On the applicability of simplified state-to-state models of transport coefficients

Chemical Physics Letters 686 (2017) 161–166

Contents lists available at ScienceDirect

Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

Research paper

On the applicability of simplified state-to-state models of transport coefficients E. Kustova, M. Mekhonoshina, G. Oblapenko ⇑ Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia

a r t i c l e

i n f o

Article history: Received 18 July 2017 In final form 19 August 2017 Available online 22 August 2017 Keywords: State-specific Thermal conductivity Bulk viscosity

a b s t r a c t Thermal conductivity and bulk viscosity coefficients are studied in the state-to-state approximation to assess the importance of accounting for rovibrational coupling and increasing diameters of vibrationally excited molecules. Transport coefficients are computed in binary mixtures for a wide temperature range, and compared to those obtained for the rigid rotator model. It is shown that accounting for rovibrational coupling leads to a twofold decrease in the bulk viscosity coefficient and a 5–7% decrease in the thermal conductivity coefficient; accounting for variable diameters has no effect on the bulk viscosity, but leads to a larger decrease in the thermal conductivity. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Various approaches to modeling of strongly non-equilibrium flows exist, such as the one-temperature, multi-temperature, and state-to-state approximations. Of these, the state-to-state approximation, which assumes that characteristic times of vibrational energy transitions and chemical reactions are of the same order of magnitude as the gas-dynamic timescale, is the most detailed, since it can describe arbitrary vibrational energy distributions. For strongly non-equilibrium flows, the state-to-state description, while computationally expensive, provides the most accurate results and the best agreement with experimental data [1,2]. Due to significant increases in computational power, state-specific approaches have begun to receive more interest lately, including simulations which account not only for state-specific vibrational distributions, but also non-equilibrium rotational distributions [3–5], where the role of vibrational–rotational transitions may have a noticeable effect on the transport coefficients [6]. However, considering all the rovibrational states is very expensive from a computational point of view, and various energy binning approaches for the rovibrational levels have been proposed, both for CFD (Computational Fluid Dynamics) and DSMC (Direct Simulation Monte Carlo) simulation methods [7–10]. Despite this, the state-to-state simulations of viscous flows require the computation of state dependent transport coefficients obtained by solving large linear transport systems. While applying the state-to-state approx⇑ Corresponding author. E-mail addresses: [email protected] (E. Kustova), [email protected] (M. Mekhonoshina), [email protected] (G. Oblapenko). http://dx.doi.org/10.1016/j.cplett.2017.08.041 0009-2614/Ó 2017 Elsevier B.V. All rights reserved.

imation to inviscid 2-dimensional flows has been shown to be feasible, coupling the state-to-state approximation transport equations with the state-dependent transport coefficients computation can lead to an enormous increase in computational time. Given the growing interest in performing state-specific simulations of rarefied gas flows, it is of interest whether the rovibrational coupling significantly affects transport coefficients and whether simplified algorithms can be used to compute them, saving a large amount of computational effort. Simplified state-to-state models for transport coefficients have been developed [11], which are based on two assumptions: (1) the molecular diameters are independent of the vibrational level, and (2) the molecules are rigid rotators, that is, their rotational energy is independent of the vibrational state. The influence of variable molecular diameters on the transport coefficients has been studied in [12,13] for gases with vibrational excitation and in [14,15] for electronically excited gases. It has been shown that accounting for the change in molecular diameter for vibrationally excited molecules has a small influence on the transport coefficients. For electronically excited gases, the influence of the variable diameter is more important [14,15]. However, up till now, no study of the importance of the second simplifying assumption (that the molecules are rigid rotators) has been performed. The objectives of the present work are to compute thermal conductivity and bulk viscosity coefficients in the state-to-state approximation, taking into account the coupling between rotational and vibrational degrees of freedom and increasing of molecular diameter with vibrational energy, and to compare the results to those obtained in the frame of a commonly used simplified model of rigid rotator with fixed molecular size and constant

