State-to-State Kinetic Description
of Non-Equilibrium Radiative Gas Flow
A. Chikhaoui*, E.V. Kustovaf
*IUSTI - MHEQ, Universite de Provence 13453 Marseille, Cedex 13, France
t Math, and Mech. Dept, Saint Petersburg University, 198904) Bibliotechnaya pi. 2, Saint Petersburg, Russia
Abstract. The state-to-state kinetic theory approach is applied for the investigation of radiative reacting
gas mixture flows with strong vibrational and chemical non-equilibrium. A closed system of macroscopic
equations taking into account the coupling of vibrational relaxation, chemical reactions and radiative transitions is derived. The state-to-state model is compared with the quasi-stationary ones.
I
INTRODUCTION
The state-to-state vibration-chemical kinetics in various gas flows has been studied by many authors during
the last decade. The growing interest to the state-to-state approach is explained by the fact that in high temperature and high enthalpy flows there exists a strong coupling between the processes of vibrational relaxation
and chemical reactions. Under these conditions the multi-temperature models based on the quasi-stationary
vibrational distributions are not valid, and the master equations for vibrational level populations must be
solved for a correct prediction of gas flow parameters. This approach has been used to investigate many nonradiative flows, see [1-6]. The transport theory in the state-to-state approach has been developed in [6,7], the
state-to-state reaction rates have been considered in [8].
The objective of the present contribution is to generalize the state-to-state model elaborated in [6,7] in order
to take into account radiative processes. Radiative transitions are of great importance for laser technology,
photochemistry, shock tube measurements; the estimation of heat transfer caused by radiation is necessary
for the calculation of thermal protection systems. The macroscopic equations of aerothermochemistry taking
into account radiative effects are given, for instance, in [9—11]. However, the greatest part of equations for
radiative transfer is based on the usual assumption about local thermal equilibrium. A kinetic model of a
strongly non-equilibrium radiative reacting gas flow is elaborated in [12].
In the present paper a closed description of a non-equilibrium flow of gas mixture with coupled vibrational
relaxation, dissociation, recombination, exchange reactions, absorption and emission of photons on the basis
of the rigorous kinetic theory is proposed.
It is known that the kinetic scheme depends essentially on the relation between the characteristic times of
various relaxation processes. Numerous experimental results [13] show that translational and rotational modes
of molecules equilibrate much faster than vibrational relaxation, chemical reactions and radiative processes
proceed. The characteristic times of the relaxation processes are connected by the following relation:
Tel < Trot «
Tvibr < Treact ~ Trad ~ 9,
(1)
Here r e j, r ro t, rV2-tr, r reac t, rrad are respectively the mean times between the elastic collisions, those with
change of rotational and vibrational energy, chemical reactive collisions, and the ones leading to absorption
and emission of a photon; 9 is the macroscopic time. Moderate gas temperatures are considered when ionization
and electronic excitation of atoms and molecules may be neglected. Condition (1) provides the basis for the
state-to-state description of a non-equilibrium flow.
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
680
II
KINETIC EQUATIONS
The gas flow is simulated on the basis of the kinetic equations for distribution functions. The motion
of particles is considered in a six-dimensional phase space, its coordinates are the three components of the
position r and three components of the momentum p. A radiation field may be regarded as being equivalent to
a collection of particle-like entitles called photons [14]. Photons are characterized by the following properties:
they all move with the speed of light c, their energy is determined by their frequency v\ e^ = hv (h is the
Planck constant), and they carry a momentum p^ = hvjc - O^ (O^ is the unit vector defining the direction of
travel of the photon) .
The distribution functions fdj (r, pc, t) are introduced for every chemical species c, vibrational i and rotational
energy level j, and for photons /^(r, p^ 3 1). The kinetic equations for distribution functions have the next form:
mc
Or
(3)
mc is the mass of a material particle. For further development it is useful to determine the distribution functions
of material particles in the phase space (r, u) where u is the microscopic velocity of a particle. Hereafter the
same nomenclature for the distribution functions in different phase spaces is retained.
The integral operators Jcij and J^ describe all the collisions leading to the change of distribution function.
