On the simplified state-to-state transport coefficients

Chemical Physics 270 (2001) 177±195

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On the simpli®ed state-to-state transport coecients E.V. Kustova * Department of Mathematics and Mechanics, Saint Petersburg University, Bibliotechnaya pl. 2, 198904 Saint Petersburg, Petrodvoretz, Russia Received 6 December 2000; in ®nal form 21 March 2001

Abstract Simpli®ed expressions for the state-to-state transport coecients are proposed. The main stress is on the di€usion of vibrational energy. The state-to-state di€usion coecients are compared to the commonly used di€usion coecients, and the role of non-equilibrium vibrational distributions in the di€usion process and heat transfer is estimated. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 51.10.‡y; 51.20.‡d; 82.20.Mj; 82.20.Rp Keywords: State-to-state vibrational kinetics; Transport properties; Di€usion coecients

1. Introduction Recent advancement of state-to-state kinetic theory can be explained by the growing interest to di€erent kinds of non-equilibrium ¯ows. It is well known that in many real gas ¯ows the strongly non-equilibrium conditions arise, so that widely used quasi-stationary distributions over vibrational energies are not valid. Excellent examples are the ¯ows behind shock waves [1±3], expanding ¯ows [4±7], ¯ows in the boundary layer under reentry conditions [8,9]. In the papers cited above the master equations for the vibrational level populations have been solved and noticeable deviations from the Boltzmann vibrational distribu-

*

Fax: +7-812-428-6649. E-mail addresses: [email protected], elena.kustova@ pobox.spbu.ru (E.V. Kustova).

tion have been found in some regions of the ¯ow ®eld. Another important aspect of the state-to-state kinetic theory is the evaluation of transport properties under non-equilibrium conditions. The formal transport kinetic theory in the case of strong vibrational and chemical non-equilibrium has been developed in Ref. [10]. In the more recent papers [11±14] the e€ect of non-equilibrium vibrational distributions on the heat transfer and di€usion in di€erent ¯ows is estimated. In particular, it has been shown that the vibrational energy transport by di€usion makes a signi®cant contribution to the total heat ¯ux behind shock waves and in the boundary layer. The investigation of transport properties in papers [11±14] has been performed using an approximate approach. First, the macroscopic parameters (vibrational distributions, species concentrations, temperature, pressure) are computed either in the Euler approach [11,13] or on the basis

0301-0104/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 1 ) 0 0 3 5 2 - 4

178

E.V. Kustova / Chemical Physics 270 (2001) 177±195

of essentially simpli®ed models for transport coecients [12]. In the paper [14] the state-to-state distributions are derived by the DSMC method. Then the calculated macroscopic parameters are used as input for the transport properties evaluation. The reason for invoking such an approximate scheme is that the practical implementation of the state-to-state transport coecients in the computational ¯uid dynamics (CFD) is extremely dicult, re-calculation of these coecients at the each step of numerical code leads to very high consumption of computational time. The computational cost of the linear transport systems solution is discussed in Section 3. The aim of the present paper is to derive the simpli®ed linear transport systems and thus reduce noticeably the time for the transport coecients calculation. The obtained systems consist of few equations, it makes possible to solve them at the each step of CFD code, and therefore to get more self-consistent results. Another objective is to check the validity of some commonly used models for the di€usion coecients.

2. Transport terms in the state-to-state approach The state-to-state transport kinetic theory is developed in Ref. [10] on the basis of the generalized Chapman±Enskog method. An experimentally observed relationship between the characteristic times of main processes [15] sel < srot  svibr < sreact  h;

…1†

(sel , srot , svibr , sreact are respectively the mean times of translational, rotational and vibrational relaxation and chemical reactions, h is the macroscopic time) provides a strong vibrational±chemical coupling. In this case no quasi-stationary vibrational distributions exist, and the zero order distribution …0† function fcij (c, i, j denote chemical species, vibrational and rotational quantum numbers) is obtained under the form of Maxwell±Boltzmann distribution over velocities and rotational energy depending on the non-equilibrium vibrational level populations and chemical species concentrations:

…0† fcij

  m 3=2 nci sci c j ˆ exp ci 2pkT Zrot

mc c2c 2kT

ecij kT

 ;

…2†

mc is the molecular mass, k is the Boltzmann constant, T is the gas temperature, nci are the nonequilibrium vibrational level populations, cc ˆ uc v is the peculiar velocity, v is the macroscopic gas velocity, scij , ecij are the statistic weight and energy of the jth rotational state corresponding to ci the vibrational level i of chemical species c, Zrot is the rotational partition function. The ®rst order distribution functions are given by the expression [10]: …1†

1 X dk D
ddk n dk cij  1 1 Fcij r
v Gcij ; n n

1 Acij
r ln T n

…0†

fcij ˆ fcij

1 Bcij : rv n

…3†

n is the total number density, dci are the di€usion driving forces for each chemical and vibrational species:  n  n qci ci ci dci ˆ r ‡ r ln p; …4† n n q P P qci ˆ mc nci , q ˆ c mc i nci , p is the pressure. Functions Acij , Bcij , Ddk cij , Fcij and Gcij are found from the linear integral equations derived in Ref. [10]. A closed set of macroscopic equations for the macroscopic parameters nci , v, T is obtained in Refs. [10,16] and consists of the conservation equations for the momentum and total energy coupled with the equations of detailed vibrationchemical kinetics for the vibrational level populations nci . The ®rst order transport terms are determined by the ®rst order distribution functions. The pressure tensor P has been obtained in the following form: P ˆ …p

prel †I

2lS

gr
vI:

…5†

Here I is the unit tensor, S is the tensor of deformation velocities, l, g are the shear and bulk viscosity coecients, prel is the relaxation pressure:

E.V. Kustova / Chemical Physics 270 (2001) 177±195

lˆ

kT ‰B; BŠ; 10

g ˆ kT ‰ F ; F Š;

prel ˆ kT ‰ F ; GŠ: …6†

The bracket integrals ‰ A; BŠ introduced in Refs. [10,16] are determined by the cross-sections of the most frequent collisions, i.e. the elastic collisions and those leading to the rotational energy exchange (see Appendix A.1 for de®nition). Therefore these bracket integrals di€er from the ones de®ned in the classical thermal equilibrium onetemperature approach. In the latter case the bracket integrals depend also on the cross-sections of vibrational energy exchanges. The di€usion velocity of the molecules of each chemical and vibrational species has been found in Ref. [10] under the form: Vci ˆ

