Strong Non-Equilibrium Quasi-Stationary Model for
Dissociation-Recombination in Expanding Flows.
A.Chikhaoui*, E.A.Nagnibeda1", E.V.Kustova1", T.Yu.Alexandrova*
*IUSTI - MHEQ, Universite de Provence 13453 Marseille, Cedex 13, France
^ Math, and Mech. Dept, Saint-Petersburg University, 198904, Saint-Petersburg, Russia
Abstract. A quasi-stationary model for dissociation, recombination and vibrational kinetics in expanding
flows is presented. The model is based on strongly nonequilibrium vibrational distributions different from the
Boltzmann and Treanor ones. Different mechanism of energy exchanges at low, middle and high vibrational
levels is taken into account. The equations of nonequilibrium gas dynamics are derived and applied to nozzle
flows of (N2jN) and (02,0) gas mixtures. The gas parameters and vibrational distributions in the nozzle
are computed and compared with the ones obtained using approximate models and the influence of kinetic
models on flow parameters is investigated.
INTRODUCTION
Modeling of vibrational and dissociation-recombination kinetics in expanding flows is important for many
practical problems of physical gas dynamics. The peculiarity of the rapid expansion of an initially heated
gas mixture is that chemical reactions proceed in a strongly vibrationally excited gas. Indeed, the vibrational
energy occurs much higher than the translational one because the gas temperature decreases more rapidly
than the vibrational one. Such a situation requires adequate models of vibrational-chemical coupling in the
flow. The most rigorous approach consists in considering the state-to-state vibrational and chemical kinetics.
Another approach is based on the quasi-stationary vibrational distributions. Such models are much more
simple compared to the state-to-state one and therefore important for applications. Actually, in this case the
equations for level populations can be reduced to the equations for a less number of gas flow parameters. The
most often used quasi-stationary models are based on the nonequilibrium Boltzmann and Treanor distributions
which are not sufficiently good for a nozzle flow because of strong vibrational excitation. The experiments and
numerical calculations of vibrational distributions in nozzles [1], [2] show that populations of intermediate and
high levels differ from the Boltzmann or Treanor ones. Therefore the aim of this paper is the elaboration of
the model for the reacting flow in nozzles using non-Boltzmann and non-Treanor vibrational distributions [3].
NONEQUILIBRIUM DISTRIBUTIONS AND GOVERNING EQUATIONS
We consider a flow of a binary mixture of diatomic molecules and atoms with dissociation, recombination
and VT(TV) and VV vibrational energy transitions. Due to rapid translational-rotational relaxation the
distributions over translational and rotational energies are supposed to be in equilibrium with the gas temperature T. The equations for macroscopic gas flow parameters follow from the equations of the state-to-state
kinetics coupled with the gas dynamic equations using quasi-stationary vibrational distributions [3], [4]. The
last ones are obtained in [3] taking into account the peculiarities of vibrational energy exchanges in a strongly
excited gas. Actually, the dominant mechanism of vibrational relaxation in this case is different at various
groups of vibrational levels. Thus, at lower levels a non-resonant exchange of vibrational quanta dominates, at
intermediate levels the resonant exchange between neighbouring levels occurs more probable, at high levels all
vibrational energy transitions have comparable probabilities.
Under these conditions the quasi-stationary solution of the equations for level populations have the form [3]:
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
613
*-*!
~.
,
Q <
t*<t<t**,
.
(1)
Here the coefficients Ci, 62, F are found from the normalizing and continuity conditions, Q^ r (T,TI), Q^ibr(T)
are the truncated Treanor and Boltzmann partition functions correspondingly, TI is the vibrational temperature
of the 1-st level connected with the total number of vibrational quanta W = X^n«? £i '18 the vibrational
energy of anharmonic oscillator calculated from the energy of the 0-th level, k is the Boltzmann constant, the
expressions for i*, $** are given in [4], L is the total number of excited levels.
Level populations (1) depend on two temperatures T,Ti. They have a form of the Treanor distribution at
low levels, slopping plateau at intermediate levels and one-temperature Boltzmann distribution at high levels.
Ai i = i* Treanor level populations have minimum values, ^** is taken from the conditions that for i > i**
probabilities of all vibrational energy transitions are of the same order. At TI = T expressions (1) come to
the one-temperature thermal equilibrium Boltzmann distribution, if TI < T the Treanor distribution is valid
practically for all levels because ^* is close to L. In nozzle flows TI > T and populations of the levels i > i*
differ from the Treanor ones.
