Hybrid Rocket Fuel Regression Rate Data and Modeling

Transcription

1 42nd AIAA/ASME/SAE/ASEE Joint Proplion Conrn & Eiit 9-2 Jly 2006, Saranto, Caliornia AIAA AIAA Hyrid Rot Fl Rgrion Rat ata and Modling rg Zillia NASA A Rar Cntr, Mot Fild, Montain Viw, CA And M. Ari Karayogl anord Univrity, anord, CA Atrat Hyrid rot l avrag rgrion rat i on o t ot iportant val to aratly dtrin in t yrid rot dign pro and or rot proran prdition. Yt tr i no oprniv tory tat an d to rlialy prdit ti qantity. Additionally, rgrion rat data i diilt to ar. Mard data otn ontain a ig dgr o attr, r ro al t and i gnrally a loly ld rt y to proring t print and tror i navailal or any propllant oination. Ti papr prnt a rgrion rat odl tat a n dvlopd ad on t rlt o vral prvio tdi. T odl i applial to vaporizing l in a ylindrial grain onigration tat do not or igniiant ar or lt layr. It aont or t prn o a pr-otion ar ptra o t l grain and alo or varial ga proprti (to a liitd dgr). T odl i opard wit iting plid rgrion rat data. T rlt o t oparion ar raonal givn t lvl o appriation in t odl t additional wor i rqird or odl o ti typ will pplant rgrion rat arnt or rot dign prpo. a A = Rgrion rat oiint = roping o varial p = Spii at at ontant prr E a H v = Port diatr = Ativation nrgy = oal intantano a l = Oidizr a l = Cannl igt = Entalpy or at tranr oiint = Etiv at o gaiiation = Condtivity or R ponnt Nonlatr l M & = rain lngt or aratriti lngt = Tpratr ratio ponnt = Mollar wigt = ngt ponnt or a la grain = Ma low rat & = Oidizr a low rat n O/F Q & = Fl ponnt or φ ponnt = Oidizr to l ratio = Rat o at tranr # = Spatially-avragd qantity #ˆ = Tporally-avragd qantity Rar Sintit, NASA A Rar Cntr/M.S. 260-, Mr AIAA. Rar Aoiat, pt. o Aronati and Atronati, anord Univrity, Mr AIAA. Arian Intitt o Aronati and Atronati Ti atrial i dlard a wor o t U.S. ovrnnt and i not jt to opyrigt prottion in t Unitd at.

2 #ˆ R = Avragd patially and tporally = Univral ga ontant r = Radial oordinat ( ig. ) r& T t = Intantano loal l rgrion rat = Tpratr = Ti (rn dration), v = Aial and noral vloity, rptivly, y = Coordinat yt ( ig. ) α β δ ε η θ = Tral diivity = Blowing oiint = Bondary layr tin = Eiivity = Bondary layr oordinat = Bondary layr ont tin = Aolt vioity σ = an-boltzann ontant = Fl dnity Sript = Bl proprty l = Bondary layr i o = Prtaining to t la = Edg o t vloity layr = Final or l = Initial = Man = Rrn or witot lowing = Oidizr = Sra I. Introdtion In a yrid rot ngin, t l rgrion rat i t rat tat t l ra rd ovr t or o a rn and ti qantity a a irt ordr ipat on t onigration (i.. otion ar lngt and diatr) and tror t proran o a otor. For apl, in t pii ipl o any ydroaron l rnd wit a givn idizr ar o iilar lvl, a ig rgrion l will rlt in a otion ar dign tat i ortr and o gratr diatr in oparion wit a otor tat ploy a low rgrion rat l (or a ingl port otor). In ordr to opar propllant, iz a l grain, prdit t proran o a yrid otor, and avoid rn-trog, arat rgrion rat data i o paraont iportan. A ajority o t iting tori d to analyz yrid rot l otion ploy a l rgrion rat law tat rlat t loal intantano l rning rat to t loal intantano a l trog t l port in tr o an pirially-ad powr law. Ti a n t a in Maran drivd an prion in t irt oprniv tortial tratnt o yrid rot l otion in 963. Maran t al., 2 owd tat t loal intantano rgrion rat r& o a l la rd in an idizr tra i rlatd to t loal intantano a l y: & () r = v Wr r& i t intantano loal l rgrion rat, i t intantano loal a l, i t ditan along t port, i t vloity ratio o t ga in t ain tra to tat at t la, v t ratio o t total ntalpy dirn twn t la and l ra to t tiv at o vaporization o t l and i t vioity ain tra ga low (not tat t oiint i or Engli nit a originally drivd). For a givn propllant yt, it i typially ad tat r&, and ar varial and t raining qantiti in qation ar ontant. i 2 Arian Intitt o Aronati and Atronati

