Heat Transfer and Diffusion in Mixtures Containing CO2
E.V. Kustova , E.A. Nagnibeda and A. Chikhaoui†
Department of Mathematics and Mechanics, Saint Petersburg State University,
198504, Universitetsky pr. 28, Saint Petersburg, Russia
†
IUSTI – MHEQ, Université de Provence, 13453 Marseille, Cedex 13, France
Abstract. In this paper the non-equilibrium vibration-chemical kinetics, diffusion and heat transfer in polyatomic gas
mixtures containing CO2 molecules are studied using state-to-state and quasi-stationary multi-temperature kinetic theory
approaches. The influence of mixture composition on the gas dynamic parameters and heat flux is discussed, the role of
different internal modes in the heat transfer is estimated.
INTRODUCTION
Investigation of non-equilibrium vibration-chemical kinetics and transport properties in reacting gas mixtures containing CO2 is important for prediction of gas dynamic parameters and heat and mass transfer in the flow near a spacecraft
entering Mars or Venus atmosphere. Consideration of strong deviations from thermal and chemical equilibrium in
polyatomic molecules with several vibrational modes requires adequate models for non-equilibrium distributions and
transport properties in multi-mode systems. During the last decade vibrational relaxation and dissociation in CO 2 mixtures have been studied experimentally [1, 2, 3] as well as theoretically. Several theoretical models of non-equilibrium
vibrational kinetics in pure CO 2 and in mixtures containing CO 2 molecules have been elaborated: some of them are
based on various quasi-stationary distributions [4, 5, 6]; other important results have been obtained in the frame of the
detailed state-to-state model [7, 8, 9, 10]. Recently the effect of vibration-dissociation coupling in mixtures of CO 2
and N2 behind shock waves on the gas temperature and vibrational temperatures of different modes has been estimated
[10]. The results have been compared with the ones obtained experimentally in [3], a comparison of theoretical and
experimental results is given in [11].
Another important problem is the influence of non-equilibrium kinetics on the transport properties. Up to now
most of existing results concerning dissipative processes in CO 2 are obtained for weak non-equilibrium conditions or
without taking into account real molecular spectra. Strong deviations from thermal equilibrium as well as anharmonic
effects have been considered in [5, 6], where various multi-temperature distributions for pure CO 2 have been derived
from the kinetic equations, and heat conductivity coefficients have been calculated on the basis of these distributions.
The aim of the present paper is evaluation of transport properties in mixtures containing dissociating CO 2 molecules.
We generalize the kinetic theory models developed in [6] for pure CO 2 , to the case of polyatomic gas mixtures taking
into account dissociation of vibrationally excited CO 2 molecules, and apply these models for the conditions behind
shock waves.
KINETIC MODELS
In this paper we consider two mixtures: CO 2 /N2 /Ar and CO2 /N2 , and take into account CO 2 dissociation (with
formation of CO and O), TV exchanges of translational and vibrational energy in all vibrational modes, VV exchanges
of vibrational quanta within each mode and VV’ exchanges of vibrational energy between different modes of CO 2 and
between CO2 and N2 . The vibrational spectra of symmetric, bending and asymmetric CO 2 modes and N2 are simulated
as anharmonic oscillators.
