nasa technical note
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N A S A TECHNICAL NOTE N A S A TN D-3921
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PROGRAMS FOR COMPUTING EQUILIBRIUM THERMODYNAMIC PROPERTIES OF GASES
by Hawy E, Bailey Ames Research Center Moflett Field Cdl$
N A T I O N A L A E R O N A U T I C S A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. A P R I L 1967
https://ntrs.nasa.gov/search.jsp?R=19670013971 2019-04-13T06:25:03+00:00Z
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. NASA TN D-3921
PROGRAMS FOR COMPUTING EQUILIBRIUM THERMODYNAMIC
PROPERTIES OF GASES
By Harry E. Bailey
Ames Research Center Moffett Field, Calif.
N AT I ON AL A E RON AUT ICs AN D SPACE ADM I N I ST RAT I ON
For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00
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, PROGRAMS FOR COMPUTING EQUILIBRIUM THERMODYNAMIC
PROPERTIES OF GASES
By Harry E. Bailey
Ames Research Center
SUMMARY
This r e p o r t p r e s e n t s a s e r i e s of subroutines, w r i t t e n i n FORTRAN I V , f o r c a l c u l a t i n g the equi l ibr ium thermodynamic p rope r t i e s of any mixture of r e a c t - ing diatomic gases i f the temperature, dens i ty , and molar concentrat ions of the spec ie s a r e given. One t r i a t o m i c gas, carbon dioxide, may a l s o be included i n t h e mixture. Two a d d i t i o n a l subrout ines permit t h e c a l c u l a t i o n of t he molar concentrat ions of t h e spec ie s i f the gas mixture i s a i r . A sample main program i s included which uses a l l t hese subrout ines t o compute the equ i l ib r ium composition and thermodynamic p r o p e r t i e s o f a i r f o r a given d e n s i t y over a range of temperatures.
INTRODUCTION
So lu t ions t o problems involving the flow of a chemically r e a c t i n g gas r e q u i r e the computation of the equi l ibr ium thermodynamic p r o p e r t i e s of t he gas mixtures over wide ranges of temperature and d e n s i t y . These p r o p e r t i e s a r e a v a i l a b l e i n t a b u l a r form f o r a i r ( r e f s . 1, 2, and 3) , i n g raph ica l form f o r carbon d iox ide ( r e f . 4), and f o r t h ree n i t rogen carbon dioxide mixtures ( r e f . 5 ) . However, n e i t h e r t he t a b u l a r nor g raph ica l form i s convenient f o r use i n e l e c t r o n i c machine c a l c u l a t i o n s without e l a b o r a t e and tedious curve
u s e f u l f o r t h e s o l u t i o n of many problems i n the flow of a chemically r e a c t i n g gas.
I f i t t i n g of t h e da t a . Therefore, the subroutines presented he re should prove
F ive d i f f e r e n t subrout ines are presented i n t h i s r e p o r t . I n add i t ion , a main program designed t o compute the equi l ibr ium thermodynamic p r o p e r t i e s of a i r i s p resen ted t o i l l u s t r a t e t h e use of t h e f i v e subrout ines . Two of the subrout ines , SPECIE and ENTH, can be used t o o b t a i n the pressure, enthalpy, s p e c i f i c h e a t a t cons t an t pressure (with frozen composition), entropy, gas constant , and equ i l ib r ium cons tan t s of the components of a gas mixture. The va r ious chemical and phys ica l constants needed as inpu t by subrout ine SPECIE may be ob ta ined from re fe rences 6 and 7. A t h i r d subroutine, MTCON, gives the r e a c t i o n r a t e s f o r e i g h t chemical r eac t ions i n a i r . This subrout ine would r e q u i r e modif icat ion i f some o t h e r gas mixture were used. t i o n a l subrout ines , GUESS and ITERA, c a l c u l a t e t he equ i l ib r ium molar concen- t r a t i o n s o f each spec ie s f o r a i r a t a given temperature and d e n s i t y . Although t h e s e subrout ines would r e q u i r e major modif icat ion f o r any mixture o t h e r t han a i r , they do provide a s u i t a b l e basis from which s imilar programs f o r o t h e r mixtures could be w r i t t e n e a s i l y .
Two addi-
Each ind iv idua l spec ie s i s assumed t o behave as an ideal gas. The thermodynamic p r o p e r t i e s of the diatomic species a r e approximated by the harmonic-osci l la tor r i g i d - r o t a t o r model with r o t a t i o n a l and v i b r a t i o n a l constants appropr i a t e t o the lowest e l e c t r o n i c s t a t e . a r e given f o r two a i r models. Model A contains the spec ie s N2, 02, N, 0, NO, NO+, O', N+, and e-.
Computational r e s u l t s
Model B contains the same nine spec ie s as model A p lus two
C P
P i C
En
*i
gn
H
Hi
HOT
Ki
Mi
n i
P
P i
R
S
si
T
T V
2
add i t iona l species , O++ and N++.
SYMBOLS
s p e c i f i c h e a t a t constant p re s su re (wi th composition frozen)
s p e c i f i c h e a t a t constant pressure f o r t he i t h spec ie s
energy of the nth energy l e v e l
f r e e energy of t he i t h spec ie s
degeneracy of the n th energy l e v e l
t o t a l enthalpy of the mixture
enthalpy of the i t h spec ie s
enthalpy of formation of t he i t h spec ie s
equi l ibr ium constant f o r the i t h r e a c t i o n
molecular weight of t he i t h spec ie s
number of atoms i n the i t h spec ie s
t o t a l pressure of t he mixture
standard p res su re of one atmosphere
p a r t i a l pressure of t he i t h spec ie s
un ive r sa l gas constant
t o t a l entropy of the mixture
entropy of t he i t h spec ie s
temperature, OK
c h a r a c t e r i s t i c r o t a t i o n a l temperature
c h a r a c t e r i s t i c v i b r a t i o n a l temperature
t
ZN mass concentration of nitrogen atoms in cold mixture
' ZO mass concentration of oxygen atoms in cold mixture
Pi sum of Bij values over the index j
Bi j difference of the stoichiometric coefficients of the jth species in the ith reaction
concentration of the ith species in moles/gm Yi
SAMPLE MAIN PROGRAM
This is a typical program (see appendix A) which uses all the following subroutines to compute the thermodynamic properties of equilibrium air. It is intended as a guide for the use of the subroutines SPECIES, mTH, RATCON, GUESS, and ITERA. It by no means exhausts the possibilities for their use.
