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2 AIAA h Inflnc of Mdim Comrssibility on Chmical-Raction Rats Part II: mrical Exrimnts. D. D Ambrosio Politcnico di orino - Di. di Inggnria Aronatica Sazial Corso Dca dgli Abrzzi, 24, 29 orino, Italy D. Giordano ESEC - Arothrmodynamics Sction Klrlaan, 22 AG oordwijk, h thrlands A. Viviani II Univrsità di aoli - Di. di Inggnria Arosazial Mccanica Via Roma 29, 83 Avrsa (A, Italy Abstract mrical xrimnts ar carrid ot to qantify th inflnc xrtd by comrssibility corrctions on chmical-raction rats basd on th law of mass action. In articlar, th xansion insid a hyrsonic nozzl is considrd in th hyothsis of a qasi-ondimnsional inviscid flow. h obtaind data show non-ngligibl diffrncs btwn rslts obtaind with or withot th alication of comrssibility corrctions and stimlat frthr invstigations on th sbjct. c,i E h h i h f i h o K c k κ f k,κb k L M n o r R G S t t fd t ck omnclatr comonnt constant-rssr scific hat total nrgy r nit volm mixtr nthaly comonnt nthaly comonnt formation nthaly stagnation nthaly chmical-ilibrim constant (concntrations forward, backward raction constants rfrnc lngth comonnt molar mass avrag molar mass nmbr of comonnts rssr xit rssr stagnation rssr nmbr of chmical ractions nivrsal gas constant srfac ntroy tim charactristic flid dynamics tim scal charactristic chmical kintics tim scal tmratr Assistant Profssor, Mmbr AIAA,(domnic.dambrosio@olito.it. Rsarch nginr, Mmbr AIAA, (Domnico.Giordano@sa.int. Associat Profssor, (antonio.viviani@nina2.it. Coyright c 23 by th Amrican Institt of Aronatics and Astronatics, Inc. All rights rsrvd. o rfrnc tmratr vlocity x-comonnt v scific volm v vlocity vctor x x-coordinat α j scis mass fraction α i comonnt mass fraction α j scis mass fraction divrgnc orator Θ D charactristic dissociation tmratr ki global stoichiomtric cofficint r ki, ki stoichiomtric cofficints of ractants, rodcts ξ k chmical-raction rat ρ total-mass dnsity τ v vibrational rlaxation tim for diatomic comonnts i comonnt rodction rat comonnt rodction rat (law of mass action only ma i Introdction h ida that comrssibility ffcts may inflnc chmical-raction rats is not a nw on. In th ast, varios thortical invstigations wr carrid ot by diffrnt athors, 6 bt a final conclsion on th sbjct was nvr rachd. h contrasting oint was not whthr comrssibility ffcts can altr chmical kintics rats or not, bt whthr sch an ffct, whos xistnc is confirmd by all th stdis, is sfficintly strong to b obsrvd in ractical alications or not. In th ar that accomanis th rsnt on with thortical considrations on th sbjct, 7 Giordano ts in vidnc th inconsistncy introdcd by sing th law of mass action as it is in high-tmratr flid dynamics comtations whr th gas mixtr is not macroscoically at rst. h aim of this ar is to carry ot nmrical comtations whr comrssibility corrctions to th law of mass action ar imlmntd, and to comar th rslts with thos obtaind withot corrctions. o achiv sch a goal in a rlativly siml way, th bhavior of high-tmratr Amrican Institt of Aronatics and Astronatics

3 gas mixtrs that xand to hyrsonic sd insid nozzls is invstigatd. In ordr to focs on th ffct of th comrssibility corrction alon, th simlifid modl of qasi-on-dimnsional inviscid flow has bn sd. h ar rns as follows. In th following sction, information abot flid dynamics and thrmochmical modls is rovidd and th form of two diffrnt comrssibility corrctions is dfind. xt, th sd nmrical mthod is shortly dscribd and th gomtrical configration and th tst matrix ar rsntd. hn, rslts obtaind with and withot th adotion of comrssibility corrctions ar comard and th diffrncs ar discssd. Conclsions ar finally drawn in th last sction. 