4-temperature approaches. Two mixtures: 2000ppm CO2 + 10%^ + Ar
and 2000ppm GO*} + ^2 have been considered. Equilibrium conditions behind the shock are Teq = 2495K,
peq = 0.78atm. The initial vibrational distributions are assumed to be thermal equilibrium.
First, let us estimate the influence of the mixture composition on the macroscopic parameters. The time
dependence of the gas temperature and the temperatures of the first level in each mode behind the shock
calculated using the 4-temperature model is presented in Fig. 1. Fig. la presents the results for the mixture
CO2/N2/Ar, Fig. Ib gives the temperatures in the mixture CO2/N2. It is seen that in the first mixture
vibrational relaxation in CO2 proceeds faster because of two reasons: a higher efficiency of AT in VT relaxation
of CO2, and a lower influence of the VV ( 3 ~ 4 ) exchange in this case. With increasing of the N2 concentration
the role of this exchange becomes more important, it leads to a slower equilibration of the asymmetric mode
because of the loss of vibrational energy during the near-resonant interaction with N2. In the case of CO2/N2
mixture the temperatures TS and T^ are much closer one to another. However, one can see that a commonly
used assumption TS = T^ is not valid in the general case. One can also notice that the mixture composition
effects weakly vibrational relaxation in the symmetric and bending modes as well as in N2. In both cases
the first and the second CO2 modes come to equilibrium much faster than the remaining modes, the slowest
process is VT relaxation of N2.
20
40
60
80
100
0
t, (IS
20
40
60
80
100
t, \HS
FIGURE 1. Gas temperature and vibrational temperatures as functions of time, a - CO^/N^/Ar, b
The vibrational distributions in the symmetric and bending modes behave similarly in both mixtures. Concerning the asymmetric mode, in the mixture CO2/'N2/'Ar excitation of its high vibrational states begins
earlier, populations of high levels of the third mode in this mixture exceed those in the mixture CO2/N2. This
fact can lead to the higher dissociation rate in the mixture CO2/'N2/'Ar at the same initial conditions.
Now we will discuss in more details the state-to-state vibration-dissociation coupling in the mixture CO2/N2.
In order to estimate the role of different vibrational transitions on the level populations and macroscopic parameters the state-to-state calculations have been performed using the complete kinetic scheme and neglecting
a) VV exchange within each CO2 mode; b) VV^l^~3), VV^2~3) exchange between CO2 modes; c) VV^3~4)
exchange between CO2 asymmetric mode and ^2 •
An interesting peculiarity of CO2 flows behind a shock is an inversion of some vibrational level populations
just behind the shock front. This fact has been reported in [13,14] for several mixtures of polyatomic gases.
It is important to emphasize that this feature of shock heated CO2 flows cannot be described in the frame
of multi-temperature models. The inversion of populations in the quasi-stationary approaches has been found
only in the case of very high storage of vibrational energy (for instance, in a nozzle flow) [5]. Fig. 2 represents
the inversion of populations between the first level of asymmetric mode (00°1) and levels (20°0) and (04°0) of
the remaining modes. In Fig. 2a the values (n 2 ooo — ^oo°i)/ n and (^04°o —noo°i)/n are given as functions of
time. It is seen that inversion arises just behind the shock front and disappears at t ~ 10 — 12//s. It is found
that inversion between (04°0) and (00°1) exists only in the case of complete kinetics and does not appear if one
neglects some of VV or VVf exchanges. In Fig. 2b the values (?i20 0 o — HQQ°I)/^ are given for different kinetic
624
1.E-04
1.E-04
2
4
6
8
10
10
12
15
20
25
30
1 - complete kinetics
2 - no VV34
3 - no VV23
4 - no VV
1.E-05
1.E-05-
1.E-06
1.E-06
t,|lS
FIGURE 2. Inversion of vibrational level populations as a function of time, a — complete kinetics; b — comparison of
different kinetic schemes.
schemes. A neglect of VV exchange within GQ<z molecules increases the time of existing of the inversion up to
20 — 25//s, this time becomes shorter if one neglects the exchange between CO^ and N%.
4.E-06
3.E-06
3.E-06
2.E-06
2.E-06 1.E-06
5.E-07
O.E+00
10
15
20
25
30
8.E-05
7.E-05
6.E-05
5.E-05
4.E-053.E-05 2.E-05
1.E-05
O.E+00
10
15
20
25
30
t,|is
FIGURE 3. Reduced populations of selected GQ<2 levels as functions of time. Curves 1 — complete kinetics, 2 — no
VV ( 3 ~ 4 ) exchange, 3 - no VV ( 1 > 2 » 3 ) exchange, 4 - no VV exchange.
The reduced populations of selected CO^ levels are presented in Fig. 3. The influence of various VV and
VV1 exchanges on the populations can be estimated from this figure. One can see that a neglect of any kind
625
of vibrational transitions can lead to a noticeable error in the vibrational level populations.
Vibrational distributions in the CO^ asymmetric mode calculated using the state-to-state and 4-temperature
approaches are shown in Fig. 4 for different time values. The initial Boltzmann distribution is also plotted
in the figure. The quasi-stationary 4-temperature model gives a lower excitation rate of the high vibrational
states of the third mode compared to the state-to-state model. This may be a reason for the lower dissociation
rate found in the beginning of the relaxation zone in the 4-temperature approach. This result is presented in
the next figure.
1.E+00
1.E-01 -
1.E-02
1.E-03
1.E-04 J
FIGURE 4. Reduced populations of GQ<2 asymmetric mode as functions of is at different t. Solid curves — state-to-state
approach, dashed curves — 4-temperature approach. Curves 1,1' — t = 10/us, 2,2' — t = 50/us.
Fig.5 plots the oxygen atom concentration as a function of time. One can notice that the state-to-state
model using the complete kinetic scheme gives the maximum O production in the beginning of the relaxation
zone. The effect of the different energy transitions on the dissociation rate is also significant. Redistribution
of the vibrational energy as a result of VV and VV1 exchange provides the flux of vibrational quanta to the
high vibrational levels and therefore increases the dissociation rate.
1 - complete kinetics
0.12- 2 - no VV34
0.1
3 - no VV23
4 - no VV
5 - 4-temperature
0.08
0.06 0.04
0.02
0
20
40
60
80
100
FIGURE 5. Oxygen atom concentration as a function of time. Solid curves - state-to-state model, dashed curve 4-temperature model.
626
VI
CONCLUSIONS
The vibration-dissociation kinetics of CO^/N^ mixtures has been investigated on the basis of the state-tost ate and quasi-stationary models. The influence of the mixture composition and different vibrational energy
exchanges on the vibrational distributions, flow field parameters and CO^ dissociation rate is estimated. It
is shown that the gas temperature and vibrational temperature distributions behind the shock wave depend
significantly on the mixture composition. Increasing of N% concentration leads to slower relaxation of the GO*}
asymmetric mode because of the VV ( 3 ~ 4 ) vibrational energy exchange. The inversion of the vibrational level
populations has been found in the very beginning of the relaxation zone. The CO^ dissociation rate in the
shock heated flow is affected by numerous vibrational energy transitions, neglecting some of them leads to an
underestimation of the dissociation rate.
ACKNOWLEDGMENTS
This study has been supported by INTAS (99-00464).
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