AIAA 2008–1263

Entropy Relations for Nonequilibrium Gas Mixtures:

Monatomic and Diatomic Gases

Tahir Gokcen†ELORET Corporation, NASA Ames Research Center, MS 230-2, Moffett Field, CA 94035

Approximate relations for computation of entropy in chemically reactingflows are described. Several nonequilibrium thermodynamic formulations ofgases (one-, two-, and three-temperature models) frequently employed inhypersonic computational fluid dynamics codes are considered. The presentanalysis is limited to a multi-component mixture of monatomic and diatomicgases because the intended applications are hypersonic flight simulations ofair and arc-jet simulations of air/argon mixtures. For six species of air (N2,O2, NO, N, O, Ar), the computed standard molar entropy for each speciesis compared against that of NASA Glenn curve fits under thermal equilib-rium conditions. A sample computed entropy field for a two-temperaturechemical nonequilibrium flow is also presented. The entropy expressions andprocedures consistent with the thermodynamic formulations used in thesesimulations can be implemented in the post-processor of a reacting flowsolver.

Nomenclature

e = molar energyei = molar energy of species ietr = translational molar energyer = rotational molar energyev = vibrational molar energyee = electronic molar energyev−e = vibrational-electronic molar energygn = degeneracies for electronic energy levels (n)ho = total enthalpy per unit massk = Boltzmann constant, 1.380 650 5×10−23 J/KM = molecular weightMi = molecular weight of species im = molecular or atomic massmu = atomic mass constant, 1.660 538 86×10−27 kgN = number of particlesp = pressurepi = partial pressure of species iQ = total partition functionQint = partition function for internal energy modeQtr = partition function for translational energy modeQr = partition function for rotational energy modeQv = partition function for vibrational energy modeQe = partition function for electronic energy modeR = universal gas constant, 8.314 472 J/(mol K)

† Senior Research Scientist, Senior Member AIAA

1American Institute of Aeronautics and Astronautics

46th AIAA Aerospace Sciences Meeting and Exhibit7 - 10 January 2008, Reno, Nevada

AIAA 2008-1263

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc.The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.

AIAA 2008–1263

Sc = Sackur-Tetrode constants = entropy per unit masss = molar entropysi = molar entropy of species is◦ = standard molar entropys◦i = standard molar entropy of species istr = translational molar entropysr = rotational molar entropysv = vibrational molar entropyse = electronic molar entropysv−e = vibrational-electronic molar entropyT = translational or translational-rotational temperatureTr = rotational temperatureTv = vibrational or vibrational-electronic temperatureTe = electronic or electron-electronic temperatureV = volumeXi = mole fraction of species iθen = characteristic temperatures for electronic energy levels (n)θr = characteristic temperature for rotational energyθv = characteristic temperature for vibrational energyσ = symmetry factor for diatomic molecules (2 - homonuclear, 1 - heteronuclear)

I. Introduction

Entropy, one of the thermodynamic properties of the gas, is not commonly used in the mathemati-cal formulation for computational simulations of chemically reacting flows. However, there are instanceswhere knowledge of the entropy field or distribution is desirable and sometimes required. Knowledge ofthe entropy field is useful, for instance, for investigations of entropy layer-swallowing effects in a hyper-sonic blunt-body flow (or to investigate the effect of nose bluntness on the development of the boundarylayer). 1 For the analysis of steady-state flowfields over hypersonic vehicles, the boundary layer equa-tions and the results of boundary layer theory are often employed. For non-isentropic reacting boundarylayer simulations, entropy at the boundary layer edge, along with other thermodynamic properties, isa required input. For example, the boundary layer code BLIMPK (Boundary Layer Integral MatrixProcedure - Kinetic), which has been extensively used by the Space Shuttle Orbiter program over thelast 20 years, was recently employed to analyze the localized catalytic heating effects on the Shuttlethermal protection system. 2 Since BLIMPK requires knowledge of the entropy at the boundary layeredge, in the work of Ref. 2, it was approximated using equilibrium curve fits.