X

Dcidk ddk

DTci r ln T ;

The expressions for the heat ¯ux and di€usion velocity contain not only the gradient of the gas temperature but also the gradients of all level populations of di€erent molecular species and number densities of atoms with corresponding di€usion and thermal di€usion coecients. In the general case these coecients are di€erent for various vibrational and chemical species. Following the procedure proposed in Ref. [10], the functions Acij , Ddk cij and Bcij are expanded into ®nite series of Sonine and Waldmann± Tr ubenbacher orthogonal polynomials over reduced translational and rotational energy correspondingly:    ci  ej mc cc X mc c2c …r† …p† Acij ˆ aci;rp S3=2 Pj ; 2kT rp 2kT kT …11†

…7†

dk

where Dcidk and DTci are the di€usion and thermal di€usion coecients for every chemical and vibrational species: Dcidk

1  ci dk  D ;D ; ˆ 3n

DTci

1  ci  D ;A : ˆ 3n

…8†

The expression for the total heat ¯ux in the ®rst order approximation takes the following form: X q ˆ k0 rT p DTci dci ‡

X5 ci

2

ci

 kT ‡ hecij ir ‡ eci ‡ ec nci Vci ;

179

…9†

Ddk cij

  mc cc X dk …r† mc c2c ˆ d S ; 2kT r ci;r 3=2 2kT

Bcij ˆ

mc 2kT

The transport coecients can be easily expressed in terms of the ®rst non-vanishing coecients of expansions (11)±(13): k0 ˆ

X 5 nci X k nci k aci;10 ‡ crot;ci aci;01 ; 4 n 2 n ci ci

Dcidk ˆ

k k0 ˆ kt ‡ kr ˆ ‰A; AŠ 3

DTci ˆ

is the coecient of thermal conductivity. The coecients kt and kr express thermal conductivity connected with elastic and inelastic translational± rotational TR energy transfers. It is important to emphasize that vibrational energy does not contribute to the thermal conductivity in this approach.

   1 2 X mc c2c …r† cc I bci;r S5=2 : 3 2kT r …13†

where hecij ir is the mean rotational energy of c species at the vibrational level i, eci , ec are the vibrational energy of the ith level and energy of formation of chemical species c, …10†

 cc cc

…12†

lˆ

1 ci d ; 2n dk;0 1 aci;00 ; 2n

kT X nci bci;0 : 2 ci n

…14†

…15†

…16†

…17†

Here crot;ci is the dimensionless (divided by the factor k=mc ) rotational speci®c heat of c species at the ith vibrational level:

180

crot;ci ˆ

E.V. Kustova / Chemical Physics 270 (2001) 177±195

mc oErci ; k oT

qci Erci ˆ

XZ j

ecij fcij duc :

Using the polynomial expansions, the integral equations for functions Acij , Ddk cij , Bcij can be reduced to the systems of linear algebraic equations. Thus the system of equations for the expansion coecients aci;rp has the following form [10]: XX 15kT nci dr1 dp0 Kcidk rr0 pp0 adk;r0 p0 ˆ 2 n dk r0 p0 nci crot;ci dr0 dp1 ; ‡ 3kT …18† n c ˆ 1; . . . ; L;

i ˆ 0; 1; . . . ; Lc ;

r; p ˆ 0; 1; . . .

Here L is the number of chemical species, Lc is the total number of vibrational energy levels in molecular species c. The coecients of system (18) are expressed in terms of level populations and partial bracket integrals: i0 X nci nbl h p rp r 0 p0 d Kcidk ˆ m m d Q ; Q 0 0 c d cd ik rr pp cibl n2 bl ! i nci ndk h 0 0 00 ‡ 2 Qrp ; Qr p ; …19† cidk n r    ci  ej mc mc c2c …r† …p† Q ˆ cc S Pj : 2kT 3=2 2kT kT rp

Eq. (18) occurs not to be linearly independent in the case r ˆ p ˆ 0, and therefore must be solved using a constraint obtained in Ref. [10]: Xq ci aci;00 ˆ 0: …20† q ci dk Similarly the equations for the coecients dci;r can be written:   XX qci cidk bl crr0 ddk;r0 ˆ 3kT dcb dil …21† dr0 ; q dk r0

b; c ˆ 1; . . . ; L;

i; l ˆ 0; 1; . . . ; Lc …Lb †;

r ˆ 0; 1; . . . ; where ccidk represent the particular case of the rr0 cidk 0 cidk bracket integrals Kcidk rr0 pp0 at p ˆ p ˆ 0: crr0 ˆ Krr0 00 .

A linearly independent system is obtained taking dk : into account constraints for the coecients dci;r Xq ci dk dci;0 ˆ 0; d ˆ 1; . . . ; L; k ˆ 0; 1; . . . ; Ld : q ci …22† The equations for the coecients bci;r are found in the form: XX 2 nci dr0 ; Hrrcidk …23† 0 bdk;r0 ˆ kT n dk r0 c ˆ 1; . . . ; L;

i ˆ 0; 1; . . . ; Lc ;

r ˆ 0; 1; . . . ;

where Hrrcidk ˆ 0

i X nci nbl h 0 0 Qr ; Qr 2 cibl n bl ! i nci ndk h 0 00 ‡ 2 Qr ; Qr ; cidk n

2 5kT

dcd dik

  mc …r† mc c2c S Q ˆ cc cc 2kT 5=2 2kT r

…24†

 1 2 cI : 3 c

This is the general method for the calculation of state dependent transport coecients based on the rigorous kinetic theory approach [10]. Until this point no additional assumptions have been introduced to simplify this procedure. 3. Simpli®ed expressions for transport coecients Let us remind the main problems which arise during the practical calculation of the state-tostate transport coecients. First, several kinds of bracket integrals for each chemical and vibrational species must be computed. It requires the data on the cross-sections of elastic collisions and TR processes depending on the vibrational states of colliding particles. Another diculty encountered in reaching this goal is the great amount of linear algebraic equations which have to be solved for every value of temperature and each distribution over vibrational energies and chemical species. Let Ntot be the total number of chemical and vibraP mol tional species (Ntot ˆ Lcˆ1 Lc ‡ Lat , Lmol is the

