Non-Equilibrium Kinetics and Transport Properties in
Reacting Flows in Nozzles
T.Yu. Alexandrova , A. Chikhaoui , E.V. Kustova† and E.A. Nagnibeda†
†
IUSTI – MHEQ, Université de Provence, 13453 Marseille, Cedex 13, France
Department of Mathematics and Mechanics, Saint Petersburg State University,
198504, Universitetsky pr. 28, Saint Petersburg, Russia
Abstract. Non-equilibrium vibration-dissociation kinetics, transport properties and dissociation rate coefficients in expanding
flows are studied on the basis of different kinetic theory approaches: state-to-state, multi-temperature and one-temperature
ones. The limits of validity of more simple models are discussed, the influence of non-equilibrium vibrational distributions,
initial conditions, nozzle profile on the macroscopic parameters, heat transfer and reaction rates is investigated.
INTRODUCTION
In the present paper the non-equilibrium distributions, gas dynamic parameters and transport properties in reacting
flows in nozzles are studied using different approaches of the kinetic theory. For the accurate prediction of real gas flow
parameters, gas dynamic equations should be coupled with the equations of non-equilibrium kinetics. An important
practical problem is the choice of an adequate kinetic model, which depends on specific flow conditions and on the
relations between characteristic times of non-equilibrium processes under consideration.
The peculiarity of nozzle flows is that fast cooling of an expanding flow results in strong vibrational non-equilibrium,
the vibrational energy occurs much higher than the translational-rotational one. Under such conditions vibrational
level populations are far from the equilibrium ones and cannot be described by the thermal equilibrium Boltzmann
distribution or the non-equilibrium quasi-stationary distributions like the Boltzmann or the Treanor ones. The peculiar
features of state-to-state populations in a nozzle flow have been recently studied in [1, 2], and in [3, 4] the transport
properties in nozzle flows have been evaluated on the basis of state-to-state distributions.
A detailed state-to-state kinetic theory approach developed in [5] gives a good accuracy for the investigation of
distributions and gas flow parameters but its practical realization requires a lot of computational time. It is because
of the fact that many equations for vibrational level populations of all molecular components of a mixture should
be solved together with the gas dynamic equations and, moreover, for evaluation of transport properties one should
calculate many diffusion coefficients which are different not only for various chemical species but also for molecules
at different vibrational levels [5].
Much more simple models are based on the quasi-stationary solutions of master equations for level populations.
In this case, the non-equilibrium distributions are expressed in terms of few macroscopic parameters. The equations
for level populations are reduced to a much less number of equations for macroscopic parameters, and the transport
algorithms are also noticeably simplified [6, 7]. There exist many papers where nozzle flows have been studied on
the basis of quasi-stationary vibrational distributions. However, the accuracy of quasi-stationary models has not been
estimated. Up to now, it is not sufficiently understood when the rigorous state-to-state kinetic theory gives principle
new results for macroscopic gas flow parameters and transport properties in nozzle flows and under what conditions
more simple and, therefore, more cheap models can be used with a good accuracy. This question is very important for
the practical purposes. The main attention of this study is focused on the influence of non-equilibrium distributions on
such important flow parameters as temperature, total energy flux and averaged reaction rates in nozzles.
In the present paper for the calculation of non-equilibrium distributions, gas dynamic parameters, transport properties and dissociation rates in expanding flows four kinetic theory approaches have been used: the state-to-state one and
three models based on quasi-stationary distributions. A comparison of the results obtained in different approaches is
presented.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
77
STATE-TO-STATE MODEL
We consider a flow of a binary mixture of diatomic molecules and atoms with dissociation, recombination
A2 (i) + M
VT (TV) and VV vibrational energy transitions
A2 (i) + M
A2 (i) + A2 (k)
*) A
*) A
*) A
+A+M
0
2 (i ) + M
0
;
(1)
;
(2)
0
:
2 (i ) + A2 (k )
(3)
Here A2 (i) is a molecule at the ith vibrational level, and M is an inert partner which can be a molecule or an atom A.
Due to rapid equilibration of translational and rotational energies the relaxation times are assumed to satisfy to the
condition
τtr τrot τvibr τdiss rec θ
(4)
<
; ; ;
;
where τtr τrot τvibr τdiss rec are the characteristic times of translational, rotational, vibrational relaxation and
dissociation-recombination processes, θ is the macroscopic characteristic time.