They can be written as a sum of several terms corresponding to different processes:
/
_ jel
Jcij — Jcij
i J jinel \ J /react i J rrad
' dj
' dj
' dj '
( A\
W
The integral operators of elastic and inelastic collisions J®/ • and J^f?1 correspond to the collisions of material
particles which do not result in the change of chemical species. Elastic collisions lead to the change of only the
particle velocities, and in the inelastic collisions the internal state of molecules varies also. Operator J^f?1 can
describe the inelastic RT rotational-translational exchange, VT vibrational-translational and VV exchange of
vibrational quanta. The expressions for the collision operators J^- and J^f?1 are given in [15,16].
The integral operator of chemical reactive collisions J^f^ describes exchange reactions and dissociationrecombination process. Therefore
/react _
J
cij
jexch •
— Jcij
jdiss — rec
T Jcij
'
/pr\
\°)
The expressions of J^ch and J^s~rec can be found in [16-19].
Among the radiative collisions the ones leading to absorption, emission and scattering of photons may be
distinguished. In an absorption process, an entire photon is removed from the incident beam and the photon's
energy is deposited in the material particle. The inverse processes is the induced emission, in this case photons
of a particular frequency in the incident beam stimulate an excited particle to emit a photon in the same
direction as the beam and at the same frequency. These two processes are described by the transition
Adj + hi/ <— ¥ Adiji + 2Ai/.
(6)
An isolated excited particle is also capable of emitting a photon, even in the absence of stimulating radiation.
This process is referred to as spontaneous emission and has no corresponding inverse process:
Adj — ¥ Aci>j> + hi/.
(7)
Scattering of photons by a material particle occurs as a result of the following process:
Adj + hv<^ Ad^i + hi/1.
(8)
In this study the gas flow at moderate temperature is simulated, when the degree of ionization and electronic
excitation is negligibly small (T < 6000 — 8000K). In this case one may take into account only the bound-bound
radiative transitions between internal energy levels of molecules and neglect the transitions between electronic
states and also bound- free and free- free radiative transitions. Under this condition the collision operator of
radiative processes is obtained in the form:
jrad __ jem — as
cij — Jcij
J
The expressions for the collision operators J^~a
s
, /sc
scat
' Jcij
and «/*/?*
/*/ have been derived in [12].
681
{Q\
Ill
DISTRIBUTION FUNCTIONS AND MACROSCOPIC EQUATIONS
The dimensionless kinetic equations are given by
„
dt
Oi
'
at
c
L'BBl/
Q
t/r
——
£
t/jy
T «/j/
•
Here J™?, J™p and «7*L, ^ are ^e collision operators of rapid and slow processes, e = Trap/rsi is the small
parameter, r ra p, r 5 j are the average times between the frequent and rare collisions respectively. For the solution
of the kinetic equations the Chapman-Enskog method generalized for rapid and slow processes [16], [20] is used.
The solution is found as an expansion of the distribution functions in a power series of the parameter e.
It is known that the zero order distribution function is determined by the summational invariants of the
most frequent collisions (eigenfunctions of the linearized collision operator of rapid processes J™f) [16]. The
collision invariants are found under specific flow conditions.
Under condition (1) the operators of rapid and slow processes may be written in the form:
J
jrap __ jel
cij
~~ Jcij
,
jrot
' Jcij 5
J jrap
v
J
jsl
_
jvibr •
cij ~ Jcij
__ n
— U,
rreact ,
' Jcij
'
J
rrad
cij >
rsl __ rrad
Jv — J^
^19^
\LZi)
(^ o\
^lOJ
In the conditions of slow radiative processes the kinetic equation for the zero order distribution function of
photons /j) ^ is satisfied identically. It means that the formalism of asymptotic methods of small parameter
cannot be applied for determination of the photon distribution functions. Therefore /^ are to be found directly
from microscopic equations (11).
