NASA-TR-R-50-1959 Approximations for the Thermodynamic and Transport Properties of High-Temperature Air《高温空气的热力学和运输性能近似值》.pdf

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1、_r_NASA TR R-50NATIONAL AERONAUTICS ANDSPACE ADMINISTRATIONt / ,y.2 _, . - .,., . . n energyof 7.37 electron volts l)er nlole(ule and the otherto (,).76 ele(tron volts per molecule. At first, (helower value was widely accepted as the mostprobable one (lierzberg, rcf. 3). Krieger andWhite (ref. 4) an

2、d Ilirsehfehh,r and Curtiss (ref.5) have published tables of thermodynamic prop-erties of high-tenH)eraturc air based on this value.(blydon (r(,f. 6) was l)erhaps one of the first advo-(ales of the view that (he higher value was the(orrect one. Su/)sequently a ntmll)er of exl)eri-ments were performe

3、d whi(h (onlirmed (,aydonsO)illi()l)_ l)lIl()g helll he )neaslllelll(ql,_ of sl)o)gsho(k waves in nitrogeu made 1)y (hrislian, Duff,and Yarger (ref. 7) and the delom)tion studiesmade by isiiakowsky, Knight, am Malin (ref.S). This reu(h,r(d the work of references 4 am| 5()bs0et(, but :barfly lhl,reaf

4、t(r (;ihnore (ref. 9)(.omputed the ehemi(al (.omposition, em,rgy,entropy, compressibility, and pressure of qir asfun(.tions of temperalur( and (l(,nsity based (in thehivher value for the (liss()(iati()n ()f nilro_en.t_ater, lilsemath and Be(k(tt 0ef. 10) 1)u)lish(la similar tal)le of these t)rot)erl

5、ies, I)ut in muchsmaller in()eme)lts ()f (ql)p()alllF( ll.lld (h,nsily.The (ahulations in Iiolh of th(,se r(,feren(es (9 aml10) are highly refined in the sense (hat they no()nly account for tile major (Iml)(me)ts 20)- T - -+“ In Qp(O)-ln Q,(O_) (20atIn K_(No2N)= -113_00+2 In Qp(N)-ln Q,(N2) (20b)158

6、,000In/kp(O-+O+-) =“ T fin ,(O+ t+ln Q/e-)-ln Qp(O) (20c)lt;8,g00,hlKp(N-_N+e-)- - T tm Qp(N*)+ln Qp(e-)-ln Qp(N) (20d)The concentration equilibrium constant is de-fined byK,=iin% (A 0 (21)where *_(At) and n(li_) are, respectively, theconcentrations of the chemical reactants andproducts. This quanti

7、ty will also be needed forsubsequent calcubttions, and it is obltdned byiephlcing the pressure standaMized liar(it(onfunctions, Qv, with the correspondillg conc(mtla-(ion stan(hudizod partition fun(tions, Q_, inequation (19). From equation (6) it is seenlhatI(,- I(v( ll)=“,-b_ (223The logarithmic de

8、rivatiws of the equilibrhmlconstants will also be required later. Fromequations (5), (10), and (19) these beconleT d In K, .XEo (E-If,|,IT - lit F T_, b _-:?T-/,-_, a_ -_,_- (23) 1,7 /.,5/,d In Ap_ T d hi K,dT 17 _-_ b,-_ (t_ (24)The equilibrium constants and their logarithmicderivatives for the rea

9、ctions represented t)y equa-tions (20at through (20(l) are listed as functionsof temperature in table III. A populationweighted average quantity is given for the oxygenand nitrogen iouizatimt reaction. These quantitieswill lit)iv 1)e its(,( ill eah,ulatitlg the component nlolfractions and their deri

10、vatives.CALCULATION OF THE EQUILIBRIUM MOL FRACTIONS ANDTHEIR DERIVATIVESTim possil)ility /hat apltroximaie solutions inclosed form couht be obtained for the propertiesof air suggests itself upon examination of theresults of Gilmore (ref. 9). IIis tal)les of thecompositot_ of air show that there are