The definition of entropy strictly depends on the thermodynamic formulation of a nonequilibriumgas. Multi-temperature thermochemical models are increasingly used to explain the experimental dataobtained in high enthalpy facilities, 3−5 and the models so developed are then implemented in compu-tational fluid dynamics (CFD) codes for hypersonic flight simulations. 4,6 It is outside the scope of thispaper to cover all of the nonequilibrium thermochemical models used in hypersonic flow CFD solvers.However, two primary examples of such CFD codes developed at NASA are the DPLR code 7,8 of theAmes Research Center and the LAURA code 9,10 of the Langley Research Center. Nonequilibrium ther-mochemical models employed in these codes include one-, two- and three-temperature models for air.For these thermochemical nonequilibrium flows, the expressions for the computation of entropy are notreadily available.

The objective of this paper is to present analytical expressions and procedures to compute theentropy of a thermochemical nonequilibrium gas that are consistent with the models commonly em-ployed in hypersonic CFD flow solvers. The present analysis is limited to a multi-component mixtureof monatomic and diatomic gases, because the intended applications are hypersonic flight simulations ofnonequilibrium air and arc-jet simulations of air/argon mixtures.

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II. Entropy Relations

Single-Species Gas

In statistical mechanics, partition functions are used to characterize thermodynamic information ofgas models. All of the thermodynamic properties of a gas can be calculated from its partition function.

The relationships to determine entropy (and other thermodynamic properties) from partition func-tions can be obtained from any statistical mechanics textbook (e.g., see Vincenti and Kruger 11 orAnderson 12). The molar entropy and internal energy of a gas in terms of its partition function Q aredefined by

s = R

[

lnQ

N+ 1 + T

∂(lnQ)

∂T

]

, (1)

e = R T 2∂(lnQ)

∂T, (2)

and pressure is also defined by the partition function

p = N k T∂(lnQ)

∂V. (3)

Determination of the partition functions is usually based on quantum mechanical calculations. Whenthe energy levels and degeneracies of these levels are known accurately from theoretical/experimentalspectroscopic data, the quantum mechanical calculations provide accurate computations of the thermo-dynamic properties.

Here, based on the thermal equilibrium assumption (equilibrium at some level such that temper-ature(s) of the gas can be defined), approximate descriptions of partition functions and subsequentanalyses are used for the computation of thermodynamic properties. An important assumption madeis that there is no coupling among the various modes of energy. The validity and limitations of thisassumption are well known and discussed in various textbooks. 4,11,12 This assumption provides a basicframework for analysis, and it allows us to express the partition function in the form

Q = Qtr Qint = Qtr Qr Qv Qe. (4)

Substituting Eq. (4) into Eq. (1), the molar entropy can then be expressed as the sum of thecontributions from each energy mode,

s = str + sr + sv + se, (5)

where the molar entropies for each mode are defined as

str = R

[

lnQtr

N+ 1 + T

∂(lnQtr)

∂T

]

,

sr = R

[

lnQr + T∂(lnQr)

∂T

]

,

sv = R

[

lnQv + T∂(lnQv)

∂T

]

,

se = R

[

lnQe + T∂(lnQe)

∂T

]

.

(6)

Similarly, the internal energy in Eq. (2) is expressed as the sum of all internal modes:

e = etr + er + ev + ee, (7)

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and making use of the internal energy components defined in Eq. (2), Eq. (6) can be written as

str = R lnQtr

N+ R +

etr

T,

sr = R lnQr +er

T,

sv = R lnQv +ev

T,

se = R lnQe +ee

T.

(8)

The partition functions Qtr, Qr, Qv Qe for diatomic gases of air are available in the literature 4,11

as

Qtr = V

(

2 π m k T

h2

)3/2

,

Qr =1

σ

(

T

θr

)

,

Qv =1

1 − e−θv/T,

Qe =

∞∑

n=0

gn e−θen/T ,

(9)

where Qr and Qv are computed using rigid-rotor and harmonic-oscillator approximations. Since there areno rotational and vibrational modes of energy for a monatomic gas (or species), the partition functionsQr and Qv are set to unity.