= Tn
The collision invariants of any collision of material particles are: ^-j = 1, ^ij
cUcfj, (ft = 1 5 2 , 3 are
the spatial indices), ip^A = ra c t^/2 + £^. For photons ipv
= j?^ (fj, = 13 2 3 3), ^ = hv. Besides that there
exist additional invariants of the most frequent collisions. In the conditions of slow vibrational relaxation and
chemical reactions, any variable aC2- independent of the velocity and the rotational energy level j and depending
a
arbitrary on c and i is conserved: ^ij
di (^ = 1> "-5 ^> where N = J^c« -0The macroscopic parameters corresponding to the collision invariants in the generalized Chapman-Enskog
method are determined by the zero order distribution function:
C 2 3
Em
3 * * * j C j
jr(0)j
If Ucfrijduc
= /?v,
c
cij
3 * * * 3 3
\ /
/1C\
(15)
J
pEvibr + pEt = pU.
(16)
cij
Here nci is the level population of ith level of chemical species c, L is the number of chemical species, Lc is
the number of vibrational levels in c chemical species, v is the macroscopic gas velocity, p = J^c mcnc is the
gas density, Cc = uc — v is the peculiar velocity, n = J^c nc is the total number density of particles, ej is the
rotational energy of a molecule c chemical species at ith vibrational level, £? is the vibrational energy, ec is the
energy of formation of particles of c species, Erot, Evibr and Ef are, respectively, the rotational, vibrational
energy and energy of formation per unit mass, U is the total specific energy of material particles.
On the basis of the system of collision invariants and using the normalizing conditions (14)—(16) the zero
order distribution function of particles /^-- is obtained in the form of the Maxwell-Boltzmann distribution over
velocity and rotational energy depending on the non-equilibrium level populations of chemical species [20], [7]:
682
(0) _ / _meJ|»» **
/c
« ~ \2wkT)
exp -
-
V
Zfot
\ 2kT
(
kTj'
(17)
'
s™ is the rotational statistic weight, Zfot is the rotational partition function. The distribution function (17) is
expressed in terms of macroscopic parameters n C2 -(r,£), v(r,tf), T(r,tf).
Following reference [14] it is conventional to define a specific intensity /^ of the radiation field in such a way:
Ivdvdnv = chvfvdp^,
(18)
dOjx denotes an element of solid angle defined by the unit vector O^. Consequently Eqs.(ll) for fv may be
rewritten in terms of !„ :
ldlv
, _
A4!/3
dlv
jrad
J
--HT
c at + ""-*ar = ~^~
cd ™ »
, 10 ,
^ = ^1,^2,...,^-
(19)
In the general case there is an infinite set of frequencies v at which a photon can appear. In practical calculations
the number of equations R corresponds to the number of characteristic frequencies in a flow. The equations
for the specific intensity in the form (19) are equivalent to the equations of radiative transfer [14].
For further development it is convenient to introduce a mean momentum of photons per unit volume, TT, and
a mean radiative energy per unit volume, £radT = fpjvdpv
=1/
C
J
JO
/
Ivnvdvdtov,
£rad = f /^/^ = - /
J47T
C
J
JO
/
Ivdvdflv.
(20)
J47T
Note, that for these quantities there are no normalizing conditions similar to Eqs.(14)— (16) for macroscopic
parameters of material particles.
Now let us derive the macroscopic equations. For the vibrational level populations n C2 -(r,£), similarly to [7]
one can obtain:
+ n c f V - v + V.(n c f V c f ) = .Rcf, c=l,..,L, i = 0,...Lc.
(21)
Here VC3- is the diffusion velocity of c species at the ith vibrational level:
/
3
the production terms in these equations are defined by all slow processes:
*.
(22)
Equations (21) represent the equations of detailed vibrational-chemical kinetics taking into account radiative
transitions.
The conservation of momentum in the general case may be expressed in the form [12]:
•
In this equation P, Prad
are
7,
at
J
r\,
ut
'
v
'
v
TQid
'
\
)
the tensors of pressure for material particles and photons:
P = V" / mcCcCcfcijduc,
^A
cij
'
Prad = / p^cO^dp^ = - /
J
C
JO
/ J^O^O^
J^K
It should be noted that the transfer of momentum in a photon-particle collision is very small, because usually
the momentum carried by photons is much less compared to the momentum of material particles. Therefore,
the contribution of TT and Prad to the conservation equation of momentum may be neglected, and Eq.(23) takes
the usual form:
683
P + V - P = 0.