E.V. Kustova / Chemical Physics 270 (2001) 177±195

number of molecular species, Lc is the total number of vibrational energy levels in molecular species c, PLLmolat is the number of atomic species), Nv ˆ cˆ1 Lc denotes the total number of vibrational states in a mixture. In order to deduce the thermal conductivity coecient k0 one must solve a system consisting of 3Nv ‡ 2Lat algebraic equations, the system for the shear viscosity coecient calculation contains Ntot equations. The worst case is the computation of di€usion coecients: there 2 are Ntot coecients Dcidk in the mixture, for each coecient the system of Ntot algebraic coecients has to be solved. Strictly speaking, the number of independent di€usion coecients is Ntot …Ntot ‡ 1†=2 because Dcidk ˆ Ddkci . However, despite this fact the total number of di€usion coecients remains tremendous. A simple example: in a mixture (N2 =N) taking into account 46 vibrational levels of nitrogen, one obtains 1128 independent di€usion coecients, each of them is found solving the system of 47 algebraic equations. In the same mixture, for the computation of thermal conductivity and shear viscosity coecients the systems of 140 and 47 equations should be considered respectively. It is obvious that such a computational scheme cannot be implemented to any CFD code, re-calculation of all transport coecients at the each step of numerical code will make it too much time consumable. In order to maintain the advantages of the state-to-state transport theory and simplify the procedure of transport coecients calculation some additional assumptions are required. In the present work the following assumptions are introduced: (1) The rotational motion of a molecule is described by the rigid rotator model. It allows to consider the rotational energy independently from the vibrational state …ecij ˆ ecj † and therefore to put crot;ci ˆ crot;c . This fact simpli®es noticeably the thermal conductivity computation. (2) Usual assumptions of Monchick et al. [17] for the calculation of bracket integrals are introduced: ®rst, all complex collisions are neglected, and second, the internal and translational motions are considered as the uncorrelated ones. These assumptions allow one to express all bracket integrals as linear combinations of the elastic colli-

181

sion integrals and the integrals depending on the change of rotational energy Decidk rot at the inelastic collisions. (3) All collision integrals depending on Decidk rot are supposed to be much smaller than the elastic collision integrals. This assumption was discussed in details and proved in Refs. [10,18]: the contribution of the collision integrals depending on Decidk rot to the thermal conductivity coecients is found to be less than 1±2%. (4) The cross-sections of elastic collisions are assumed to be independent of the vibrational states of colliding particles. This assumption is usually accepted in the transport kinetic theory despite the fact that molecules in vibrationally excited states may have a larger e€ective radius than ground state molecules (see, for instance, Ref. [19]). The validity of this assumption is discussed in Ref. [14]. It is shown that taking into account the dependence of elastic cross-sections on the vibrational quantum number provides only a very small correction to the transport coecients. Applying these assumptions one can simplify signi®cantly the procedure of transport coecients calculation. First, assumptions (2)±(4) permit to express all the bracket integrals in terms of the vibrational level populations, atomic number densities and reduced elastic collision integrals …l;r† Xcd (see Refs. [20±23] for de®nition and Ref. [24] for the most reliable data on these integrals). The ®nal expressions for the bracket integrals are obtained in Ref. [10] and are given in Appendix A.2 in the simpli®ed form ((A.2)±(A.12)). The next result concerns the simpli®cation of systems (18) and (20)±(23). Let us consider ®rst the most simple system (23) for the coecients bci;r . Due to assumption (4) the bracket integrals Hrrcidk 0 depend on the vibrational level i only through the vibrational level populations nci (see Appendix A.2). Then, after some algebra, from system (23) it follows that coecients bci;r do not depend on the vibrational level: 8i ˆ 0; . . . ; Lc bci;r ˆ bc;r . Keeping only the ®rst non-vanishing terms of expansion (13) one can reduce the calculation of the shear viscosity coecient to the simple formula: lˆ

kT X nc bc;0 : 2 c n

…25†

182

E.V. Kustova / Chemical Physics 270 (2001) 177±195

Coecients bc;0 are found from the simpli®ed system: X d

cd H00 bd;0

2 nc ; ˆ kT n

c ˆ 1; . . . ; L:

…26†

As it is seen, system (26) contains only L equations instead of Ntot in system (23). This solution coincides formally with the one obtained in the quasi-stationary local thermal equilibrium onetemperature approach described in Refs. [16,23], cd the bracket integrals H00 for the one-temperature solution are given in many classical works on the …l;r† transport theory in terms of standard Xcd integrals and ®ctitious viscosity and heat conductivity coecients [21±23]. However, the principal di€erence exists: as the transport coecients are determined by the cross-sections of rapid processes, the …l;r† Xcd integrals in the state-to-state model depend on the cross-sections of elastic collisions and TR exchanges; in the one-temperature approach they depend also on the cross-sections of vibrational energy exchanges. Finally one can conclude that shear viscosity does not depend on the vibrational distributions. It is reasonable because this coecient describes the transfer of momentum which depends signi®cantly on the velocities but not on the internal state of particles. The bulk viscosity coecient as well as relaxation pressure are determined by the integrals depending on Decidk rot and, consequently, cannot be calculated using assumption (3) proposed above. In some cases these coecients can be expressed in terms of the rotational relaxation times. For a pure diatomic gas bulk viscosity in the state-to-state approach is obtained in Ref. [25]. Now, let us consider the coecients making part of the di€usion velocities and total energy ¯ux. Proceeding in a similar way and invoking also assumption (1), the expression for the thermal conductivity coecient has been simpli®ed: k0 ˆ

X 5 nc X k nc k ac;10 ‡ crot;c ac;01 ; 4 n 2 n c c

…27†

coecients ac;10 and ac;01 are found from the system:

X d


cd cd Kcd 0000 ad;00 ‡ K0100 ad;10 ‡ K0001 ad;01 ˆ 0;

X


15kT nc cd cd ; Kcd 1000 ad;00 ‡ K1100 ad;10 ‡ K1001 ad;01 ˆ 2 n d X
nc cd cd crot;c ; Kcd 0010 ad;00 ‡ K0110 ad;10 ‡ K0011 ad;01 ˆ 3kT n d

c ˆ 1; . . . ; L:

…28†

System (28) must be solved together with the constraint obtained from Eq. (20) after summation over i: Xq c ac;00 ˆ 0: …29† q c Again, compared to systems (18) and (20) the number of equations is signi®cantly reduced: there are only 3Lmol ‡ 2Lat equations instead of 3Nv ‡ 2Lat . This result is also close to the one given by the one-temperature model [16,23] except two points: the ®rst one, concerning the inelastic cross-sections is discussed above, and the second point is that in the state-to-state approach only rotational degrees of freedom make contribution to the internal thermal conductivity whereas in the one-temperature approach both rotational and vibrational energy contribute to this coecient. Therefore in the latter case the internal speci®c heat cint;c appears in Eqs. (27) and (28) instead of rotational speci®c heat crot;c . The reason for this is that thermal conductivity as well as viscosity are determined by the rapid processes. Finally in the state-to-state approach the thermal conductivity coecient does not depend on the vibrational distribution. The transport of vibrational energy is described by introducing di€usion coecients for every vibrational state. Due to the fact that aci;00 ˆ ac;00 , 8i, the thermal di€usion coecients are also independent from the vibrational state …DTci ˆ DTc ; 8i†, and the term P P total heat ¯ux is equal to Pci DTci dci entering to the D d , where d ˆ c c Tc c i dci . The situation is more complicated for the diffusion coecients. Actually, system (21) cannot be simpli®ed in the same way as the previous ones because of the structure of its right-hand sides. However, it is easy to show that the main part of

E.V. Kustova / Chemical Physics 270 (2001) 177±195 bl coecients ddk;r 0 do not depend on the vibrational bl quantum number except the coecients dbl;r 0. bl b bl Therefore, ddk;r0 ˆ dd;r0 , 8 d 6ˆ b, 8 k, l, and dbk;r0 ˆ b db;r 0 , 8 l, 8 k 6ˆ l. Keeping the ®rst non-vanishing terms in expansion (12), i.e., the terms corresponding to r ˆ 0, system (21) can be written in the reduced form:

for c ˆ b; i ˆ l: X X X bl b b cblbl cblbk dd;0 cbldk 00 dbl;0 ‡ 00 db;0 ‡ 00 

k6ˆl

ˆ 3kT 1 for c ˆ b; X i6ˆl

ˆ

bl cbibl 00 dbl;0 ‡

3kT

…30†

i ˆ 0; . . . ; l

qb

q

for c ˆ 1; . . . ; b

X k6ˆl

qbl

k

d6ˆb

 qbl ; q

1; l ‡ 1; . . . Lb :

b cbibk 00 db;0 ‡

X d6ˆb

b dd;0

X k

;

! cbidk 00 …31†

i ˆ 0; . . . ; Lc : ! X X X X cibl bl cibk b b cidk c00 dbl;0 ‡ c00 db;0 ‡ dd;0 c00 i

ˆ

q 3kT c ; q

b ˆ 1; . . . ; L;

1; b ‡ 1; . . . L;

k6ˆl

d6ˆb

k

…32† l ˆ 0; . . . ; Lb :

The constraint (22) is replaced by the following equation: X qd qbl bl q qbl b dbl;0 ‡ b db;0 ‡ d b ˆ 0; q q q d;0 d6ˆb b ˆ 1; . . . ; L; l ˆ 0; 1; . . . ; Lb :

…33†

System (30)±(33) contains only L ‡ 1 equations instead of Ntot in the case of systems (21) and (22). Finally the system of di€usion coecients is reduced to the following: there are Nv coecients Dcici which are di€erent for any vibrational level i of species c, Lmol coecients Dcc ˆ Dcick , 8 i, 8 k 6ˆ i, and L2 coecients Dcd ˆ Dcidk , 8d 6ˆ c, 8 i, k. The total number of independent di€usion coecients is Nv ‡ Lmol ‡ L…L ‡ 1†=2. Coming back to the …N2 =N† mixture mentioned at the beginning of this section, after simpli®ca-

183

tions proposed above, one obtains only 49 independent di€usion coecients, each of them is found from the system of three equations. For the computation of thermal conductivity and shear viscosity coecients the systems of ®ve and two equations are to be solved respectively. Finally, computation of the transport coecients is reduced to the solution of linear algebraic systems (26), (28), (29) and (30)±(33). Coecients of these systems represent the partial bracket integrals determined by the cross-sections of elastic collisions and translation±rotation energy exchange. The expressions for the bracket integrals cd cd , Kcd H00 rr0 pp0 , c00 are summarized in Appendix A.3 ((A.13)±(A.22)). These formulas are close to the classical ones [21±23]. Nevertheless, one should keep in mind that in the state-to-state approach …l;r† Xcd integrals as well as the bracket integrals depend on the cross-sections of elastic and TR collisions (while in the well known one-temperature case they depend also on the cross-sections of vibrational energy transitions). However, for the …l;r† practical calculations of Xcd integrals, the inelastic cross-sections are usually substituted by the elastic ones. The in¯uence of the inelastic crosssections on these integrals occurs to be weak (see, for instance, Ref. [26]). Various methods of solution of transport linear systems have been elaborated since the work of Chapman and Cowling [20]. The most evident way is the direct solution of algebraic systems by the Cramer's rule which allows one to express the transport coecients explicitly as ratios of determinants [20±22,27]. Nevertheless, solving the linear systems by this method is extremely computationally expensive. Another direct method using the Gaussian elimination is considered, for instance, in Ref. [28]. This method is noticeably less time consumable but still not satisfactory as a majority of direct methods. Recently the mathematical properties of transport linear systems have been studied comprehensively in Ref. [23] and new ecient iterative algorithms have been proposed. This technique provides very fast and accurate evaluation of transport properties. It should be emphasized that the resemblance of systems (26), (28) and (29) to the classical onetemperature ones remains only formal, it is seen

184

E.V. Kustova / Chemical Physics 270 (2001) 177±195

from the de®nition of partial bracket integrals (see Appendix A.1) and from the fact that vibrational degrees of freedom do not contribute to the transport coecients. Strictly speaking, the independence of some transport coecients of the vibrational distributions is not an evident fact, and application of the state-to-state transport theory must in essence be based on the complete systems (18) and (20)±(23). Thus, the analysis of transport properties performed in Refs. [11±14] is carried out in this way, and it was a rather lengthy procedure, even using an approximate method mentioned in Section 1. A possibility of getting simpli®ed transport linear systems in the state-to-state approach is discussed in Ref. [10], but their validity has never been proved before. The derivation of transport linear systems (26), (28)±(33) is the ®rst step to the implementation of state-to-state transport theory to the computational ¯uid dynamics.