Substituting Eqs. (9), individually, into Eq. (2), the molar internal energies corresponding to eachmode can be expressed as

etr =3

2R T,

er = R T,

ev =R θv

eθv/T − 1,

ee =R

Qe

∞∑

n=1

gn θen e−θen/T .

(10)

Note that the molar energy for vibrational mode is relative to the zero-point energy level. In calculationof energy levels for partition functions, the levels are computed relative to the lowest energy quantumstate.

The molar entropies can also be computed using Eqs. (8). However, computation of str needs to beclarified with additional algebra. Note that str is a function of two thermodynamic variables, temperatureand pressure (or volume), while all of the other entropy components are functions of temperature only.Consider the translational molar entropy from Eq. (6),

str = R lnQtr

N+ R +

etr

T. (11)

Making use of the ideal gas law from Eq.(3), p = N k T/V , and noting that

lnQtr

N=

5

2lnT − ln p + ln

[ (

2 π m

h2

)3/2

k5/2

]

, (12)

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the translational molar entropy can be written as

str/R =5

2+

5

2lnT − ln p + ln

[(

2 π m

h2

)3/2

k5/2

]

. (13)

This is the so-called Sackur-Tetrode equation for the absolute translational entropy.In order to put Eq. (13) in a more recognizable form, we make use of the Sackur-Tetrode constant,

Sc, defined as

Sc =5

2+ ln

[ (

2 π mu k T1

h2

)3/2k T1

p◦

]

, (14)

where T1 = 1 K and p◦ is the standard pressure taken as either 1 atm or 1 bar. The National Institute ofStandards and Technology (NIST) website 13 provides the most up-to-date physical constants includingthe Sackur-Tetrode constant: Sc = −1.1517047 at p◦ = 1 bar, and Sc = −1.1648677 at p◦ = 1 atm.Note that the present NIST and JANAF (Joint Army, Navy, and Air Force) thermodynamic tables usestandard pressure of p◦ = 1 bar while the JANAF tables published prior to 1982 were based on thestandard pressure of p◦ = 1 atm.

Making use of the Sackur-Tetrode constant Sc and noting that molecular mass is related to theatomic mass constant by m = M mu, the molar translational entropy in Eq. (13) can be expressed as

str = R[

Sc +3

2lnM +

5

2lnT − ln (p/p◦)

]

. (15)

In order to separate the pressure dependence in the molar entropy, it is customary to define a standardmolar entropy, s◦, as the molar entropy at standard pressure p◦. For the translational entropy,

str = s◦tr − R ln (p/p◦). (16)

For internal molar entropies, since there is no pressure dependence, the molar and standard molarentropies are equivalent, i.e., s◦int = sint.

In summary, all standard molar entropies are now given as follows:

s◦tr = R[

Sc +3

2lnM +

5

2lnT

]

,

s◦r = R ln Qr + R,

s◦v = R ln Qv +R θv/T

eθv/T − 1,

s◦e = R ln Qe +R

Qe

∞∑

n=1

gn θen/T e−θen/T ,

(17)

and total standard molar and molar entropies are given by

s◦ = s◦tr + s◦r + s◦v + s◦e,

s = s◦ − R ln (p/p◦).(18)

Multi-Component Gases

As defined by Eq. (18), the entropy of species i is given by

si = s◦i − R ln (pi/p◦). (19)

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The molar entropy of a gas mixture is given by the sum over all species:

s =

ns∑

i=1

Xi si =

ns∑

i=1

Xi s◦i − R

ns∑

i=1

Xi ln (pi/p◦). (20)

Using the relationship between mole fractions and partial pressures, pi = Xi p, and rearranging theterms, the molar standard entropy and molar entropy for the mixture can be expressed as follows:

s◦ =

ns∑

i=1

Xi s◦i ,

s = s◦ − R ln (p/p◦) − R

ns∑

i=1

Xi ln Xi,

(21)

where the last term in s is called the “mixing entropy.”