The sample program requires an input AT (DELTA), a density (RHO), and nine species concentrations for the cold mixture of nitrogen and oxygen (G(N)). The program will then call the appropriate subroutines for the computation of the thermodynamic properties pressure, gas constant, enthalpy, entropy, specific heat at constant pressure, equilibrium constants, reaction rate con- stants (as defined under subroutine RATCON), and the species concentrations. These computations will be performed at the given density for 25 temperatures starting at AT and ending at 25 L1T. A sample output is given for air at a density of 0.01288 gm/cm3 and at 4 temperatures from 1000° K to 4000° K.
SUBROUTINE SPECIE
This subroutine (see appendix B) will compute the pressure (P),' enthalpy (H(21)), specific heat at constant pressure (with composition frozen) (CP( 21) ) , entropy (FE( 21) ) , equilibrium constants (BK( J) ) , reaction rate constants (AK(J)), and the gas constant (GCONST) for a given temperature (T) , density (R) , and the molar concentrations (GA( J) ) of each species. On the first entry to the subroutine, all necessary chemical and physical constants are read from cards and stored in the program. made for the computation of 20 species and 40 reactions.
Provision has been
This subroutine uses the following equations to compute the thermodynamic properties of the mixture from the thermodynamic properties of the components.
~ ~~
'Characters in parentheses are those used in the FORTRAN listing in the appendixes. There is an unavoidable inconsistency between these characters and those defined under Symbols.
3
The en tha lpy i s given by
RT
The s p e c i f i c h e a t a t constant pressure (wi th frozen composition) i s given by
v c = Y p p i P
The entropy of t he mixture i s given by
R RT P O
The t o t a l p ressure of t he mixture i s
P = pRT 1 y i
The equi l ibr ium cons tan t f o r each r e a c t i o n i s
The p a r t i a l p ressure of each spec ies i s computed from
- Y i Pi - - CYi
The en t ropy of each spec ies i s computed i n t h i s subrout ine as
S . = Hi - Fi 1
( 3 )
(4)
The d a t a t h a t must be input t o t h i s subrout ine a r e the number of r eac t ions ( I L ) , t he number of spec ie s ( J L ) , t he s to i ch iomet r i c c o e f f i c i e n t s f o r each r e a c t i o n ( B ( I , J ) ) , t he number of e l e c t r o n i c energy l e v e l s for each spec ies (NL(J) ) , t he energy (E(J ,N)) and degeneracy ( G ( J , N ) ) of each of t hese e l e c t r o n i c energy l e v e l s , t he number of atoms i n each molecule ( E N J ( J ) ) , t he v i b r a t i o n a l (TV( J) ) and r o t a t i o n a l (TR) temperatures of each diatomic spec ie s , the h e a t of formation (ENTA(J ) ) o f each spec ie s , t he molecular weight (EMWT(J)) of each spec ies , and f i n a l l y the percent by volume (GX(J)) of each spec ie s i n the cold mixture, A l l t h i s information i s s t o r e d i n the subrout ine and used on a l l subsequent e n t r i e s f o r t he computation of t he thermodynamic p r o p e r t i e s .
Each time the subrout ine i s c a l l e d i t must be given the temperature (T) , i n degrees Kelvin, t he d e n s i t y ( R ) i n gm/cm3, and t h e mass concent ra t ions (GA(1) ), i n moles/gm. The subrout ine w i l l then compute a l l t he q u a n t i t i e s
4
Y
l i s t e d i n the f i r s t paragraph of t h i s sec t ion . I n add i t ion , t he en tha lpy ( H ( J ) ) , entropy ( F E ( J ) ) , and s p e c i f i c h e a t a t cons t an t pressure (CP(J ) ) for each ind iv idua l spec ies (J = 1, J L ) w i l l be computed.
'
Subroutine SPECIE uses two subrout ines , ENTH and RATCON, which a r e d iscussed i n subsequent s ec t ions .
Sample input d a t a requi red by subrout ine SPECIE a r e shown i n appendix F. The inpu t d a t a a r e shown f o r model A and model B.
SUBROUTINE ENTH
This subrout ine ( s e e appendix C ) w i l l compute the en tha lpy ( H ) , s p e c i f i c h e a t a t cons tan t p re s su re (CP), and f r e e energy (FE) of a given spec ie s for a given temperature (TX). the h e a t of formation (ENTAX), t he number of atoms i n one molecule of t he spec ies (ENJX) , the v i b r a t i o n a l temperature (TVX) , t he number of e l e c t r o n i c s t a t e s ( N U ) , as wel l as the energy (EX) and degeneracy (GX) of each e l ec - t r o n i c s t a t e and a cons tan t (EMWTX) which depends on the molecular weight of t he spec ie s and the r o t a t i o n a l temperature of the spec ies .
For each spec ies , th is subrout ine must be given
The computations performed by t h i s subrout ine a r e based on the harmonic- o s c i l l a t o r r i g i d - r o t a t o r model f o r a l l diatomic molecules. It i s a l s o pos- s i b l e t o compute the thermodynamic func t ions f o r one t r i a tomic molecule, carbon d ioxide . I n t h i s computation i t i s assumed t h a t t he re i s only a ground e l e c t r o n i c s t a t e for t he carbon dioxide molecules.
