Heat Equation for Gax Mixtures

7/31/2019 Heat Equation for Gax Mixtures

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N A S A T E C H N I C A LR E P O R T

A GENERAL STAGNATION-POINTCONVECTIVE-HEATING EQUATIONFOR ARBITRARY GAS MIXTURESby Kenneth Szltton und Randokh A. Graues, Jr.Lungley Reseurch CenterHampton, Va. 23365N A T I O N A L A E R O N A U T IC S A N D S P AC E 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 . NO VEM BER 1971


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TECH LIBRARY KAFB, NM

..___ _ _2. Government Accession No.I. Report No.NASA TR R-376t. Title and SubtitleA GENERAL STAGNATION-POINT CONVECTIVE -HEATINGEQUATION FOR ARBITRARY GAS MIXTURES

.- . . ..7. Author(s)

Kenneth Sutton and Randolph A. Gra ves , Jr.- -. - - - . - . -3. Performing Organization Name and AddressNASA Langley Re se ar ch Cen terHampton , Va. 23365

-. ..____ . -~ . . ~ . ~.2. Sponsoring Agency N ame and AddressNational Aeronautics and Space AdministrationWashington, D.C. 20546 .~_ _ -. --.5. Supplementary Notes

3. Recipient's Catalog No .

5. Report DateNovember 19716. Performing Organization Code

. . ~8. Performing Organization Report No.

L-788510. Work Unit No.

117-07-01-01- .11. Contract or Grant No.

1". of Report and Period CoveredTechnical Report~-14. Sponsoring Agency Code.._____ __ . ~ _ _ - - .6. Abstract

Th e stagnation-point convective heat transfer to an axis ymme tric blunt body for a rbi -tr ar y gases in chemical equil ibrium was invest igated. The gases considered were basegas es of nitrogen, oxygen, hydrogen, helium, neon, argon, carbon dioxide, ammonia, andmethane and 22 ga s mixt ure s composed of the bas e gases. In the study, enthalpies rangedfr om 2.3 t o 116.2 MJ/kg, pr ess ur es ranged from 0.001 to 100 atmo sphe res, and the wall tem-peratu res wer e 300 and 1111 K.

A gener al equation for the stagnation-point convective heat tran sfe r i n base g ase s andgas mixt ures was derived and is a function of the m as s fraction, the mol ecular weight, and atrans port par ame ter of the base gases. The relation comp ares well with pres ent boundary-layer computer results and with other analytical and experimental results. In addition, theanalysis verified that the convective heat t r ansfer in g as mixt ures can be determined from asummatio n relation involving the heat-transfer coefficients of the base gases. The basictechnique developed for the prediction of stagnation-point convective heating to an axisym-met ric blunt body could be applied to other he at-tr ansfe r problems.

- -7. Key- Words (Suggested by A uthoris) )Heat transfe rGas mixturesConvective heat tra nsf er

- - . __ - - - __ -T.istribution StatementUnclassified - Unlimited

19. Security Classif. (o f this report) 20. Security Classif. (of this page) 21 . No. of Pages 22. Price'Unclassified Unclassified._

Fo r sale by the Nati onal Technical In form atio n Service, Spr ingfie ld , Virg in ia 22 15 1


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A GENERAL STAGNATION-POINT CONVECTIVE-HEATING EQUATIONFOR ARBITRARY GAS MIXTURES

By Kenneth Sutton and Randolph A. Graves, Jr.Langley Rese arc h Center

SUMMARYThe stagnation-point convective heat tra ns fe r to an axisymmetric blunt body for

arb itr ary g ases in chemical equilibrium was investigated. The gas es considered wer ebase gas es of nitrogen, oxygen, hydrogen, helium, neon, a rgon, carbon dioxide, ammonia,and methane and 22 ga s mixt ures composed of the ba se gases. In the study, enthalpiesranged from 2.3 t o 116.2 MJ/kg, pr es su re s ranged fro m 0.001 to 100 atmospheres, andthe wall temp era ture s we re 300 and 1111K.

A genera l equation for the stagnation-point convective heat tran sfer in base ga sesand gas mixtu res was derived and is a function of the m as s fracti on, the molecula r weight,and a transport parameter of the base gases. The relation compares well with presentboundary-layer computer r esu lts and with other analytical and experimental results. Inaddition, the analysis ver ified that the convective heat trans fer in gas mixtures can bedetermined from a summation relation involving the heat- tran sfe r coefficients of the basegases. The basi c technique developed fo r the prediction of stagnation-point convectiveheating to an axis ymm etri c blunt body could be applied to othe r hea t-tr ans fer problems.

INTRODUCTIONConvective heating during planetary entry depends on the atmosp heri c gas composi-

tion surrounding the planet. For the planets presently considered for entry, the atmo-sphe res a r e prim aril y binary mixtu res of nitrogen and oxygen (air) or Earth; carbondioxide and nitrogen fo r M ar s and Venus; carbon dioxide and argon for Mercu ry; andhelium and hydrogen for Jupiter. (See refs. 1 o 7.) Trace amounts of neon, methane,and ammonia have also been considered fo r se ver al of the planets. Except fo r Earth, theactual composition of the atmo spheres a r e not accurat ely known. Consequently, numerouscalculations will be required to determine the convective heating during planetary entryunless a general relation can be determined. Generally, existing analytical (refs. 8 to 16)and experimental (refs. 17 to 27) stud ies have emphasized only particula r gas es or gasmixtures and a solution for ar bi tr ar y gas es was not obtained. In references 14, 16,

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and 28, general relations for convective heating are presented but the analyses weredeveloped for a limited number of gases and gas mixtures.

The purpose of the pre sen t study is to develop a general rela tion for calculating theconvective heating to the stagnation point of a blunt axisy mmetric body for gas es or gasmixtures in chemical equilibrium, especially those gas es encountered during high-velocityplanetary entry. In the analytical study, the approach is to derive a simple approximaterelation fro m the basic theor etical definition of heat transf er in gases. Also, the convec-tive heating for numerous ga ses is calculated by an existing computer code (computerstudy) to obtain resul ts for use in developing the approximate general relation and as aba sis of comparison with the approximate relation. For the computer study, stagnationenthalpies ranged from 2.3 to 116.2 MJ/kg, stagnation pr es su re s ranged from 0.001 to100 atmospheres (1 atmosphere equals 101.325 kN/m2), and wa l l temperatu res were 300and 1111K. The gases considered wer e base g ase s of nitrogen, oxygen, hydrogen, helium,neon, argon, carbon dioxide, ammonia, and methane and 22 mixtures composed of the base

C

cP-DDeffDijEFhho

jK2

gases. Radiative heating w a s neglected i n the present study,SYMBOLS

mass fractionspecific heat at constant pressure, J/kg-Krefe rence diffusion coefficient, se e equation (6), m2/s

effective diffusion coefficient defined by equation (16)binary diffusion coefficient, m2/senergy facto r, se e equation (C2)diffusion fac tor , see equation (6)enthalpy, MJ/kgstanda rd enthalpy of formation, MJ/kg .species mass flux, kg/ma-shea t-tr ans fer coefficient defined by equation (33), kg/s-m

I .

'2-atm '2


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k

M molecu lar weightNLeN p rNsc

Boltzmann constant, 1.380622 X 10-23 J/K

Lewis number defined by equation (15)Prandtl number defined by equation (13)Schmidt number defined by equation (14)

P pressure , a tm (1 atmosphere equals 101.325 kN/m2);I convective heating rate, W / m 2R equivalent body radius , md? univ ersal ga s constant, 8.31454 x lo3 J/kmole-Kr body coordinate, mS distance along body s ur face fro m stagnation point, mT temperature, K

T* nondimensional temp eratu re defined by equation (B2)U velocity component para lle l to body su rf ace , m/s

Y normal d istance through boundary layer fro m wal l surface, m

a constant, see equations (36) and (38), 2.2621 X 10-6P velocity gradient defined by equation (23), s-1

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transport parameter defined by equation (37),maximum energy of attraction, Jtransformed coordinate defined by equation (20)

K- .15YE

rlx ther mal conductivity, J/m-s -KI-L viscosity, N-s/m2

tP density, kg/m30

transformed coordinate defined by equation (21)

molecular collision diameter, A (1 angstrom equals 10-10meter),2)* reduced collision integral for viscosity

Subscripts:ei

jmax0

S

W

co

4

external edge of boundary layerith spe cies or componentjth speciesmaximuminitial composition

stagnat ion condition at boundary -layer edgewall valuefree-s ream value


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DESCRIPTION OF COMPUTER PROGRAMAs par t of th e presen t study, the convective heating of the bas e,g ase s and gas mix-

tures was calculated by an existing computer code which is a numerical solution of themulticomponent boundary-layer equations for ga ses in chemical equilibrium. Resu ltsfro m these computations were used i n developing the approximate general relation and asa ba si s of comparison with the approxim ate relation. Descript ions of the computer codeand method of solution of the boundary- layer equations are presented in referenc es 29to 32. The routines for the thermodynamic and tra nsp ort proper ties are so constructedthat any arbit rar y gas mixture can be considered. Hereinafter, these solutions arereferred to as computer results.