162

E. Kustova et al. / Chemical Physics Letters 686 (2017) 161–166

specific heat. The calculations are performed for different mixture compositions in a wide temperature range (up to 40,000 K), so as to assess the role of the coupling under various conditions, including high-temperature conditions typical for high-velocity atmospheric re-entry problems [16,17]. The work is done within the framework of the modified Chapman-Enskog method [18], applicable for description of slip and continuum regimes, i.e., for Knudsen numbers in the range 0.0001–0.25, which corresponds to most atmospheric re-entry problems. 2. State-to-state approximation The state-to-state model, valid when the rates of vibrational and chemical relaxation are of the same order as the rate of change of macroscopic flow variables, gives the following set of transport equations [18]: dnci dt

þ nci r  v þ r  ðnci Vci Þ ¼ Rci ; i ¼ 0; . . . ; Lc ; c ¼ 1; . . . ; L;

ð1Þ

q

dv ¼ r  P; dt

ð2Þ

q

dU ¼ r  q  P : rv ; dt

ð3Þ

where nci is the number density of molecules of species c at vibrational level i; v is the flow velocity, Vci is the diffusion velocity of molecules of species c at vibrational level i; Rci is the relaxation term describing the change in the number density of species c with vibrational state i due to vibrational energy transitions and chemical reactions, P is the pressure tensor, U is the specific total energy, q is the heat flux. Lc is the number of vibrational levels of molecular species c; L is the number of species in the mixture. The specific total energy is given by the following expression:

qU ¼

X X 3nkT X ci  e rot nci þ eci nci þ ec nc : þ 2 c ci ci

ð4Þ

Here q is the density of the mixture, n is the number density of the mixture, k is the Boltzmann constant, T is the gas tempera  ture, eci rot is the average rotational energy of molecule of species c at vibrational level i; eci is the vibrational energy of level i of molecular species c; ec is the formation energy of species c; nc is the number density of species c. The average rotational energy is defined as

 ci 

e

rot

! ecij 1 1 X ci ci ; ¼ s e exp  r Z cirot j j j kT

ð5Þ

where ecij is the rotational energy of level j of molecular species c at vibrational level i; scij ¼ 2j þ 1 is the degeneracy of rotational state j; r is a symmetry factor, equal to 1 for heteronuclear and 2 for homonuclear molecules, correspondingly, and Z cirot is the rotational partition function:

Z cirot

¼

1X

r

j

scij

exp 

ecij kT

!

:

ð6Þ

   1 jðj þ 1Þ; ¼ Bce  ace i þ 2 hc

ecij

where h is Planck’s constant, c is the speed of light, and

xce ; xce xce ; xce yce ; xce zce ; Bce ; ace are spectroscopic constants. The number of vibrational levels of molecular species c is determined from the condition that the vibrational energy must not exceed the dissociation energy of the species Dc :

ecLc < Dc 6 ecLc þ1 ;

   2  3  4 1 1 1 1  xce xce i þ þ xce yce i þ þ xce zce i þ ; ¼ xce i þ 2 2 2 2 hc

eci

ð7Þ

ð9Þ

while the number of rotational levels Lrot;ci at each vibrational level i is determined in a similar manner, but taking into account the vibrational energy:

eciLrot;ci < Dc  eci 6 eciLrot;ci þ1 :

ð10Þ

If we assume that ace ¼ 0 and that the number of rotational levels is independent of the vibrational level (Lrot;ci ¼ Lrot;c0 ¼ Lrot;c ), the rotational degrees of freedom are independent of the vibrational degrees of freedom, and the molecule is a so-called ‘‘rigid rotator”. Constitutive equations for the diffusion velocities, stress tensor and heat flux in the first-order approximation of the the generalized Chapman-Enskog method are as follows [18]:

X Vci ¼  Dcidk ddk  DT ci r ln T;

ð11Þ

dk

P ¼ ðp  prel ÞI  2gS  fr  vI;