2 Govrning ations h inflnc of th gas mixtr comrssibility on th law of mass action is tstd throgh nmrical simlations of qasi-ondimnsional inviscid flows insid hyrsonic nozzls. Whn writtn in intgral consrvation form, th qasi-ondimnsional Elr ations rad lik t x2 x x2 ρsdx + [( + ρ 2 S ] x2 t x ρsdx + (ρs x2 x = (a x2 x x ds dx dx = (b x2 ESdx + [(E + S] x2 x t = (c x x2 ρα i Sdx + (ρα i S x2 x t = x i =,, n x2 h total nrgy r nit volm E is dfind as x i Sdx (d E = ρh + 2 ρ2 (2 and th mixtr nthaly h is obtaind throgh smmation of ach comonnt nthaly n h = α i h i (3 i= h govrning ations systm is finally closd by th stat ation = ρ R Ḡ M (4 whr th avrag molar mass M dnds on th chmical comosition of th mixtr according to th rlation ( n α i M = ( M i= i In this work, w assm for simlicity that th intrnal dgrs of frdom of ach comonnt ar in ilibrim. hrfor, th comonnt nthaly is dfind as a fnction of th translation tmratr according to th formla h i ( = h f i( o + o c,i dθ (6 Comonnt constant-rssr scific hats c,i ar assmd to dnd on tmratr according to th crv fits roosd by Prabh and Erickson. 8 Formation-nthaly information has bn takn from th tabls by McBrid t alii Chmical modl Comonnt chmical raction rats i ar dfind as i = r ξ k k= ki (7 In comtational flid dynamics alications, it is common ractic to idntify chmical raction rats ξ k as thos rovidd by th law of mass action ξ ma k = κf k i= n ( ραi r ki κ b k k =,, r n ( ραi i= ki Howvr, it is notd in Rf. 7 that this assmtion introdcs an inconsistncy in th hysical modl, bcas in th cas of a comrssibl flow vry clos to chmical ilibrim conditions, chmical rodction rats shold tnd to zro on th on hand, bt, on th othr hand, th chmical comosition is xctd to chang d to variations of thrmodynamics variabls. h inconsistncy is rmovd if comrssibility corrctions ar incldd in th comonnt rodction rats. In this cas, th law of mass action is still rsnt in th i sorc trms, bt it is sidd or altrd by trms roortional to th divrgnc of th vlocity vctor, as it will b shown in th following sbsction. hroghot th ar, only binary mixtrs of dissociating nitrogn or oxygn ar considrd. hs, th chmical ractions that com into lay ar for dissociating nitrogn and for dissociating oxygn. h choic of th chmical-kintics modl is not fndamntal for th aim of this ar, bcas th comrssibility corrction formlas sd hr do not dnd on th form of th raction rats. Anyway, w chos to s th forward raction rats sggstd by Park in Rf.. Backward raction rats ar obtaind via th chmical-ilibrim constants rovidd by thrmodynamics as dscribd in Rf. 7 κ f k κ b k 2.2 Comrssibility corrctions (8 = K c k( (9 wo diffrnt comrssibility corrctions to th law of mass action ar tstd in th rsnt work. h first on, that in th following will b addrssd to as corrction #, is th on tntativly roosd by Giordano in th comanion ar ddicatd to thortical considrations. 7 It consists in sing comonnt rodction rats dfind as i = r k= ξ ma k ki + v ( α i v ρ v (,α 2 Amrican Institt of Aronatics and Astronatics

4 h scond addndm on th right-hand sid rrsnts th comrssibility corrction, roortional to th divrgnc of th vlocity vctor, to th scific volm and to th drivativ of th virtal ilibrim comosition with rsct to th scific volm itslf. h mthod for comting th drivativs ( α i v,α is dscribd in Rf. 7, whr it is also shown that, in th cas of atomic oxygn or nitrogn, th sign of sch drivativs is always ngativ. hs, it can b concldd that, at last for th considrd binary mixtrs, th comrssibility corrction roosd by Giordano will romot rcombination in xanding flows and dissociation in comrssing flows. h scond corrction ( corrction #2 is similar to th on roosd by Viviani in Rfs., 2. It consists in mltilying th forward raction rats (for dissociation by an xonntial factor that dnds on th divrgnc of th vlocity vctor. Comonnt rodction rats ar now dfind as i = r k= ξ k ki ( that ar rsnt at ach latral srfac d to th discrtization, is solvd hr in an aroximat way: shock wavs that cold b gnratd by th collas of th initial discontinitis ar aroximatd by comrssion wavs. h consrvativ form of th ations insrs that th corrct jm and ntroy conditions ar satisfid. Scond ordr accracy in sac and tim is achivd following th gidlins of th Essntially on scillatory schms for shock catring tchniqs, which ar caabl of avoiding srios oscillations at flow discontinitis. 4 Gomtrical configration and tst matrix mrical tsts ar carrid ot considring th xanding flow insid a hyrsonic nozzl. In articlar, th gomtrical configration of th Scirocco lasma wind tnnl of th Italian Cntr for Arosac Rsarch (CIRA is considrd in this cas (Fig.. whr th corrctd chmical raction rats ξ k rad lik n ( ( r ξ k = ραi ki 2 Θ D κf k 3 τ v v n ( κ ραi b k i= k =,, r i= ki (2 Charactristic dissociation tmratrs Θ D ar takn al to 97 K for oxygn and to 32 K for nitrogn. h vibrational rlaxation tim τ v aaring in Eq. (2 is hr comtd with th siml formla τ v 2 = 32 ( /3 (3 y/l ozzl gomtry y/l. Zoom of th throat rgion x/l for oxygn and with τ v 2 = 32.. ( 4. /3 ( x/l for nitrogn. Eithr Eq. ( and Eq. ( rmov th inconsistncy that wold aris if th law of mass action alon wr sd in sitations clos to chmical ilibrim. In addition, as shown in Rf. 7, Eq. ( is xact in th limiting cas of chmical ilibrim conditions. 3 mrical mthod Govrning ations ar discrtizd in sac according to a cllcntrd finit volms aroximation and thy ar intgratd in tim sing a forward xlicit mthod for th convctd flxs and an imlicit mthod for th chmical sorc trms. Flxs at cll intrfacs ar valatd sing an wind Flx- Diffrnc Slitting (FDS Rimann solvr. 3 Uwind mthods rsnt th aaling fatr of introdcing th lmnts of th wav roagation hnomnology in th intgration of th consrvation laws. In articlar, th adotd FDS tchniq consists in dfining and solving a scial initial-val roblm, calld th Rimann roblm, 4 at ach latral srfac. h Rimann roblm, that is, th voltion in tim of th flow discontinitis Fig. Hyrsonic nozzl gomtry h sd nmrical grid is comosd of 78 clls, clstrd in articlar at th cornr whr th nozzl bgins and in th throat rgion. A rliminary grid convrgnc stdy dmonstratd that nmrical rslts do not chang if th msh is frthr rfind. o vrify th ffct of th comrssibility corrction in a sfficintly wid rang of sitations, varios nmrical tsts wr mad considring diffrnt stagnation conditions. In this ar, rslts rlatd only to a singl, bt rrsntativ, st of conditions ar rortd (abl..c. h o [MJ/kg] o [bar] [Pa] L [m] gas [m] [m] [m] [m] 2 - abl st matrix 3 Amrican Institt of Aronatics and Astronatics

5 Rslts. h ffct of comrssibility corrction # W start with dscribing th rslts obtaind in th conditions of tst cas #.. with and withot th corrction roosd by Giordano. Looking at tmratr, vlocity and atomic oxygn concntration distribtions (Figs. 2a and 2b, no arciabl diffrnc can b noticd. Howvr, a closr obsrvation of th varios contribtions to th atomic oxygn rodction rat (Fig. 2c dmonstrats that, dsit th fact th th global chmical sorc trms ar idntical with or withot comrssibility corrction, thir constittiv lmnts ar vry diffrnt. h oint hr is that comrssibility corrction # is vry strong in a rgion whr th law of mass action is also vry activ. U to x =.4 m, in fact, vlocity is incrasing vry raidly and, in aralll, tmratr is sfficintly high to allow for th