3 T igniian o Maran tory i tat it idntii any o t ator tat inln l rgrion rat and ow ty ar rlatd. Maran odld t tady tat low in t port o a yrid rot otor a a diion la witin a trlnt ondary layr tat or ovr t rgring l ra. Inldd in i analy ar t t o ondtiv and onvtiv at tranr to t l ra and t at tranr liiting loing t tat rlt ro a dparting t l ra into t ondary layr. A radiativ at tranr tr (not own in q. ) wa alo inldd in an ad o aion t ti tr i togt to all or l tat do not ontain tal. Ovr t yar, t rlt o any rgrion rat tt av provn tat t ntional or o Maran rgrion rat law i valid. Unortnatly, t law a originally drivd i not arat nog or rot dign prpo in tat it prdit an avragd a l ponnt tat i too ig and an aial dpndn tat i too grat. Ti i not rpriing a t law wa dvlopd or a yrid onigration wrin t l i a la and ot pratial yrid otor dign ploy a ylindrial l onigration. Hn, t tandard prati today i to invo t or o Maran rgrion rat law, agntd y oiint and ponnt drivd ro al tt. Hn, t patially and tporally avragd rgrion rat law i writtn a: r ˆ = a ˆ ˆ & n (2) In qation 2, t oiint a and ponnt n and ar propllant dpndnt ontant tat tat ar dtrind printally. Not tat t avraging pro a it poil to pr t avrag rgrion in tr o t idizr a l (an aily ard qantity) intad o t total a l. Ti prion a tat t rgrion rat i prr indpndnt. (Not tat t oiint a i not nitl tror ar t tan to onitnt nit. For t data prntd rin, t nit o t a l ar g/ 2 - and t rgrion rat r& i in /.) Rgrion rat data, d to dtrin t rgrion rat powr law rlationip, ar typially otaind ro -al tting o a l-idizr oination in a grond-ad aility. Spatially and tporally avragd rgrion rat, dtrind y aring t l a and port diatr ang, ar plottd againt t idizr a l and a nonlinar rgrion algorit i tn ployd to opt t rgrion rat law oiint() and ponnt. Mltipl otion tt o ingl-port grain ar otn rqird to ontrt an avrag rgrion rat rv ovr a rang o avrag idizr a l. In o odrn tting ailiti, t intantano rgrion rat i ard ing ltraoni or -ray tniq to ar t intantano rgrion rat t rding t nr o tt rqird. Uing t traditional approa, a ini o two (or tr i 0 ) tt ar rqird to ontrt a rgrion rat rv t tn or or ar diral or raonal aray. A opliation tat ari ro printally dtrind rgrion rat law ad on a ri o tt i tat otn ty ontain t o tporal and patial avraging tat an a iportant loal rgrion rat avior. Frtror, t idizr-to-l ratio i ally not ld ontant ro tt to tt n t data old O/F ratio orrtd. Rrn 3 dri t t and ital rgrion rat data rdtion tniq to otain an arat and naigo avrag rgrion rat law. Evn tog tniq it to ar rgrion rat, t tt ar otn otly and typially t data a a gratr dgr o attr tan i diral. A inor ang to propllant orlation otn nitat a nw tt ri. An rging rot to otaining l rgrion rat i trog t o optational lid dynai od opld wit odl tat odl t l pyrolyi and ial ration. ood agrnt wa otaind y Srin 4 twn t ard and optd rgrion rat o HTPB rnd wit OX ing t orially availal CF-ACE od. Evn o, rrntly, vral liitation prld t o CF in li o otion tt inlding trln odling, and t opliti o init rat ial ration and l pyrolyi. 3 Arian Intitt o Aronati and Atronati