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
100
State-to-state model
The most rigorous state-to-state model of vibration-dissociation kinetics and transport properties is needed under the
condition of strong vibrational and chemical coupling when the characteristic times of all vibrational energy transitions
and dissociation τvibr and τdiss are of the same order as the mean time θ of the gas dynamic parameters changing, and
exceed significantly the times of translational and vibrational relaxation τtr and τrot :
τvibr
τtr < τrot
<
τdiss θ
(1)
Using the generalized Chapman-Enskog method for the solution of kinetic equations, for mixtures containing CO 2 and
N2 in each approximation one can express the distribution functions in terms of vibrational level populations ni il i ,
1 2 3
ni , chemical species number densities and gas temperature T (im , m = 1; 2; 3; 4 are the vibrational levels of CO 2
4
symmetric, bending and asymmetric modes and N 2 respectively, l is the additional quantum number describing the
projection of the momentum of bending vibrations onto the axis of the molecule). The closed set of equations for the
quantities ni il i , ni , T , gas density ρ , atomic number densities and macroscopic velocity v follows from the kinetic
4
1 2 3
equations and contains the equations for vibrational level populations coupled with the conservation equations:
dρ
dt
+ρ∇
v = 0
;
(2)
ρ
dv
+ ∇ P = 0;
dt
(3)
ρ
dU
dt
dni
+∇
l
1 i2 i3
dt
dni
4
dt
q + P : ∇v = 0
+n
+ ni
4
i1 il2 i3
(4)
;
∇ v + ∇ (ni
l
1 i2 i3
∇ v + ∇ (ni Vi
4
4
vibr
) = Ri ;
4
dnO
+ nO ∇ v + ∇ (nOVO ) =
dt
dnAr
dt
+ nAr ∇
=R
l )
1 i2 i3
Vi
∑
i1 ;i2 ;i3 ;l
v + ∇ (nArVAr ) = 0
vibr
diss
+R l ;
i1 il2 i3
i1 i2 i3
im = 0; :::; Lm ;
m = 1; 2; 3;
(5)
i4 = 0; :::; L4 ;
(6)
Rdiss
i il i
(7)
;
1 2 3
(8)
:
Here U is the total energy per unit mass, nO is the oxygen atom number density, P and q are the pressure tensor
and the total energy flux, Vi il i and Vi are the diffusion velocities of CO 2 and N2 at the corresponding vibrational
4
1 2 3
states, VO , VAr are the diffusion velocities of oxygen and argon atoms. The right hand sides of master equations for
level populations (5) and (6) describe the change of population of each vibrational state as a result of different kinds
of vibrational energy exchanges and dissociation and depend on the state-to-state rate coefficients of all considered
processes. The expressions for the production terms are reported in [12]. It should be noted that CO molecules
appearing as the result of CO 2 dissociation are supposed to be in thermal equilibrium. Also we neglect in this study
dissociation of N2 , recombination and the simultaneous transitions of translational, rotational and vibrational energy.
The state-to-state transport theory for a reacting mixture of diatomic gases has been developed in [13]. In this study
we generalize it for mixtures containing polyatomic gases. In CO 2 /N2 /Ar mixture with dissociating CO 2 the expression
for the heat flux in the first Chapman-Enskog approximation has the form:
p ∑ DTc dc +
q = λ ∇T
+
∑
i4
5
2
c
kT
∑
5
i1 ;i2 ;i3 ;l
h i
N
+ εrot2 + εi
4
2
CO2
i+εi
kT +hεrot
l
1 ;i2 ;i3
n i Vi
4
4
+
101
5
2
kT
h
i
+ε
CO2
CO
CO
+ εint + ε
ni
l +
1 ;i2 ;i3
Vi
l
1 ;i2 ;i3
nCO VCO +
+
5
5
kT + ε O nO VO + kT nAr VAr ;
2
2
(9)
where λ = λtr + λrot is the heat conductivity coefficient of translational and rotational degrees of freedom, p is the
pressure, DTc is the thermal diffusion coefficient of chemical species c (c = CO2 , N2 , CO, O, Ar), dc is the diffusive
c i and hε c i are the mean rotational and internal energies
driving force for c species, k is the Boltzmann constant, hεrot
int
of c species, εi ;il ;i and εi are the vibrational energies of CO 2 and N2 at the corresponding vibrational state, ε c is
4
1 2 3
the energy of formation of c species. For CO2 /N2 mixture the last term in Eq.(9) should be eliminated. The diffusion
velocities Vc of atomic species and species in thermal equilibrium ( c = CO, O, Ar) have the classical form:
Vc =
∑ Dcd dd
DTc ∇ ln T ;
d
Dcd are the multi-component diffusion coefficients, whereas Vi
and Vi look much more complicated:
l
1 ;i2 ;i3
Vi
l
1 ;i2 ;i3
=
∑
Di ;il ;i k ;km ;k dk ;km ;k
1 2 3
k1 ;k2 ;k3 ;m 1 2 3 1 2 3
Vi
4
here Di
l
m
1 ;i2 ;i3 k1 ;k2 ;k3
and Di
4 k4
∑ Di4 k4 dk4
=
k4
∑
d 6=N2
(10)
4
∑ Dcd dd
d 6=CO2
Dcd dd
DT
CO2
∇ ln T ;
(11)
DT ∇ ln T ;
(12)
N2
are the diffusion coefficients for different vibrational states CO 2 and N2 , di
are the diffusive driving forces for the corresponding vibrational states depending on the gradients of ni
l
1 ;i2 ;i3
l
1 i2 ;i3
;
and di
4
and ni .