ð12Þ

 X X5   q ¼ k0 rT  p DT ci dci þ kT þ eci rot þ eci þ ec nci Vci : 2 ci ci ð13Þ where Dcidk ; DT ci are vibrational state-specific diffusion and thermal diffusion coefficients, p ¼ nkT is the pressure, prel is the relaxation pressure, g is the shear viscosity coefficient, f is the bulk viscosity coefficient, k0 is the partial thermal conductivity coefficient, I is a unit tensor, S is the rate-of-shear tensor, and ddk are diffusive driving forces. Since in the present work we are interested in the effect of the coupling between rotational and vibrational degrees of freedom on the bulk viscosity coefficient f and the thermal conductivity coefficient k0 , we shall forego the definitions of other transport coefficients, for more detail and methods of their calculation, the reader is referred to [18]. The generalized Chapman-Enskog method allows one to construct a numeric scheme for the calculation of the transport coefficients, by expanding the first-order correction to the distribution function in series of Sonine and Waldmann-Trübenbacher polynomials, and expressing the transport coefficients via the expansion coefficients. The thermal conductivity coefficient k0 , which describes energy transfer due to elastic collisions and translational-rotational and rotational-rotational energy exchanges in the state-to-state approach, can be expressed in terms of these expansion coefficients aci;10 ; aci;01 in the following way [21]:

k0 ¼

In the present work, we use the following expressions for the vibrational and rotational energies [19,20]:

ð8Þ

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

ð14Þ

where mc is the mass of molecular species c, and the specific heat of rotational degrees of freedom is defined as

D  2 E ci

crot;ci ¼

e

rot



mc kT

2

 ci 2

e

rot

:

ð15Þ

163

E. Kustova et al. / Chemical Physics Letters 686 (2017) 161–166

It is worth mentioning that in the accurate state-to-state transport model, the rotational specific heat has to be calculated for each vibrational state; in the present work we compute it by direct summation over rotational states. For the rigid rotator model, crot;ci ¼ crot;c 8i since the rotational energy in Eqs. (5) and (6) does not depend on i. Moreover, for further simplification, the rotational specific heat crot;c is commonly assumed to be equal to k=mc . However it has been shown in [22] that at high temperatures, the specific heat crot decreases and can no longer be assumed constant. The bulk viscosity coefficient f, which arises due to the presence of fast inelastic processes (in the state-to-state approximation, these processes are the rotational-translational energy transitions), can be expressed in terms of expansion coefficients as follows:

f ¼ kT

X nci n

ci

f ci;10 :

ð16Þ

XX cidk Krr0 pp0 adk;r0 p0 ¼ 15kT 2 dk r 0 p0

nci n

dr1 dp0 þ 3mc T

nci n

crot;ci dr0 dp1 ;

ð17Þ

c ¼ 1; . . . ; L; i ¼ 0; . . . ; Lc ; r; p ¼ 0; 1; . . . ;  XX cidk 1 brr0 pp0 f dk;r0 p0 ¼ ctr þc  nnci crot dr1 dp0 þ qqci crot;ci dr0 dp1 ; rot dk r 0 p0

ð18Þ

c ¼ 1; . . . ; L; i ¼ 0; . . . ; Lc ; r; p ¼ 0; 1; . . . ; 0

0

0

0

pp rr pp where Krr cidk ; bcidk are the bracket integrals introduced in [21], c tr is the specific heat of the translational degrees of freedom of the gas mixture:

3 kn ; 2 q

ð19Þ

and crot is the specific heat of the rotational degrees of freedom of the gas mixture:

crot ¼

Xq

ci

q

ci

X cidk nci cidk K1000 adk;00 þ Kcidk crot;ci ; 1100 adk;10 þ K1001 adk;01 ¼ 3mc T n dk c ¼ 1; . . . ; L; i ¼ 0; . . . ; Lc :

crot;ci :

ð20Þ

ð24Þ

For the coefficients f ci;rp , the system takes on the following form (taking into account the auxiliary condition (22) for f ci;00 ):

X cidk nci crot b1100 f dk;10 þ bcidk ; 1001 f dk;01 ¼  n ctr þ crot dk q X cidk ci b0110 f dk;10 þ bcidk 0011 f dk;01 ¼ dk

The expansions coefficients can be found as solutions of the following systems of linear equations [21]:

ctr ¼

15kT n X cidk ci cidk K1000 adk;00 þ Kcidk ; 1100 adk;10 þ K1001 adk;01 ¼ 2 n dk

crot;ci

q ctr þ crot

;

c ¼ 1; . . . ; L; i ¼ 0; . . . ; Lc :