III. Computation of Entropy for Nonequilibrium Thermochemical ModelsAs mentioned earlier, the definition of entropy as a thermodynamic property depends on the thermo-

dynamic model of the gas. In this section, various thermochemical models employed in hypersonic CFDcodes are considered. These models are somewhat loosely classified as one-, two-, and three-temperaturemodels, and for these models, expressions and procedures for computation of the entropy are presented.

One-Temperature Model

In one-temperature models, excitations of all energy modes are assumed to be equilibrated, and theyare characterized by one temperature, T . Thus, only chemical nonequilibrium effects are included in themodel. For this case, the entropy relations presented in Section II, i.e., Eq. (17), Eq. (18), and Eq. (21),are directly applicable. However, the CEA (Chemical Equilibrium with Applications) program curvefits, also known as the NASA Glenn curve fits, 14−16 are often employed in CFD codes for computation ofthe thermodynamic properties (e.g., internal energy or enthalpy). McBride and Gordon 14 also providea curve fit for species standard entropy, s◦i /R, in the following functional form

s◦i (T )/R = −a1

T−2

2− a2 T−1 + a3 ln T + a4 T + a5

T 2

2+ a6

T 3

3+ a7

T 4

4+ a9. (22)

The coefficients a1 through a9 are given by the CEA program for three temperature ranges: T =200 − 1000 K, T = 1000 − 6000 K, and T = 6000 − 20, 000 K. The CEA curve fits are applicable onlyunder thermal equilibrium conditions, and they are expected to be more accurate representations ofthermodynamic data than those presented in Section II because the partition functions used also includevarious corrections (e.g., rotational stretching, and anharmonic effects due to vibration and rotationalmode coupling).14 The comparisons with the CEA curve fits will be made later.

For the one-temperature model, the molar standard entropy and molar entropy for the mixture arecomputed using the expressions in Eq. (21):

s◦(T ) =

ns∑

i=1

Xi s◦i (curve fit or analytical),

s(T ) = s◦(T ) − R ln (p/p◦) − R

ns∑

i=1

Xi ln Xi.

(23)

Two-Temperature Model

In the two-temperature model presented by Park, 3,4 excitations of internal degrees of freedom aredivided into two classes, and it is assumed that the excitations are equilibrated within each class. The

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translational and rotational modes of energy make up one class, and it is characterized by a translational-rotational temperature, T . The vibrational and electronic modes form the other class, characterized bya vibrational-electronic temperature, Tv. There are variations of this model depending on whetherthe molecular species have individual vibrational temperatures or whether there is just one vibrationaltemperature for the mixture. Also, when ionization is included in the formulation, the translationalenergy of electrons is lumped into the vibrational-electronic energy pool, and the translational energy etr

includes only the contributions from heavy particles. Most hypersonic CFD codes today have a versionof Park’s two-temperature thermochemical model implemented. Here, it is assumed that the speciesand gas mixture have two temperatures, i.e., only one vibrational temperature for the mixture. Thecomputational approaches for the two cases described here differ in how the internal energy componentsare computed in the flowfield formulation.

For the first case, all internal energy components are modeled using the partition function-derivedexpressions in Eq. (10), i.e.,

etr(T ) =3

2R T,

er(T ) = R T,

ev(Tv) =R θv

eθv/Tv − 1,

ee(Tv) =R

Qe(Tv)

∞∑

n=1

gn θen e−θen/Tv .