4 In t rrnt papr, Maran rgrion rat law i rviitd and d a a tarting point in an attpt to driv a rgrion rat odl o igr idlity. rain onigration and port ntran t ar inldd in t odl. Fator tat inln t agnitd o t avrag a l ponnt n ar onidrd. Prvioly plid rgrion rat data ar tn opard wit t odiid odl. A. inionl Paratr o Hyrid Cotion II. Analyi Svral non-dinional nr ar o intrt in t odling o t diion la and ty ar arizd in Tal. Tal. inionl paratr. Nondinional Nr Prandtl Sidt wi anton inition Pyial Maning Typial Val p Pr = g S = 2 S = = Pr p 2 N = = R Pr p ont diivity tral diivity ont diivity a diivity tral diivity a diivity at tranrrd lid tral apaity Intrprtation Appriatly t ratio o t vloity ondary layr tin to t tral layr tin. Pr = S << Modiid Nlt nr Not tat in o rrn Nlt aolr Rynold N = τ t a = τ R = total at tranr ondtiv at tranr lid dynai ti al ial ration ti al inrtial or vio or >> inionl tpratr gradint at t ra. >> a >> >> ipli diion ontrolld otion B. Fl Rgrion Modl A ntiond prvioly, Maran rgrion rat law gnrally ovr prdit t l rgrion rat. An ption to ti tatnt i or low rgrion rat l in a la onigration. In ti tion, w will a o optation and printal vidn ro vral or to pport a w odiiation to Maran tory in ordr to iprov t rgrion rat prdition or ylindrial l grain onigration. T l-ra tady-tat nrgy alan ntral to t odl i own atially in igr. 4 Arian Intitt o Aronati and Atronati

5 Fig.. Conptalization o t low and nrgy alan witin yrid rot otor wit a ylindrial port. T n o t odl i tat a tin la t or witin a ondary layr on t l ra. T la t i d ro low y vaporizd l and ro aov y t port idizr low. T l i vaporizd priarily y onvtiv at tranr ro t la t to t l ra, altog or o l (.g. l ontaining tal) radiation an ontrit to t vaporization pro. T la t or at a loation wr t idizr to l itr ratio i nar t l tan toiiotri. It i ad tat t ration rat i ininitly at (i.. a>>) and tror t ial inti o t ration ar not pliitly onidrd and t ration rat i liitd y diion o idizr and l into t la t. T la-t appriation ipli tat t all ial ration ar onind to an ininitly tin t. In rality, t la t a n orvd to appriatly 0% o t ondary layr tin. Altog t l port vloity proil i ipatd y t l lowing, (i.. tranpiration ro t l ra) ang to t vloity proil ad y t prn o t la t ar ngligil 5. It i a airly a aption tat t ondary layr low in t port i trlnt ro inption a o t tranpiration o l ro t ra. To ipliy t analyi, ondary and tral layr iilarity i ad rlting in l and idizr onntration proil tat ar linarly dpndnt on t vloity proil. At t l ra, t tady-tat nrgy alan a own in igr i: Q & onvtion+ Q & radiation in= Q & ondtion ot+q & pa ang+ Q & radiation ot tat an writtn pr nit ra ara a: g T y T y αε 0 gσt = y + y 0 rg + ε σt = = & (3) Wr g i t tiv at (ntalpy) o gaiiation. For any non-talizd l, radiation at tranr an ngltd. At t l ra, t rat-o-at tranr pr nit ara onvtd ro t la t to T y = g y= 0 t ra i qal to tat ondtd, + alan an writtn a: Q &. Tror t ipliid l ra nrgy Q & = r& (4) g trining t at tranr rat Q & i t r o t rgrion rat prol. In gnral, Q & i dpndnt on any gotrial, otion and low rlatd ator. An additional opliation in a yrid rot otor i tat t vaporizd l laving t l ra tantially dra t rat o at tranrrd to t ra and tror t rgrion rat (aa t loing t ad y l lowing). 5 Arian Intitt o Aronati and Atronati