Under condition (4) using the generalized Chapman-Enskog method one can derive the closed set of equations for
vibrational level populations ni , atomic number density nat , gas velocity v and gas temperature T describing a nonequilibrium gas flow in the state-to-state approximation [6, 5, 8]. For A 2 /A mixture these equations have the form:
dni
dt
v + ∇ (niVi ) = Ri ; i = 0; 1;:::; L;
+ ni ∇
dnat
dt
+ nat ∇
v + ∇ (nat Vat ) = Rat ;
ρ
dv
+∇P = 0
dt
ρ
dU
dt
+∇
(5)
(6)
;
(7)
q + P : ∇v = 0:
(8)
Here Vat and Vi are the diffusion velocities of atoms and molecules at the ith level, L is the number of vibrational
levels for molecular species, ρ is the density, P is the tensor of pressure, U is the total energy per unit mass, q is the
total energy flux.
The production terms Ri describe the change of ni as a result of VT, VV vibrational energy transitions, dissociation
and recombination:
dis rec
Ri = Rvibr
+ Ri
(9)
i
;
Rat describes the atom number density change:
Rat
=
2 ∑ Ridis
i
The expressions for Rvibr
and Ridis
i
Rvibr
=
i
∑
k;k ;i (k
0
;
Ridis
rec
0
0
rec
6=k i 6=i)
;
dis
rec
= Ri + Ri
;
:
(10)
have the form:
k0 k
(ki0 i ni0 nk0
0
rec
mol
kiikk ni nk ) + nmol ∑ (kki
nk
0
0
k6=i
mol
at
Rdis
i = ni (nmol kdis;i + nat kdis;i )
;
mol
at
kik
ni ) + nat ∑ (kki
nk
k6=i
at
kik
ni )
;
2
mol
at
Rrec
i = nat (nmol krec;i + nat krec;i )
(11)
:
(12)
0
mol kat are the rate coefficients of VT transitions at the collisions with molecules or atoms correspondingly, kkk
Here kik
ik
ii
M are
are the rate coefficients of VV exchanges (TRV transitions are neglected because of their low probability), kdis
;i
M are the rate coefficients of
the rate coefficients of dissociation from the ith level at the collision with a partner M, krec
;i
recombination to the ith level, nmol = ∑i ni is the molecular species number density.
0
78
The total energy flux q in the first order Chapman-Enskog approximation in the state-to-state approach can be
written as follows [4]:
q = λ ∇T
p DT dmol + DTat dat
mol
+
∑
i
5
5
kT + hε ij ir + εi ni Vi +
kT + ε at nat Vat
2
2
;
(13)
p is the pressure, k is the Boltzmann constant, εi and ε at are the vibrational energy and the energy of atom formation,
hε ij ir is the rotational energy averaged over rotational spectrum. The coefficient of heat conductivity λ = λtr + λrot is
due to translational and rotational degrees of freedom, vibrational modes do not contribute to thermal conductivity in
this approach. The transport of vibrational energy is described in terms of diffusion velocities Vi for each vibrational
state. Coefficients DT and DTat are the thermal diffusion coefficients of molecules and atoms correspondingly, the
mol
diffusive driving forces for molecules at different vibrational levels di , for atoms dat and molecules dmol depend
on the gradients of ni , nat and nmol . Expressions for Vi , Vat are given in [5, 4], they are determined by the diffusive
driving forces, di , dat , dmol , diffusion coefficients Dcidk for each vibrational and chemical species and thermal diffusion
coefficients.
One can notice that the total energy transfer in the state-to-state approximation is determined by heat conductivity,
thermal diffusion, mass diffusion of molecules and atoms and diffusion of vibrationally excited molecules. The last
mentioned effect manifests itself only in the state-to-state approximation.
In order to evaluate the total energy flux in the reacting flow using the rigorous state-to-state approximation described
above one has to compute a big amount of state-to-state diffusion coefficients at each step of numerical integration.
Because of complexity and high computational cost of such a scheme we divide it to two separated steps. First, the
non-equilibrium level populations and other flow parameters are calculated in the Euler approximation of non-viscid
flow (in the zero order Chapman-Enskog approximation). Then the values of ni , nat , T are inserted to the state-to-state
transport kinetic theory algorithms [5]. In result all transport coefficients and the heat flux have been evaluated.