law of mass action to qickly ract to flow-fild variations. If th comrssibility corrction is switchd on, it drivs th gas mixtr towards rcombination (dottd lin with circls in Fig. 2c. his hans bcas th divrgnc of th vlocity vctor has a ositiv larg val in th throat rgion (th flow is acclrating and in th mantim th drivativ ( α v,α is ngativ. Howvr, this is xactly what th law of mass action wold do if th comrssibility corrction wr switchd off. hs, th law of mass action racts to th rsnc of th comrssibility corrction, that ncorags rcombination, by driving th mixtr towards dissociation (dottd lin with triangls in Fig. 2c, in sch a way that th rslting rodction trm (dottd lin in Fig. 2c b th sam that wold aris withot corrction (solid lin in Fig. 2c. h ictr changs whn th law of mass action is not sfficintly activ to ract raidly to th ffct of th comrssibility corrction. his hans, for instanc, whn th gas mixtr is changd to - 2 maintaining th sam stagnation conditions (tst cas #... itrogn raction rats ar slowr than th oxygn ons, so that chmical ilibrim is rachd lss asily. mratr, vlocity and concntration distribtions (Figs. 3a and 3b t in vidnc a crtain diffrnc btwn rslts obtaind with or withot alying th comrssibility corrction. In articlar, rcombination is nhancd by th ffct of comrssibility. A clos look at th atomic nitrogn rodction rat (Fig. 3c in th throat rgion xlains th diffrnc with rsct to th oxygn mixtr cas. Also in th rsnt sitation th law of mass action racts to th inflnc of th comrssibility corrction by romoting dissociation in ordr to comnsat th comrssibility rcombination ffct. Howvr, th lowr tmratr that charactrizs th tst cas and th slowr raction rats tyical of nitrogn wakn sch a contr-acting ffct (dottd lin with triangls in Fig. 3c, so that th rslting rodction trm (dottd lin in Fig. 3c is diffrnt from th on that wold b givn by th law of mass action alon (solid lin in Fig. 3c. h sitation dictd abov is frthr magnifid if th gomtrical scal of th nozzl is rdcd, as in tst cas #.. (Fig. 4, whr th rfrnc lngth is on hndrd tims smallr than in tst cas #... A smallr rfrnc lngth imlis a smallr rsidnc tim for flid articls insid th nozzl and thrfor a largr difficlty for th chmical comosition to follow flid dynamics variations that dictat continosly changing ilibrim conditions. vrthlss, comrssibility corrction # is not affctd by this roblm and its ffct rmains th sam as in tst cas #... h lattr statmnt can b xlaind rarranging th mass-comonnt balanc ation (Eq. (d in th form ( Dα i Dt = ma i α + v i v ( ρ v,α An stimation of th ordr of magnitd of th thr trms aaring in Eq. ( shows that ( Dα i Dt = (6a t fd ma ( i = (6b ρ t ck ( ( α v i v = (6c v,α t fd whr t fd is th tim scal charactristic of flid dynamics hnomna and t ck is th tim scal that charactrizs chmical kintics. Eqation (6 ts in vidnc that, if t fd bcoms small bcas th gomtrical rfrnc lngth is rdcd, than th chmical rodction trm ma i ρ bcoms small with rsct to th othr two, sinc th charactristic chmical rlaxation tim t ck dosn t dnd on th gomtrical rfrnc lngth. hs, in tst cas #.. th law of mass action is not sfficintly strong to fficintly ract to th ffct of th comrssibility corrction, whos inflnc dominats th total atomic nitrogn rodction trm (Fig. 4c. h ffct is so strong that, at th vry bginning of th xansion has, th rcombination rocss romotd by rodcs an incras in tmratr that ovrwhlms th tmratr dcras indcd by th flid dynamics xansion, with th final rslt that th xansion in th nozzl starts with a tmratr incras ( Fig. 4a. o strngthn th abov dscribd intrrtation, th ffct of comrssibility corrction # is tstd again, this tim incrasing th gomtrical