11、 fourchemical reactions of major importance. Theseare the, dissociation of nmlecular oxygen andmolecu_:ar nitrogen, and the ionization of atoinicoxygen mid of atomic nitrogen.02 20 (25a)N2 2N (25)O - O _+e- (25c)N -_ N+e - (25(t)With (:he exception, all other reactions whichcccur 3Md component conce

12、ntrations which arethe or(l,_r of 0.1 percent, or less. The exceptionis the forination of ifitric oxide, NO, which at sealevd (hnsily lnay become as much as 10 pert(mrof the _,ir aroun(l 5000 K. lh)wever, even thismuch nit, rio oxi(lc does not strongly inlhlen(e theresultin ,_thernmdynanlic properti

13、es of air, and atdensitie_ less than 0.01 noImal set_ lew_l density,where tae NO is less than 1 percent at its inaxi-InUre, 1,71ceffects arc very slnall.Two distinctive features of the chemical reac-tions gi_,en above are (d)servat)h; front Gilmoresresults. The tirst is thai at, all I)r(,ssures the

14、dis-sot(alto t of oxygen is csst,ntia, lly comph,le t(,forethe (lis.,oci_iion of nit.rogen |)(,gins. This meanstha.t th(se two roact,ions cmt be treated independ-ently fi,r the imrposes cf approxinlation. Thesccon(l :eat ure is l hat nitrogen and oxyg(n atomsionize at about the s,tme telnperatur(_ a

15、nd withabout t m same energy changes. Consequently,it is possible to assume that once air i,-. completelydissocia ed, all atoms constitut(, t_ single specieswhich has the populatior weight.ed average prol)-ert, ies oi the nitrogen and oxygen atoms.Provided by IHSNot for ResaleNo reproduction or netw

16、orking permitted without license from IHS-,-,-THERMODYNAMIC AND TRANSPORT PROPERTIES OF AIRThe equation of state will be definedp_ZRTp- Mo (26)where Z is the compressibility. To the approxi-mation that all of the partich, s obey lhe ideal gaslaw, Z represents the total number of tools formedfrom a t

17、ool of initially undissociated air. It isalso equal to the ratio ef the initial nmlecula.rweight of undissociated air to the mean molecularweight, 3_Io/_ll. If _1 is the fraction of moleculeswhich dissociate into oxygen aloms, E2the flactionof molecules which dissociate into nitrogen atoms,and ea th

18、e fraction of atoms which arc ionized,then the compressibility is given byZ-1 -_ _l-_ E2-_ 2_a (27)The reactions arc now assumed to be independ-ent and, in view of the order of approximationbeing ccnsidered, the ratio of nitrogen to oxygenhas simply been taken as 4 to 1. Then at rela-tively low temp

19、eratures only three major com-ponents exist sinmltaneously: molecular nilrogen,molecular oxygen, and atomic oxygen. The par-tial pressures for these three components may beexpressed0.8p (N2) = a: (N_)p = 1 -_-(_1) (283)p(O2)=x(O2)p=_2-+-_! p (28b)p(O) =, _u)p=l_N+ q e-)(37)The component tool fiaclio

20、lis in air me lhmi givenI)y,(Oa)- 02-lJo/p)_2 _ 8(46b)- (K_a.llo/p) -vI(_a.llo/p)_ 8(K_aMo/p)ca= 4(46c)All the other calculations follow as before, onlythe quantities _l, _:, and _a fronl eqmltions (46a),(4(it), llid (4(ic) replac(, llios(, fronl equiliions(:_0), (: _), and (:_O).With the preceding

21、relations in hand, we are in1)ositiol_ to calcuhite the energy, (,ltirol)y , sp(,cificheat, al d speed of sound for it it.ENERGY, SPECIFIC HEAT, ENTROPY, AND SPEED OF SOUNDFOR AIR IN EtlUILIBRIUNIThe (,nergy per nlol of air is sinli)ly lhe sumt,:= 52_,E,i(47)Provided by IHSNot for ResaleNo reproduct