The equat ions used by t h i s subrout ine f o r t h e computation of t he thermodynamic func t ions of each ind iv idua l spec ies a r e as fol lows. The f r e e energy i s computed from
F. 1 = -2n RT -En/T - [ ( n i - 1) + 2.512n T - 1 .5 2n M. + 3.6649516
1
( 8 )
The en tha lpy of each spec ie s i s
E e-En/T (ni - 1 ) ~ ~ H O i 1 a n n + + - RT
H . -&= [ ( n i - 1) + 2.51 + Cgn e -En/T T(eTVlT - 1)
( 9 )
5
The s p e c i f i c h e a t a t cons tan t pressure (wi th f rozen composition) i s
SUBROUTINE RATCON
This subrout ine ( s e e appendix D ) w i l l compute the r a t e cons tan ts f o r a given temperature according t o the formulas and cons tan ts of r e fe rences 8 and 9 f o r the r e a c t i o n s l i s t e d below.
KF
KF
KF
% KF
KB
5 KB
02 + 02 2 20 + 02
N 2 + N 2 P 2N + N2
NO + M z N + 0 + M
N + 02 2 NO + 0
0 + N 2 2 NO + N
+ - O + O t O + e
N + N t N + e - +
+ N + O ~ N O + e -
The symbol KF a f t e r a r e a c t i o n i n d i c a t e s t he forward r a t e cons tan t ; KB i nd ica t e s t he backward r a t e cons tan t .
This r o u t i n e r equ i r e s on ly the temperature as inpu t . It i s a simple mat te r t o a l t e r t h i s subrout ine t o g ive the r a t e cons t an t s f o r any r e a c t i o n des i r e d .
SUBROUTINES GUESS AND ITERA
For any mixture of n i t rogen and oxygen, two a d d i t i o n a l subrout ines ( s e e appendix E ) a r e a v a i l a b l e t h a t permi t t h e computation of t he equ i l ib r ium spec ie s concentrat ions a t a given temperature and d e n s i t y . These two
6
I
subrout ines r equ i r e a s i npu t the temperature ( T ) , t he d e n s i t y (RHO), t he atom concent ra t ions of t h e cold mixture (ZN,ZO), and e i g h t equi l ibr ium cons tan ts ( EK( I) ) .
The f irst subrout ine GUESS es t imates the r e l a t i v e magnitude of t he var ious spec ie s concent ra t ions . The second subrout ine ITERA then r e f i n e s these guessed values u n t i l they s a t i s f y the t h r e e conservat ion equat ions, t h a t i s , conservat ion of oxygen atoms, n i t rogen atoms, and charge, as we l l a s s i x equi l ibr ium equat ions. These two subrout ines r e q u i r e a block of common s torage c a l l e d /COM4/ which contains a r rays f o r 20 equi l ibr ium cons tan ts (EK(20)), 20 mass concent ra t ions (B(20) ) , 20 f irst guesses ( C ( 2 0 ) ) of the mass concent ra t ions , t he d e n s i t y (RHO), and t h e cold mass concent ra t ions of n i t rogen atoms (ZN) and oxygen atoms (ZO) .
The equat ions t h a t must be solved by subrout ine ITERA a r e l i s t e d below.
PY o2 yo2
K1 = -
"OYN K3 = - yNO
PY e -Yo+ YO
K G =
K8 = 'e -'NO+
'OYN
The convergence of t he i t e r a t i o n process sed i n subrout ine ITERA I r e q u i r e s that equat ion (19) be solved f o r the l a r g e s t of yo, yo,, or yo+, and
tha t equa t ion (20) be solved f o r t he l a r g e s t of i n g y i va lues a r e then found from equat ions (22) through (27r. The value of ye- i s always found from equat ion (21). The pa th taken i n subrout ine
YN, YN,, o r y +. The remain-
7
S!
ITERA depends therefore on the relative magnitudes of m- lllc *.---+<-.. 1~~~~~~~~ VI - 0 su'ijruutine GUESS is to provide an initiai
guess of these relative magnitudes. This initial guess is based on the temperature and density Of the mixture and on the relative values of the equilibrium constants K1 and K 6 for oxygen and K2 and K7 for nitrogen.
y o , yo2., 7~ and of ,v-- ,l\l, yq2, aEc *--* ll\lT.
r
Based on these guessed values, subroutine ITERA then computes the species concentrations by iterating equations (19) through (27) until all the equations are satisfied. At each step in the iteration the newest value of each concentration is computed as the average between the value obtained on the previous iteration and the value computed on the present iteration. The iteration is terminated when the change in the values for two successive iterations is less than a specified amount model B) and when all the equations are satisfied to within a specified amount (5~10-~ for model A and
for model A and lom5 f o r
for model B) . Appendix E gives two listings for subroutine ITERA and two listings f o r
0'' subroutine GUESS. or
In each case one listing is for model A (contains no N+') and one listing is for model B (contains 0" and N + + ) .
POSSIBLE USES OF TKE PROGRAMS
A series of programs for computing the thermodynamic properties of a variety of gases is presented. computation of the equilibrium species concentrations of air or any other mixture of nitrogen and oxygen, as well as the thermodynamic properties of the mixture.
In addition, two of the programs permit the
The programs are presented in a form which can be modified easily to For instance, the thermodynamic properties of any suit the user's needs.
diatomic or monatomic gas may be obtained simply by altering the input data to subroutine S P E C I E . Subroutine RATCON can be easily altered to give any rate constant desired. Subroutines GUESS and ITERA may be rewritten to give the equilibrium concentrations of gas mixtures other than air. A l l the sub- routines as presented may be used in a new main program to compute the equilibrium flow properties behind a normal shock wave.
DISCUSSION OF RESULTS
The main program described in this report has been used to compute the thermodynamic properties of air. model A the species assumed to be present are and e-. Model B contains and O++ in addition to the species in model A. The values obtained are compared to the values presented in references 2 and 3 (table I) and to the values presented in reference 1 (table 11).