6 Sin a dirt arnt o Q & i narly ipoil, on a to rort to iilitd and analogi to tiat t at tranr rat. Maran ployd t Rynold analogy to tiat t at tranr rat or a trlnt ondary layr in t an o lowing. H tn dvlopd an prion or t rdtion in at tranr rat in a ondary layr wit lowing. W will ollow a iilar or. A anton nr an dind a = Q & wr i t ntalpy dirn twn t la t and t l ra. T nrgy qation an tn writtn in tr o t anton nr a: = r& (5) g T goal now o to ind a ipl way to dtrin t anton nr or t rating low nvironnt witin t l port. Pr. Ti appriation allowd i to oo t = ovr or opl orrlation (.g. Cilton-Colrn analogy, In Maran original odl, ad tat Rynold analogy, / 2 2 Pr 3 = / 2 tat i valid or 0.6 < Pr < 60 ). Ipliit in t Pr aption i tat t tral and vloity ondary layr ar o t a tin. ata 2,5 ow tat t la t in a yrid rot otor rid witin t ondary layr (a own atially in igr ) at a y / δ ranging ro 0. to.0 (dpnding on t ditan along t grain and t idizr oi) and tror t tral layr i tinnr tan t vloity layr and ga low wr tr i a larg tpratr dirn twn t ra and t l low rlting in proprti tat vary igniiantly. A rlation rondd y aill 6 (q. 6) tat i applial or at l lvl o 600 W/ 2 or igr or a trlnt t low trog a t and i valid ovr a rang o Prandtl nr i t ot appropriat analogy idntiid and o ti i t orrlation tat w will. ining a Nlt nr a R = Pr >. Frtror, t Rynold and Cilton-Colrn analogi ra down or N = wr, tn N = and: R Pr i t loal at tranr oiint and a Rynold nr. Pr 6 ( T / T l ) 0. 02R 0 = (6) In qation 6,, Pr and R ar t loal val (i.. at ) wit proprti valatd at t loal l ga tpratr T. T l tpratr i an nrgy-avragd lid tpratr aro a t tat will appriatd a t il tpratr T ( T + T )/ 2. In rrn 6, = 0. 2 and l = ( ) t in t rrnt ontt, and l will rain a r paratr dring t drivation. Fndantally, qation 6 rlat t onvtiv at tranr twn a ga and a ra to t lid rition or a low trog a t. Eqation 6, li all onvtiv at tranr orrlation or trlnt T = T low, i pirial. It old notd tat i, qation 6 i narly qivalnt to t Cilton-Colrn analogy or low in a t wi in trn i narly qivalnt to t Rynold analogy wn Pr =. T dnoinator o qation 6 opnat or varial ga proprti ad y larg tpratr gradint. In ot yrid rot otor o pratial intrt, t grain onigration i ylindrial wit lngt-toport diatr ratio in t rang o 5 to 50 and a l grain or-nd gotry tat rl a orward-aing 6 Arian Intitt o Aronati and Atronati

7 tp. Tror in t Rynold nr rang o 0 4 to 0 6 (ad on port diatr) typially nontrd in yrid rot otor, it i ptd tat ntran t old play a rol. T HTPB-OX ylindrial-port yrid optational rlt own in igr 2-4 illtrat ti point. T rlt wr otaind ing an qiliri-itry Navir-o od on a annl onigration wit a pr-otion ar. Clarly, t low i not lly dvlopd ntil / H = 4. 2 wi i lo to t port it. Tror, qation 6 t odiid to aont or ntran t. Fig. 2. Coptd port vloity ditrition, = 3. =0 4 / (Fro rrn 4). Fig. 3. Coptd port tpratr ditrition, T = =0 33 K (Fro rrn 4). Fig. 4. Coptd port dnity ditrition = 43. =0 3 g/ 3 (Fro rrn 4). 7 Arian Intitt o Aronati and Atronati