Substituting vibrational distributions (1) into the master equations for level populations and summing them
over i after multiplying, correspondingly, by 1 and ^, we reduce the last ones to the equations for molecular
number density nmoi and vibrational quantum number W(T,Ti,n mo /). These equations are coupled with
the equations for atomic number density nat and impulse and total energy conservation equations. Finally
we obtain a closed set of equations for macroscopic parameters n mo j, n a t, TI, T, v in the quasi-stationary
approximation (v is the gas velocity).
For a steady-state one-dimensional flow in a nozzle these equations are written as follows:
dx
d(natvS)
= -2SRd
dx
dv _ dp
dE
here S(x) is the nozzle cross-section, #, v are the co-ordinate and velocity along the nozzle axis, p is the
pressure, E is the total energy per unit volume, ftdls-rec ? R^s~rec and R™1 are the source terms.
The term Rdis~rec has the form:
gdis-rec = _nmol
nM
fc
(T, T X ) + n
M
M
Here fe^(T,Ti), kr^c(T) are the dissociation and recombination rate coefficients, M is the collision partner
(molecule or atom). These coefficients are defined as follows:
614
where k^is\ and &£ ^ are the state-to-state rate coefficients of dissociation from the ith level and recombination
to the ith level, U{ are given by Eqs.(l). Dissociation rate coefficients have been calculated using two models:
the ladder-climbing model and the Treanor-Marrone one. According to the ladder-climbing model dissociation
is permitted only from the last level. In this case the dissociation rate is determined by the rates of vibrational
energy transitions. In the frame of the Treanor-Marrone model [5] dissociation happens from any level. The
coefficients k
can be written as follows:
T).
(5)
Here k^'eg(T) is the Arrhenius dissociation rate coefficient, Z\M'(T) is the non-equilibrium factor for each
level i. Using the Treanor-Marrone model Z\M'(T) can be found in the form:
1
where Qvibr(T) is the total vibrational partition function, U is the parameter of the model which can be found
fitting to experimental data or results of more exact numerical calculations. Recently the evaluation of U
values for (N^^N) mixture has been done on the basis of the results obtained by trajectory method [6].
The expression (6) for Zi has been given previously in [7].
Using (3), (5), (6) we obtain the expression for two-temperature dissociation rate coefficients:
Hi are defined by (1).
For state-to-state recombination rate coefficients we use the detailed balance principle and finally derive from
Eq.(4) the following expressions:
k(r^(T) = k(dfs)eq(T)K(T),
K(T) =
where
has the form [8]:
\ 3/2
'
/
~\™lt)
/
*
™texp^
— 7~)\
kj ,
J
Here Qrot is the rotational partition function, ra mo j, mat are the masses and D is the dissociation energy.
Similarly, R^s~rec is presented using Z^ KI and U{ given by (1):
M
Here
i,
Grec(T) =
The source terms R™1 are calculated using the rate coefficients of vibrational energy transitions for anharmonic
oscillators taken from [1],
615
T,T
K
T,T11r, K
T,T
1, K
TJi.K
5000
5000
4500
3500
3500
4000
3000
3000 -
3500
T lad-cl
^Tlad-cl
T1 lad-cl
lad-cl
~-T1
T VV,VT
^TVV.VT
T1 VV.VT
VV,VT
--T1
Tr-Mar
TT Tr-Mar
T1 Tr-Mar
Tr-Mar
T1
2500
2500
T
T1
3000
T Bol
2000
2000
T1 Bol
2500
T Trin
1500
1500
T1 Trin
2000
1000
1500
500
1000
x/R
500
500
x/R
0
00 22 44 66 88 10
10 12
12 14
14 16
16 18
18 20
20
fig.1(a)
fig.l(a)
0 2 4 6 8 10 12 14 16 1820
18 20
fig.1(b)
fig.l(b)
FIGURE 1.
1. T
T,Ti
versus x/R
; T1 versus
x=R
FIGURE
(a): for
for dierent
different vibrational
vibrational distributions,
(a):
distributions, (N%,N),To
(N2 ; N ); T0 == 5375K,po
5375K; p0 == 17.4atm,
17:4atm,
(b): for
for dierent
different models
(b):
models of
of dissociation,
dissociation, (O2,O),To
(O2 ; O); T0 == 4468AT,po
4468K; p0 == 182aim.