The thermodynamic properties are calculated by th e method descr ibed i n refer-ence 33, and, for the present study, the necessary inputs were obtained from ref erenc es 34to 37. The inputs a r e for a common bas is, and a comparison of this method with othe rmethods is given in re feren ce 33.

The mass diffusion is calculated by using the bifurca tion approximation as presentedin reference 38. The accura cy of th e method depends on the bifurcation diffusion fac tor s,and, fo r the present study, the fac tors a r e calculated by the method given in reference 39.For a few species, the basic data required were not available, so the diffusion fac tor swere taken fro m the correlat ion of r efer ence 38. Comparisons of th e bifurcation diffusionapproximation with exact multicomponent diffusion are given in ref erenc es 30, 38, and 39.

The viscosity and th erm al conductivity values are calculated by approximate r el a-tions of the Sutherland-Wassiljewa type as presented in reference 38. The methodreq uir es the previously mentioned bifurcation diffusion facto rs and also th e self-diffusionfacto rs calculated by the exact method presented i n referen ce 34. Comparisons of thetra nsp ort prope rties calculated by this method and other methods ar e given inreference 34.

The computer program provided solutions for a rbi tra ry gas mixtur es and thus gavea consistent bas e for developing and evaluating the approximate relation. In the presentstudy, the computer pro gram was used to obtain nume rica l solutions for the stagnation-point convective heating for nine base gas es and 22 gas mixtures. They a r e as follows -

Base gases:

HeN eAr

5


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G as mixtures (composition by mass fraction):0.2320 02-0.7680 N2 (a ir)[email protected] Ar0.1345 Ca-0.8655 N20.5000 Ca-0.5000N20.8500 Ca-0.1500 N20.7500 02-0.2500 N20.5000 02-0.5000 N20.1500 H2-0.8500 He0.3500 H2-0.6500 He0.6500 H2-0.3500 He0.5000 H2-0.5OOOAr0.1500 N2-0.8500 H20.5000 N2-0.5000 H20.2000 [email protected] H20.4000 C%-0.6000 Ha0.6000 C%-0.4000 H20.8000 [email protected] H20.3626 [email protected] N2-0.3297 Ar0.1339 CO2-0.8525 N2-0.0136 Ar0.3000 Ne-0.3000 Ar-0.4000 He0.3500 H2-0.3500 CO2-0.3000 N20.352 H2-0.423 He-0.176 Ne-0.024 CHq-0.025 NH3

6

.... -.. .. . .


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The 81 chemical species considered in the computer study are as follows:Ar c 3 NO c 0 2 -C c302 N02 He+CH c 4 NO3 HeuCHNO C4N2 N2 H+CH2 c 5 N2H2 H-CH2O He N2 0 HCO'

H 0 H ~ O +HCN OH Ne+

C2H HCO 0 2 Ne++C2H2 HNO 0 3 N+C2H4 HNO2 e N++C2H40 HNO3 Ar+ NO+CN H02 Ar++ NO2-CNNCN2co

H2 C+ N2+H20 C++ O+Ne C- 0-

For each base gas o r g as mixture, the heating was calculated for a combination of conditions of -

Stagnation enthalpy: 2.3, 4.6, 9.3, 13.9, 18.6, 23.2, 27.9, 32.5, 37.2, 46.5,58.1, 69.7, 81.4, 93.0, 104.6, and 116.2 MJ/kg

Stagnation pressure: 0.001, 0.01, 0.1, 1.0, 10.0, and 100.0 atmWall temp erature: 300 and 1111K

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ANALYSISA simple general expression for the convective heating to the stagnation point of a

blunt axisy mmet ric body for high-enthalpy ga ses in chemical equilibrium was derivedfrom the basic theo retical definition of heat tra nsfe r in gases. To meet the criterion ofsimplicity, it was necessary to approximate seve ral of the mor e complex expression s thatoccu rred in the theoretical development with mo re tr acta ble expressions.

From reference 40, he convective heating to a wall fro m a gas is

The first te rm on the right-hand sid e of the equation is the heating due to conduction andthe second te rm is the heating due to diffusion. The mass diffusion due to ther mal gr adi-ents is smal l in comparison with m as s diffusion due to concentration grad ients (ref. 40)and has been neglected. The specif ic heat and enthalpy of a mixture a r e defined as

where

hi = cp,i dT + hfWith these definitions, equation (1)becomes

The species mass flux ji is based on a bifurcation approx imation to binary diffusioncoefficients.shown to be i n good agreement with exact multicomponent diffusion. The presen t fo rm ofexpressed as

This approximation has been used in re feren ces 29, 30, 38, and 39 and w a she b ifurcation approximation is taken from refer ence 38. The approximation is

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where 5 is a ref ere nce diffusion coefficient and Fi and Fj are diffusion fact ors fo rthe ith and jth spec ies, respectively.

As given in re feren ce 38, the species mass flux for the bifurcation approximationcan be exp ressed as

For unequal diffusion, the quantity zi lies between a mass and a mole fraction and isdefined by

and /31 and & are sys tem quantities defined by

From equation (8) , ueglectedThe last two te rms on the right-hand sid e of equation (11) are neglected in the prese ntanalysis. The neglected ter ms involve variations in syste m quantities and the variationsof these quantities should be much less than the var iat ion s of individual spec ies. Thus,in the present analysis, it is assumed that

With the following nondimensional parameters -Prandtl number: (13)

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where

Schmidt number :

Lewis number:

and equations (5), (7), and (12) , the convective heating can be expressed as

In the presen t study, an analogy between energy and ma ss tran sfe r is used in theform

This analogy is based on the analyse s presented in references 30, 38, 40, 41, and 42. Byusing equation (18), equation (17) becomes

The present equation is transf ormed into the usual boundary-layer coordinate sys -tem for a blunt axisy mmetric body by the Lees-Dorodnitsyn trans forma tion (ref. 43):

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For a stagnation-point solution

where the velocity gradi ent p is

Thus, the transforma tion from the y coordinate to the 77 coordinate for the stagnation-point solution of a blunt axi symm etri c body is

and applying this transform ation, equation (19) becomes f or th e stagnation point

At the stagnation point, the velocity gradien t for Newtonian flow is

ForPe ,

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the velocity gradient can be approximated by

With the u se of the ideal gas equationP R Tp =- M

and the boundary-layer approximation at the stagnation point

equation ( 2 5 ) becomes

An expression is now needed which includes the enthalpy gradient at the wall, and itw a s necessary to use a correlation of the re su lt s from the computer study. A discussionof the enthalpy gra dient at th e wa l l is given in appendix A. A cor rel ati on which is reason-ably valid for all the base gases and gas mixtures is presented (eq. (A3)) as follows:

Special note should be taken of the pa ra me te r Po,e, which is the viscosity evaluated atthe temperature at the outer edge of th e boundary layer (Te) fo r a mixture based on theinitial composition of the cold mixtu re or the ambient composition of the a tmosphere su r-rounding a planet.,0.60N2-0.20Ar-0.20 C02 , the viscosity po,e is based on this composition rat her thanthe chemical equilibrium composition at the outer edge of the boundary layer.

For example, in determining the heat tra nsf er i n a gas mixture of

Fr om the enthalpy-gradient corr ela tio n of equation (3 1), the heat transfer fromequation (30) becomes

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The stagnation-point convective heating to a blunt axis ymm etri c body is often expres sedin t e rms of a heat-transfer coefficient. This coefficient is considered to be essentiallyconstant over a range of conditions for a particular gas or gas mixture. As presented inreferences 16, 25, 28, and 44, the heat-transfer coefficient K is defined as

When equations (32) and (33) are compared, it is seen that the heat-transfer coefficientcan be written

2/3 2 hi,w(zi,e - zi,w)+ (NLe,w) hs - hw (34)

The viscosity is a dominant factor for the convective heat tran sfe r of gase s; thus,in or der to find a general expression for all gase s, the viscosity must be express ed inte rm s of basic parameters. For gas mi xtur es, th e exact calculation of the viscosity istoo complex to be used in any simple expression for the heat trans fer. For example, seereferences 45 to 48. In the present analysis, a simple summation is proposed for mix-tur e viscosity which agr ees adequately with experimental r esu lts and theoretic al models.The viscosit y of t he pur e components is based on the analysis presented in reference 47.A discussion of the presen t viscosity relation s is presented i n appendix B.