Two different models have been used. In N2, Opt NO, N, 0, NO', N', O',
8
I
The dimensionless enthalpy, H/RT, computed for model A agrees with the , values presented in reference 2 to within 1 percent except at 15,000° K, where
the error rises to 1.39 percent at 100 times standard density and to 8.46 per- cent at times standard density. The dimensionless entropy, S/R, computed for model A, agrees with that computed in reference 2 to within 0.5 percent except at 15,000° K and 2.54 percent. The pressure (in atmospheres), for model A, a rees with that computed in reference 2 to within 3 percent except at 15,000 standard density where the error is 3.27 percent.
times standard density where the error is
% K and loe7 times
0 The increase in error at 15,000 K for enthalpy, entropy, and pressure is attributed to inadequacies in model A. The neglect of species begins to be felt at 15,000° K. comparing the results computed for model B to the results of reference 2. 15,000° K and 8.46 to 0.13 percent, the error in dimensionless entropy drops from 2.54 to 0.25 percent, and the error in pressure drops from 3.27 to 0.28 percent.
and O++ That this is indeed the case may be seen by
At p/po = 10-7, the error in the dimensionless enthalpy drops from
The highest temperature available in references 2 and 3 is 15,000° K. Therefore, for comparisons at higher temperatures, it is necessary to use the results of reference 1. The comparison between values in reference 1 and the present calculations is presented in table 11. both the model A and model B air of the present report. 25,000' K and at However, for model B, which includes points are drastically reduced. in the low temperature region.
Again, comparison is made with At 18,000° K and
p/po = the errors in model A are seen to be enormous. and O++, the errors at these
In fact, they become comparable to the errors
CONCLUSIONS
A series of subroutines written in FORTRAN IV is presented. These programs may be used to compute the thermodynamic properties at a given den- sity and temperature of any diatomic gas plus the one triatomic gas, CO2, if the molar concentrations are given. subroutines are included that permit the computation of the molar concentra- tions themselves. The thermodynamic properties of two models for air are compared to the results obtained by other investigators.
In the special case of air, additional
Ames Research Center National Aeronautics and Space Administration
Moffett Field, Calif., Dee. 28, 1966 129-01-08-20
9
APPENDIX A
I I B F T C C A I N NCREF C C A I N CHECKOUT B A I L E Y
C O C C C N / C O C 4 / E K l 2 0 l ~ B l 2 O l ~ C l 2 O ~ . R H O ~ Z N ~ ~ O D I Y E N S I O N A K l 2 O l ~ B K l 2 O l r H I 2 1 ~ ~ C P l Z l ) r S o . G ( 2 1 )
12
2
4
1
13
9
10 e
11
5
c 3
7
R E A 0 1 5 1 1 2 ) hCAX FCRCATI I31 R E A O I 5 r 2 ) DELTA FCRYAT lF15 .6 I REAC15.4) I G I N I r h = l r 9 1 F O R C A T l 5 F 1 5 . 6 1 ZN=2.*G I 5) LC=2.*G14) R E A C l 5 . 2 1 Rl-C TECP=O. YR I T E I 6.13 I F O R M A T l l 2 H l T E M P E R A T U R E / ~ 9 H CENSITYIPRESSUREVGAS CONSTANT.ENTHALPVI
1ENTROPY.CONSTANT PRESSURE S P E C I F I C H E A T l 9 H E N T H A L P Y l 8 H t N T R O P Y l 3 2 H 2 CCNSTANT PRESSURE S P E C I F I C H E A T l 2 l H E O U I L I B R I U C C O N S T A h T l 1 4 H REAC 3 T I O N R A T E l l 9 H MASS CCNCENTRATIONl34H O I N I E - I O ~ ~ N ~ , N O ~ N O ~ , O ~ , N ~ , O ~ ~ 4 . N t t ) DO 3 M = l r N C A X TEMP=TECP+OELTA CALL S P E C I E ~ T E M P ~ R H O I P R E S S I G . H I C P . S I A K I B K I G ~ O N S T I 00 8 N = l . l l I F I B K I N I -82 . I E K l N I = l O . * * B K l h l GO T O 8 E K I N l = l . E + 3 6 CONTINUE CALL GUESS CALL I T E R A 00 11 N=1.11 G I N I = B I N 1 CALL S P E C I E I T E L P I R H O I P R E S S I G I H I C P ~ S ~ A K ~ ~ K ~ G C O N S T I W R I T E l 6 r 5 1 T E M P ~ R H O ~ P R E S S ~ G C O N S T ~ H l 2 l l ~ S l 2 l l ~ C P l 2 l l F O R C A T l 1 H O ~ E 1 5 . 6 / 1 0 X 1 6 E 1 5 ~ 6 ~ W R I T E I 6 . 6 1 l H I N I . N = l . 1 1 1 W R I T E l 6 . 6 ) l S I N l l N = l r l l )
YR ITEI6.61 I B K I h ) .N=l . 10 I W R I T E l 6 . 6 1 l ~ K l ~ l ~ N = l ~ l O l W R I T E 1 6 . 6 1 I G l N l r N = 1 ~ 1 1 1 F C R C A T I L H r l l E l l . 4 1 CONTINUE W R I T E 1 6 . 7 1 FORCATI lHOs13HEhC OF RUN1 cc IC 1 END
9 1 9 9 1.2
W R I T E l 6 . 6 1 I C P l h l r N = l r l l l
10
I
TIN- 000
m N - 0 0 0
m N - 4 0 0 0
W W W m N d u m o u m n d N N
000 0
C ) N - N O 0 0 O b - l
. . . . W W W - m u e m 0 m m m
W W W m O N N 9 0 m U v I 4 N N
0 0 0 0 . . . . NNN
000 0 . . . .