8 In rrn 7, print wr prord to invtigat t ipat o a paratd zon on t at tranr in a pip. T print ar o intrt a o tir iilarity to t low at t inlt o a ylindrial l grain. In t print, a rtrition wa d witin a atd pip to rat a paratd zon. T t o t rtrition on t at tranr wr ard ovr a rang o Nlt, Rynold and Prandtl nr or a w rtrition ratio. Prntd in igr 5 i t ratio o loal anton nr to t lly dvlopd anton nr vr ditan downtra o t rtrition (not tat t anton nr ratio own in igr 5 wa allatd ro Nlt nr data prntd in rrn 7). T data plottd i or a rtrition diatr ratio o d / = 2 / 3 wi i onidrd to rprntativ o t ntran t (i.. rigt trnd t agnitd i too ig at / 2 ) ptd nar t or-nd o a ylindrial l grain. A on an in igr 5, a inor paration zon tantially inra t at tranr or / l tan aot 0 ovr a rang o Rynold nr. It old notd tat t ondary layr i trlnt and lly dvlopd ptra o t rtrition, and tror t inrad at tranr i rlatd to t rtrition and not lainar or tranitional low. Fig. 5. T ntran t on anton nr (Pr=3, Nlt nr data or i rrn 7). It wa diilt to ind a ipl ntion tat wold it t data o igr 5 aro t oplt / rang. Eqation 7 wa on a it i wll avd at larg / and do a raonal jo or / gratr tan. For t oparion own in igr 5, C = t ad on data own in otr rrn and alo t at tat t l port ntran o rondd dring t rn, it i livd tat C = 2. 0 will ttr rprnt t atal gotrially-indd ntran t nontrd in a l port and o ti val will d in t rgrion rat odl. / 0.4 = + C R (7) 8 Arian Intitt o Aronati and Atronati

9 In t paragrap tat ollow, w will a o vral rlt tat wr drivd or a zro prr gradint trlnt ondary layr (.g. lat plat ondary layr) wit a tin δ and a aratriti vloity at t ondary layr dg (ript l i d to ditingi ondary layr qantiti ro to prd in tr o otr aratriti vloiti and lngt al,.g. a t low). In low trog ylindr witot a addition, t aratriti vloity typially ployd i t an vloity wi along wit R ar ad to ontant. In a yrid l grain port, t vloity inra, t dnity R tat i not ontant wit. For R, ad on port ntrlin qantiti, dra wit or t optational rlt prntd dra and t tpratr inra wit inraing rlting in intan, in igr 2-4. Maran dind a non-dinional lowing oiint ( v) β l = to qantiy t a / 2 addition to a ondary layr on a lat plat. H tn drivd a vry ndantal rlt, naly, an prion or t vloity proil o a lly-dvlopd trlnt ondary layr wit a injtion. Bondary layr iilarity i ad and t rlt i valid or any ondary layr wit a no-lowing vloity proil tat ollow a powr law o t or φ = η n l = wr << η = y /δ and or a t, n = 9 wit η = 2y / wn β = 0 Maran drivd i: ylindr i φ l φ = φ = n β + l n β + l η η 2 2 φ and n = 9 =. Uing t dinition o l l n. Typially, or a lat plat, n = 7 wit. T vloity proil wit lowing tat. T vloity ratio dird or low trog a, it an own tat.22 or a t-low vloity proil wit a addition in t rang o 5 < β < 00 wr i t t ntrlin vloity. For a t, aing t dnity do not vary igniiantly wit y, t lowing oiint an dind a t: ( v) β =, tror β β l =. 22β l / 2 φ = β 9.22η.22 + η 2 β o =. Hn, or a A ntiond prvioly, l lowing rd t at tranr to t l. Ti t an alo qantiid in tr o a anton nr ratio wr o i t anton nr witot lowing. Maran 2 invod t Rynold analogy opld wit t von Karan ont intgral qation to ow tat o ln + = β ( β ) l l 4 5 I I o + β l 5 wr θ I = = δ 0 dη or a trlnt ondary layr wit a injtion.,2,8 In proring t intgration, a linar dnity variation wa ad twn t ra and t dg o t ondary layr. H alo owd tat ti rlt an appriatd y t iplr (8) 9 Arian Intitt o Aronati and Atronati