QUASI-STATIONARY MODELS
Three next models are based on the quasi-stationary distributions. First of them is the non-Boltzmann strongly nonequilibrium distribution for anharmonic oscillators which has a form of the Treanor distribution at low levels, the
slopping plateau at intermediate levels and the Boltzmann distribution at high levels. This distribution has been
proposed first for a one-component gas as an approximate solution of master equations for level populations [9],
then it has been derived from the kinetic theory in [10]. Recently this distribution has been used for modelling of
non-equilibrium reacting mixture flows [8].
Using this distribution, equations (5) for ni are reduced to two equations for molecular number density nmol and for
the temperature of the first vibrational level T1 . The populations ni are then calculated on the basis of the numerical
solution of the set of equations for nmol , nat , T , T1 , v.
Two other models are much more simple and based on the Boltzmann quasi-stationary vibrational distributions: the
non-equilibrium Boltzmann distribution with vibrational temperature Tv different from T (this model is valid only for
harmonic oscillators) and the Boltzmann thermal equilibrium distribution. In the first case the equations for nmol , nat ,
T , Tv , v have to be solved whereas in the second case the equations for nmol , nat , T , v should be considered.
The transport terms in the quasi-stationary approaches are defined by gradients of macroscopic parameters mentioned above. Formulas for the total energy flux q in the quasi-stationary approximations are given in [4] and contain
gradients of nmol , nat , T and T1 in the case of complex distribution for anharmonic oscillators; nmol , nat , T and Tv in the
case of non-equilibrium Boltzmann distribution for harmonic oscillators; nmol , nat and T for the thermal equilibrium
case. The number of independent diffusion coefficients is much less in the quasi-stationary approaches compared to
the state-to-state model. Moreover, the heat conductivity coefficients are different in each case and include contribution
not only translational and rotational degrees of freedoms but also the contribution of vibrational modes.
In the next section we present a comparison of vibrational distributions, temperatures, transport properties and
dissociation rates obtained in the state-to-state and quasi-stationary approaches.
79
RESULTS
Equations (5)–(8) for ni , nat , v, T with production terms (9)-(12) for O 2 /O and N2 /N mixtures expanding in a nozzle
have been solved numerically in the Euler approximation for three nozzle profiles:
(1) the axis-symmetric conic nozzle with a semi-angle 21 and throat radius R = 1 mm;
(2) the nozzle studied in [1] with the profile of the divergent part:
S (x )
S(x )
S is the section area, x is the throat location;
(3) F4 nozzle [2]:
= 1+x
:
0 0908exp( 4 9204(x
:
0 2277exp( 0 1884(x
r(x) = 0 0959
r(x) = 0 3599
2
tan θ
x
2
;
(14)
: ;
:
:
0 5 )2 )
:
:
0 5 )2 )
x
:
(x
:
0 0184
0 5)2 + 0 1447
:
:
;
< 0:5m
(15)
:
x 0 5m
Several test cases with different initial conditions in the reservoir have been studied. State-to-state dissociation rate
M
have been found using the Treanor-Marrone model [11] modified for the state-to-state approach
coefficients kdis
;i
in [12]. For the recombination rate coefficients calculation the detailed balance principle has been applied. The rate
coefficients for all vibrational energy transitions are calculated using the interpolating formulas proposed in [13] for the
rate coefficients obtained in [14, 15] by means of molecular dynamics methods. The results are compared with the ones
obtained using the first order perturbation theory of Schwartz, Slawsky, Herzfeld [16] for the transition probabilities.
In all cases two models for transition probabilities give a noticeable discrepancy in vibrational distributions, gas
temperature and dissociation rates. Therefore elaboration of accurate models for energy transitions remains a very
important task.
O 2/O
reduced level populations
0,01
1E-3
1E-4
1E-5
1E-6
1E-7
O 2/O
0,1
nozzle (1), x/R=5
nozzle (1), x/R=50
nozzle (2), x/R=5
nozzle (2), x/R=50
nozzle (3), x/R=5
nozzle (3), x/R=50
0,01
reduced level populations
x/R=0
x/R=1
x/R=2
x/R=5
x/R=10
x/R=20
x/R=30
x/R=40
x/R=50
0,1
1E-3
1E-4
1E-5
1E-6
1E-7
1E-8
1E-8
0
5
10
15
20
vibrational quantum number
25
0
30
5
10
a
15
20
vibrational quantum number
FIGURE 1. Reduced level populations ni =n versus i in different cross sections x=R. Mixture O2 /O, T
(a): Conic nozzle (b): Different nozzle profiles.