4
We would like to note that in the state-to-state approach the total energy flux is defined by gradients of all level
populations with corresponding diffusion coefficients. Practical implementation of this model for transport properties
of mixtures containing polyatomic molecules with several vibrational modes calls significant computational difficulties
due to a large number of diffusion coefficients: the total amount of diffusion coefficients is about N 2 where N is the
total number of chemical and vibrational species in a system. Therefore it is important to develop more simple models
for transport properties based on the quasi-stationary distributions.
Quasi-stationary models
The experiments show that in a wide range of temperatures the characteristic times of various vibrational energy
exchanges in CO2 and N2 satisfy the relation:
k
τVV
1 2
123 4
12
τVV
τVV
τVV
; ;
0
;
0
3
0
<
τVk T
<
τdiss θ ;
(13)
k = 1; :::; 4:
k , τk m
τVV
VV 0
are the characteristic times of vibrational energy exchanges within the same k-th mode and between different
k-th and m-th modes, τVk T is the mean time of vibration-translation exchange in the k-th mode. It means that along with
fast VV exchanges within all vibrational modes, the exchange between symmetric and bending CO 2 modes proceeds
rapidly due to Fermi resonance.
Under condition (13) the vibrational temperatures of the first levels of all four modes can be introduced, and the
relation connecting temperatures of the first and the second CO 2 modes (T1 and T2 ) due to Fermi resonance can
be written. Finally all level populations are expressed in terms of combined vibrational temperature of symmetric
and bending CO 2 modes T12 = T2 , and temperatures T3 and T4 of the first levels of asymmetric CO 2 mode and N 2 ,
gas temperature T and species concentrations [10]. The equations of detailed vibration-chemical kinetics (5)–(6) are
reduced in this approach to 5 equations for the concentrations of CO 2 and N2 and for temperatures T12 , T3 and T4 .
Consequently, the set of macroscopic equations is essentially simplified.
The expression for the total energy flux in this case reads:
q = λ ∇T
λv;12 ∇T12
p ∑ DTc dc + ∑
c
c
5
2
λv;3 ∇T3
λv;4 ∇T4
c
kT + hεint
i + ε c ncVc :
102
(14)
The coefficient λ = λtr + λrot + λvt is the coefficient of thermal conductivity due to the translational, rotational energy
exchange and the non-resonant character of rapid VV (k) and VV (1 2) exchanges (in all these transitions some part
of vibrational energy transfers to the translational energy); vibrational heat conductivity coefficients λv;12 , λv;3 and
0
λv;4 describe the transport of vibrational energy due to the fast VV (1 2) , VV(3) and VV(4) exchange. The diffusion
velocities of all chemical species are given now by Eq. (10). One can see that in the quasi-stationary approach q
depends on the gradients of temperatures T12 , T3 , T4 instead of gradients of level populations, and only three additional
vibrational heat conductivity coefficients appear instead of numerous diffusion coefficients for different vibrational
states. After some simplifications all diffusion and heat conductivity coefficients are expressed in terms of T , T12 , T3 ,
T4 , number density of species and elastic collision integrals.
0
RESULTS AND DISCUSSION
In this section we present the results of calculations of energy fluxes in two mixtures: 2000ppm CO 2 + 10%N2 + Ar and
2000ppm CO 2 + N2 . Non-equilibrium flows of these mixtures behind a plane shock wave have been considered in the
four-temperature approximation discussed in section 2.2. The initial conditions are: for the CO 2 /N2 /Ar mixture, M0 =2,
T0 = 1258 K, p0 = 0:167 atm; for the CO 2 /N2 mixture, M0 =2.4, T0 = 1364 K, p0 = 0:118 atm. Equilibrium conditions
behind the shock are the same for both cases considered: Teq = 2495K, peq = 0:78 atm. The initial vibrational
distributions are assumed to be thermal equilibrium. The rate coefficients of various vibrational energy exchanges
and dissociation are the same as those used in [10] and [11].
The values of T , T12 , T3 , T4 and number densities of species have been found from the equations of a one-dimensional
stationary flow behind a shock in the Euler approximation. The time dependence of the gas temperature and the
temperatures of the first level in each mode behind the shock calculated using the four-temperature model is presented
in Fig. 1. It is seen that in CO2 /N2 /Ar vibrational relaxation of CO 2 proceeds faster because of two reasons: a higher
efficiency of Ar in VT relaxation of CO 2 , and a lower influence of the VV (3 4) exchange in this case. Increasing of N 2
concentration leads to a slower equilibration of the asymmetric mode because of the loss of vibrational energy during
the near-resonant interaction with N 2 . In the case of CO2 /N2 mixture the temperatures T3 and T4 are much closer one to
another. In both cases the first and the second CO 2 modes come to equilibrium much faster than the remaining modes,
the slowest process is VT relaxation of N 2 .