The relation proposed for the mixture viscosity is (eq. (B6)) as follows:

(3 5)

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For a pure gas component

where the transpo rt parame ter yi is

and the constant Q! isQ! = 2.2621 X

With the use of the above viscosity relations, equation (34) becomes

As previously mentioned, the heat-transfer coefficient is used as a constant ove r arange of conditions. In the prese nt analysis, the Lewis number augmentation factor(appendix C ) ,

C hi ,w h i , e - zi,w)hs - hwCFihi,w(Zi,e - zi,w) + ( ~ L e , w ) ~ / ~p2 ,w1 - hs - hw

is assumed to be approximately equal to 1. This approximation is based on a comparisonwith the equivalent expre ssion for equal binary diffusion coefficients. (See refs. 8 and 49.)Fur ther more, the following approximation is made

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The te rm s in equation (40) have small exponents and will have a smal l variation over arange of conditions. Also, Me will decre ase with incr eas ing Te due to dissociationand ionization. The value used in the approximation wa s determined from the computerresults . The effect of the par am ete rs in equation (40) on the hea t-tr ans fer coefficientwill be discussed further in the next section.

With the above approximat ions and evaluation of the constants, equation (39) for theheat-tra nsfer coefficient can be expressed as

The Prandtl number at the wa l l varied from 0.39 to 0.71 for the various base gases andgas mixture s considered in the pre sen t study. However, the Prandtl number at the wal lonly varied from 0 . 6 7 to 0.71 for those base gases and gas mixtures which are presentlyconsidered for planetary entry. For Npr,w = 0.69,

/, \- 1/2K = O . l 1 0 6 ( I Mooiiyo ,i,)

The pre sent analys is has shown that the approximate stagnation-point convective heatingfor gases has a general relation involving the mass fraction (Co,i), molecular weight(Mo,i), and transport par ameter yThe nece ssary values for calculating yo,i from equation (37) re given in r eferen ce 50for numerous gases . Listed in table I a r e the values for the base gase s considered in thepresen t study. The approximate convective heating to the stagnation point of a blunt axi-symmetric body can be calculated by the use of equation (33) and either equation (41)or (42) for base gases and gas mixtures in chemical equilibrium.

transfer coefficients (Ki) of the base ga ses a r e known. By manipulating equation (4l) , itcan be shown tha t

of the base gas or the base gases of a mixture.( 0,i)

The heat-tra nsfer coefficient for gas mixtures can also be calculated i f the heat-

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-=I-co,i2 y2

(44)

The present analysis is for the stagnation-point convective heating to an axisym-metric blunt body and the developed relation is only applicable to this problem. However,the basic technique used in the present development could be applied to other heat-transferproblems as well.

RESULTS AND DISCUSSION

The present analysis has shown that the convective heating for gas es can beexpres sed by a genera l relation involving the mass fraction co,i , molecula r weight(Mo,i), and transport parameter (yo,l ). These parame ters a r e known for most gases.The resu lts of this general expression are compared with the re sul ts of the numericalsolution to the bouddary-layer equations from the pre sen t computer study and with theresults of analytical and experimental studies presented in the literature .

0

Results fro m the presen t computer study a r e presented in figure 1 in the fo rm ofb w E as a function of hs - hw. The line ar curve shown in figure 1 for a particular gaso r gas mixture is based on a single heat-tra nsfer coefficient for the gas as determined bya least-square f i t of the values of K fro m each data point. The hea t-tr ans fer coefficientfor each gas could be correlated with good accuracy by a single coefficient fo r a range ofenthalpies, pre ssu res, and wall temperature s. The heat-trans fer coefficients for thegases and the average er ro r and maximum e rr or for the correlation ar e listed in table II.

The presen t analysis showed that the heat-trans fer coefficient for ga ses can beexpressed as (see eq. (41))

K =

and for Npr,w = 0.69 (s ee eq. (42))

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/

The analytical expre ssions (eqs. (41) and (42)) for the heat-trans fer coefficient are com-pared with the present computer resu lts i n figure 2. As shown by the comparison, thegeneral expression is i n good agreement with the present computer results. Furthe rmore,the data pre sen ted i n figu re 2(b) show that equation (4l), which includes the wall Pr and tlnumber, is in better agree ment with the computer results. Heat-transfer coefficientswere determined for variou s gas es from experimental and analytical studies presentedin the literature; and, as shown by the comparison in figu re 3 , there is good agreementbetween the general re lation and the r esu lts f rom other studies (refs. 13 to 22, 24 to 26,and 28). In references 14 and 15 the convective heating for sev era l bas e g as es and gasmixtures was calculated by the same method. The heat-tr ansfer coefficients determinedfrom these references are also compared separ ately with the general relation in figure 4.As shown by the data, the hea t-trans fer coefficients from re feren ces 14 and 15 ar e l inear

functions of the parameter (1 MI;to,)-1'2; however, the slope is slightly gre ate r thangiven by the prese nt re sult s,

The previ ous discussions have been concerned with the validity of the final, ove ral lexpre ssion fr om the present analysis. The trend of the variations of the heat-tr ansfe rcoefficient can. also be explained from the analysis. Fro m the computer res ult s presentedin figure 1, the reduced-heating par ame ter qnia) app ear s to have an "apparent" wall temper ature effect. This apparent temperatureeffect is actually a wall Prandtl number effect. The analytical analysis, s ee equation (4l) ,shows such an eff ect and it can be substantiated fro m ana lysis of the computer resu lts.For the gases which showed a wall temperature effect, there w a s a variation in the wallPrand tl number for the different wall tempera ture values.

- or some gases (for example, ammo-[ W E , )

The computer re sult s in figure 1 also show a decrea se in reduced-heating param eterwith decreased pre ssu re for a given enthalpy. Th is effect can be explained by examiningequation (40). The value of 1.81 was based on an average over a range of conditions. Fo ra given enthalpy, the degree of di ssocia tion and ionization w i l l be greatest at the lowerpr es su re s, and the molecular weight Me will be decreased. Also, this effect will cause

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the temperatu re Te to be lower at the lower pr ess ur es. Without the approximation ofequation (40), quation (42) an be express ed as follows for Np r ,w = 0.69:

The trend of Me and Te with pr ess ur e, from the computer res ults , is shown in f ig-u re 5. Also, in figure 5, the heat-transfe r coefficient as calculated by equation (45),withthe value of Me and T e taken fro m the computer solution, is compar ed with the heat-transfer coefficient from the computer results. As shown by the com parison, the pre sentanalysis predicts co rrectly the effect of pre ssu re.

Equation (45) lso shows the trend of the heat- tran sfe r coefficient with enthalpy.The heat-trans fer coefficients from the computer study and from equation (45) re com-pared in figure 6 for several cases. The ter m T,0*075 gives the predominanttrend of the coefficient with enthalpy and, as shown in figu re 6, good agreement generallyexists between the presen t analy sis and the computer results. The present analysis givesgood final res ult s and also gives the cor rec t trend of intermed iate steps.

The present re sul ts of the analytical and computer studie s were compared withth re e othe r methods for calculating the convective heating of gases. These methods arepresented in referenc es 14, 16, and 28 and are different from each other and fro m thepresen t analysis. These methods wer e based on a sma lle r number of gase s than thepresent study.

A simp le method for determin ing the convective, stagnation-point heating of gasmixtures is presented in ref ere nce 28. In this method the h eat- tran sfer coefficient ofgas mixtures is calculated by knowing the coefficient of the base g ase s by the relation

The relation wa s derived by empirical means and w a s based on mixt ure s of C0 2, N2, andAr. Similar expressions (s ee eqs. (43) nd (44))were derived for a wider range of gasmixtures in the present analysis:

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A comparison between equations (43), (44),and (46) nd the present computer results isgiven in figu re 7 for s ome binary gas m ixtures and in table III for all the gas mixturesconsidered in the present study. In comparing the summation relations, the heat-transfercoefficients for the base gases are taken fro m the computer res ult s. The summationrelations from the prese nt studies, equations (43) nd (44), ave good agre emen t with thecomputer res ult s for the ir resp ecti ve conditions. Equation (43)has the best agreementfor all the ga ses ; however, equations (44) s equally valid for the g ase s when the P rand tlnumbers at the wall are approximately equal for both the mixt ures and the componentbase gases. The empi rica l method of refere nce 28, equation (46),has good agreementwith the computer resul ts for the mixtu res where the Prandtl numbers are approximatelyequal for the mixture s and component base gase s. However, this method is not as accu-ra te as equation (43)when the Prandt l numbers for the mixtures are different from thosefor the component base ga ses. The method of re fere nce 28 was based on empi rica lmeans, whereas the present rel ations were derived from analytical considerations. Fromequations (44) nd (46) nd analysis of the computer res ult s, it can be s een that differenti-ation between a first and second power for the heat-transfer coefficient would be difficultby emp iric al means.