I W W W W W QtnOldO m U N m 0 P N O t n O Y l N N - 0
I W W W W W VIOlCl-0 0 - o m 0 outnr-0 m N N C O
W W W W W
- N O ~ O N d C N O
O n v I N O N N N - 0 . . . . . . o o o ? o o
. . . . . . 000000
I 1 - 8 m N d m P 0 0 0 0 0 -
. . . . . . 0 0 0 0 0 0
I I - ~ N - N O I 00000-
*NNd&bN 0 0 0 0 0 - N
d N N - N ( T Q 0 0 0 0 0 4 4
I I W Y W W W W W m m u o m o c - m e c . n o ~ o n m m ~ o m 4 V I N N U O U u...... ~ 0 0 0 0 0 0
I 1 O N N - - O ~
or>oo- - W W W W W W
I
m m ~ t n t n m . a O U - A N - ~
m t n m m - m W U N N C l - m e...... C O O O O O O 0. I Y I N N - N S C goooo-0
I W W W W W W
m m N O o O W N N N - 0 m...... m O O O O O O
m o 9 m a o O N N ~ ~ O
I W W W W Y W m m ( P N C 0 m m m o u o 4-4"LndO m r ( N N D 0 m . . . . . , u 0 0 0 0 0 0
I 1 W W W W W W W C C d d . t O N Q O O N Q O C s c m m r t o t n U C N N Q O O u... . . . u 0 0 0 0 0 0
I 1 O m N d N C
0 0 0 0 -
I 1 O N N d N C C 0 0 0 0 4 y
W W W W W
O O Q O l - m ~ N ~ O N N
m - m m a
W W W W W W m . n m o m ~ . n o m m o u + r ( 9 m m - 0 -
W 6 N N d N a
W W W W W W
o o u o c m m
e......
m c o - - Y t n . n
W S N N ~ - - y \ u m m o o
~ 0 0 0 ~ 0 0
W - 8 N N r . L n I n . . . . . . m o o o o o o 0 I
- 0 0 0 0 - e . W W W W W
C l m - m s c d N d m P N m m Q I D
~ 0 0 0 0 0 0
utoooo- 9 - W W W w W
N ~ N - N ~
- ~ m r - r l d . . . . . .
W N N - N ~
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AF'PENDIX C
S I E F T C T C l Z l O NOREF S U E R O U T I N E E N T H I T X I E h T A X ~ E N J X t T V X . E M U T X ~ E X ~ G X . N L X . H ~ C P ~ F E I D I M E N S I O N E X l Z C ) ~ G X I Z O I N X - I F I X I E N J X I
1
4
Z
6
8
1
9
- 3
5
GO T O I 1 . 2 . 3 ) ~ N X SUM1 - 0 . SUM2=0. SUM3=0. 00 4 N I l t N L X X X = G X I N I ~ E X P I - E X I N I / T X I S U N l = S U C l + X X S U M 2 = S U M Z + X X * E X l N l S U M 3 = S U M 3 + E X I N l ~ E X l H ) r X X F E = - A L O G I S U C 1 ) - 2 . 5 ~ A L O G ( T X I - E M U T X t E N T A X / T X H = 2 . 5 + S U C Z I l T X ~ S U M l ) + E N T A X / T X C P = Z . 5 + I S U M 1 ~ S U C 3 - S U ~ 2 ~ ~ 2 l / l l T X ~ S U ~ l ~ ~ ~ ~ ~ GO TC 5 SUM 1=0. SUM Z 4. SUM3=0. 00 6 N = l r N L X X X = G X l N l * E X P I - E X I N I I T X I S U M l = S U M l + X X S U M Z = S U M Z + X X * E X I N ) S U M 3 = S U M 3 + E X l N l * E X l N ) . X X F E = - A L O G I S U C 1 l - 3 . 5 ~ A L O G O - E ~ U T X + E N T A X / I X H = 3 . 5 + S U M Z / l T X ~ S U M 1 1 t E N T A X / T X C P = 3 . 5 + l S U M 1 ~ S U C 3 - S U C Z ~ ~ 2 l / l l T X ~ S U M 1 l ~ ~ Z 1 I F ( T V X / T X - 8 8 . 0 1 l r l r 0 V I B H = O . V I BCP.0. RZ-0. GO TO 9 A 1 = E X P I T V X / 1 x 1 R 2 * A L O G l l . - E X P I - T V X / T X ) ) A l l = A l - l . V I B H = I T V X I T X I I A ~ ~ V I B C P = l T v X / T X ) . . 2 V I 8 C P = l V I 8 C P / A 1 1 l ~ l A l / A l l ~ C P = C P + V I E C P H = H + V I B H F E - F E t R Z
V I 8 1 = 1 9 3 2 . 1 / T X V I B 2 = 9 6 0 . 1 / T X V I B 3 = 3 3 8 0 . / T X VIE11~1./11.-EXPl-VIEl~l VIR22=1./l1.-EXPl-VI82l~ VIB33=1./Il.-EXPI-VIE3)l V I 8 l c = V I 8 1 . . 2 . E X P l V I E l ~ / l E x P l v I e l l - l . ~ ~ ~ 2 V I 8 2 C ~ V I 8 Z ~ ~ 2 ~ E X P l V I @ Z l / l E X P l V I ~ Z l - l ~ l ~ ~ Z V I 8 3 C = V 1 8 3 . . 2 . E X P l V 1 @ 3 ) / 0 - 1 . ~ ~ ~ ~ V I B 1 ~ V I B 1 / I E X P I V I E 1 ) - 1 . 1 VIe2=VIBZ/lExPlVIBZl-l.l VI83=VIB3/IEXPlVIE3l-l.l H = 3 . 5 + V I 8 1 + 2 . ~ V I e Z + V I 8 3 + E N l A X / T X F E ~ - 3 ~ 5 ~ A L C G l T X l - A L O G l V I ~ l l ~ V I 8 2 Z ~ V I 8 Z 2 * V I B 3 3 l ~ E M U T X + E N T A X / T X CP=3.5+VIE1C+2.*VIB2CtV183C RETURN END
GO T o 5
S I B F T C T C l Z O B NOREF SUBROUTINE R A T C O N I A K I T X I D I M E N S I O N A K I 1 0 1 T 1 5 = T X * S C R T l T X l AKlll=3.6EtZ1.EXPI-5938O./lXl/Tl5 A K l Z 1 = 1 . 5 E + Z O / T 1 5 A K l 3 1 = 5 . 1 8 E t Z l . E X P I - 7 5 4 9 O . / T X ) / T l 5 A K l 4 l ~ 1 . G E + 1 2 ~ E X P I - 3 l Z O ~ / l X l ~ S C R ~ ~ l X ~ A K l 5 1 ~ 5 . G E t 1 3 ~ E X P l - 3 8 0 1 6 ~ / T X ~ A K l b ~ ~ b ~ O E + 2 4 / l T l 5 ~ T X l AK I 7 l = A K I 6 1 A K l 8 ) = 1 . 8 E t Z l / T 1 5 RETLRN END
14
APPENDIX D
0 N
z N
0 m
m d
c d
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- z N . N U . d -
- m m - -
- u 0 .