ð25Þ

We see that the system for determination of the thermal conductivity coefficient is of order 3N v ibr þ 2La , where N v ibr ¼ P c ðLc þ 1Þ is the total number of vibrational levels in the gas, and La is the number of atomic species. For the bulk viscosity coefficient, the system is of order 2N v ibr þ La . For a N2/N flow, if we assume that the nitrogen molecule has 48 vibrational levels, the systems for computation of the thermal conductivity and bulk viscosity coefficients consist of 146 and 97 equations, correspondingly. Solving systems of such size at each time step in each cell in real-life CFD codes is unfeasible. However, if one assumes that the molecules are rigid rotators (i.e. the rotational degrees of freedom are independent of the vibrational degrees of freedom), the systems can be simplified and their size reduced considerably [11]:

X cd cd K0000 ad;00 þ Kcd 0100 ad;10 þ K0001 ad;01 ¼ 0; d

15kT n X cd c cd K1000 ad;00 þ Kcd ; 1100 ad;10 þ K1001 ad;01 ¼ 2 n d

These systems are degenerate, and have to be augmented by equations following from the Chapman–Enskog normalization conditions for the first-order correction to the distribution function:

X cd nc cd K1000 ad;00 þ Kcd crot;c ; 1100 ad;10 þ K1001 ad;01 ¼ 3mc T n d

Xq

c ¼ 1; . . . ; L;

ci

q

ci

aci;00 ¼ 0;

f ci;00 ¼ 0; c ¼ 1; . . . ; L; i ¼ 0; . . . ; Lc ;  Xnci q ctr f ci;10 þ ci crot;ci f ci;01 ¼ 0: n q ci

ð21Þ ð22Þ ð23Þ

If we assume that (1) simultaneous transitions of rotational and vibrational energy are improbable and can be neglected, (2) the collision cross-sections are independent of the vibrational state of the colliding particles (which is more or less equivalent to assuming that the diameter of an excited molecule does not change with the vibrational state), then all the bracket integrals 0

0

0

0

pp rr pp Krr cidk ; bcidk needed can be expressed in terms of elastic collision integrals and rotational relaxation times. If there is no significant difference in the masses of the mixture components, it is sufficient to only consider the lowest-order decomposition terms, that is, terms with r ¼ 0; p ¼ 0; r ¼ 1; p ¼ 0; r ¼ 0; p ¼ 1. The system for aci;rp then becomes

X cidk cidk K0000 adk;00 þ Kcidk 0100 adk;10 þ K0001 adk;01 ¼ 0; dk

ð26Þ

X cd nc crot b1100 f d;10 þ bcd ; 1001 f d;01 ¼  n ctr þ crot d q X cd c b0110 f d;10 þ bcd 0011 f d;01 ¼ d

c ¼ 1; . . . ; L;

crot;c

q ctr þ crot

; ð27Þ

Thus, for a gas flow where molecules are modeled by rigid rotators with constant diameters, determination of the thermal conductivity coefficient requires solution of a system of order 3Lm þ 2La , and for the determination of the bulk viscosity — of order 2Lm þ La , where Lm is the number of molecular species in the mixture. For the N2/N mixture, the systems (26) and (27) include 5 and 3 equations respectively. In the next section, we study the (1) applicability of the rigid rotator approximation and how it affects the thermal conductivity and bulk viscosity coefficients; (2) the applicability of the simplified model with constant rotational specific heat crot;c ¼ k=mc for computation of transport coefficients; and (3) the influence of variable molecular diameters coupled with a vibrational statedependent rotational spectrum in the state-specific approach.

164

E. Kustova et al. / Chemical Physics Letters 686 (2017) 161–166

3. Results and discussion The thermal conductivity and bulk viscosity coefficients were computed in a temperature range of 500–40000 K for binary mixtures of N2/N, O2/O and H2/H using the full state-specific systems (24) and (25) and using the simplified systems (26) and (27). When performing the calculations using the full state-specific systems, the rigid rotator approximation was not used, and the rotational spectrum was computed with a dependence on the vibrational level. When computations were done using the simplified systems, they were performed with the constant and variable rotational specific heats, the latter is obtained assuming in Eqs. (5), (6), and (15) ecij ¼ ecj 8i. The constant specific heat was not used for the H2/H mixture, where due to the large rotational quanta of hydrogen, it is inapplicable even at low temperatures. For computation of the elastic collision integrals, the phenomenological potential proposed in [23,24] (valid for temperatures up to 50000 K), was used, for computation of the rotational relaxation times, Parker’s formula was used [25]. It should be noted that the use of Parker’s formula for non-rigid rotator molecules is not entirely correct, since it does not account for the dependence of the rotational spectrum on the vibrational level. Nevertheless, introducing a state-specific rotational relaxation time based on a kinetic theory definition [18] and its evaluation using some model of rotationally inelastic collisions, such as the one presented in [26], is outside the scope of this study and will be carried out in the future work. The vibrational levels were assumed to be populated according to a Boltzmann distribution at atmospheric pressure. To assess the influence of variable molecular diameters, we use the method proposed in [12] and assume that the elastic collision integrals can be presented in the following form: 2