182atm:
RESULTS
RESULTS
The numerical
numerical calculations
calculations have
The
have been
been performed
performed for
for aa (N%
(N2,;N)
N )mixture
mixtureflow
owexpanding
expandingininaaconic
conicnozzle
nozzlewith
with
21° angle
angle for
for the
the following
following reservoir
aa 21
reservoir conditions:
conditions: TT00 == 5375K,
5375K, 7525K,
7525K, pp00 == 1.74atm,
1:74atm, 17.4atm,
17:4atm, 174atm,
174atm,the
the
radius of
of the
the critical
critical cross-section
cross-section R
radius
R=
= I1 mm,
mm, and
and also
also for
for (O
(O22, ;0)O)mixture
mixturewith
withthe
theconditions:
conditions:TT0 0==4468K,
4468K,
= 182atm
182atm and
and T
T00 =
= 4490K,
pPo0 =
4490K, pp00 =
= 18.3atm.
18:3atm. In
In both
both the
the reservoir
reservoir and
and critical
critical cross-sections
cross-sectionsthe
themixture
mixture
supposed to
to be
be in
in equilibrium.
equilibrium. The
isis supposed
The parameters
parameters nnat,, n-moi,
n , T,
T , TI
T1 have
have been
been found
found asas aa numerical
numericalsolution
solution
of equations
equations (2)
(2) and
and also
also from
of
from the
the equations
equations obtained
obtained using
using the
the Treanor
Treanor distribution
distribution (up
(uptotothe
thelevel
leveli*iand
and
neglecting
populations
at
i
>
i*)
and
Boltzmann
nonequilibrium
distribution
with
T
^
T
(for
neglecting populations at i > i ) and Boltzmann nonequilibrium distribution with Tv 6= T (forharmonic
harmonic
oscillators). The
The comparison
comparison of
oscillators).
of the
the results
results shows
shows the
the influence
inuence ofof different
dierent distributions
distributions on
on the
the flow
owfield
eld
parameters
as
well
as
the
effect
of
anharmonicity
of
vibrations.
The
calculations
have
been
done
also
neglecting
parameters as well as the eect of anharmonicity of vibrations. The calculations have been done also neglecting
the dissociation
dissociation and
and recombination
the
recombination in
in order
order to
to show
show the
the role
role ofof these
these processes
processesininaanozzle.
nozzle.
The
results
show
a
noticeable
influence
of
vibrational
distributions
The results show a noticeable inuence of vibrational distributions and
and the
the anharmonicity
anharmonicity effect
eect on
on TI,
T1 ,
this
effect
on
T
and
n
/,
n
t
is
very
weak.
Figs.l(a,b)
plot
T,
TI
versus
x/R
in
(]V
,iV)
and
(O^^O)
mo
a
2
this eect on T and n , n is very weak. Figs.1(a,b) plot T; T1 versus x=R in (N2 ; N ) and (O2 ; O) mixmixtures. Fig.1(a)
Fig.l(a) shows
shows the
tures.
the influence
inuence of
of vibrational
vibrational distributions
distributions on
on TT and
and TI
T1 in
in (]V
(N22,iV)
; N ) mixture.
mixture. Boltzmann
Boltzmann
distribution for
for harmonic
harmonic oscillators
distribution
oscillators leads
leads to
to aa noticeable
noticeable overestimation
overestimation ofof TI
T1 and
and toto an
an underestimation
underestimationofof
the gas
gas temperature.
temperature. TI
the
T1 values
values obtained
obtained using
using the
the Treanor
Treanor distribution
distribution exceed
exceedTIT1 calculated
calculatedononthe
thebasis
basisofof
(1).
The
influence
of
dissociation-recombination
processes
on
T
and
TI
is
presented
in
Fig.l(b)
for
(O<2,O)
(1). The inuence of dissociation-recombination processes on T and T1 is presented in Fig.1(b) for (O2 ; O)
mixture. It
It plots
plots T
T,, T
TI computed
computed using
mixture.
using the
the ladder-climbing
ladder-climbing and
and Treanor-Marrone
Treanor-Marronemodels
modelsand
andalso
alsoneglecting
neglecting
1
dissociation-recombination
(only
with
VV
and
VT
processes).
dissociation-recombination (only with VV and VT processes).
Figs.2(a,b) illustrate
illustrate the
Figs.2(a,b)
the influence
inuence of
of vibrational
vibrational distributions
distributions on
on the
the ratio
ratio Ti/T.
T1 =T . InIn fig.2(a)
g.2(a)Ti/T
T1 =T isisprepreo
at
mol
v
mol
at
616
sented for ( N z , N ) mixture using distribution (1) with complete kinetics (dis-rec), neglecting dissociation and
taking into account only recombination reaction (rec), neglecting recombination (dis) and neglecting dissociation and recombination (considering only VV and VT processes). In the same figure one can see Zi/T
calculated on the basis of the Treanor and Boltzmann distributions (with complete kinetics). Fig.2(b) depicts
Ti/T for (O 2 ,O) mixture obtained using the Treanor-Marrone model, the ladder-climbing one and neglecting
dissociation-recombination for different conditions in the reservoir.