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m + r . I w * u I X W L u -
N + N .-. - u - I d " m
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- u 0 .
c r .
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c In *
m z m a
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MODEL A
S I B F T C T C 1 2 0 6 NCREF SUBRCUTINE I T E R A C O ~ P C N / C O H 4 / E K l Z 0 l ~ 8 l 2 O l ~ C l Z O ~ ~ R H O ~ Z N ~ Z O DATA O E L l r D E L Z / 1 ~ E ~ 0 l r 5 ~ E ~ O E / DO 9 N = l t 2 0 0 I F l E l 4 l - E l l l l 1.192
E l 4 1.1 B I 4 I * C I 4 I I / 2 . Clll=SPRTlEl4l~EKl1l/RHOl
2 Cl4l~l~C-8l1l-8l6l-Elll-ElBll/2.
1 5
B I 1 1 = I B I 1 1 * c I 1 1 1 / 2 . C l B l = l B 1 1 l / E l 3 l l ~ l E K l 6 E l E l = 1 B l E l * C l 8 l l / 2 . N r E S T l = l GO 10 3 I F l E l 1 l - E l E l ) 4.4.5 C l 1 l ~ Z C - 2 ~ ~ E I 4 1 - ~ l 6 1 - ~ B I 1 1 = I B l 1 1 * C I 1 1 1 / 2 . C14 I = R H O * E I l I *E I11 I E K I B l 4 1 = l ~ 1 4 1 * C 1 4 1 1 / 2 . C l E l = l E 1 1 l / E l 3 l l ~ l E K l 6 E 1 8 1 = 1 8 1 E l ~ C 1 E l 1 / 2 . NTEST1=2 GO T O 3
I RHO I
7l-ElEl
1
/RHO I
4 ClEl~ZO-Elll-2~*El4l-El6l-Elll E l E l = l E l E l ~ C 1 E l 1 / 2 . C l l l = R H C ~ E l 3 1 ~ E l E I / E K O 8 l l l = l B l l l * c l l l l / 2 . C l 4 1 = R H O ~ E l 1 l ~ E l 1 1 / E K l l l 8 1 4 l = l E l 4 1 t C l 4 I 112. N l E S T l - 3
3 I F 1 E l 5 I - E l 2 1 1 6 . 6 9 1 1 C l 5 1 ~ l 2 N - ~ l 2 1 - 8 1 6 1 - ~ ( 7 ) - 8 ( 9 ) ) 1 2 ~
6 l 5 1 = 1 6 l 5 1 * C 1 5 1 1 / 2 . C ( 2 I = S P R T I E l 5 1 * E K ( Z l/RHOl E l 2 1 = 1 B I 2 1 + C 1 2 1 1 / 2 . Cl9l=lElZl/@13ll~lEKll~/RHOl E l 9 l = l E l 9 l * C l 9 l l / Z . N T E S T 2 = 1 GO T O E
6 I F l E I Z l - E 1 9 1 1 3 1 9 3 1 ~ 3 2 3 2 C 1 2 1 - Z N - 2 . ~ ~ l 5 1 - ~ l 6 1 - 8 o - B I P )
E l 2 l = l E l Z I * C I Z I I / Z . C 1 5 1 = R H O * E l 2 I ~ E 1 2 1 / E K l 2 1 8 t 5 1 = l 8 l 5 1 * t l 5 1 I/Z. Cl91=1~12l/@l31l~IEKlll/RHOl 8 1 9 1 = 1 8 l 9 1 ~ C l 9 1 1 / 2 .