4. Di€usion coecients In this section the di€usion coecients are considered in some more details. The di€usion coecients for each vibrational state have been introduced recently in Ref. [10], they are required for the estimation of vibrational energy transport in the absence of quasi-stationary vibrational distributions. For a mixture of diatomic gas A2 and atoms A the analytical expressions for the di€usion coecients can be derived from Eqs. (30)±(33). Finally all di€usion coecients are expressed in terms of mass fractions of species cd ˆ qd =q and binary and self-di€usion coecients: 1=2

3 …2pmcd kT † Dcd ˆ ; …1;1† 16nmcd p r2cd Xcd Dc ˆ

3 …pmc kT †1=2 8nmc p r2c Xc…1;1†

mcd is the reduced mass, rcd is the collision diameter. Thus for the (N2 =N) mixture the di€usion coecients are given by:

 DN2 iN2 i ˆ DNN2

mN q=n

cN DN 2

2 ‡ DNN



2  2

cN2 2DN2

1

cN

cN2 i

N ‡ DcNN

1

 ;

2

i ˆ 0; . . . ; 45;

…34† 

DN2 N2 ˆ DN2 iN2 k 

DNN2 ˆ

cN DN2

cN2 2DN2

DNN2

DNN ˆ DNN2

mN ˆ DNN2 q=n 2 … c ‡ 1† N DNN 2

N ‡ DcNN

mN mN2 …q=n†2

mN mN2 …q=n†

2



2

;

; 1 cN

8k 6ˆ i;

…35†

2

…36†  1 :

…37†

One can notice that expressions for DNN2 and DNN are similar to the well-known formulas given by Ferziger and Kaper [22] for a binary mixture (again, the di€erence is in the cross-sections de…1;1† termining the Xcd integrals). The state speci®c di€usion coecients DN2 iN2 i depend on the mass fractions of chemical species and population of the corresponding level. Due to assumption (4) of the previous section, these coecients are the only ones determined by the non-equilibrium stateto-state distributions. In the limit of a pure diatomic gas …cN ˆ 0† coecients DN2 iN2 i and DN2 N2 take the form obtained in Ref. [25]:   1 DN2 iN2 i ˆ DN2 1 ; DN2 N2 ˆ DN2 : c N2 i The state-to-state vibrational distributions as well as transport coecients and heat ¯ux have been investigated in the previous papers [11±13] for three di€erent ¯ows of a …N2 ; N† mixture: (1) ¯ow behind a strong shock wave (initial conditions are: M0 ˆ 15, T0 ˆ 293 K, p0 ˆ 100 Pa), (2) ¯ow in a hypersonic boundary layer (temperature and pressure at the edge of the boundary layer are Te ˆ 5000 K, pe ˆ 1000 Pa, the wall temperature is Tw ˆ 300 K), and (3) expanding ¯ow in the F4

E.V. Kustova / Chemical Physics 270 (2001) 177±195

nozzle used in ONERA (reservoir conditions are: T0 ˆ 6000 K, p0 ˆ 105 Pa). The analysis carried out in these papers is based on a simpli®ed scheme: ®rst, the vibrational distributions, species concentrations, temperature, pressure, velocity are computed either in the Euler approach of non-viscous and non-conductive gas ¯ow [11,13] or on the basis of very simple models for transport coecients [12]. The numerical methods are described in details in Refs. [6±9,11]. Then the calculated macroscopic parameters are used as input for the transport coecients and heat transfer evaluation. This approach permits to estimate qualitatively the in¯uence of state-to-state vibrational kinetics on the transport properties. Certainly, in order to understand the mutual e€ect of non-equilibrium kinetics and state dependent transport properties, it is necessary to use the equations of viscous and heat conductive gas ¯ow and compute the state-to-state coecients at the each time and space cell. Up to now, the computational cost of such a scheme was extremely high, and the practical incorporation of the state-tostate linear transport systems to the numerical codes has not been accomplished. The implementation of simpli®ed systems proposed in the present paper makes this goal attainable. A simple illustration: in the present study the transport coecients and heat ¯uxes in …N2 ; N† mixture ¯ows are calculated using the same macroscopic parameters as in Refs. [11±13] but applying reduced systems (26), (28)±(33) in contrast to the previous papers where complete systems (18) and (20)±(23) have been solved. The computational time for each step is now 100±150 times less, the accuracy of calculations remains the same. The gain of time is expected to be much higher for multi-component mixtures. This can encourage one to consider simultaneously the state-to-state kinetics and transport theory and by this means get more comprehensive results. The following discussion serves as a warning against an over-simpli®cation of the di€usion coecients. The main factor which a€ects the state speci®c di€usion coecients Dcici is the non-equilibrium vibrational kinetics, Dcici appear to be approximately inverse proportional to nci (see Refs. [11±13]). The similar dependence of usual self-

185

di€usion coecients on the concentration of corresponding chemical species has been predicted by Ferziger and Kaper [22]. It is known that vibrational level populations can vary within several orders of magnitude, therefore, the inverse proportional dependence leads to very strong variations of di€usion coecients as far as one computes them as functions of i. For the practical calculations sometimes it is more convenient to deal with slowly varying coecients. Consequently, it is useful to introduce the coecients e cici in the following way: D e cici ˆ nci Dcici ; D n

…38†

e cici to the macroscopic equations inand insert D stead of Dcici . Various de®nitions of di€usion velocity are discussed in Ref. [22], and it is shown that de®nitions like (7) are more self-consistent. e cici vary much slower with However, coecients D the vibrational quantum number compared to Dcici , this makes easier to use them for rapid estie cici allows mations. Moreover, weak variation of D one to suppose that these coecients do not depend on the vibrational state and can be replaced e which, for a mixture by a unique coecient D, …N2 ; N†, is connected with a simple binary di€usion coecient: e ˆD e N iN i ˆ nN2 DNN ; D 2 2 2 n

8 i:

…39†

Such an assumption is often used in the stateto-state calculations in order to incorporate the vibrational energy transport by di€usion to the master equations and to treat it in the most simple way [8,9,29,30]. By this means the constant Lewis and Schmidt numbers can be introduced, it facilitates signi®cantly the numerical algorithms [8,9]. However, assumption (39) requires further veri®cation. The limits of its validity as well as the impact of vibrational distributions on the di€usion coecients are discussed hereafter. In Figs. 1±7 the vibrational distributions and state speci®c di€usion coecients (38) are given for di€erent conditions. For a comparison the e (39) independent on the values of coecient D vibrational state are also plotted.