scal of th nozzl by a factor with rsct to tst cas #... As it cold b xctd, th law of mass action now is sfficintly strong to contrast th action of th divrgnc of th vlocity vctor, as it can b vincd from Figr. h analysis of th abov rsntd rslts shows that comrssibility corrction # can xrt an imortant inflnc on flows xrincing a strong xansion in th hyrsonic rgim. Comrssibility corrction # tnds to favor rcombination in acclrating flows and dissociation in dclrating flows. Its ffct is thrfor orintd in th sam dirction of th law of mass action, which is drivn by dcrasing tmratrs dring xansions and incrasing tmratr dring comrssions. Howvr, sinc information abot th chmical ilibrim comosition is containd insid th law of mass action only, th lattr attmts to contrast th coorativ, bt xcssivly strong contribtion of corrction #, in ordr to crat an ovrall chmical rodction rat caabl of lading to th chmical ilibrim comosition dictatd by chmical kintics. In thos occasions whn advrs flid dynamics conditions hindr th law of mass action from fficintly racting to comrssibility corrction #, th lattr rvails ovr th formr, giving ris to xotic sitations sch as an incrasing tmratr d to vry fast rcombination in an xanding flow..2 h ffct of comrssibility corrction #2 Comrssibility corrction #2 acts on forward (dissociation raction rats (s Eq. (2 containd in th law of mass action. Its 4 Amrican Institt of Aronatics and Astronatics

6 ffct is to rdc dissociation in acclrating flows (ths, romoting rcombination and to incras it in dclrating flows. h inflnc of both th comrssibility corrctions # and #2 on chmical-raction rats is shown in Figr 6 for tst cas #... h choic of this articlar tst cas was mad to t in vidnc th fact that th ffct of comrssibility corrction #2, thogh lss intns than th on of corrction #, gos in th sam dirction. Any othr tst condition listd in abl rodcd ffcts too wak to b hlfl in dscribing th inflnc of corrction #2. In Figs. 6a and 6c, it can b noticd that th ffct of corrction #2 is in fact vry similar to th on d to corrction #. Again, rcombination is romotd at a rat that is fastr than th on that th law of mass action alon wold rodc, so that th charactristic ffct of an initially incrasing tmratr d to dissociation in th first art of a strong xansion is rsnt whn sing corrction #2 also. h xonntial factor that mltilis th forward raction rats in Eq. (2 dnds on th divrgnc of th vlocity vctor and on th vibrational rlaxation tims of th diatomic comonnts. h smallr τ v is, th lss ffctiv is th comrssibility corrction. his xlains why comrssibility corrction # and #2 bhav in a qalitativly similar way; in fact, th flid dynamics conditions that favor a raid raction of th law of mass action to comrssibility corrction # ar th sam that giv ris to small vals for th vibrational rlaxation tims that aar in comrssibility corrction #2. 6 Conclsions mrical invstigations abot th inflnc of two diffrnt comrssibility corrctions on chmical-raction rats hav shown not ngligibl ffcts. In articlar, th diffrncs with rslts obtaind sing th law of mass action alon ar rominnt whn flid dynamics conditions do not favor th xistnc of fast chmical rlaxation tims. Both th comrssibility corrctions tnd to romot rcombination whn th flow is acclrating and dissociation whn th flow is slowing down. In som cass, th rcombination hnomna d to th comrssibility corrctions wr so strong that th formation nrgy rtrnd to th flow rondratd ovr th kintic nrgy sbtractd by th xansion rocss, rodcing tmratr riss in th first art of th xanding nozzl. Howvr, th stdy rortd in this ar is jst rliminary. For instanc, as admittd in Rf. 7, comrssibility corrction # is