22、ion or networking permitted without license from IHS-,-,-THE:RMODYNAMIC AND TRANSPOtT PROPERTIES OF AIR 11il)-_ (57)The dimensionless sl)eed of sound parameter,a2p/p, is listed in table IV(f) and is plotted as afunction of teml)ertdure in figure 6. The see(rodterm on the right side of equation (57)

23、is generallynear unity, so that figure 6 is also indicative ofthe variation in 7 with temperature.AERODYNAMIC CONSIDERATIONSThe thermodymunic properties obtained at thispoint are those required to perform calculationsof inviscid air-flow problems. These propertiesare given for a range of lelnperatme

24、 from 500 to 15,000 K and of pressure from 0.0001 to 100atmospheres. It is of interest now to examinethe altitude and velocity at which these condi-tions will occur in flight. A grid of the pressureand teml)erature at, the stagnation point of a bodyin flight is shown in figure 7 as a function ()f fl

25、ightaltitude and veh)city. The stagnation euthalpyper unit mass was simply taken as one half thevelocity squared, and the stagnation pressure wasrelated to the static pressure (and thus to altil u(le)572920 .60 ._Provided by IHSNot for ResaleNo reproduction or networking permitted without license fr

26、om IHS-,-,-16 TECHNICAL REPORT R-50-NATIONAL AERONAUTI,S AND SPACE ADMINISTRATION17161_ 1,5E 1.42SI! Ii-/, -f5 4 5 6 7 8Tempera ftlre, T_KFmURE 6.-Speed of sound parameteri i9 I0 _ 12for air a,_ “_ function of temperature.13 14 15xlO 3with the results of Feldman (ref. 21), who com-puted the pressure

27、 ratio developed across normalshock waves in air at various altitudes. Generallylower temperatures and pressures will be attainedat regions other than the stagnation region, sothe range of variables will be adequate for such400 xtO 3360520280240,u200r6o12o8O40go o _ oo o oI 0,/, , / I I / ,/ ; /“ I

28、7Vi P/ , /_ or 71, i .-7-,I/ i _“ I .I I I ._,/ 2_ _ ; / _, _i t /_-_/_:_-_,_/ / ,i-.-.1“- Io / / , / ! _f/_/ ,; ,/ _ / /1./I /jfJ-_ooo,ml/ I z“ I I I I I4 8 12 J6 20 24 28 32 3603Velocity, fl/secFnc, vRs 7.-Stagnation temperature and pressure in airas a function of altitude and velocity.cases also.

29、 It can be seen that the thermody-namic properties of air can be approximated inclosed form over the range of conditions of currentinterest in aerodynamics. Some of the results ofthese _pproximations will next be used in estimat-ing th_ transport properties of high-temperatureair.TRANSPORT PROPERTIE

30、SCOLLISION CROSS SECTIONSCondder ill,t, some qualitative aspeels of thecollisions 1)etween gas particles. The particletrajeclories arc influenced by a pottmtial U whichis negl giblc at long range, which may be attractiveor relulsive at intermediate range, but whichMway, becomes strongly iepulsiw, at

31、 very shortrange. A particle with ldnetic energy kT will notbe gre _.tly deflected if it passes only through thatpart o! the pottmtial field where IUIkT. Thedirection of the deflection is unimportant, so faras lrm_sport properties are concerned, for it. is theabsoh;le w_lue of the dettect ion angle

32、which deter-mines the change in mass, nmmentum, and energyfluxes caused by the collision. To a first approx-imati(,n, the absolute value of this deflection isindependent of the sigm of the potential and theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-

33、,-,-THERMODYNAMIC AND TRANSPORT PROPERTIES OF AIR 17effective collision diameter a is the order of thelargest distance where U=kT. The effectivecollision cross section S will be defined as _ra_.In the rigorous treatment of the transportproperties of gases, the effective collision crosssection S is f

34、ound to be an integral of the deflectionangles produced by collisions and this integral is afunction of the relative velocity of the collidingparticles. The so-called “collision integral“ is afunction of temperature only which is S times avelocity function integaated over all velocities.Thus the col