186

E.V. Kustova / Chemical Physics 270 (2001) 177±195

Fig. 1. Vibrational distributions ni =n behind a shock wave at di€erent distances x from the shock front. (1) x ˆ 0:015 cm; (2) x ˆ 0:05 cm; (3) x ˆ 0:1 cm; (4) x ˆ 1 cm; (5) x ˆ 2 cm.

Fig. 2. Di€usion coecients DN2 iN2 i , m2 /s, behind a shock wave at di€erent distances x from the shock front. Solid lines ± state-to-state e 1,10 ± x ˆ 0:015 cm; 2,20 ± x ˆ 0:05 cm; 3,30 ± x ˆ 0:1 cm; 4,40 ± x ˆ 1 cm; 5,50 ± x ˆ 2 cm. approach, dashed lines ± coecient D.

The process of vibrational relaxation behind a shock wave is illustrated by Fig. 1. The initial Boltzmann distribution over vibrational energies evolves towards the ®nal equilibrium state through

the series of non-equilibrium state-to-state distributions. They are formed by translation±vibration TV excitation of low levels and consecutive vibrational energy re-distribution by VV exchange

E.V. Kustova / Chemical Physics 270 (2001) 177±195

187

Fig. 3. Vibrational distributions ni =n in a boundary layer at di€erent distances g from the wall. (1) g ˆ 0; (2) g ˆ 1; (3) g ˆ 2; (4) g ˆ 3; (5) g ˆ 4.

Fig. 4. Di€usion coecients DN2 iN2 i , m2 /s, in a boundary layer at di€erent distances g from the wall. Solid lines ± state-to-state e 1,10 ± g ˆ 0; 2,20 ± g ˆ 1. approach, dashed lines ± coecient D.

between vibrational states. The high levels are slightly depleted because of dissociation process. Similar shapes of the distributions behind a shock wave are found also in Refs. [2,3]. It is shown in

Refs. [2,11] that in the vicinity of the shock front the quasi-stationary Boltzmann distributions remain higher than the corresponding stateto-state ones. With distance from the shock front

188

E.V. Kustova / Chemical Physics 270 (2001) 177±195

Fig. 5. Di€usion coecients DN2 iN2 i , m2 /s, in a boundary layer at di€erent distances g from the wall. Solid lines ± state-to-state e 3,30 ± g ˆ 2; 4,40 ± g ˆ 3. approach, dashed lines ± coecient D.

Fig. 6. Vibrational distributions ni =n in a nozzle at di€erent distances x from reservoir (xthroat ˆ0.5 m). (1) x ˆ 0; (2) x ˆ 0:5 m; (3) x ˆ 0:51 m; (4) x ˆ 0:55 m; (5) x ˆ 0:6 m; (6) x ˆ 2 m.

all quasi-stationary and state-to-state distributions e N2 iN2 i behind converge. The di€usion coecients D the shock are presented in Fig. 2 as functions of the vibrational quantum number i. One can notice

e is signi®cantly higher than that the coecient D state-to-state coecients for low vibrational states …i < 5±7†, especially in the beginning of the relaxation zone where the quasi-stationary distributions

E.V. Kustova / Chemical Physics 270 (2001) 177±195

189

Fig. 7. Di€usion coecients DN2 iN2 i , m2 /s, in a nozzle at di€erent distances x (xthroat ˆ 0:5 m). Solid lines ± state-to-state approach, e 1,10 ± x ˆ 0; 2,20 ± x ˆ 0:5 m; 3,30 ± x ˆ 0:55 m; 4,40 ± x ˆ 0:6 m; 5,50 ± x ˆ 0:8 m; 6,60 ± x ˆ 2 m. dashed lines ± coecient D.

do not exist. Approaching to the thermal equilibrium state this discrepancy decreases noticeably. Concerning the high levels, the di€erence between e N2 iN2 i and D e does not exceed 10% even in the D range of strong vibrational non-equilibrium. It can be connected with the fact that the shape of distributions remains close to the Boltzmann one and, therefore, high states are not populated suf®ciently to give a noticeable contribution to the di€usion coecients. Quite di€erent behaviour of di€usion coecients has been found in a boundary layer, where the vibrational distributions are essentially nonBoltzmann. The mechanism of formation of these distributions is the following: dissociation in the gas ¯ow near the high-temperature edge of the boundary layer produces nitrogen atoms which di€use towards the cool surface. Recombination near the surface pumps the populations of the high vibrational states. As the result of this process and VV and VT exchange the strong non-equilibrium distribution with a plateau part at the intermediate and upper levels appears just near the wall (see Fig. 3). This e€ect has been studied in Refs. [8,9,29] and the role of recombination and VV

exchange is found to be particularly important in this ¯ow. The non-Boltzmann character of vibrational distributions in¯uences dramatically the di€usion coecients: close to the surface, at i > 5 e N iN i = D e reaches several thousands (Fig. the ratio D 2 2 4). The over-population of high states leads to very large values of corresponding di€usion coecients. Again, coming to equilibrium at the external edge of boundary layer, this ratio decreases, and e N iN i and D e for ®nally the discrepancy between D 2 2 high levels tends to 5±10% (Fig. 5). Another example of non-equilibrium ¯ow is a nozzle expansion. Characteristic vibrational distributions and di€usion coecients in the F4 nozzle [7,13] are shown in Figs. 6 and 7. In the initial part of the nozzle the distributions are close to the Boltzmann ones and then a plateau develops at the intermediate and upper levels as a result of VT deactivation, VV exchange and recombination. The region of major change of the level populations is in the beginning of the expanding part just after the throat …x < 0:6 m†. Then the populations (except very high levels) change weakly and approach the constant frozen values. This character of the distributions has been found in Ref. [7], in

190

E.V. Kustova / Chemical Physics 270 (2001) 177±195

e N2 iN2 i and D e (curves 1 and 2). Curve Fig. 8. Heat ¯ux q, kW/m2 , behind a shock wave as a function of x calculated using coecients D 3 ± ¯ux Fourier.