Xðl;rÞ cidk ¼

dcd;ik 2

dcd;00

Xðl;rÞ cd ;

ð28Þ

where dcd;ik ¼ ðdc;i þ dd;k Þ=2; dc;i is the diameter of molecule of species c in vibrational state i; dc;0 is the diameter of molecule of species ðl;rÞ

c in the ground state, Xcd is the elastic collision integral. When the partner in a collision is an atom, its diameter dc is set constant since the atoms are considered in the ground electronic state. The vibrational state-dependent diameters were computed using the Morse potential [27,13]. Figs. 1–3 (left) show the ratios of the thermal conductivity coefficient computed using various models to the coefficient kRR obtained using the rigid rotator approximation; the ratios are given

as functions of temperature T at fixed atomic molar fraction xc ¼ nc =n. We see that the difference between the state-specific (with constant diameters) and simplified approaches does not exceed 7%; with an increase in the molar fraction of atomic species, the discrepancy between the simplified and the full state-specific approaches decreases. Therefore, the simplified model can be used to compute the thermal conductivity coefficient of translational and rotational degrees of freedom in a wide range of temperatures, if one assumes constant molecular diameters. However, if the molecular diameters are assumed to be dependent on the vibrational level, the discrepancy increases and reaches up to 15–20%. This is especially noticeable in hydrogen, and to a lesser extent in oxygen — even at temperatures in the range of 10000– 15000 K, the state-specific model with variable diameters gives values of the thermal conductivity which are 10–15% smaller than those given by the rigid rotator approximation. In hydrogen, the effect of the vibrational state-dependent diameters is more pronounced due to their more rapid increase with the vibrational level. Conversely, the use of a constant specific heat leads to an overestimation of the thermal conductivity coefficient, especially significant in oxygen (up to a 30% increase compared to the rigid rotator approximation with a non-constant specific heat). It is worth mentioning that for the thermal conductivity coefficients, each of two studied effects weakly influences the coefficient if taken into account separately. The molecular non-rigidity has a maximum effect at temperatures about 15000–20000 K, and then the effect decreases with the temperature. However the coupled effect of non-rigidity and diameter variation on the thermal conductivity is rather important at T > 20; 000 K. Figs. 1–3 (right) present the ratio of the bulk viscosity coefficient computed using various models to the coefficient fRR calculated for the rigid rotator model. One can see using this assumption leads to a significant overestimation of the bulk viscosity; the discrepancy between the simplified and full state-to-state models grows with increasing temperature. Accounting for variable molecular diameters does not have a noticeable influence on the bulk viscosity; this might be due to the fact that rotational relaxation times were computed without accounting for an increase in the molecular diameter. The use of a constant specific heat yields further overprediction of the bulk viscosity coefficient; again, in oxygen, the overestimation is much larger than in nitrogen. The difference between results obtained using the full state-tostate approach and the simplified model is due to the fact that for higher-lying vibrational levels, the number of rotational levels decreases significantly, and the rotational spectra also differ from

Fig. 1. Ratio of thermal conductivity coefficients k=kRR (left) and bulk viscosity coefficients f=fRR (right) computed in a N2/N mixture for various models at fixed atomic molar fraction xN as functions of T. Black line: rigid rotator model with constant crot ; colored lines: full state-specific model; solid and dashed lines correspond to models with constant and variable molecular diameters respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

E. Kustova et al. / Chemical Physics Letters 686 (2017) 161–166

165

Fig. 2. Ratio of thermal conductivity coefficients k=kRR (left) and bulk viscosity coefficients f=fRR (right) computed in a O2/O mixture for various models at fixed atomic molar fraction xO as functions of T. Black line: rigid rotator model with constant crot ; colored lines: full state-specific model; solid and dashed lines correspond to models with constant and variable molecular diameters respectively.