=
=
25
30
b
= 4000 K, p = 1 atm.
Figure 1a shows the reduced vibrational level populations ni n (n is the total number density) of O2 molecules
versus i at different values of the dimensionless distance from the throat x R (R is the throat radius) in the conic
nozzle. The conditions in the throat are: T = 4000 K, p = 1 atm. One can see the decreasing of all level populations
(except two first ones) just near the throat. Then, with increasing x, pumping of intermediate levels and formation
of stable non-Boltzmann and non-Treanor distributions with a long plateau part is observed. In N 2 /N mixture the
shape of vibrational distributions is different, the plateau part is much shorter than in oxygen. The important role
of recombination in plateau formation has been found. Thus, the populations have been calculated for the complete
80
kinetic scheme and neglecting 1) recombination, 2) dissociation, 3) both dissociation and recombination. Dissociation
practically does not change the shape of level populations whereas neglecting recombination leads to disappearing of
the plateau part of distributions.
A comparison of level populations obtained for different nozzle profiles is presented in Figure 1b (it should be noted
that the flow in the convergent part of nozzles (2) and (3) is supposed to be equilibrium). It is seen that qualitatively
the shape of distribution is similar for all nozzles, however some quantitative difference exists.
In Figure 2 the state-to-state distributions are compared with the ones calculated using the quasi-stationary approximations. The figure gives the evolution of vibrational distributions for O 2 /O mixture in the case T = 4000K,
p = 1atm. Curves (1) plot the state-to-state distributions, curves (2) correspond to strongly non-equilibrium distributions of anharmonic oscillators, curves (3) depict non-equilibrium Boltzmann distributions of harmonic oscillators
with vibrational temperature Tv 6= T and curves (4) correspond to thermal equilibrium distributions with Tv = T . A significant distinction between the distributions growing with increasing the distance from the the throat can be noticed.
Thermal equilibrium distributions give an essential underestimation of the populations of all levels. For low levels
quasi-stationary models for both harmonic and anharmonic oscillators lead to populations close to the ones obtained
by means of the state-to-state approach while for other levels the difference between distributions provided by various
models is rather high.
1
1
x/R=5
0,01
0,01
1E-3
1E-3
1E-4
1E-5
1
1E-6
1E-7
1E-8
1E-9
2
1E-10
4
1E-11
x/R=50
0,1
reduced level populations
reduced level populations
0,1
1E-4
1E-5
1
1E-6
1E-7
2
1E-8
3
1E-9
1E-10
4
1E-11
3
1E-12
1E-12
1E-13
1E-13
0
5
10
15
20
vibrational quantum number
25
30
0
a
5
10
15
20
vibrational quantum number
25
30
b
FIGURE 2. Reduced level populations ni =n versus i at (a) x=R = 5 and (b) x=R = 50. Mixture O2 /O, T = 4000 K, p = 1 atm,
conic nozzle. Curves 1 – state-to-state model; 2 – two-temperature anharmonic oscillator model; 3 – two-temperature harmonic
oscillator model; 4 – one-temperature model.
For practical applications it is important to understand how the discrepancy in distributions influences the gas
dynamic parameters and transport properties. This effect is shown in the next figures. Figure 3a presents the change of
gas temperature T and temperature of the first vibrational level T1 along the nozzle axis, calculated on the basis of four
approaches described above. As it could be expected, the thermal equilibrium model leads to an underestimation of
the gas temperature compared to the state-to-state model. Both quasi-stationary models for harmonic and anharmonic
oscillators give close values of temperature and vibrational temperature compared to the ones obtained using the stateto-state approach. The influence of the nozzle profile on the temperatures is shown in Figure 3b. One can notice that in
the conic nozzle the difference between T and T1 remains higher than in other nozzles. It is explained by a more sharp
gas expansion in the conic nozzle that leads to a higher storage of vibrational energy in the nozzle (1).