(24)

The molar standard entropy is then computed by

s◦(T, Tv) = s◦tr(T ) + s◦r(T ) + s◦v(Tv) + s◦e(Tv) (25)

where the functional expressions for Qint and s◦int are those given in Eq. (9) and Eq. (17), respectively.For the second case, some of the internal energy modes are computed in the flowfield simulation

using polynomial curve fits (e.g., see Gnoffo et al. 9):

etr(T ) =3

2R T,

er(T ) = R T,

ev−e(Tv) = ev(Tv) + ee(Tv) = etot(Tv)(curve fit) − etr(Tv) − er(Tv).

(26)

The molar standard entropy for vibrational and electronic modes is computed by

sv−e(Tv) = sv(Tv) + se(Tv) = stot(Tv)(curve fit) − str(Tv) − sr(Tv), (27)

where stot(Tv)(curve fit) indicates the curve fit entropy is evaluated using the vibrational temperature.The molar standard entropy of the two-temperature gas is given by

s◦(T, Tv) = s◦tr(T ) + s◦r(T ) + s◦v−e(Tv). (28)

The rationale for using the curve fits for vibrational and electronic energy is that, in the limit of thermalequilibrium, thermodynamic properties such as internal energy and entropy approach those given by theequilibrium curve fits.

As in the one-temperature model, once the molar standard entropies for all species are obtainedthis way, the molar standard entropy of the mixture and molar entropy of the mixture are computedusing Eq. (21):

s◦(T, Tv) =ns∑

i=1

Xi s◦i (T, Tv),

s(T, Tv) = s◦ − R ln (p/p◦) − R

ns∑

i=1

Xi ln Xi.

(29)

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Three-Temperature Model

In a manner analogous to the approach for the two-temperature model described above, the excita-tions of internal degrees of freedom are divided into three classes, and the excitations are assumed to beequilibrated within each class. Two different three-temperature models are considered here, depending onhow the internal energy modes are split: (1) translational, rotational, vibrational (vibrational-electronic),and (2) translational (translational-rotational), vibrational, and electronic (electron-electronic) temper-atures. As mentioned earlier, when ionization is included in the formulation, translational energy ofelectrons is usually lumped into the electronic energy pool.

The first case is analogous to the two-temperature model, and the standard entropy of the three-temperature gas can be computed using

s◦(T, Tr, Tv) = s◦tr(T ) + s◦r(Tr) + s◦v−e(Tv). (30)

For the second case, when the curve fits are not used, the standard entropy can be computed similarlyusing

s◦(T, Tv, Te) = s◦tr(T ) + s◦r(T ) + s◦v(Tv) + s◦e(Te). (31)

The use of curve fits can be implemented in two ways. Either vibrational energy or electron-electronicenergy can be obtained from the curve fits:

ev(Tv) = etot(Tv)(curve fit) − etr(Tv) − er(Tv) − ee(Tv),

sv(Tv) = stot(Tv)(curve fit) − str(Tv) − sr(Tv) − se(Tv),(32)

oree(Te) = etot(Te)(curve fit) − etr(Te) − er(Te) − ev(Te),

se(Te) = stot(Te)(curve fit) − str(Te) − sr(Te) − sv(Te).(33)

However, both options may pose some problems in implementation. The first option could be problematicif there are no molecular species in the thermochemical model. The second option may significantlyoverestimate the electronic energy at relatively low temperatures because of the curve fits. Again, forboth cases, equations analogous to Eq. (29) can be used to compute the mixture entropies.

IV. Computed ResultsThe entropy relations described in Sections II and III are valid for any mixture of monatomic

and diatomic gases. In this section, applications of these procedures to compute the nonequilibriumentropy are presented. As mentioned earlier, for thermal equilibrium cases, the CEA curve fits areavailable to compute the species standard molar entropies, and they are used for verification of thepresent approach. For this purpose, six species (N2, O2, NO, N, O, Ar) of air are considered. Thecomputed standard entropy for each species is compared against that of the CEA curve fits in Fig. 1.For the partition function computations, the spectroscopic constants are obtained from Refs. 4, 11 andthe DPLR database, and are listed in Table 1. For the temperature range considered, from 200 K to20,000 K, the computed values using partition functions are within 0.5% of those from the CEA curvefits for N and O, 0.9% for Ar, and as much as 3% different for O2, 3.4% for NO and 4% N2 . Thelarger differences for molecular species are expected, because no coupling effects among various internalmodes are considered in the present partition functions. Nevertheless, considering the larger differencesare observed at the higher end of the temperature range, at which the molecular species are likely to bedissociated, the overall differences in computed mixture entropies are expected to be much less than 4%(for flows in thermal equilibrium).