Fig. 3. Ratio of thermal conductivity coefficients k=kRR (left) and bulk viscosity coefficients f=fRR (right) computed in a H2/H mixture for various models at fixed atomic molar fraction xH as functions of T. Solid and dashed lines correspond to models with constant and variable molecular diameters respectively.

those in the ground vibrational state; this leads to a noticeable decrease in the specific heat capacity of rotational degrees of freedom for the high-energy vibrational states. Because of this, the influence of the rigid rotator approximation on the thermal conductivity coefficient for real-life non-equilibrium flows (where vibrational levels of molecules are not populated according to a Boltzmann distribution) can be expected to be even lower: highlying vibrational levels of molecules in non-equilibrium flows behind strong shock waves are scarcely populated; while in nozzle flows, where the upper vibrational levels are more densely populated, the temperatures are not very high. 4. Conclusion The thermal conductivity and bulk viscosity coefficients are studied in the state-to-state approximation, and are compared to the thermal conductivity and bulk viscosity coefficients computed using a simplified model, which assumes that the molecules are rigid rotators with a fixed size. It is shown that use of the simplified algorithm leads to an overestimation of the thermal conductivity coefficient. This effect is especially prominent in hydrogen and to a lesser extent in oxygen; accounting for increasing diameters of vibrationally excited molecules leads to an even further decrease of the thermal conductivity coefficient. Using the rigid rotator model causes a twofold increase in the bulk viscosity coefficient; accounting for variable molecular diameters does not have any significant effect on the bulk viscosity coefficient. Assumption of a constant specific heat of rotational degrees of freedom breaks down at temperatures higher than 10,000 K and leads to an overestimation of transport coefficients, especially in oxygen flows.

Acknowledgments This study is supported by the Russian Science Foundation(project 15-19-30016).

References [1] I. Armenise, P. Reynier, E. Kustova, Advanced models for vibrational and chemical kinetics applied to mars entry aerothermodynamics, J. Thermophys. Heat Transf. 30 (4) (2016) 705–720. [2] O. Kunova, E. Kustova, M. Mekhonoshina, G. Shoev, Numerical simulation of coupled state-to-state kinetics and heat transfer in viscous non-equilibrium flows, AIP Conf. Proc. 1786 (1) (2016) 070012, http://dx.doi.org/10.1063/ 1.4967588. http://aip.scitation.org/doi/abs/10.1063/1.4967588 . [3] M. Panesi, R.L. Jaffe, D.W. Schwenke, T.E. Magin, Rovibrational internal energy 4 transfer and dissociation of N2(1Rþ g Þ  Nð Su Þ system in hypersonic flows, J. Chem. Phys. 138 (2013) 044312. [4] J. Kim, I. Boyd, State-resolved master equation analysis of thermochemical nonequilibrium of nitrogen, Chem. Phys. 415 (2013) 237–246. [5] M. Panesi, A. Munafò, T.E. Magin, R.L. Jaffe, Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method, Phys. Rev. E 90 (2014) 013009. [6] C. Nyeland, G. Billing, Transport coefficients of diatomic gases: internal state analysis for rotational and vibrational degrees of freedom, J. Phys. Chem. 92 (1988) 1752–1755. [7] T.E. Magin, M. Panesi, A. Bourdon, R.L. Jaffe, D.W. Schwenke, Coarse-grain model for internal energy excitation and dissociation of molecular nitrogen, Chem. Phys. 398 (2012) 90–95. [8] A. Munafò, M. Panesi, T.E. Magin, Boltzmann rovibrational collisional coarsegrained model for internal energy excitation and dissociation in hypersonic flows, Phys. Rev. E 89 (2) (2014) 023001. [9] N. Parsons, D.A. Levin, A.C. van Duin, T. Zhu, Modeling of molecular nitrogen collisions and dissociation processes for direct simulation Monte Carlo, J. Chem. Phys. 141 (23) (2014) 234307. [10] E. Torres, Y.A. Bondar, T. Magin, Uniform rovibrational collisional N2 bin model for DSMC, with application to atmospheric entry flows, AIP Conf. Proc. 1786 (1)