Figure 4 gives the heat flux calculated in different approaches and at various initial conditions. The total energy
flux decreases with x due to decreasing of gradients of macroscopic parameters. Again, the thermal equilibrium
model underestimates noticeably the heat flux, the deviation may reach many tens percents. Non-equilibrium twotemperature models give the heat flux values significantly closer to the rigorous state-to-state ones, the anharmonic
approach provides a slightly better accuracy. In all test cases the effect of non-equilibrium distributions on the heat
transfer is found to be weak. It is interesting to note that in many non-equilibrium flows (behind shock waves, in
boundary layers) the role of vibrational kinetics in the heat transfer is much higher [17, 18]. It is connected with the
fact that in expanding flows only few low vibrational states contribute to the total heat flux [4] while in other flows the
contribution of intermediate and high levels is more significant. Figure 4b represents the total energy flux calculated
by means of state-to-state approach for different gases and various initial conditions. It is seen that qualitatively the
81
O 2/O
4500
4000
4000
1,1' - state-to-state
2,2' - 2-temperature, anh.osc.
3,3' - 2-temperature, harm.osc.
4 - 1-temperature
3500
3000
3000
2500
2000
1500
1'
1000
2',3'
2500
2000
1500
1,2,3
nozzle (1), T
nozzle (1), T 1
nozzle (2), T
nozzle (2), T 1
nozzle (3), T
nozzle (3), T 1
1'
1000
4
500
1
1'
2
2'
3
3'
3500
temperature
temperature
O 2/O
4500
2
1
3'
2'
3
500
0
0
0
10
20
30
40
0
50
10
20
a
x/R
30
40
50
b
x/R
FIGURE 3. Temperature and vibrational temperature versus x=R in (a) different approaches, conic nozzle and (b) different
nozzles, state-to-state approach. Curves 1–4 correspond to the gas temperature; 1’–3’ correspond to the vibrational temperature T1 .
O 2/O
10
1
2
3
4
5
state-to-state
2-temperature, anh. osc.
2-temperature, harm. osc
1-temperature
heat flux
heat flux
10
10
6
10
5
10
4
10
3
1
2
3
4
N 2/N, T *=7000K, p *=1atm
N 2/N, T *=7000K, p *=100atm
O 2/O, T *=4000K, p *=1atm
O 2/O, T *=4000K, p *=100atm
1
2,3
4
4
10
10
6
3
2
4
1
3
0
10
20
30
40
50
0
10
a
x/R
20
30
x/R
40
50
b
FIGURE 4. Heat flux q (W/m2 ) versus x=R. (a) Mixture O2 /O, T = 4000 K, p = 1 atm, conic nozzle. Different approaches. (b)
Different mixtures and various initial conditions, state-to-state approach.
behaviour of fluxes is similar for all test cases, the heat flux decreases with rising the pressure in the nozzle throat.
Using the vibrational distributions and gas temperature obtained by numerical solution of macroscopic equations
one can compute the averaged dissociation rate coefficients:
M
kdis
=
1
nmol
M
∑ nikdis
i
;
(16)
i
mol are given in Figure 5 for four kinetic models and three nozzle profiles. One can see a quite strong
The results for kdis
influence of the kinetic model on the global dissociation rate coefficient. It is also seen that the dissociation rate in
the conic nozzle is much lower compared to other nozzles. It is explained by rapid flow freezing due to the sharp
expansion.
Finally, one can conclude that thermal non-equilibrium multi-temperature quasi-stationary models provide a good
approximation for the evaluation of gas dynamic parameters and heat flux in a nozzle flow, while for the correct
82
-22
1x10
-24
10
-26
10
-28
10
-30
1x10
-32
10
-34
10
-36
10
-38
10
-40
1x10
-42
10
-44
10
-46
10
-48
O 2/O
O 2/O
1E-23
1
1E-24
dissociation rate coefficient
dissociation rate coefficient
10
2
3
4
1
2
3
4
state-to-state
2-temperature, anh. osc.
2-temperature, harm. osc
1-temperature
1
2
3
3
1E-25
2
1E-26
nozzle (1)
nozzle (2)
nozzle (3)
1E-27
1E-28
1
1E-29
1E-30
1E-31
1E-32
1E-33
0
10
20
30
x/R
40
0
50
a
10
20
30
40
50
x/R
b
mol (m3 /s) versus x=R. Mixture O /O, T = 3000 K, p = 1 atm. (a) Different
FIGURE 5. Averaged dissociation rate coefficient kdis
2
approaches, conic nozzle. (b) Different nozzle profiles, state-to-state approach.
prediction of vibrational distributions and dissociation rates the more rigorous state-to-state approach is required.
Acknowledgement. This study is supported by INTAS (grant 99-00464).
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