this paper a comparison of the calculated distributions with the experimental ones [31] is given and a qualitative agreement is observed. In the vicinity of the throat, where the distributions have an approximately Boltzmann shape, the state dee pendent di€usion coecients are not far from D. The di€erence increases with x from 5% at x ˆ 0:5 m up to 50% at x ˆ 2 m. It should be noted that for the calculation of di€usion coecients only the lowest 20 levels have been retained. At higher levels the distributions in this kind of nozzle take more complicated form [7,13], it can change also the di€usion coecients. The general tendency can be formulated in the e overestimates following way: average coecient D state-to-state di€usion coecients at low vibrational levels, and underestimates them for the high states. The discrepancy between these coecients decreases signi®cantly approaching to the thermal equilibrium conditions. Let us consider now a qualitative behaviour of the total heat ¯ux calculated for the ¯ows discussed above using the state dependent and con-

e N2 iN2 i and D. e Figs. 8± stant di€usion coecients D 10 present respectively the heat ¯uxes behind a shock wave, in a boundary layer, and in a nozzle. Curves denoted as ``1'' in these ®gures correspond e N iN i , to the results obtained with coecients D 2 2 ``2'' are calculated with the state independent coe ecient D. The role of di€usion in a shock heated gas ¯ow is found to be very important [11], even the qualitative behaviour of the total energy ¯ux and Fourier ¯ux caused by heat conductivity …qF ˆ k0 rT † di€ers in the relaxation region. Curve ``3'' in Fig. 8 denotes the Fourier ¯ux. Under these conditions the employment of averaged coee gives only a very rough approximation for cients D the heat ¯ux. Such a discrepancy can be explained in the following way. The only positive di€usion velocity in this case is that of the ground vibrational state V0 (the population of the ground state is the only one which decreases during the relaxation process, remaining, however, much higher than the populations of other levels). Consequently, noticeable overestimation of the di€usion

E.V. Kustova / Chemical Physics 270 (2001) 177±195

191

e N2 iN2 i and D e (curves 1 and 2). Fig. 9. Heat ¯ux q, W/m2 , in a boundary layer as a function of g calculated using coecients D

e N2 iN2 i and D e (curves 1 and 2). Fig. 10. Heat ¯ux q, W/m2 , in a nozzle as a function of x calculated using coecients D

e N 0N 0 (see Fig. 2) leads to overesticoecient D 2 2 mation of the V0 contribution to the heat ¯ux. In the case of state dependent di€usion coecients, the contribution of di€erent levels to di€usion of vibrational energy is of the same order of magnitude, and negative di€usion velocities decrease

signi®cantly the total energy ¯ux compared to the previous case. Looking at Figs. 4 and 5 one can expect a strong disagreement between the heat ¯uxes calculated in a boundary layer using two models for di€usion coecients. Nevertheless, the di€erence

192

E.V. Kustova / Chemical Physics 270 (2001) 177±195

between two ¯uxes does not exceed 30% (see Fig. 9). The reason is that in the boundary layer with non-catalytic surface the contribution of the diffusion processes to the total energy ¯ux is much less than in the shock heated ¯ow, the heat transfer in this case is determined mainly by thermal conductivity. The heterogeneous reactions on the surface increase dramatically the role of di€usion [32], and in the case of catalytic wall the e€ect of di€usion coecients on the heat ¯ux should be greater. Fig. 9 plots the heat ¯uxes computed in a nozzle. It is not surprisingly that the in¯uence of state dependent di€usion coecients on the heat ¯ux is weak (the discrepancy is within 15% in a very narrow interval just near the throat and does not exceed 5% in other regions). This is explained by the fact that in this ¯ow only the low vibrational levels contribute considerably to the heat transfer at x > 0:6 m (the highly located states are populated very poorly). For low levels the discrepancy e N iN i and D e is small, therefore, coebetween D 2 2 e cient D gives a satisfactory description of vibrational energy di€usion in a nozzle. One should keep in mind that the present results can be treated only as the qualitative ones because of approximations adopted for the heat ¯ux calculation. A comparison with experimental data is necessary for the estimation of accuracy of the proposed method. Unfortunately, the lack of data on the heat transfer in the ¯ows discussed above makes this task unattainable. Nevertheless, it can be pointed out that only state-to-state description gives an adequate shape of vibrational distributions in strongly non-equilibrium conditions. For instance, in papers [30,33,34] the experimentally measured vibrational distributions in optically pumped CO and other diatomic gases are compared with the state-to-state and quasistationary distributions. An excellent correlation between the state-to-state and measured distributions is observed, whereas all quasi-stationary models give a very poor agreement. This fact is a good reason to believe that the results of state-tostate simulation of the heat transfer in extremely non-equilibrium conditions will be much more accurate compared to the results obtained by means of quasi-stationary one-temperature or multitemperature models.

5. Conclusions The transport kinetic theory in the state-to-state approach is considered. The simpli®ed algorithm for the viscosity, thermal conductivity and di€usion coecients calculation is proposed. This algorithm reduces essentially the time of transport coecients computation and thus encourages one to insert the obtained linear transport systems to the numerical codes for the solution of master equations. The results concerning state speci®c di€usion coecients which describe the transport of vibrational energy are presented, and the in¯uence of non-equilibrium vibrational distributions on these coecients is estimated. A comparison with the results obtained on the basis of the simple intuitive model is also given. This model gives a satisfactory agreement with the state-to-state approach only for weak deviations of vibrational distributions from the Boltzmann ones. Under strongly nonequilibrium conditions, state-to-state di€usion coecients di€er signi®cantly from the average values. Using the constant di€usion coecients instead of the state dependent ones may lead to an essential error in the values of di€usion velocities and total energy ¯ux.

Acknowledgements This study is supported by INTAS (99-00464). Appendix A A.1. De®nition of bracket integrals ‰ A; BŠ ˆ

 X nci ndk  0 00 ‰ A; B Š ‡ ‰ A; B Š ; cidk cidk n2 cidk

…A:1†

the partial bracket integrals are introduced as follows: 0

‰ A; BŠcidk ˆ

Z 1 X …0† …0† fcij fdkl …Bcij 2nci ndk jlj0 l0  …Acij

0 0

Bcij0 †

l 2 Acij0 †grjcidk; jl d Xduc dud ;

E.V. Kustova / Chemical Physics 270 (2001) 177±195 00

‰ A; BŠcidk ˆ

XZ

1 2nci ndk

jlj0 l0

 …Adkl

…0† …0†

fcij fdkl …Bcij

Bcij0 †

Kcidk 1100 ˆ

0 0

l 2 Adkl0 †grjcidk; jl d X duc dud :

0 0

l rjcidk; jl is the cross-section of collision of two molecules c and d chemical species at i, k vibrational and j, l rotational states leading to the change of translational and rotational energy, d2 X is the solid angle where the relative velocity after the collision can appear, g is the relative velocity before the collision, j0 , l0 are the rotational levels after collision.