166

[11] [12]

[13]

[14]

[15]

[16]

[17]

[18] [19]

E. Kustova et al. / Chemical Physics Letters 686 (2017) 161–166 (2016) 050010, http://dx.doi.org/10.1063/1.4967560. http://aip.scitation. org/doi/abs/10.1063/1.4967560 . E. Kustova, On the simplified state-to-state transport coefficients, Chem. Phys. 270 (1) (2001) 177–195. E. Kustova, G.M. Kremer, Effect of molecular diameters on state-to-state transport properties: the shear viscosity coefficient, Chem. Phys. Lett. 636 (2015) 84–89. O.V. Kornienko, E.V. Kustova, Influence of variable molecular diameter on the viscosity coefficient in the state-to-state approach, Vestnik Saint Petersburg Univ. Ser. 1 Math. Mech. Astron. 3 (61 3) (2016) 457–467. V.A. Istomin, E.V. Kustova, M.A. Mekhonoshina, Eucken correction in hightemperature gases with electronic excitation, J. Chem. Phys. 140 (2014) 184311. V. Istomin, E. Kustova, State-specific transport properties of partially ionized flows of electronically excited atomic gases, Chem. Phys. 485–486 (2017) 125– 139. D. Olynick, W. Henline, L. Chambers, G. Candler, Comparison of coupled radiative flow solutions with project Fire II flight data, J. Thermophys. Heat Transf. 9 (4) (1995) 586–594. G.V. Candler, R.W. MacCormack, Computation of weakly ionized hypersonic flows in thermochemical nonequilibrium, J. Thermophys. Heat Transf. 5 (3) (1991) 266–273. E. Nagnibeda, E. Kustova, Nonequilibrium Reacting Gas Flows Kinetic Theory of Transport and Relaxation Processes, Springer-Verlag, Berlin, Heidelberg, 2009. M. Capitelli, G. Colonna, D. Giordano, L. Marraffa, A. Casavola, P. Minelli, D. Pagano, L. Pietanza, F. Taccogna, Tables of Internal Partition Functions and

[20]

[21]

[22]

[23]

[24] [25] [26]

[27]

Thermodynamic Properties of High-Temperature Mars-Atmosphere Species from 50 K to 50,000 K, Esa str-246, ESA Publications Division, ESTEC, Noordwijk, The Netherlands, October 2005. D. Pagano, A. Casavola, L.D. Pietanza, G. Colonna, D. Giordano, M. Capitelli, Thermodynamic properties of high-temperature Jupiter-atmosphere components, J. Thermophys. Heat Transf. 22 (3) (2008) 434–441. E. Kustova, E. Nagnibeda, Transport properties of a reacting gas mixture with strong vibrational and chemical nonequilibrium, Chem. Phys. 233 (1998) 57– 75. R.L. Jaffe, The calculation of high-temperature equilibrium and nonequilibrium specific heat data for N2, O2 and NO, in: 22nd Thermophysics Conference, Honolulu, 1987, AIAA Paper 87-1633, 1987. D. Bruno, M. Capitelli, C. Catalfamo, R. Celiberto, G. Colonna, P. Diomede, D. Giordano, C. Gorse, A. Laricchiuta, S. Longo, D. Pagano, F. Pirani, Transport properties of high-temperature Mars-atmosphere components, ESA STR 256, ESA, Noordwijk: ESA Publications Division, 2008. D. Bruno et al., Transport properties of high-temperature Jupiter atmosphere components, Phys. Plasmas 17 (11) (2010) 112315. J. Parker, Rotational and vibrational relaxation in diatomic gases, Phys. Fluids 2 (1959) 449–462. K. Koura, Statistical inelastic cross-section model for the Monte Carlo simulation of molecules with continuous internal energy, Phys. Fluids A 5 (3) (1993) 778–780. J. Kunc, F. Gordillo-Vazquez, Rotational-vibrational leveles of diatomic molecules represented by the Tietz–Hua rotating oscillator, J. Phys. Chem. 101 (1997) 1595–1602.