The method presented in ref eren ce 14 an d later extended in reference 15 is basedon knowing the low-temperature properties of the gas. In using the method, a correlationwould have to be made for each particular gas, and, for air, N2, CQ, Ar, H2, and severalNz -C e- Ar mixtures, the necessary correlative parameter a re l is ted in reference 15.Basically the method of reference 14 and the present theoretical analysis a r e in agree-ment concerning low-temp erature prop erti es. In the pres ent analysis, the heat-tran sfercoefficient is a function of the par ame ter s Npr,w, yo, Mo, and co. These par ame ter sa r e low-temperature properties of the gases. The low-temperature properties give themajor component of the convective heating, and the boundary-layer -edge pro per tie s (high-temperature properties) give minor variations to the heating. As previously shown in

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figure 4, heat-transfer

and 15 and were shown

coefficients can be determined from the re sul ts of refer ence s 14f , 112

to be linear functions of the pa ram ete r

is i n agreement with the present analysis. The pres ent summation relation of equa-tion (44)was applied to the data of ref ere nce s 14 and 15. As shown in table IV , the heat-transfer coefficients determined by the present relation is in good agreeme nt with thosedetermine d with the use of r efe ren ces 14 and 15. The Prandtl number at the wa l l forgases considered for planetary entry is approximately 0.69; herefore, the present methodis easi er to us e than the method presented in refer ence 14.

The results of reference 16 are presented as a corr elat ion of the heat -tra nsfe rcoefficient with molecular weight in the form

K = 0.0323 + 0.00233Mo (4 7)o r

f \ -1K = 0.0323 + 0.00233("2)

A comparison of the results presented in reference 16 and the present computer resu ltsis shown in figure 8. The molecular weight corr elat ion of refe renc e 16 was based on alimited number of gase s and for the se ga ses the cor relation is valid. However, as shownby the present resu lts, this method is not valid for ar bit rar y gases.

CONCLUSIONSThe prese nt analys is has resu lted i n the development of an approximate method for

calculating the convective stagnation-point heating to an axisymmetric blunt body for gasesin chemical equilibrium. The heating is expressed by a general relation of the ma ss fr ac-tions, molecular weights, and tra nsp ort param ete rs of the base gases . Compariso ns withthe mor e exact computer study and with other analytical and experimental st udies haveshown the method to be valid. Also, it has been shown that a summation relation derivedfrom the analysis is valid for calculating the heating of gas mixtures from the data of thebase gase s comprising the mixtures. For the ga ses presently considered for planetaryentry, the approximate convective heating can be easily calculated. Since the analy siswa s based on a wide range of bas e ga ses and gas mixtu res, the r es ul ts should be valid20


23/67

for most gas es . The bas ic technique developed for the prediction of stagnation-pointheating to an axisymmetric blunt body could be applied to other heat-transfer problemsas well.Langley Research Center,

National Aeronautics and Space Administration,Hampton, Va., October 8, 1971.

2 1


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APPENDIX AENTHALPY GRADIENT AT THE W A L L

The resu lts presented in referen ces 51 and 52 have shown that th e enthalpy gradientat the wa l l for the stagnation point of a blunt axisymmetric body in high-enthalpy air canbe expres sed by

where a and b are given as follows:

a Determinedfrom reference -0.51 0.975.49 .938

5152

Also, fro m analysis of the heat-transf er relation s presented in refere nce s 8 and 53,

where a, b, and c are given as follows:a Determinedfrom reference -C

0.54 0.900 0.4.54 .980 .4

853

The interest in the present study was whether a single expression si mila r to equation (A2)nent for the density-viscosity product was 1, then the heat-transfer equation (eq. (30))in reference 14, but a single correlation could not be determined for air, nitrogen, hydro-

could be determ ined and whether it would be reasonably valid for all gases. If the expo-could be made sim pler by the elimination of ter ms . A sim ila r approach was attemptedgen, carbon dioxide, and argon. Preli minary work in the present study showed that sucha corre lation was feasible and additional studies wer e performed.

22


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APPENDIX A - ContinuedThe data used to cor rel ate the enthalpy gradient were taken from the r esu lts of the

pres ent computer study. The initial corre latio n is shown in fig ure A1 in the forma t ofequation (A2) with b = 1 fo r the primary base ga ses and air. The data of the variousgases are in good ag reem ent with each other; however, a stra ight li ne could not be fittedthrough the z er o point and the re is a slight curvature to the data. Also, the relation was

LhA

o Ai r0 Nitrogen0 OxygenA HydrogenD He l i umD ArgonD Carbon dioxide

O k i I I I I I I I I I I I I I I I I I I I I I I I I I I l l I I I l l I l l I I I I I I I I , I 1 I I I I I I I I I I lllll,llllllllllllllllllllll0 . I -2 - 3 .Y - 5 .6 .I .8 .2 1 .o

F i g u r e AI..- I n i t i a l c o r r e la t io n of t h e e n t h al p y g r a d i e n t.

not in good agreement for all gas es considered in the present study. Variations to thebas ic for m of equation (A2) were tried and the best correlation obtained to the data was

As shown in figure A2(a), the final correlation, equation (A3), has good agreement fo r thegas mixtures which are presently considered in planetary entry and for the ba se ga seswhich comprise these mixtures. Representative data for all the g ases considered in thepresent study are also compared with equation (A3) in figure A2(b). The greate st devia-tion from equation (A3) is for the g ase s with a significan t variation of the molecular weightat the wall for a range of conditions, especia lly with stagnation enthalpy. The variation ofmolecular weight is affected by chemical reac tion and diffusion.

23


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APPENDIX A - Continued

( a ) Gases considered for p l a n e t a r y e n t r y .

c m "

I I I I I I I I I I I I I I.?

o A i ro Nitrogeno oxygena Hydrogenn Heliumn Argon0 Carbon dioxide0 0.35 Hp - 0.65 He0 0.65 Hp - 0.35 He@ 0.85C02-0.15Np0 0.3626COp-0.3077 N 2- 0.38 7A rA 0.1339COp- 0.8525N2-0.0136Ar

n 0.50 CO2 - 0.50 NpQ 0.6714~02-0.3286Ar

o A i ro Nitrogen0 Oxygena Hydrogentl HeliumD Argon0 Carbon dioxideo 0.35 H2 - 0.65 He0 0.65H2-0.35Hen 0.50C02-0.50N2lil 0.6714C02- 0.3286Ar@ 0.85 Cop - 0.15 N2

0 0.3626C02-0.3077 N2-0.3297ArLA 0.1339 COP- 0.8525 N2 - 0.0136Arrh Neonra Methane

Ammoniae 0.50H2-0.50Ar@ 0.30Ne-0.30Ar-0.40He@ 0.60CO2-0.40Hp

( b ) All g a s es c o n si d er e d i n p r e s e n t s tudy.F igure A2.- F i n a l c o r r e l a t i o n o f t h e e n t h al p y g r a d i e n t .

24


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APPENDIX A - ConcludedThe corre lat ion of equation (A3) for the enthalpy gradient at the wall w a s based on

boundary-layer results for a wide ra ng e of flow conditions and a wide ran ge of bas e gase sand ga s mixtures. The bas e gas es included monatomic, diatomic, and polyatomic gases.The gas mixture s we re composed of base g as es of only high molecul ar weights, of onlylow molecular weights, of high and low molecular weights, of basically iner t gases, andof reactive gases. Since the correlation was in agreement for a wide range of base gasesand gas mixtures, the correlation should be equally valid for most gases.

25


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APPENDIX BVISCOSITY RELATIONS

The viscosity equation fo r a pure component from reference 47, is

where

p iT

Mi'51

viscosity, N -s/m2temperature, Kmolecula r weightcollision diameter , Areduced collision integral fo r viscosity

The reduced collision integral 51(2,2)* is based on the Lennard-Jones (6-12) potentialand is a slowly varying function of t he nondimensional t em pera tu re T* defined as

T * = 7T~kwhere E is the maximum energy of at tra cti on and k is the Boltzmann constant. Theva lues of f2(2,2)* as a function of T* are presented in referen ce 47 and are shown infigure B1. F or the species and temp era ture s of int ere st in the pre sent study, the reducedcollision integral can be expressed as

51(2,2)*= 1.18(T*) -0.15 033)By combining equationsexpressed as

p i = (2.2621

(Bl), (B2), and (B3), the viscosity for a pure species can be

x 10-6)yiMi 1/ZTO.65

26

.~.. . - . .. ..


29/67

APPENDIX B - Continued

- rom reference 47

I 1 ~~ I10-1100 lo1 1O2 103

T"Figure B1.- The reduced collision integral for viscosity as a functionof the nondimensional temperature.

where the transport parame ter is given as

Jktabulation of the collision dia meter ai and the maximum energy of a ttr acti on Efor numerous species are given in reference 50.A simple summation expression is proposed for the mixture viscosity. This

expression is given by

The exact trea tmen t of mixtur e viscosi ty involves mor e complicated express ions. (Forexample, see refs. 45 to 48.) The simple relation given by equation (B6) has been com-pared with experimental resu lt s and with mor e exact calculations. Some re su lt s of thecomparison are shown in table BI and figur e B2. (Also see refs. 54 to 60.) A s shown bythese results, equation (B6) is in good agreemen t with experim ental and calculatedvalues.