0
(a)
T12
2400
T
2600
T3
2200
T4
2000
temperature
temperature
(b)
2800
T
2600
1800
1600
2400
T12
2200
T3
2000
T4
1800
1600
1400
1400
1200
1200
0
20
40
60
80
100
0
t, µs
FIGURE 1.
20
40
60
80
100
t, µs
Temperatures as functions of time. (a) CO2 /N2 /Ar; (b) CO2 /N2 .
The oxygen atom concentration calculated in both mixtures is presented in Fig. 2. It is seen that the rate of
dissociation is considerably higher in CO 2 /N2 mixture, it is explained by the fact that nitrogen is more efficient in
CO2 dissociation compared to Ar. Another reason is that in the present conditions the gas temperature just behind the
shock front is higher in CO 2 /N2 mixture.
103
nO x 10 -12cm -3
0,10
CO 2/N 2
0,05
CO 2/N 2/Ar
0,00
0
50
100
t, ms
FIGURE 2.
Oxygen atom concentration as a function of time.
At the next step the macroscopic parameters computed in flows considered above have been inserted to the kinetic
theory code for the the energy flux evaluation. Fig. 3 reports the total energy flux as well as its part determined only
by heat conductivity qF = λ ∇T λv;12 ∇T12 λv;3 ∇T3 λv;4 ∇T4 . This allows one to estimate the contribution of
diffusion and thermal diffusion to the heat transfer. The total heat flux is rather small in the mixture CO 2 /N2 /Ar, in
the second mixture it has significant absolute value just close to the front and then decreases with rising the distance
x from the shock front. It is explained by the gas temperature behavior. The values of q and qF are very close to each
other, in the mixture CO 2 /N2 they practically coincide. Therefore, in our cases, the heat flux is determined mainly
by heat conductivity, the role of diffusion and thermal diffusion processes is weak. It means that the influence of
dissociation process on the energy transfer is not important under conditions considered: otherwise the gradients of
species concentrations would make a higher contribution to the diffusion velocities. Nevertheless, at higher initial
velocities one can expect more significant effect of dissociation and diffusion on the energy flux.
0
heat flux
-100
-200
1'
1
2,2'
-300
-400
0,00
0,01
0,02
0,03
0,04
0,05
x, m
FIGURE 3. Heat flux q (W/m2 ) as a function of x. Curves (1,1’) – CO2 /N2 /Ar; (2,2’) – CO2 /N2 . (1,2) – total heat flux; (1’,2’) –
heat flux due to heat conductivity.
104
0.00
(a)
(b)
3000
2000
qtr
1000
heat flux
heat flux
300
250
200
150
100
50
0
-50
-100
-150
-200
-250
-300
qvibr
0
-1000
qvibr
-2000
-3000
0.01
0.02
0.03
0.04
-4000
0.00
0.05
x, m
FIGURE 4.
CO2 /N2 .
qtr
0.02
0.04
x, m
Fluxes of translational-rotational and vibrational energy qtr and qvibr (W/m2 ) as functions of x. (a) CO2 /N2 /Ar; (b)
Fig. 4 presents a comparison of the translational-rotational and vibrational energy fluxes qtr = λ ∇T , qvibr =
λv;12 ∇T12 λv;3 ∇T3 λv;4 ∇T4 calculated for two mixtures. It is seen that qtr and qvibr have very close absolute
values and different signs, therefore these fluxes compensate one another, and finally the total heat flux is found to be
low. In the beginning of the relaxation zone the absolute values of qtr and qvibr are rather high; energy fluxes in the
CO2 /N2 mixture exceed considerably those in the CO 2 /N2 /Ar mixture; however with the distance from the shock front
they tend to zero in both cases. Calculations show that qvibr is defined mainly by the gradient of T4 because the nitrogen
concentration is much higher than that of CO 2 and consequently the heat conductivity coefficients for CO 2 appear to
be small. That is why the flux qvibr is considerably higher in the CO 2 /N2 mixture; the translational-rotational energy
flux in this case exceeds qtr in the CO2 /N2 /Ar mixture because of the difference in the gas temperature T gradient
behavior (see Fig. 1).
Acknowledgement. This study is supported by INTAS (grant 99-00464).
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