3 1
E
1 0 11 1 2 13 1 4 1 5 1 6 11 1 8 1 9
22 23 2 4 25 2 6 2 1 2 6 2 9 30 9
3 4
3 3
NTESTZ-2 GO T O R Cl9l=LN-E12~-2~*Bl51-El6l-Blll E l 9 1 = 1 8 l 9 I + C l 9 ~ 1 / 2 . ClZl=RHO~8l3l~E19l/EKlll B l 2 1 = I B l Z ) + C l 2 l l / Z . Cl5l=RHC~BlZI~ElZl/EKl2l B I 5 1 = ( 8 ( 5 1 + c ( 5 1 1 / 2 . N T E S T 2 = 3 C 1 3 1 = 8 l 7 1 ~ ~ l 8 1 + @ l 9 1 8 l 3 1 = l 8 l 3 1 * c l 3 1 1 / 2 . C l 6 1 ~ R H C ~ ~ l 1 1 ~ ~ 1 2 1 / E K ( 3 1 E 1 t l = l E l 6 l * C l 6 l l / Z . C l 7 ~ = ~ ~ 1 1 * ~ 1 2 1 ~ E K l 8 1 / 6 l 3 1 8 1 1 1 = 1 8 1 1 1 * c l 1 1 1 / 2 . I F I A B S I l C l l l - E l 1 l I / @ l 1 1 l ~ D E L 1 I 1 1 * 1 1 * 9 I F I A B S l l C l Z l - E I Z 1 l / E l Z ~ I ~ D E L l 1 1 2 ~ 1 2 ~ 9 I F I A B S l l C l 3 1 - 8 l 3 1 1 / 8 l 3 1 1 - 0 E L 1 1 1 3 . 1 3 1 9 I F I A E S l l C l 4 1 - ~ 1 4 1 1 / ~ l 4 1 1 ~ D E L 1 1 1 4 . 1 4 * 9 I F I A E S l l C 1 5 ~ - ~ l 5 1 1 / ~ l 5 1 1 ~ 0 E L 1 1 l 5 r 1 5 r 9 I F I A E S l ( C l 6 1 - E l 6 1 1 / 8 l 6 1 1 - O E L 1 1 1 6 . 1 6 1 9 I F I A E S l l C l 1 1 - 8 l 7 1 1 / @ 1 1 1 1 ~ C E L 1 1 1 7 i 1 7 r 9 I F I A E S l l C l E l - E l E l l / E l 8 l l ~ O E L 1 I 16.18rS I F I A E S l l C l 9 1 - ~ l 9 1 1 / ~ 1 9 1 1 - C E L 1 I 1 9 r 1 9 r 9 0 1 ~ 2 C - ~ l 1 1 - 2 . ~ 8 l 4 1 - ~ ( 6 ) - 8 0 - 8 ( 8 1 0 2 ~ ~ N - ~ l 2 1 - 2 ~ ~ ~ l 5 1 - B ( 6 ) - 8 ( 7 ) - 8 1 9 ) 0 3 ~ B l 3 1 - E l l l - E 1 B l ~ ~ l 9 l O 4 = E l 4 1 ~ E K l l l - R H C * E l l ~ ~ ~ ~ l l O 5 = E l 5 ~ ~ E K l 2 l - R H C ~ E l 2 l * E l 2 l D 6 ~ ~ l 6 l ~ E K l 3 l - R ~ C ~ 8 l l ~ ~ B l ~ l 0 l = E l 1 l o E K l 6 1 - R H C ~ 8 l 3 l ~ E l E l O E = E l 2 ~ ~ E K l l l - R H C ~ 8 l 3 l ~ E l 9 l D 9 = ~ l 1 1 ~ ~ l 2 1 ~ E K 1 ~ 1 - ~ 1 3 1 . 8 ( 7 ) I F I A E S l O X ) - C E L Z I 2 2 1 2 2 1 9 I F I A B S 1 0 2 1 - D E L 2 I 2 3 , 2 3 1 9 I F I A E S l O 3 1 - O E L 2 I 24.24.9 I F I A E S l D 4 l - C E L 2 I 2 5 1 2 5 . 9 I F I A E S I O 5 l - C E L Z 1 2 6 1 2 6 . 9 I F I A E S 1 0 6 l - C E L Z 1 2 1 r 2 7 . 9 I F I A E S 1 0 1 1 - D E L 2 I 2 6 1 2 8 9 9 I F I A E S l D E l - C E L 2 I 2 9 , 2 9 9 9 I F I A E S l 0 9 1 - O E L Z 1 3 0 1 3 0 1 9 GO IC 33 CONTINUE W R I T E l 6 ~ 3 4 l h T E S T l ~ N T E S ~ 2 ~ l E l J l ~ J ~ l ~ 9 l F O R C A T l l H O t Z Z H N O CONYERGENCE N T E S T 1 = 1 3 e
C A L L E X I T RETURN END
l H N T E S T . ? = 1 3 / 1 H . l P 9 € 1 4 . 6 )
16
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@PENDIX F
MODEL A 8 9
2. 2.
1. 1. 1. -1. -1. 1. -1. 1.
-1. 1. -1. -1. 1.
5 1 5.0 3 -0 1 .o 5 -0 1.3 1 .o
4 .O t .o 4 .O 4 .O 2 .o
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5 4 3 -0 2 .o 1 .o 3 -0 3 -0 2.0 4 5
1 .o 3 .O 6 .O 1 .o 2 .o
2 .o 2 .o 2 .o 4 -0
4 6
2.0 4 1 1.0 3 -0 6 -0 6 -0 2.0 5 8
4 - 0 6 -0 4 -0 4.0 2.0 1 .o 1 .o 6 9
3.0 5.0 5.0 1.0 5.0 1.0
0 . 0 .
I L AN0 JL
-1.
-1. 1. -1. 1.
-1.
-1.
1. 1.
1.
0.0 159.0
227.0 15868.0 33792.0 1.0
0.0 19228.C 19281.0 28840.0 28840.0
N L AN0 J 0.0
0.c
1 .o
0 .o 7918.0 13195.0
36096.0 49802.0 2256.0
0 .o 50256.0
59626.0 6CCOC . C 3314.0
0.0 121.0 43966.0 45918.0 2119.0
0.0 20000.0 3100C.O
3397.0 36000.0
0 . 0 26808. 26829.0
40467.0 40468.0 1 .o
0 . 0 49.0 131.0
32687.0 47168.0
0.
15316 .O
1 .o
0.
5 8980. C
11259C.O ELECTRON
0.0
0.0
0.0
21477.0
234970 -0
372900.0
447700 .O
0 . 0 .
OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM
1c.coo 1 .o NITCGEN ATOM NITOGEN ATOM NITOGEN ATOM NITCGEN ATOC: N ITCGEN ATOM
NITROGEN ATOM
14.007 1 .o G AND E ELECTRON
CXYGEN MCLECULE .00054847 1.0
0
N
E-
OXYGEN MOLECULE OXYGEN MCLECULE CXYGEN MCLECULE OXYGEN MCLECULE CXYGEN MCLECULE
NITROGEN MOLECULE NITROGEN MOLECULE NITROGEN MOLECULE NITROGEN MOLECULE NITROGEN MOLECULE 28.C14 5.784 N 2 N I T R I C O X I D E N I T R I C O X I D E N I T R I C O X I D E N I T R I C O X l O E N I T R I C O X I D E 30.007 2.453 NO
32.000 4.160 02
h I T R l C C X I O E I O N N I I R I C C X I D E I O N N I T R I C C X I O E I O N N I T R I C C X I O E I O h N I T R I C C X I D E I O N 30.001 2.880 NO+
OXYCEh I O h OXYGEN I O h OXYGENIOh OXYGErt IOh OXYGENIOh OXYGENIOh
lt.OOO 1 .o o+ h I T R O G E h ICN h I T R O G E h I C h N I T R O G E h I C N NITROGEh ICN NITROGEh I C h N ITROGEh I t h h l T R O G E h I C N
14.007 1 .o N * -21153 .18841
0.
18
010011 2. -1.
2. -1. 1. 1. -1. 1. -1. -1. 1.
-1. 1. -1. 1. -1. 1.
-1. 1. -1. -1. 1.
1.c 1 .c
5 1 5.0 3 -0 1 .o 5 -0 1.0 1 .o 4 -0 6 .O 4 .O 4 -0 2 .o
5 2
1.0 1 3
2 .o 1.
5 4 3.0 2 .o 1 .o 3 .O 3.0 2.0 4 5
1 .o 3.0 6 -0 1 .o 2 .o 2.0 2 .o 2 .o 4 .O 2.0 4 7
4 6
1 .o 3 -0 6 .O 6.0 2.0 5 8
4 .O 6 .O 4 -0 4 .O 2.0 1 .o
1.0 6 9
3.0 5.6 5.0 1.0 5.0 1.0
006010 1 .o 3 .O 5 .O 5.0 1 .o 5 .O 1 .o
005011 2 .o 4 .O 2 .o 4.0 6 -0 1 .o 0. 0 . 0.
0.0 159.0
227.0 15868.0 33792.0 C.O
0.0 19228.0 1928l.C
28840 - 0
NL ANC J 0 . c
0.0
28840 .o 0.7
0 .o 7918.0 13195 -0
36096.0 49802.0 2256.0
0.0 50256.0
59626.C 6COOO.O 3374.0
0 .o 121.0 43966.0 45918.@ 2719.0
0.0 2COOO .o 31000.0
3397.0 36000 .O
0.0 26808.
40467.0
0 .o
26829.0
40468.0
0.0 49.0 131.0
32687.0 153 16 -0
4716a.c 0.0
J .0 113.4 306.8 20271.0 43183.5 60312.1
0.0
0.0 174.5
57192.1 57252.0 57333.2
0.0 0. 0.
MODEL B
1. 1.
-1.0 1.c 1.
-1.0 1.3 OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM OXYGEN ATCM OXYGEN ATOM
58980.0
112590.0 ELECTRON
0 . 0
0.0
0.0
21477 .O
234970 -0
~ 16.000 1 .o NITROGEN ATOM
NITCGEN ATOM N I T C G E N ATOM N I T C G E N ATOM NITCGEN ATOM NITCGEN ATOM
14.007 1 .o C A N 0 E ELECTRON
OXYGEN MOLECULE CXVGEN MOLECULE OXYGEN MCLECULE OXYGEN MCLECULE
. O C O 5 4 8 4 7 1.0
CXYGEN MCLECLLE OXYGEN WCLECLLE 32.coo 4.160
NITROGEN MOLECULE NITROGEN MOLECULE k I T R O G E h MOLECULE
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N ITROGEN MOLECULE h ITROGEN MOLECULE 28.C14 5.784 N I T R I C O X I D E N I T R I C O X I O E N I T R I C O X I D E N I T R I C O X I O E N I T R I C O X I D E
N I T R I C C X I O E I O N N I T R I C C X I O E I O N N I T R I C C X I O E I O N h I T R I C C X I D E I O N N I T R I C C X I O E I O N
3C.CO7 2.453
3C.CO7 2.880 NO+ OXYGENION OXYGENION OX YGEN I ON OXYGENION OXYGENION OXYGENION
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447700 .C 14.007
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REFERENCES
1. Gilmore, Forrest, R.: Equilibrium Composition and Thermodynamic Properties of Air to 24,000' K. RAND Rep. RM-1543, Aug. 24, 1955.
2. Hilsenrath, Joseph; Klein, Max; and Woolley, Harold W.: Tables of Thermodynamic Properties of Air Including Dissociation and Ionization from 1,500° K to 15,000° K. AEDC-TR-59-20, Dec. 1959.
3. Hilsenrath, Joseph; and Beckett, Charles W.: Tables of Thermodynamic Properties of Argon-Free Air to 15,000° K. AEDC-TN-56-12, Sept. 1956.
4. Bailey, H. E.: Equilibrium Thermodynamic Properties of Carbon Dioxide. NASA SP-3014, 1965.
5. Bailey, H. E.: Equilibrium Thermodynamic Properties of Three Engineering Models of the Martian Atmosphere. NASA SP-3021, 1965.
6. Herzberg, G.: Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules. D. Van Nostrand Company, Inc., 1950.
7. Moore, Charlotte E.: Atomic Energy Levels as Derived From the Analysis of Optical Spectra. Vol. I. NBS Circular 467, June 15, 1949.
8. Marrone, Paul V.: Inviscid, Nonequilibrium Flow Behind Bow and Normal Shock Waves. Part I. General Analysis and Numerical Examples. CAL REP. &~-1626-~-12(1), May 1963.
9. Hall, J. Gordon; Eschenroeder, Alan Q.; and Marrone, Paul V.: Inviscid Hypersonic Airflows with Coupled Nonequilibrium Processes. CAL AF-1413-A-2, May 1962.
20
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