2 7


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APPENDIX B - ContinuedTABLE BI.- COMPARISON OF MIXTURE VISCOSITIES

Gas composition(composition by mole fraction)

0.1754 He - 0.5556 Ne - 0.2670 Ar0.5175 He - 0.3210 Ne - 0.1615 Ar0.3594 He - 0.3193 Ne - 0.3213 Ar0.3333 H2 - 0.3333 Ne - 0.3333 C0 20.2500 H2 - 0.2500 Ne - 0.2500 C02 - 0.2500 CC12F20.1754 He - 0.5576 Ne - 0.2670 Ar0.3594 He - 0.3193 Ne - 0.3213 Ar0.5429 He - 0.2189 Ne - 0.2382 Ar0.5000 He - 0.5000 Ar0.5000 He - 0.5000 Ar0.4000 N - 0.6000 N2

0.2500 H2 - 0.2500 N2 - 0.2500 C02 - 0.2500 CC12F2

0.6000 0 - 0.4000 0 20.4828 C2H4 - 0.5172 NH30.6719 H2 - 0.3281 N2

2

2

2

2

21N5 I 8zi

16

I4

12

10

8

_ " 10-6 . . ..o Experimental fron, ref. 45---- Theoretical- - resent. eq. 186)

T = 4 2 3 Ka.LMole fraction of first gas.2 .4 .6 .8 I .( a ) Comparison withr e f e r e n c e 45.

32

30

28

26

24NEz 22si1

20

I 8 '16

14I

12

remp.,K

293293473298298298473473293

500010 00010 000LO 000

523523

Reference

545446464646

55 and 5755 and 5755 and 57

lo-6 _ _ _ .0 Experimental. ref. 55Experimental. ref. 56A Experimental. ref. 56

.

- resent. eq. (86)

585859595660

iT = 298 K

L . 1 1 ~ . I ...2 .4 .6 .8Mole fraction of first ga s

( b ) Comparison withr e f e r e n c e s 55 and 56 .

Fromreference27.34 X25.8636.0019.0215.9314.6737.9035.7425.04

186.00316.00234.00144.0017.6422.02

IJ.,N-s/m2Computed (eq. (B6))with pi fro m reference

27.05 X26.4236.6518.7216.6915.5538.0836.7725.90

200.30344.20241.80240.70

17.4323.25

38 x io k6 _ _ _ ----- Theoretical. ref. 54Present, eq. IB6136 c -3234i1 8 L - i i J0 .2 .4 .6 .8 1.0

Mole fraction of f i rst ga s(c) Comparison withr e f e r e n c e 54.

Figure B2 . - A c ompa ri son o f t he p r e s e n t p r opose d r e l a t i o n f o r mi x t ur e v i s c os i t yw i t h o t h e r s t u d i e s .

28


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APPENDIX B - ConcludedWhen equations (B4), (B5),and (B6)are combined, the mixture viscosity can be

expressed as(2.2621 X 10-6)T .65

P = M1/2

29


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APPENDIX CLEWIS NUMBER AUGMENTATION FACTOR

In the present analysi s, the Lewis number augmentation fac tor

is assumed to have a value of 1. This approximation is based on a compar ison with theequivalent exp ress ion for equal diffusion coefficients. The equations fo r the bifurcationapproximation of multicomponent diffusion will reduce to the usual equation for binarydiffusion when the binary diffusion coefficients ar e ass ume d equal.

When the binary diffusion coefficients are assumed equal, then Fi = 1, p2 = M ,and zi = ci, and the approximation fo r the Lewis number augmentation fact or is

The approximation is exact for a Lewis number of 1 o r if the species concentrations donot va ry through the boundary la yer . Also, the approximation is reasonably valid overa range of other conditions.

Let an energy factor E be defined as

o r

Now,hi,e 2 hi,w

From equations (C3) and (C4) , the maximum value of E will be= 1hs - hwE m a x - h s - hw

30


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APPENDIX C - Continuedwher eas t he minimum value of E will be zero. Therefor e,

O S E Z l (C )Expressio ns fo r the Lewis number augmentation factor are presented in refe r-

ences 8 and 49 for binary diffusion as follows:Ref erence Lewis number augmentation factor

Present

49

8

E + ( N L ~ 1)EIoa6

The Lewis number augmentation factors a r e presented in table CI f a r a range of Lewisnumbers and energy fac tor s. As shown by the data, the Lewis number augmentation

TABLE C1.- LEWIS NUMBER AUGMENTATION FACTORS

E0.10

0.25

0.50

0.75

1 oo

N L e0.6.8

1 o1.21.40.6.8

1.01.21.40.6.8

1 o1.21.40.6

.81 o1.21.40.6

.81 o1.21.4

.~Le w i s numbe r augme ntat i on fac torP r e s e n t0.953,978

1.0001.0191.0360.881,946

1.0001.0471.0890.763.892

1.0001.0941.1790.644,838

1 ooo1.1411.2680.526.784

1 ooo1.1881.357

..R e f e r e n c e 490.976.9881.0001.0121.0240.939,9701.0001.0301.0590.875.9391.0001.0591.1160.807.9071.000

1.0871.1700.736.a751.0001.1161.224

Re fe r e nc e 80.977.9891.0001.0101.0190.942,9731.0001.0251.0480.883,945

1.0001.0491.0960.825,918

1 ooo1.0751.1430.767.8901.0001.0991.191

31


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APPENDIX C - Concludedfactor is approximately 1 except fo r the ca se s where the Lewis number is significantlydifferent fro m 1 and the energy factor approaches th e maximum value. The energy fac-to r will approach the minimum f or high enthalpies at the ou ter edge of the boundary laye runle ss the wall enthalpies of th e speci es prese nt at the boundary-layer edge a r e signifi-cantly gr ea te r than the wall enthalpies of the s pec ies pres ent at the wall.

Fro m analysis of the pres ent computer res ults, it was noted that the Lewis numberat the wall did not va ry significantly from 1 fo r the base gas es and gas mixtures whichare presently considered fo r planetary entry. Significant variation from 1was noted fo rconcluded that the approximation of 1 for the Lewis number augmentation fac tor is rea-sonably valid for a wide range of conditions and gase s.the ga ses i n which significant dissociation or reactions occurred at the wall. Thus, it is

32


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REFERENCES1. Sodek, Benard A., Jr.: Jovian Occultation Experiment. J. Spacecraft Rockets, vol. 5,

no. 4,Apr. 1968,pp. 461-463.2. Vaughan, Otha H., Jr.: Model Atmo spheres of Me rcu ry. J. Spacecraft Rockets,

vol. 6,no. 10,Oct. 1969, pp. 1171-1175.3. Pritc hard, E. Brian; and Harrison, Edwin F.: Multipurpose Entry Vehicle Require-

ments for Unmanned Landings on Bodies Having Tenuous Atmospheres. J. Space-craft Rockets, vol. 5, no. 4,Apr. 1968,pp. 413-418.

4.Evans, Dallas E.; Pitts, David E.; and Kraus, Gary L.: Venus and Mars NominalNatural Environment fo r Advanced Manned Plan etar y Mission Pro gra ms. Seconded., NASA SP-3016, 1967.

5. Anon.: Models of Venus Atmosphere (1968). NASA SP-8011, 1968.6.Michaux, C. M.: Handbook of the Physic al Pro pe rt ie s of the Plan et Jup ite r. NASASP-3031, 1967.7. Glasstone, Samuel: The Book of M a r s . NASA SP-179, 968.8. Fay, J . A.; and Riddell, F. R.: Theory of Stagnation Point Heat Tr an sfer in Disso-

ciated Air. J . Aeronaut. Sci., vol. 25, no. 2, Feb. 1958,pp. 73-85, 121.9.Fay, J am es A,; and Kemp, Nelson H.: Theory of Stagnation-Point Heat Transfer in

a Par tia lly Ionized Diatomic Gas. AIAA J ., vol. 1, no. 12,Dec. 1963,pp. 2741-2751.

10.Hoshizaki, H.: Heat Tra nsfer in Planetary Atmospheres at Super-satellite Speeds.ARS J., vol. 32,no. 10,Oct. 1962,pp. 1544-1552.11. Bade, W. .: Stagnation-Point Heat Tra nsf er in a High-Temperature In ert Gas.

Phys. Fl uids, vol. 5, no. 2, Feb. 1962,pp. 150-154.12. Pallone, Adrian; and Van Ta ss el l, William: Ef fe ct s of Ionization on Stagnation-Point

Heat Tra nsf er in Air and in Nitrogen. Phys. Fluid s, vol. 6,no. 7, uly 1963,pp. 983-986.

13. DeRienzo, Philip; and Pallone, Adrian J.: Convective Stagnation-Point Heating fo rRe-Entry Speeds up to 70,000 fp s Including Eff ect s of Lar ge Blowing Rates.AIAA J ., vol. 5, no. 2, Feb. 1967,pp. 193-200.