not nforcd by rigoros hysical argmntations, thogh it rrsnts th corrct limit to which chmical-rodction rats shold tnd in qasi-ilibrim conditions. n th othr hand, corrction#2, which was drawn indndntly, still nd frthr vrification. h fact that th two corrctions dislay a qalitativly similar bhavior is nvrthlss ncoraging. Frthr rogrss on th toic cold b obtaind if nw stdis on th sbjct wr carrid ot sing diffrnt aroachs. In addition, xrimntal data wold b vry hlfl to validat th comrssibility corrction modls. In th ftr, th athors lan to invstigat ossibl links btwn th two corrctions rsntd in this ar. In addition, nw nmrical xrincs will b carrid ot in th cas comrssing flows and in siml two-dimnsional cass. 7 Ackwnoldgmnts h athors wold lik to thank Profssor Marizio Pandolfi for th stimlating discssions abot th rslts that wr rsntd in this ar. Rfrncs L. aolitano, Drivation of th Basic Eqations for on- Eqilibrim Bondary Layrs, AGARD-VKI Lctr Sris on Hyrsonic Bondary Layrs, Fbrary L. Sdov, Mécaniq ds Milix Contins. om I. (Éditions MIR, Mosco, L. aolitano and. Blotsrkovski, Comtational Gasdynamics (Sringr Vrlag, Win, S. Wiśniwski, B. Staniszwski and R. Szymanik, hrmodynamics of onilibrim Procsss (Ridl Pblishing Comany, Dordrcht, Holland, 976. G. Grasimov and E. Kolsnichnko, ransort Phnomna in a onilibrim Dissociating Gas, Mkhanika Zhidkosti i Gaza (Izvstiya Akadmii ak SSSR, 9 66 (983. English translation. 6. Makashv, h Effct of Gas Motion on th Raction Kintics of Vibrationally Excitd Molcls, Mkhanika Zhidkosti i Gaza (Izvstiya Akadmii ak SSSR 6, (984. English translation. 7 D. Giordano, h Inflnc of Mdim Comrssibility on Chmical-Raction Rats - Part I: hortical Considrations, in 36th AIAA hrmohysics Confrnc, Jn 23, rlando FL, (Amrican Institt of Aronatics and Astronatics. AIAA R. Prabh and W. Erickson, A Raid Mthod for th Comtation of Eqilibrim Chmical Comosition of Air to K, (ational Aronatics and Sac Administration. ASA P- 2792, B. McBrid, S. Himl, J. Ehlrs and S. Gordon, hrmodynamic Prortis to 6 K for 2 Sbstancs Involving th First 8 Elmnts, (ational Aronatics and Sac Administration. ASA SP-3, 963. C. Park, Assssmnt of wo-mratr Kintic Modl for Ionizing Air, Jn 987. AIAA A. Viviani, Coling btwn Flow Vlocity and on- Eqilibrim Intrnal Procsss in Hyrsonic Flow Arothrmochmistry, in Procdings of th IUAM Symosim on Arothrmochmistry of Saccraft and Associatd Hyrsonic Flows, 4 Stmbr 992, Marsill, (Intrnational Union of hortical and Alid Mchanics, A. Viviani and C. Golia, A on-eqilibrim hrmochmical Modl for Hyrsonic Air Flow, in 3th hrmohysics Confrnc, 4 Jn 2, Anahim CA, (Amrican Institt of Aronatics and Astronatics. AIAA M. Pandolfi, A Contribtion to th mrical Prdiction of Unstady Flows, AIAA Jornal, 22(, 62 6 (983. Amrican Institt of Aronatics and Astronatics

7 4 E. F. oro, Rimann Solvrs and mrical Mthods for Flid Dynamics (Sringr-Vrlag, 997. A. Hartn, B. Engqist, S. shr and S. R. Chakravarthy, Uniformly High rdr Accrat Essntially on-scillatory Schms, III, Jornal of Comtational Physics, 7, (987. y, y Ο [K] st cas.. =. m st cas... =. m 6 y st cas.. y y y =. m 4 y omga omga [K] omga ma omga corr.# [K] (corr. # y (corr. # a mratr and vlocity distribtions 8 α Ο [K], α st cas... =. m st cas.. =. m 6 α st cas... α α o α = 2. bar =. m 4 α omga omga [K] omga ma omga corr.# [K] (corr. # α (corr. # b Atomic oxygn non-ilibrim and virtal ilibrim concntrations comosition Jn 23 8 y, [K] y Ο st cas... =. m st cas #2 =. m y y y y st cas.. =. m omga omga [K] # omga ma (corr. # [K] omga corr.# XV c Atomic oxygn rodction rats 6 Amrican Institt of Aronatics and Astronatics Fig. 2.C. #.. with and withot corrction #.