As an example entropy field calculation, an arc-jet flow over a blunted wedge is considered. Arc-jetflows are dominated by various thermal and chemical nonequilibrium phenomena, noting that the arc-heated test gas, produced by passing air mixed with argon through an electric arc discharge, is expandedthrough the conical nozzle to produce hypersonic velocities. The wedge model considered was testedin the Interaction Heating Facility (IHF) at NASA Ames Research Center. 17,18 The flowfield solutionsexcept the entropy field are obtained using DPLR with a six-species air model (N2, O2, NO, N, O, Ar)

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Table 1 Physical constants used for partition functions of species

Species θr, K θv, K θen, K gn

N2 2.9 3395 0 17.22316× 104 38.57786× 104 68.60503× 104 69.53512× 104 39.80564× 104 19.96827× 104 21.04898× 105 21.11649× 105 51.22584× 105 11.24886× 105 61.28248× 105 61.33806× 105 101.40430× 105 61.50496× 105 6

O2 2.1 2239 0 31.13916× 104 21.89847× 104 14.75597× 104 14.99124× 104 65.09227× 104 37.18986× 104 3

NO 2.5 2817 0 45.46735× 104 86.31714× 104 26.59945× 104 46.90612× 104 47.05000× 104 47.49106× 104 47.62888× 104 28.67619× 104 48.71443× 104 28.88608× 104 48.98176× 104 48.98845× 104 29.04270× 104 29.06428× 104 29.11176× 104 2

N 0 42.76647× 104 104.14931× 104 6

O 0 52.27708× 102 33.26569× 102 12.28303× 104 54.86199× 104 1

Ar 0 11.61114× 105 91.62583× 105 211.63613× 105 71.64233× 105 31.64943× 105 51.65352× 105 15

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45

40

35

30

25

20

so/R, 1/K

2 3 4 5 6

103

2 3 4 5 6

104

2

Temperature, K

N2

CEA curve fit Partition function

45

40

35

30

25

20

so/R, 1/K

2 3 4 5 6

103

2 3 4 5 6

104

2

Temperature, K

O2

CEA curve fit Partition function

a) N2 a) O2

45

40

35

30

25

20

so/R, 1/K

2 3 4 5 6

103

2 3 4 5 6

104

2

Temperature, K

NO CEA curve fit Partition function

35

30

25

20

15

so/R, 1/K

2 3 4 5 6

103

2 3 4 5 6

104

2

Temperature, K

N CEA curve fit Partition function

c) NO d) N

35

30

25

20

15

so/R, 1/K

2 3 4 5 6

103

2 3 4 5 6

104

2

Temperature, K

O CEA curve fit Partition function

35

30

25

20

15

so/R, 1/K

2 3 4 5 6

103

2 3 4 5 6

104

2

Temperature, K

Ar CEA curve fit Partition function

e) O e) Ar

Fig. 1 Comparison of computed standard entropies for 6 air species using CEA curve fitsand partition functions: po = 1 atm, T = 200 - 20,000 K, internal excitations are in thermalequilibrium.

for arc-jet flow. The thermal state of the gas is described by two temperatures: translational-rotationaland vibrational-electronic, within the framework of Park’s two-temperature model. Further informationon the computational approach can be found in Ref. 19. Figure 2 shows computed flowfield contours of