Using the Treanor and Boltzmann distributions gives unreal high values of Ti/T (fig.2(a)). It is interesting
to see the different effect of dissociation and recombination on Ti/T in various cases (fig.2(b)) caused by the
competition between dissociation and recombination in a nozzle.
The atomic molar fraction versus x/R is plotted in Fig.3(a,b,c). Figs. 3(a, b) show nat/(nat + nmo{) in
(N2, N) mixture obtained using the Treanor-Marrone model with complete kinetics and neglecting dissociation
or recombination for different conditions in the reservoir. One can see the dominating recombination at high
pressure and dissociation at low pressure. Fig. 3(c) gives the comparison of nat/(nat + nmoi) calculated using
the Treanor-Marrone model and the ladder-climbing model in (O 2 ,0) mixture.
4.5
VV,VT(1)
Tr-Mar(1)
lad-cl(2)
VV,VT(2)
3.5
2.5
21.5
x/R
x/R
0 5 10 15 20 25 30 35 40
10
15
20
fig.2(b)
fig.2(a)
FIGURE 2. Ti/T versus x/R]
(a) (JV 2 ,JV), TQ = 7525K, po = 174a£m, for Boltzmann, Treanor and complex distribution (neglecting dissociation-recombination, complete kinetics, with only dissociation, with only recombination),
(b) (02,0), (1) - To = 4468K, po = 182atm, (2) - T0 = 4490K, p0 = 18.3atm using different models of dissociation and
neglecting chemical processes
617
>v k a1+n
n
at+nmol)
m0i)
n
2.36E-01 -,
3.11E-02
f—
—05)
2.61 E-02
— (15)dis
^dis-rec
— (15)rec
'
— dis
2.35E-01 -
^— rec
2.11E-02
1 .61 E-02
V
t
xm
v/P
1.11 E-02
10
15
2.34E-01 0
20
fig.3<a)
~~*~-*__^
, x/R
5
10
15
20
fig.3(b)
FIGURE 3. nat/(nat + nmoi) versus x/R] (a) and (b): (Nz,N) with complete kinetic (dis-rec), neglecting recombination (dis), neglecting dissociation (rec); TO = 7525K, po = I74atm (a), po = l.74atm (b);
(c): (02,0) with different dissociation models, (1) - T0 = 4468Kp0 = 182atm, (2) - T0 = 4490Kp0 = lS.3atm .
8.20E-02 -
\
7.20E-02 -
\
6.20E-02 -
-
lad-cl(1)
- -» - Tr-Mar(1)
5.20E-02 -
—— lad-cl(2)
^^Tr-Mar(2)
4.20E-02 3.20E-02 2.20E-02 1.20E-02 -
2.00E-03 C)
* . .. _ ^ ., ., _ .. ^ ., ^ ., _ ^ ., ., .. .. .,
1
5
10
fig.3 (c)
618
15
20
X/ri
CONCLUSIONS
The influence of non-equilibrium quasi-stationary distributions and effects of dissociation-recombination
process on the expanding nozzle flow parameters are investigated in (N2,N) and (0*2,0) mixtures. There
have been considered the vibrational distributions taking into account real vibrational spectrum and strong
nonequilibrium. The anharmonism of vibrations causes the non-Boltzmann distributions, strong vibrational
excitation requires consideration of the non-Treanor distributions at intermediate and upper levels. Neglecting
the anharmonic effects and using the Boltzmann level populations leads to a noticeable overestimation of
TI and the ratio Ti/T and a weak underestimation of gas temperature. The deviations from the Treanor
distribution influence TI and T weakly. It proves that the populations of low levels occur more important for
changing of the gas dynamic parameters than the populations of intermediate and high levels. The effects of
the non-Boltzmann and non-Treanor distributions on nozzle flow parameters grow with the distance from the
critical cross-section. The vibrational distributions and gas dynamic parameters have been calculated using
the Treanor-Marrone model (permitting dissociation from each level), ladder climbing model (with dissociation
and recombination only through the last level) and neglecting dissociation and recombination. The comparison
of the values of T, TI, Ti/T and atomic molar fractions, obtained in different cases, is presented for various
reservoir conditions. A competition of dissociation-recombination processes in a nozzle is shown as well as its
influence on the gas dynamic parameters.
Acknowledgements
The work has been supported by INTAS (99-00464).
REFERENCES
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