8 y, y Ν [K] st cas #.. st cas #.. L o rf =. m = 2. bar =. m st cas #.. =. m [K] y y [K] omga y omga y omga ma omga corr.# y, Ν [K] y st cas #.. =. m a ast cas #.. =. stm cas #.. o = 2. [K] bar =. [m/s] m omga x = [K] Pa omga omga m.a. y omga corr.# y y y a mratr and vlocity distribtions mratr and vlocity 6 Jn 23 a mratr and vlocity distribtions α Ν [K], α st cas #.. =. m st cas #2 α =. m st cas #.. =. m α α α omga α omga α omga ma omga corr.# [K] [m/s] [K] Ν α, [K] α st cas #..2 =. m α αst cas #.. =. stm cas #.. o = 2. [K] bar =. [m/s] m omga x = [K] Pa omga omga m.a. α omga corr.# α α α XV b Atomic nitrogn non-ilibrim and virtal ilibrim concntrations mratr comosition and 6 Jn vlocity 23 6 Jn 23 b Atomic nitrogn non-ilibrim and virtal ilibrim concntrations y, [K] y Ν st cas #2 =. m st cas #2 =. m st cas #.. =. m y y omga y omga y omga ma [K] omga corr.# [m/s] [K] y, Ν [K] y st cas #.. =. m a ast cas #.. =. stm cas #.. o = 2. [K] bar =. [m/s] m omga x = [K] Pa omga omga m.a. y omga corr.# y y y XV c Atomic nitrogn rodction rats (zoom c Atomic nitrogn rodction rats (zoom Fig. 3.C. #.. with and withot corrction #. Fig. 4.C. #.. with and withot corrction #. 7 Amrican Institt of Aronatics and Astronatics

9 comosition sorc trms 6 Jn 23 MEGA y, y, omga [K] ma, Comrcorr st cas #2.2 st cas #..2 =. m =. m st cas #2.2 =. m y y y y [K] MEGA [K] omga ma Comrcorr MEGA omga ma Comrcorr y, Ν [K] y st cas #.. =. m st cas #.. =. m a ast cas #.. =. m [K] y y omga [K] y omga y [K] y (corr.#2 omga m.a. (corr.#2 y (corr.#2 omga corr.# (corr.#2 omga XV a mratr and vlocity distribtions sorc trms 6 Jn 23 a mratr and vlocity distribtions α Ν [K], α α α st cas #..2 stl cas rf =. #..2m h o x = 7. MJ/kg o = st 2. cas bar #..2 =. m h o x = = Pa 7. MJ/kg =. m [K] α [K] α omga α omga α omga m. a. (corr.# omga corr Ν α, [K] α st cas #.. =. m st cas #.. =. m a ast cas #.. =. m [K] α α omga [K] α omga α [K] omga m.a. (corr.#2 α (corr.#2 omga corr.# α(corr.#2 omga b Atomic nitrogn non-ilibrim and virtal ilibrim concntrations b Atomic nitrogn non-ilibrim and virtal ilibrim concntrations y, [K] y Ν a a st cas #..2 stl cas rf =. #..2m h o x = 7. MJ/kg o = st 2. cas bar #..2 =. m h= o = Pa 7. MJ/kg =. m [K] y [K] y omga y omga y omga m. a. (corr.# omga corr y, Ν [K] y st cas #.. =. m ω Ν (corr.#2 st cas #.. =. m a ast cas #.. =. m [K] y y omga [K] y omga y [K] (corr.#2 y (corr.#2 omga m.a. y(corr.#2 omga corr.# omga (corr.# c Atomic nitrogn rodction rats (zoom c Atomic nitrogn rodction rats (zoom Fig..C. #..2 with and withot corrction #. Fig. 6.C. #... Effct of corrctions # and #2. 8 Amrican Institt of Aronatics and Astronatics