14. Marvin, Joseph G.; and Deiwert, George S.: Convective Heat Transfer in PlanetaryGases. NASA TR R-224, 1965.

15. Marvin, Jose ph G.; and Pope, Ronald B.: Laminar Convective Heating and Ablationin the M a r s Atmosphere. A L U J., vol. 5, no. 2, Feb. 1967, pp. 240-248.

33


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16. Scala, S. M.; and Gilbert, L. M .: Theory of Hypersonic Lamin ar Stagnation RegionHeat Transfer in Dissociating Gases. Tech. Inform. Ser. No.R63SD40 (ContractNo. NAS 7-loo), Miss ile Space Div., Gen. Elec. Co., Apr. 1963.

17. Luikov, A. V.; Sergeev, V. L.; and Shashkov, A. G.: Ex pe ri me nt al Study of StagnationPoint Heat Transfer of a Solid in High-Enthalpy Ga s Flow. Int. J. Heat M a s sTransfer, vol. 10, no. 10, Oct. 1967, pp. 1361-1371.

18. Pope, Ronald B.: Stagnation-Point Heat Tr an sf er in Arc-Heated Helium and Argon.AIAA J., vol. 7, no. 6 , June 1969, pp. 1159-1161.

19. Collins, D. J.; and Horton, T. E . : Experimental Convective Heat-Transfer Measure-ments. AIAA J. (Tech. Notes), vol. 2, no. 11, Nov. 1964, pp. 2046-2047.

20. Nerem, Robert M.; Morgan, C. Joe ; and Gra ber , Bruce C.: Hypervelocity StagnationPoint Heat Transfer in a Carbon Dioxide Atmosphere. AIAA J. (Tech. Notes andComments), vol. 1, no. 9, Sept. 1963, pp. 2173-2175.

21. Yee, Layton; Bailey, Harry E.; and Woodward, Henry T.: Ballistic Range Me asure -men ts of Stagnation-Point Heat Tr an sf er in Air and in Carbon Dioxide at Velocitiesup to 18,000 Feet Per Second. NASA TN D-777, 1961.

22. Rose, P. H.; and Stankevics, J . 0 .: Stagnation-Point Heat-Transfer Measurementsin Pa rt ial ly Ionized Air . AIAA J . , vol. 1, no. 12, Dec. 1963, pp. 2752-2763.

23. Rose, P. H.; and Sta rk, W. I.: Stagnation Point Hea t-Tra nsfer Meas ureme nts inDissociated Air. J. Aeronaut. Sci., vol. 25, no. 2, Feb. 1958, pp. 86-97.

24. Akin, Clifford M.; and Marvin, Jo seph G.: Comparison of Convective Stagnation HeatTransfer in Air , C 0 2 - N 2 , and C02-Ar Gas Mixtu res. NASA TN D-4255, 1967.25. Horton, T. E.; and Babineaux, T. L. : Influence of Atmospheric Composition onHypersonic Stagnation-Point Convective Heating. AIAA J., vol. 5, no. 1, Jan.1967, pp. 36-43.

26. Nere m, Robe rt M.;,and Stickford, George H.: Radiative and Convective HeatingDuring Hypervelocity Re-Entry. AIAA J., vol. 2, no. 6, June 1964,pp. 1156-1158.

27. Gruszczynski, J . S.; and War ren, W. R., Jr . : Experimental Heat-T ransf er Studies ofHypervelocity Fligh t in Planetary Atmospheres. AIAA J . , vol. 2, no. 9, Sept. 1964,pp. 1542-1550.

28. Zoby, Ernest V.: Emp iric al Stagnation-Point Heat-T ransf er Relation in S evera l GasMixtures at High Enthalpy Levels . NASA TN D-4799, 1968.

34


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29. Bartlett, Eugene P.; and Kendall, Rober t M.: An Analysis of the Coupled ChemicallyReacting Boundary Lay er and Char ring Ablator. Pt. III - Nonsimilar Solution ofthe Multicomponent Lami nar Boundary Layer by a n Integ ral Mat rix Method. NASACR- 1062, 1968.

30. Kendall, Robert M.; Rindal, Roald A.; and Bart le tt , Eugene P.: A MulticomponentBoundary Lay er Chemically Coupled to a n Ablating Surface. A M . , vol. 5,no. 6, June 1967, pp. 1063-1071.

Vol. I - Boundary La yer Integra l Matr ix Proced ure (BLIMP). AFWL-TR-69-114,Vol. I, U.S. Air Force, Mar. 1970.

32. Anderson, L arr y W.; and Kendall, Robert M.: Us er 's Manual. Vol. 11 - BoundaryLay er Integ ral Mat rix Proc edure (BLIMP). AFWL-TR-69-114, Vol. II,U.S. AirForce, Mar. 1970.

31. Anderson, L ar ry W.; Bar tle tt, Eugene P.; and Kendall, Robert M.: User's Manual.

33. Kendall, Robert M.: An Analysis of the Coupled Chemically Reacting Boundary L ayerand Cha rring Ablator. Pt. V - A General Approach t o the Therm ochem ical Solu-tion of Mixed Equilibrium-Nonequilibrium, Homogeneous or Heterogeneous Sys-tems. NASA CR-1064, 1968.

34. Deblaye, Christian; and Bartlett, Eugene P.i An Evaluation of Thermodynamic andTransport Propert ies fo r U s e in the Blimp Nonsimilar Multicomponent Boundary-Layer Program. Rep. No. 69-53, Aerotherm Corp., July 15, 1969.

35. Anon.: JANAF Thermochemical Tables. P B 168370, Therm. R es . Lab., DowChemical Co.

36. Hilsenrath, Joseph; Messina, Carla G.; and Eva ns, William H.: Tables of Ideal GasThermodynamic Functions for 73 Atoms and Their First and Second Ions t o10,OOOo K. AFWL TDR-64-44, U.S. Ai r Fo rc e, Aug. 1964. (Available fr omDDC as AD 606 163.)

37. Schexnayder, Charles J., Jr.: Tabulated Values of Bond Dis soc iat ion Energies ,Ionization P otenti als , and Ele ctron Affinities f o r Some Molecules Found in High-Te mp er at ur e Chemical Reactions. NASA TN D-1791, 1963.

38. Bartlett, Eugene P.; Kendall, Robert M.; and Rindal, Roald A.: An Analysis of theCoupled Chemically React ing Boundary Lay er and Charr ing Ablator. Pt. IV - AUn i f i ed Approximation for Mixture Tra nsp ort Pro per tie s for MulticomponentBoundary- La ye r Applications. NASA CR- 1063, 1968.

39. Graves, Randolph A., Jr . : Diffusion Model Study in Chemically Reacting Air CouetteFlow With Hydrogen Injection. NASA TR R-349, 1970.

35


38/67

40. Lee s, Le st er : Convective Heat Transfer With Mass Addition and Chemical R eactions ,Presented at Third Combustion and Pro pulsion Colloquium of AGARD (Pale rmo ,Sicily), Ma r. 17-21, 1958.

41. Dorra nce , William H.: Viscous Hypersonic Flow. McGraw-Hill Book Co., Inc.,c.1962.

42. B ird, R. Byron; Stewart , Wa rr en E.; and Lightfoot, Edwin N.: Tra nsp ort Phenomena.John Wiley & Sons, Inc., c.1960.

43. Lees, Lest er: Lami nar Heat Tr an sf er Over Blunt-Nosed Bodies at Hypersonic FlightSpeeds. Jet Pro pulsion, vol. 26, no. 4, Apr . 1956, pp. 259-269, 274.

44. Pope, Ronald B.: Stagnation-Point Convective Heat Tr an sf er in Fr oz en BoundaryLayers. AIAA J . , vol. 6, no. 4, Apr. 1968, pp. 619-626.

45. Buddenberg, J. W.; and Wilke, C. R.: Calculation of Gas Mixture Visc osit ies. Ind.Eng. Chem., vol. 41, no. 7, Ju ly 1949, pp. 1345-1347.

46. Wilke, C. R.: A Visc osi ty Equation fo r Gas Mix ture s. J . Chem. Phys., vol. 18,no. 4 , Apr. 1950, pp. 517-519.

47. Hirschfelder, Jo seph 0.; Cu rti ss, Charles F.; and Bird, R. Byron: Molecular Theoryof Ga ses and Liquids. John Wiley & Sons, Inc., c.1954.tions 1964.)

(Reprinted with correc-

48. Brokaw, Richard S.: Approximate Formulas for the Viscosity and Thermal Conduc-tivity of Gas Mixtures. J. Chem. Phys., vol. 29, no. 2, Aug. 1958, pp. 391-397.

49. Ro sner, Daniel E.: Effe cts of Diffusion and Chemical Reaction on Convective HeatTransfer. ARS J., vol. 30, no. 1 , Ja n. 1960, pp. 114-115.

50. Svehla, Roger A.: Est ima ted Viscosi ties and Th er ma l Conductivities of Gases atHigh Te mp er at ur es . NASA TR R-132, 1962.