35、lision integral may be thought of asa weighted average collision cross section, and tlLetransport coefficients can be related directly tothese integrals. However, not all the interpartielepotentials have been developed which are neededto calculate the collision integrals for ah In thepresent paper t

36、hen, plausible estimates of theeffective collision cross sections will be used todetermine the mean free paths for hard elasticspheres, and for such particles the transportcoefficients can be related to these mean free paths(ref. 1). The effects of the interaction potentialswill be taken into accoun

37、t by letting the sphericalcross sections be appropriate functions of tem-perature.The effective cross sections for collisions betweendiatomic molecules can be obtained quite accu-rately by the collision integral method. Ilowever,at high temperatures, the very steep repulsiveportion of the intermolec

38、ular potential is pene-trated so that the molecules behave essentially likehard spheres. Consistent with the approximationswhich follow, it will be sufficient to use the Suther-land formula for the molecular cross section So.So C_-_- 1+ _ (58)where the constant C is 112 t( and S, the crosssection at

39、 infinite temperature, is 3.14X 10-_Scm _,for the case of air molecules.For atomic collisions, the picture is complicatedby the fact that two atoms may approach eachother along any one of a number of potentials.For example, the potentials between two normalnitrogen atoms are sho_1_ qualitatively in

40、sketch(a) (see ref. 3). The lowest lying of these poten-tials, designated U0, has the lowest total electronspin and it is the one normally responsible for thevibrational energy levels observed in the stablemolecular state. Therefore U0 can be expressedquantitatively from experimental spectroscopicda

41、ta. Unfortunately, the higher lying potentialsfor the atonls in air are not known quantitativelyat present, so we are forced to estimate an averagecollision cross section for all of the potentials bymeans of the known lowest lying potential. Forthis purpose it is assumed that the collision dialn-ete

42、rs a are given t)yUo(_)= -kT (89)and these diameters will be used to evaluatethe coefliciel_ts of nlonlentmn and energy trans-fer. A somewhat deeper penetration of the poten-tial is normally required for a collision to affectthe particle itux, so that the diameters a whichwill t)e used to evaluate t

43、he diffusion coefficientsare assumed to be given byUo(a) = -2kT (60)It may be pointed out that Hirschfelder andEliason (ref. 22) have examined the relationbetween values of tile transport coefficients givenby the hard sphere model and by the more rigor-ous collision integral method. They find thatU(

44、) and U(a) are about -0.6kT for a widevariety of attractive potentials, that U(a) isabout 0.9kT and U(a) is about 1.6kT for a similarvariety of repulsive potentials. If all the inter-particle potentials were known, it would be simpleto use these criteria to obtain a weighted averageS_.Tc_ (a).Provid

45、ed by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-iS TECttNICAL REP()H.T I 50 NATIONAL AEI_,ONAUIICS AND SPACE AI)MINISTIATI()Ncollision dianwtm. All the potential,; mc ktmwnfor two normal hydcogen atoms (rcf. 23) am it isfound for this case that equations (

46、591) and (10)yiehl values for a and a which agree with th(,weigh t(,(l _verage collision diameters witbin 8 per-(.cnt over the range of teml)craturcs from 1000 to15,000 (. Of coursc, there is no assurance thaitlwsc sam(, relations will hold as (losdy for col-lisions t)ctwcen the atoms in air. In fac

47、t theaverage collision diameters for normal oxygen andnitrogen a.toms will probably be overestimatedby equations (59) and (60), sin(,e the shnllowintermc(liate potentials (such as Lq and 1_, sketch(a) must be considered fro“ these atoms, whereasthey (to not occur for hydrogen. The elrect ofthese int

48、ermediate potentials will be partly com-pensated fro“ by the fact, that some of tim atomswill he in excited electronic stat, es which havccollision diameters the order of three times largcrthan the normal atoms (rcf. 22). The f,action ofatoms in excited states is small over most of thetetut)erature and pressure rm_gc con,ddered so ib.atcollisim_s belween ex(itcd t)art Mes are relativelyrare. The mwmmtcrs t)ctwecn an excited and ano,ma.l atom arc the om,s which significantlyin/hwme the mean flee l)aths, aml the cross sec-lions fm such c