(24)
On the contrary, the exchange of energy in the photon-particle collision is not negligible because the energy
of a photon is of the same order of magnitude as the kinetic energy of a material particle. Therefore the
conservation of total energy is written as follows:
. qr . ad + P : V v = 0 .
(25)
Here q is the heat flux for material particles, qra{^ is the radiative heat flux:
-
-V^ + ef
f
-
J
f°° f
JQ
J4TT
The expressions for all transport terms of material particles have been derived in [6,7]. In particular, it has
been shown that in this approach the heat flux is determined by the gradients of the gas temperature, number
densities of atomic species and non-equilibrium populations of all vibrational levels of molecular species. The
transport coefficients involved in the expressions for VC2-, P and q are also obtained in [7]. Concerning the
radiative flux, it can be found using the solution of equations of radiative transfer (19).
Equations (21), (24), (25) and (19) represent a closed system of equations describing the flow of a reacting
and radiating gas mixture under strong non-equilibrium conditions. It is interesting to mention that in the zero
order approximation of the Chapman-Enskog method the diffusion velocity and heat flux by material particles
vanish as a result of the Maxwellian distribution over velocities (Vci = 0, q = 0). However, the term containing
radiative flux q ra ^ should be kept in Eq.(25) in the Euler approximation, because it depends on the specific
intensity !„ (distribution function of photons /^) that is not determined by the order of approximation of the
Chapman-Enskog method.
IV QUASI-STATIONARY MODELS
Under some specific flow conditions, the set of equations (19), (21), (24), (25) can be simplified and reduced
to a less number of equations. Thus in the case of rapid VV exchange within each vibrational mode compared
to VV1 exchange between different species and VT exchange the relaxation times are related by the expression:
Tel < Trot < Tyv «
Tyv1 ~ TyT < Treact ~ Trad ~ 0.
(26)
Under this condition the non-equilibrium quasi-stationary Treanor or Boltzmann vibrational distributions with
the vibrational temperatures different from T establish in the flow [20]. The master equations for nc{ are reduced
in this case to several relaxation equations for the mean number of vibrational quanta in molecular species and
to the equations of chemical kinetics for all chemical species.
With temperature rising the relaxation times of all internal degrees of freedom become comparable:
Tel < Trot < Tvibr «
Treact ~ Trad ~ 0.
(27)
In this case nci have the form of the thermal equilibrium one-temperature Boltzmann distributions, and the
system of macroscopic equations contains only the conservation equations coupled with the equations for the
concentrations of chemical species.
Both multi-temperature and one-temperature approaches have been considered in [20]. In this paper the
closed sets of macroscopic equations and transport properties for the quasi-stationary models have been derived.
However, the radiative transitions have not been taken into account in [20]. In order to include the radiative
processes to the multi-temperature or one-temperature model, first, it is necessary to couple the corresponding
macroscopic equations with equations (19) of radiative transfer, and second, one should keep in mind that the
production terms of the relaxation equations depend on the collision operator J™d.
684
V
DIATOMIC GAS FLOW BEHIND A SHOCK WAVE
In the Euler approximation for a steady-state one-dimensional flow of diatomic gas with W, VT vibrational
energy exchange, dissociation and radiative transitions between discrete vibrational levels, equations (21) of
detailed vibration-dissociation kinetics are written in [12]:
d(Vni)
—————
_
jjVT
, r>VV
, rfdiss
dx
,, nrad
' **>i
-_ n
r
5 2 — ", ...1^,
/Oo\
(/oj
where i? is the flow velocity in the x direction, Hi is the population of iih vibrational level, na is the number density of atoms, -R^T, B%v , Rfss, R™d are the production terms due to FT, W vibrational energy
exchange, dissociation-recombination and radiative transitions, respectively. These production terms may be
written substituting the zero order distribution function (17) to the expressions for the collision integrals. The
production terms due to collisions of material particles are obtained in [21,8,22]. The rate coefficients of VV,
VT vibrational energy exchange have been calculated by many authors [23-25,2,26]. The most reliable results
on these rate coefficients are obtained using exact quantum trajectory calculations [25]; among the recent
analytical models one can mention the ones proposed in [2,26]. Dissociation-recombination reaction rates have
been also widely discussed during the last years. For the case of strong vibration-dissociation coupling two
models of the state-to-state dissociation rate coefficients are used: the ladder climbing model [4,21,22] and the
more general one [3,22] based on the well known Treanor- Mar rone model [27].