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x, m

y,m

0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

M

5.25

5.00

4.75

4.50

4.25

4.00

3.75

3.50

3.25

3.00

2.75

2.50

2.25

2.00

1.75

1.50

1.25

1.00

0.75

0.50

0.25

0.00

Computed Mach Number Contoursz = 0 Plane

a) Mach numberx, m

y,m

0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

s, kJ/(kg K)

13.6

13.4

13.2

13.0

12.8

12.6

12.4

12.2

12.0

11.8

11.6

11.4

11.2

11.0

10.8

10.6

10.4

10.2

10.0

9.8

9.6

9.4

Computed Entropy Contoursz = 0 Plane

b) Entropy

x, m

y,m

0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

T, K

11000

10500

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

Computed Temperature Contoursz = 0 Plane

c) Temperaturex, m

y,m

0 0.05 0.1 0.15 0.2

0

0.05

0.1

0.15

0.2

Tv, K

11000

10500

10000

9500

9000

8500

8000

7500

7000

6500

6000

5500

5000

4500

4000

3500

3000

2500

2000

1500

1000

500

Computed Vibrational Temperature Contoursz = 0 Plane

d) Vibrational temperature

Fig. 2 Computed flowfield contours of a blunted wedge model (rn = 0.95 cm, w =17.78cm, l = 18.42 cm, rc =0.635 cm). Freestream conditions: IHF 13-inch nozzle flow withreservoir conditions of po = 450 kPa, ho = 22 MJ/kg, and 10% Ar in air.

Mach number, entropy, temperature, and vibrational temperature for the blunted wedge model. InFig. 2a, the Mach number contours clearly show the detached bow shock wave formed in front of thewedge model. As mentioned earlier, an entropy layer is generated by the existence of this curved shockwave as shown in Fig. 2b. The entropy layer (or entropy gradient) in the flowfield affects the developmentof the boundary layer over the wedge model. Due to the entropy layer, not only do boundary layeredge flow conditions vary along the wedge streamwise, but also normal gradients are present in flowquantities such as velocity, pressure and entropy at the boundary layer edge. These effects ultimatelylimit the use of the boundary layer equations, since they are inconsistent with the basic assumptions ofboundary layer theory. Although the entropy layer is expected to be “swallowed” eventually by the coldboundary layer, at some distance downstream of the wedge nose region, 1 for the computed flowfield,

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the so-called swallowing is by no means complete, i.e., the entropy gradients are clearly present forthe entire flowfield (between the wedge surface and and shock wave). Because of the nonequilibriumexpansion process in an arc-jet nozzle, the chemical composition and vibrational energy state of the gas(vibrationally excited and dissociated gas) freeze downstream of the throat. As shown in Figs. 2c and2d, the predicted translational-rotational temperatures differ from vibrational-electronic temperaturesfor most of the flowfield over the wedge model. Starting from the freestream of the wedge model,significant departures from thermal equilibrium as well as chemical equilibrium are predicted. Note thatthe CEA curve fits would not be applicable to compute the entropy field for this flow because the flowis in thermal nonequilibrium.

V. Summary and Concluding Remarks

Approximate relations for the computation of entropy in chemically reacting flows are developed.Expressions and procedures to compute entropy have been presented in a manner consistent with variousmulti-temperature formulations of gases (one-, two-, and three-temperature models) frequently employedin hypersonic CFD simulations. The expressions are currently limited to monatomic and diatomic gases,because the intended applications are hypersonic flight simulations of air and arc-jet simulations ofair/argon mixtures. The procedures can readily be implemented in the post-processor of a reactingflow CFD solver. For six species (N2, O2, NO, N, O, Ar) of air in thermal equilibrium conditions, thecomputed standard molar entropy for each species is compared against that predicted using the CEAcurve fits. An example of a computed entropy field over a blunted wedge in an arc-jet flow is alsopresented.

Acknowledgments

This work was partially funded by the NASA Space Shuttle Return-To-Flight and CEV TPS Ad-vanced Development Projects. The support from NASA Ames Space Technology Division throughcontract NNA04BC25C to ELORET Corporation is gratefully acknowledged.

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