10 prion tat β l = β l or 5 < β < 00. H tn d t Rynold analogy to ow o g. Ti i a vry l rlt a it qantii t rdtion o at tranr to t l ra ad y lowing in a vry int prion tat i only a ntion o a vloity ratio and two trodynai proprti. Ciavrini t al. 9 ddd a iilar orrlation ad on t rlt o HTPB-OX otor tt (wn ligtly dirnt lowing oiint dinition ar ronild wit a 0.77 otr). For a t, and.4β o β =. 49 prnt analyi vn tog it wa drivd aing tat g o. T Maran rlt will d in t Pr =. Sin ti tr i a ratio o anton Pr Pr o. o nr, inlion o t Prandtl nr in t drivation wold rlt in a qantity lo to on or ot itation and tror av a inial ipat on So t anton nr o i t anton nr witot lowing, in qation 5 old rplad y t prodt o i t ntran t. Coining qation 5 and 6 rlt in: T loal total a l i o.03 φ. tat old o wr i t at tranr drnt ad y lowing and r& = Pr R l T T (9) ( / ) o g g rgrion rat prion: 0.77 =. Applying t varial dinition, titting φ = and qation 7 into qation 9 lad to t ollowing loal intantano / Pr + C ( / ) φ l T T g & (0) r = T aov qation ow, aong otr ting, tat t poition o t la t rlativ to t l ra (a indiatdφ ) a a irt ordr ipat o t at tranr rat to t l and tror t l rgrion rat. Maran 2 d ondary layr intgral tod along wit pi ontinity qation to driv an prion or t vloity ratio or wi t t low qivalnt i 0 Arian Intitt o Aronati and Atronati

11 .22( O / F ) g φ = wr O / F i t idizr to l ratio at t la and K + ( O / F + K ) g idizr a ration at t dg o t ondary layr (ally nity). Ti rlation i not idiatly lpl a t loal O / F i nnown t it do ow t ator tat inln la tand-o ditan inlding idizr oi. In lai diion la tory, t la t rid at t loation wr t O / F i toiiotri. In yrid rot otor, t diion la i ird in a trlnt ondary layr wit iing and tranport aidd y t ddy vioity o t trlnt ondary layr. In addition, t la t i orvd to rid lor to t l ra (at a loation wr O / F i l tan toiiotri) in oparion to a lai diion la. In Maran original odling o PMMA rnd wit ygn, t la t wa poitiond at a l ri loal O / F ratio o.5, (i.. ligtly l tn t PMMA toiiotri O / F o.92) witot rigoro jtiiation, (ad on o ard rlt). In t rrnt odl, t la t O / F ratio will t toiiotri val. Eqation 0 an ipliid y groping varial tat ar appriatly ontant. It an n in igr 3 tat t la and ra tpratr ar ontant along t grain. Frtror it an ad tat t la t rid at t a loal la O F along t grain, tror φ i ontant. T onidration a it poil to a tat ar rogly ontant. t: Pr, T,,, T g φ and K i t ar indpndnt o and t and A = Pr l ( T / T ) g 0.23 φ 0.77 () Tror t inal or o t loal intantano rgrion rat prion in tr o t total a l or a ylindrial grain i: A 0.4 / + r& + C (2) Sin t total a l i a qantity tat i diilt to ar, it i diral to pr t rgrion rat law in tr o t idizr a l. It ta vral tp to aopli ti goal. In qation 2, in t tr C 0.4 / and or inraing, 0. Tror: 0.4 / a at t or nd o t grain = A 0.4 / + r& + C (3) Conrvation o a writtn twn t port ntran and ditat tat. Arian Intitt o Aronati and Atronati