The production rates due to radiative transitions R™d neglecting scattering have been derived in [12]:
Here H{J is the population of the jih rotational level at the iih vibrational state, Bi'jiij, Bijiiji, Aijiiji are the
integral Einstein coefficients for absorption, induced and spontaneous emission, respectively. These coefficients
describe probabilities of all radiative transitions which contribute to the transition (ij —>• ifjf). Expression (30)
is obtained with the assumption that !„ and hv are slowly varying functions of frequency over the line width.
Besides that we have taken into account the isotropic character of spontaneous emission.
As there exists a quasi-stationary Boltzmann distribution over rotational energy, one can define the averaged
over rotational spectrum specific intensity Iv^i and Einstein coefficients B^2-, Bat, AHL The intensity Iv^i
may be interpreted as a mean specific intensity of vibrational line, B^2-, BW, Ant as the Einstein coefficients
for the radiative transitions corresponding to the vibrational transition (i —>• i ; ). In this case the production
rate R™d is given by the expression similar to the one obtained in [24]:
Rl«d = £ (nyl^yBvi - Hi (IVtii,Bw + An,)) .
(31)
••'<••
The Einstein coefficients can be calculated using the method described, for instance, in [14].
Equations of detailed vibration-dissociation kinetics (28), (28) should be coupled with the conservation
equations of momentum and total energy and equations of radiative transfer. In the Euler approximation for
a stationary one-dimensional flow conservation equations (24) and (25) are reduced to the equations
"
+
= °-
<32>
where p = nkT is the pressure.
The equations of radiative transfer neglecting the scattering take the form:
dx
= a?*, i/ = 1/1 ,1/5,, ...,!/*,
685
(34)
!„ describes the specific intensity of radiation propagating in the positive direction of the axis x. The right
hand sides of these equations are given by:
rad
(35)
where &2>j>^-, 6,-ji/j/, Oiji/j/ are the spectral Einstein coefficients for absorption, induced and spontaneous emission. After performing the averaging over rotational spectrum, one has
jrad
(36)
+—
4?r
It should be mentioned here that these expressions are obtained without the usual assumption about local
thermal equilibrium.
The infrared radiation intensity of N% first positive band has been calculated in a flow of dissociating nitrogen
N% behind a strong shock wave (see Fig. 1). The free stream conditions are the following: MQ = 15, TO = 293 K,
PQ = 100 Pa, initially the gas is supposed to be in equilibrium. Three different approaches have been used for
the calculations: the state-to-state, the multi-temperature model based on the quasi-stationary non-equilibrium
Boltzmann distribution, and the one-temperature model based on the local thermal equilibrium assumption.
One can see that both quasi-stationary approaches give an overestimation of the intensity. This fact can be
explained by the different behaviour of the N% concentrations computed using different models (see [6,7] for
the discussion of the macroscopic parameters behaviour).
16 i
15
*14
1131
1 12
C/J
11
10
0.2
0
0.4
0.6
0.8
x, cm
FIGURE 1. Infrared radiation intensity of N% first positive band, S, kW/cm3 • sr, as a function of x. Curve 1
present approach, curves 2, 3 - multi-temperature and one-temperature approaches, correspondingly.
VI
CONCLUSIONS
A kinetic model for the simulation of a reacting and radiating gas mixture flow under strong non-equilibrium
conditions is proposed. A closed set of macroscopic equations is derived, this set contains the equations of
detailed vibration-dissociation kinetics coupled with the radiative transfer equations and conservation equations
of the momentum and total energy. Solution of these equations can give an insight of a mutual effect of
relaxation processes and radiation. A comparison of the state-to-state, multi-temperature and one-temperature
models provides a possibility to estimate the influence of non-quasi-stationary distributions on the radiative
flux.
686
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