75k 2 T xci xdk mc md 16 Acd kcd …mc ‡ md †2   55    3Bcd 4Acd ; c 6ˆ d or i 6ˆ k; 4 …A:8†

Kcici 0011 ˆ

X xb 75k 2 Txci crot;ci ; 16 Acb kcb b

Kcidk 0011 ˆ 0;

A.2. Bracket integrals depending on vibrational state [10] The expressions obtained in Ref. [10] can be rearranged to the following form: ! 2 X xb 75k Tx x ci ci cici Kcici ; 0000 ˆ c00 ˆ 16 Acb kcb Acc kcc b …A:2† Kcidk 0000

ˆ

ccidk 00

75k 2 T xci xdk ; 16 Acd kcd

ˆ

75k Txci 32

Kcici 1000 ˆ

X b

xb Acb kcb !

c 6ˆ d or i 6ˆ k;

 mb …6Ccb

5†

mc ‡ mb

 xci …6Ccc 5† ;  2Acc kcc

Kcidk 1000 ˆ

ˆ

75k 2 T xci xdk mc  6Ccd  32 Acd kcd mc ‡ md

 75k 2 Txci xci 4Acc 16 4Acc kcc X xb ‡ Acb kcb b 

15 2 mc 2

‡

25 2 mb 4

ˆ xci

…A:10†

  X 2xb mc mb 5 mb ‡ lcb …mc ‡ mb †2 3Acb mc b  ! xci 5 1 ; …A:11† 2lcc 3Acc

2xci xdk mc md ˆ lcd …mc ‡ md †2 c 6ˆ d or i 6ˆ k;



3m2b Bcb ‡ 2 …mc ‡ mb †

kcd ˆ

75k …2pmcd kT † ; …2;2† 128mcd p r2cd Xcd

lcd ˆ

5 …2pmcd kT † …2;2† 16 p r2cd Xcd

…A:12†

1=2

…A:5†

are the ®ctitious thermal conductivity and visco sity coecients. Functions Acd , Bcd , Ccd are de®ned in Refs. [21±23]:

…A:6†

Acd ˆ



4mc mb Acb

1 ;

1=2


5 ;

55 ‡ 3Bcc 4



5 3Acd

where xci ˆ nci =n is the molar fraction of c species at the vibrational state i,

…A:4†

c 6ˆ d or i 6ˆ k; Kcici 1100

cici H00

…A:9†

c 6ˆ d or i 6ˆ k;

cidk H00

…A:3† 2

193

! ; …A:7†

 Ccd ˆ

…l;r†

…2;2†

Xcd

; …1;1†

Xcd

Bcd ˆ

…1;2†

5Xcd

…1;3†

4Xcd

…1;1†

Xcd

;

…1;2†

Xcd

…1;1†

Xcd

:

Xcd integrals can be computed using the data from [24].

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E.V. Kustova / Chemical Physics 270 (2001) 177±195

A.3. Classical bracket integrals [22] cc Kcc 0000 ˆ c00 ˆ

75k 2 T X xc xb ; 16 b6ˆc Acb kcb

cd Kcd 0000 ˆ c00 ˆ

75k 2 T xc xd ; 16 Acd kcd

References …A:13†

c 6ˆ d;

…A:14†

75k 2 T X xc xb mb  6Ccb 32 b6ˆc Acb kcb mc ‡ mb

Kcc 1000 ˆ


5 ; …A:15†


5 ;

75k 2 T xc xd mc  6Ccd 32 Acd kcd mc ‡ md

Kcd 1000 ˆ

c 6ˆ d; …A:16†

75k 2 T 8

Kcc 1100 ˆ



X xc xb x2c ‡ kcc 2Acb kcb b6ˆc

15 2 mc 2

‡ 254 m2b

3m2b Bcb ‡ 4mc mb Acb

…mc ‡ mb †

!

2

;

…A:17† Kcd 1100 ˆ

Kcc 0011

75k 2 T xc xd mc md 16 Acd kcd …mc ‡ md †2   55  3Bcd 4Acd ; c 6ˆ d; 4

X xc xb 75k 2 T x2c crot;c ˆ ‡ 16 Acc kcc b6ˆc Acb kcb

…A:18†

! ; …A:19†

Kcd 0011 ˆ 0; cc H00 ˆ

c 6ˆ d;

…A:20†

x2c X 2xc xb mc mb ‡ lcc b6ˆc lcb …mc ‡ mb †2



 5 mb ‡ ; 3Acb mc …A:21†

2xc xd mc md lcd …mc ‡ md †2

cd ˆ H00

xc ˆ



5 3Acd

 1 ;

c 6ˆ d; …A:22†

P i

nci =n ˆ nc =n.

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[25]

[26] [27] [28]

98-2936, 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference. E. Kustova, F. Mallinger, Level kinetics approach in the case of strong vibrational nonequilibrium for a pure diatomic gas, Research Report No. 3557, INRIA, Rocquencourt, December 1998. Y. Gershenzon, V. Rozenshtein, S. Umanskii, Doklady Akademii Nauk SSSR 223 (3) (1975) 629. L. Monchick, K. Yun, E. Mason, J. Chem. Phys. 39 (1963) 654. R. Kee, G. Dixon-Lewis, J. Warnatz, M. Coltrin, J. Miller, A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties, SAND86-8246, SANDIA National Laboratories Report, 1986.

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[29] V. Doroshenko, N. Kudryavtsev, S. Novikov, V. Smetanin, High Temperature 28 (1990) 82. [30] E. Pl onjes, P. Palm, A. Chernukho, I. Adamovich, J. Rich, Chem. Phys. 256 (2000) 315. [31] S. Sharma, S. Run, W. Gillespie, S. Meyer, J. Thermophys. Heat Transfer 7 (4) (1993) 697. [32] I. Armenise, M. Cacciatore, M. Capitelli, E. Kustova, E. Nagnibeda, M. Rutigliano, The in¯uence of nonequilibrium vibrational and dissociation-recombination kinetics on the heat transfer and di€usion in the boundary layer under reentry conditions, Rare®eld Gas Dynamics 21, vol. 2, CEPADUES, Toulouse, France, 1999. [33] R. Deleon, J. Rich, Chem. Phys. 107 (1986) 283. [34] E. Pl onjes, P. Palm, W. Lee, M. Chidley, I. Adamovich, W. Lempert, J. Rich, Chem. Phys. 260 (2000) 353.