51. Zoby, Er ne st V.: Approximate Relations fo r Lami nar Heat -Tra nsfe r and Shear-Stress Functions in Equilibrium Dissociat ed Air . NASA T N D-4484, 1968.

52. Kemp, Nelson H.; Rose, Pe te r H.; and Detr a, Ralph W.: Laminar Heat Tr an sf erAround Blunt Bodies in Dissociated Air. J . Aero/Space Sci., vol. 26, no. 7 ,July 1959, pp. 421-430.

53. Cohen, Nathaniel B. : Boundary- Laye r Simi lar Solutions and Corr elat ion Equationsfo r Laminar Heat -Transfer Distribution in Equilibrium Air at Velocities up to41,100 Feet Per Second. NASA TR R-118, 1961.

54. Brokaw, Richard S.: Approximate F or mul as for Viscosity and Th erm al Conductivityof Gas Mixtures. NASA TN D-2502, 1964.

36


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55. Trautz, Max; and Binkele, H. E.: Die Reibung, Wzrmeleitung und Diffusion inGasmischungen. VIII. Die Reibung des H2, He, Ne, Ar und ihrer binarenGemische. Ann. Phys., ser. 5, vol. 5, 1930, pp. 561-580.

56. Tra ut z, Max; and Heberling, Robert: Die Reibung, Wgrmeleitung und Diffusion inGasmischungen. XVII. Die Reibung von NH3 und seinen Gemischen mit H2, N2,0 2 , C2H4. Ann. Phys., ser. 5, vol. 10, 1931, pp. 155-177.

57. Trautz, Max; and Kipphan, Karl Friedr.: Die Reibung, Warmeleitung und Diffusionin Gasmischungen. IV.Die Reibung biGrer und t e r s r e r Edelgasgemische. Ann.Phys., ser. 5, vol. 2, 1929, pp. 743-748.

58. Amdur, I.; and Mason, E. A.: Pr op er ti es of Gas es at Very High Te mpe ratur es.Phys. Fluids, vol. 1, no. 5, Sept.-Oct. 1958, pp. 370-383.

59. Yun, Kwang-Sik; Weissman, Stanley; and Mason, E. A.: High-Temperature Tran s-port Pr op er ti es of Dissociating Nitrogen and Dissociating Oxygen. Phys. F luids,vol. 5, no. 6, June 1962, pp. 672-678.

60. Trautz, Max; and Baumann, P. B.: Die Reibung, Warmeleitung und Diffusion inGasmischungen. 11. Die Reibung von H2-N2- und H2-CO-Gemischen. Ann.Phys., ser. 5, vol. 2, 1929, pp. 733-736.

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TABLE 1.- MOLECULAR WEIGHTS AND TRANSPORT

Base gas

N20 2H2HeNeA rc o 2NH3I CH4

PARAMETERS FOR BASE GASES

M u, A E/k, K Y MY( 4 (a) (b)

28.014 3.798 71.4 0.03654 1.02432.000 3.467 106.7 .04129 1.3212.016 2.827 59.7 .06775 .1374.003 2.551 10.22 .lo845 .434

20.183 2.820 32.8 .07449 1.50439.948 3.542 93.3 .04036 1.61244.011 3.941 195.2 .02919 1.28517.031 2.900 558.3 .04605 .78416.043 1 3.758 1 148.6 1 .03345 I -537

38


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TABLE II.- HEAT-TRANSFER COEFFICIENTS FROM CORRELATION OF PRESENT COMPUTER RESULTS

Gas composition(composition by ma ss fraction)g2'2K2HeNeArc o 2NH3CH40.232002-0.7680 N2 (air)0.6714 C02-0.3286Ar0.1345 CO2-0.8655 N20.5000 COZ-0.5000 N20.8500 CO2-0.15OON20.7500 02-0.2 500 N20.5000 02-0.5000 N20.1500 H2-0.8500 He0.3500 H2-0.6500He0.6500 H2-0.3 500 He0.5000 H2-0.5000 A r0.1500 N2-0.8500 H20.5000 N2-0.5000 H20.2000 CO2-0.8000 H20.4000 CO2-0.6000 H20.6000 CO2-0.4000 H20.8000 CO2-0.2000 H20.3626 CO2-0.3077N2-0.3297Ar0.1339 CO2-O.8525 N2-0.0136Ar0.3000 Ne-0.3000Ar-0.4000 He0.3500 H2-0.3500 CO2-0.3000 N20.352 H2-0.423 He-0.176 Ne-0.024 CH4-0.025NH3

K,kg/s-ms/a-atm 1/2

0.1112.1201.0395.0797.1474.1495.1210.0990.0807.1113.1262.1113.1131.1173.1164.1132.0657.0547.0459.0622.0473.0636.0537.0629.OW1.lo02.1218.1118.1143.0809.0662

Error for correlat ion, percentAverage

3.63.43.66.03.52.73.49.68.73.32.63.O2.73.O3.O3.14.33.33.44.95.76.47.67.68.6

10.22.72.96.39 .o6.2

Maximum10.010.714.118.810.57.58.4

18.322.6

9.86.89.47.48.58.87.7

15.813.814.213.116.717.823.528.927.017.89.19.6

19.921.516.3

39


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TABLE ID.- COMPARISON OF SUMMATION RELATIONS FOR GAS MIXTURES

Gas composition(composition by mass fraction)N20 2H2HeNeArc o 2NH3CH40.232002-0.7680N2 (air)0.6714 C02-0.328 6Ar0.1345 C02-0.8655 N20 .5000C 0~-0 .5000N~0.8500 CO2-0.1500 N20.750002-0.2500 N20.500002-0.5000 N20.1500 H2-0.8500 He0.3500 H2-0.6500He0.6500 H2-0.3500 He0.5000 H2-0.5000 A r0.1500 N2-0.8500 H20.5000 N2-0.5000 H20.2000 C02-0.8000H 20.4000 C02-0.6000H 20.6000 CO2-0.4000 H20.8000 C02-0.2000H20.3626 CO2-O.3077 N2-0.3297 A r0.1339 C02-0.8525N2-0.0136Ar0.3000 Ne-0,3000 Ar-0.4000 He0.3500H2-0.3500 CO2-0.3000 N20.352H2-0.423 He-0.176 Ne-0.024 CHq-0.025 NH3

Npr,,0.680

.685

.675

.667

.667

.667

.685,520.608,695.711.692.695,691,698.702.697.710.705.496,570,477.540.479,439,457.?lo.693.485,430.515

(a) Computer results0.1112. n o 1.0395.0797.1474.1495,1210,0990,0807,1113,1262.1113,1131.1173,1164,1132.0657.0547.0459,0622,0473,0636.0537,0629,0771.loo2.1218,1118.1143,0809,0662

Present, eq. (43

0,1117.1252,1113,1145,1187,1162,1134,0644.0536.0448.0649,0469.0649,0498,0606,0752,0937,1213,1115,1275,0795,0669

Present, eq. (44

0.1131,1285.1124,1158,1194.1177,1154.0659.0553.0460,0540,0424.0526,0436.0493.0580.0740.1246,1127,1053,0606,0571

Ref. 28, eq. (46'

0.1131,1291.1124.1159.1194.1177.1155,0691.0588,0480,0625,0437.0583.0456.0541.0662.0857.1255,1128,1104,0692.0626

40


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TABLE W.- OMPARISON OF SUMMATION RELATIONWITH REFERENCES 14 AND 15

Gas composition(composition by ma ss fraction)c o 2N2A r0.8655N2-0.1345 C020.3889 N2-0.6111 CO20.6717 CO2-0.3283 Ar0.5614 Ar-0.3149N2-0.1237 C02

K, kg/s -m3I2- atm / 2 1Refs. 14 and 15

0.1477.1259.1596.1278.1369.1513.1459

Present, eq. (44) J

0.1283..13 79.1513.14 50

41


44/67

:Bq3 100.000.IO01 .ooo10.0000-001.010.loo1 .ooo10.0001oo.ooo.001.010

1111.01111.0

(a) Key for all par t s of figure 1.

hs - hw(b) Nitrogen.

Figure 1.- Heat-transfer res ult s fro m present computer study.

42


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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ~0 20 110 w BO 100 !O

hs - hw(d) Hydrogen.

Figure 1.- Continued.

43


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0hs - hw

(f ) Neon.Figure 1.- Continued.

44


47/67

.d

'0hs - hw

(h) Carbon dioxide.Figure 1.- Continued.

45


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hs - hw

(i)Ammonia.

hs - hw(j) Methane.


0

46


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~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l l l i l l l l l l l l l l l l l60 BO 100hs - hw

(k) 0.2320 02-0.7680N2 (air).

7

I I I60 I I I30 I 1 IIJ1(11111(100

0

0

( I ) 0.6714 C02-0.3286 Ar.Figure 1.- Continued.

47


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hs - hw(m) 0.1345C020.8655 N2.

0

(n) 0.50 CO2 -0.50 N2.Figure 1.- Continued.

48


51/67

hs - hw

(0) 0.85C02-0.15N2.

hS - hw(p) 0.75 02-0.25N2 .


49


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(r) 0.15 H2-0.85 H e.Figure 1.- Continued.

50


53/67

31

hs - hw(s) 0.35 H2-0.65 He.

hS - hw(t) 0.65 H2-0.35 He.


51


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hs - hw(u) 0.50H2-0.50Ar.

(v) 0.15 N2-0.85 H2.Figure 1.- Continued.

0

J

52


55/67

8

3.U

Y

0 I l l l l r l r r r l l r r l l l l r I l r l l l l l r l l l l l l l l l l I I I I I I I I I I I I I I I I0 20 YO 60 80 100 1

hs - hw

(W ) O.50N2-0.5OH2.

!O

0hs - hw

(x) 0.20 C e - 0 . 8 0 H2.Figure 1.- Continued.

53


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hs - hw( z ) 0.60C02-0.4OH2.Figure 1.- Continued.

54


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hs - h,(aa) 0.80 [email protected] H2.

h - h,(bb) 0.3626 CO2-0.3077 N2-0.3297Ar.


55


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t

hS - hw

(cc) 0.1339 CO2-0,8525 N2-0,0136 Ar.

(dd) 0.30 Ne-0.30 Ar-0.40 He.Figure 1.- Continued.

0

56


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20 Lfo 60 EO 100 120

(ee) 0.35H2-0.35 C%-0.30N2.

9

6

3.m

3

0 1 1 1 1 ~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l l l l l l l l l l l l l l l l l0 ea 40 60 BO 100 120

(ff) 0.352 H2-0.423 He-0.176 Ne-0.024 CH4-0.025 NHQ.Figure 1.- Concluded.

57


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. ~.. _._

0 Bise gases, 0.608 2 Npr,w 2 .711+ Base gases, 0.430 2 Npy,W 2 0.5700 Gas mixtures, 0.608 2 NPr,w 2 0.711Gas mixtures, 0.430 2 Npr,w 2 .570

.10

3,--.zLazYv

I/\q- (42'0 I L0 ..2 .4 .6 .8

(a) Comparison with equation (42).

1.O 1.2 1.4

. I 5Computer resultso Base gases, .0.6082 Npr,w 2 0.7110 Base gases, 0.430 2 NPr,w 2 0.5700 Gas mixtures, 0.608 2 Npr,w 2 0.711Gas mixtures, 0.430 2 Npr,w 2 0.570

.IO -

n0 .2 .4 .6 .8 1.O 11.2 1.4

(b) Comparison with equation (41).Figure 2.- A comparison between the present analytical analysis and the present computer

resu lts fo r the heat-trans fer coefficient. (Units for K a r e k g / ~ - m ~ /~ - a t m 1 / 2. )58


61/67

.2 c

a 1 5

Y . i o

Data from literatureo Analytical (refs. 13 to 1610 Experimental (refs. 17 to 22 and 24 to 26)A Empirical (ref. 281

A

0

. . . 1 ~~ I

Figure 3 . - A comparison of the present analytical analysis with other studies.

59


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0 Present computer results0 Analyt ica l (refs. 14 an d 151

0 .21.4 I I.6 .8 1I .o II .2

Figure 4:- A comparison of the present res ult s with the res ult s ofreferences 14 and 15. (Units for K a r e kg/s-m3/2-atm1/2.)

1.4

60


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I-+0 Oc--- / - -__ - - -____- - - - -

. l l

Computer solution.1 0 L I I I I I

10-3 1o - ~ 10-1 1oo l o 1 1o2

(a) 0.232002-0.7680N2 (air). T = 1111K;hs = 81.354 MJ/kg.

Computer solution. l l c- _----- -_ - - -_ _ _ _ _ _ _ ------ K E q . 45). in L_ - L I I I I I

10-3 1 ~ - 1 1oo lo 1 1o2pS, atm

(b) 0.85 C%-0.15 N2. T = 300 K;hs = 104.596 MJ/kg.

Figure 5.- The tre nd of the heat-transfer coefficient with pressure. (Units for K are k g / ~ - m ~ / ~ - a t r n ~ / ~ . )


64/67

* 13

Y .12

.11

e--------__I I I I I20 40 60

hs, MJ / k g80 100 120

(a) 0.232002-0.7680N2 (air). T, = 300 K; (d) 0.3626 CO2-0.3077 N2-0.3297Ar .p, = 1.0 atm.

l i.06 I I I I I I0 20 40 60 80 100 120hs, MJ/kg

(b) Helium. Tw = 300 K; ps = 10.0 atm.

.1 1 I 1 I I I I I0 20 40 60 80 100 120

hs, MJ / k g(c) Carbon dioxide. T = 1111 K; ps = 0.1 atm.

T, = 300 K; ps = 1.0 atm.

I I I I Ihs, MJ / k g

20 40 60 80 100 120

(e) 0.65H2-0.35He. T, = 300 K;ps = 0.001 atm.

I ' I.12 I I I I I I80 100 12020 40 60

hs, MJ/kg(f) 0.6714Ca-0.3286Ar. Tw = 1111 IC;

ps = 10.0 atm.Figure 6. - The trend of the heat-transfer coeffic ient with enthalpy. (Units for K are kg/s-m3/2-atm1/2.)


65/67

. I 4

.I2

.IO

.08Y

.06

.04

---x- ------"U--c

o P r e s e n t c o m p u t e r r e s u l t sPresen t , eq . (43)Present , eq. (44)Ref. 28, eq. (46)

0 1 . I I I I0 .2 - 4 .6 .8 I

M a s s f r a c t i o n o f N 2

(a) c 0 2 - N ~mixtures..14

.I2

. I O

.08x

.06

.0 4

.02

0

o P r e s e n t c o m p u t er r e s u l t s- r esen t , eq . (43)Presen t , eq . (44)Ref. 28, eq. (46)

- _ _ _~-

I 1 I I

M a s s f r a c t i o n o f H 2(c )c 0 2 - S ~ ixtures.

. I 4

.I2

.10

.08 IY

.06

.04

.02

0

0 P r e s e n t c o m p u t e r r e s u l t s- resent , eq. (43)Present , eq. (44)Ref. 28, eq. (46)

_ _ - --_

I I I I0 .2 .4 .6 .8 1.0

.14

.12I

.IO

.082

.06

.0 4

.02

0

M a s s f r a c t i o n o f H p

(b) H e - H z mixtures.

0 P r e s e n t c o m p u t er r e s u l t s- r esen t , eq . (43)Presen t , eq . (44)Ref. 28, eq. (46)

_ _ _ _--

- _ I 1 I I0 .2 - 4 .6 .8 1.0

Mass f r a c t i o n of H p(d) N2-H2 mixtures.

Figure 7.- A comparison of the summation relations with the present computer re sul tsfor some binary gas mixtures. (Uni ts for K are kg/s-m3/2-atm1/2.)

63


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.08

.G4 1

C o r r e l a t i o n of ref . 16K = 0.0323 + 0.00233Mo

P r e s e n t c o m p u t e r r e s u l t s R e s u l t s f r o m r e f . 16- o r r e l a t i o n0 Base gases A GasesG a s m i x t u r e s

K = 0.0323 + 0 .00233M0I I I10 20 30 40 500 1 -0

Figure 8.- A comparison of the present computer results with the molecular weightcorrelation of reference 16. (Uni ts for K are kg/s-m3/2-,tm1/2$

64 NASA-Langley, 1971- 3 L-7885


67/67

N A T I O N A L A E R O N A U T I C S A N D S PA CE A D M I S T R A T I O NW A S H I N G T O N , D . C . 2 0 5 4 6- P O S T A G E A N D F E E S P A I D

O F F I C I A L B U S IN E S SP E N A L T Y F O R P R I V A T E U S E 8300

N A T I O N A L A E R O N A U T I C S A N DFIRST CLASS MAIL S P A C E A D M I N I S T R A T I O N

022 001 C 1 U 3 3 711105 S 0 0 9 0 3 D SOEPT OF THE A I R F O R C EAF WEAPONS L A B I A F S C )TECH L I B R A R Y / W L Q L /ATTN: E L O U ROWMANp C H I E FK f R T L A N D A F B NM 87117

If Undeliverable (SectionPOSTMASTER: Portal Man ual) Do N o t R e

T h e aeronautical and space activities of the United States shall bec ondwte d so as t o contribute , . . t o the expansion of hunzan knowl-edge of phenoiiiena in the atiriosphere and space. Th e Adminis trationshall provide for the widest prcrcticable and appropriate disseminationof iizforiiiation concerning its actizities and the restilts thereof.-NATIONALAERONAUTICSND SPACE ACT O F 1958

>

NASA SCIENTIFIC A N D TECHNICAL PUBLICATIONSTECHNICAL REPORTS: Scientific andtechnical information considered important,complete, and a lasting contribution to existingknowledge.TECHNICAL NOTES: Information less broad

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Heat Equation for Gax Mixtures