12 π ( ) = + r& d 2 π 0 4 (4) t a tat l i addd niorly along t grain. W an tn writ t a l ratio o in tr o t ( F at t nd o t grain a: O ) = K = + (5) ( O F ) = W an titt qation 3 into qation 4 and tn qation 5 to gt: 4A / + C 0 K ( O F ) = + d (6) T intgral in qation 6 an intgratd atr ing t inoial panion + K + + ( O F ) = O / > ). T rlt i: F =l ( + ) K ( O F ) = and nglting all tr (not tat + < and + 4A ( ) ( + ) K 2 ( O F ) = C 0.4 / ( ) (7) Coparing qation 5 and 7 and nglting all tr, it an n tat: ( O F ) K 4A = (8) ( ) Eqation 7 an tittd into 3 to otain t dird rlt o t loal intantano rgrion rat in tr o t idizr a l : r& A + C 0.4 / ( + )( ) A (9) Wr A i dind y qation. Mot ard rgrion rat rlt ar tporally and patially avragd rlt. Svral approa to avraging ard data ar in oon and t tniq and t vario pitall rlatd to ow t avrag idizr a l i dind ar drid in rrn 3. A will n, it i diilt to otain an at lod-or oltion or t odld avrag rgrion rat and tror iilar tniq to tat d or ard rgrion rat ar invod. A patially and tporally avragd rgrion rat an 2 Arian Intitt o Aronati and Atronati

13 dind a r ˆ & = t t 0 0 r& d dt. It i not poil to otain an at oltion o ti intgral. T intgration an prord nrially, or or ort rn ti, it an olvd appriatly. T approa tan r i to irt pror t patial avrag aing a linar aial r& variation. A patially-avragd idizr 2 a l an writtn a = 4& π wr = ( )/ 2. T intgral i: A r C + 0 = 0 + = 0.4 / + 4A + ( + )( ) d & (20) Tror, t patially avragd rgrion rat i: r A + 2.5C + 4A + ( + )( ) & (2) r ˆ & = 2t A pa-ti avragd rgrion rat i givn y ( i ) ˆ and a pa-ti avragd port diatr an dind a = ( + ) / 2, tror t patially and tporally avragd rgrion rat i givn y: i A r ˆ ˆ ˆ ˆ + 2.5C 4A ˆ ˆ ˆ + ˆ + ( + ) & (22) ˆ = 6& π i + and A i dind y qation. ) l T Wr ( ) 2 A prvioly ntiond, t ( T tr o qation 6 wa inldd to opnat or varial proprti tat ari ro t tr tpratr gradint noral to t ra. It a n rognizd y any tat ttr ig at-l orrlation rlt wit t pliit inlion o a tpratr dpndny wn proprti ar valatd at t l tpratr. T proprty ot gratly ipatd i vioity wi i trongly dpndnt on tpratr. Pyially, wn t ra i oolr tan t rtra, t ondary layr vloity proil i llr in oparion to t iotral low a t vioity i igr tan tat allatd ing t qivalnt l tpratr. For ot l grain, / i o t ordr 20 and tror w an nglt t all dpndny on / in t original ponnt l = ( ). Frtror, qation 6 wa talid witot onidration o a addition and varial pi, ot o wi av a ajor ipat on loal proprti. Nvrtl, l = will d in t rrnt odl, and all proprti rlatd to t orrlation will valatd at t l tpratr. T ponnt in qation 0 an trad a to t R ponnt o t at tranr orrlation iplntd (i.. qation 6). Ti ponnt a n onird to -0.2 y any atd-wall t-low print. Evn o, it i a it o a trt to pt tat t orrlation will wor prtly or a l port ontaining a ial ration, a la zon and a addition ro t wall. Typial val o ond printally ro rgrion rat tt rang twn and or l witot tal additiv and or o l ontaining tal, approa 0. All l ing qal, t dirn twn = 0. 2 and = 0.5 an rlt in a ator o 0 in rgrion rat. Tror t aray o ti ponnt i ritial to t o a rgrion rat tory. ˆ 3 Arian Intitt o Aronati and Atronati

14 Svral ator appar to av an ipat on inlding l opoition, idizr oi, radiation and grain onigration (i.. la or ylindrial). Yt a t tory tand, i a ontant rgardl o t ator. In rrn 23, it wa ond tat t pyrolyi Arrni ontant E a RT, wr E a i t ativation nrgy, wa trongly orrlatd wit t idizr a l ponnt a dtrind ro nrial oltion o a ially rating lainar ondary layr. A orrlation o ti typ i not rpriing a l prodtion i rlatd to t ial dgradation o t olid polyr vn tog t rat o t otion ial ration i ontrolld y t intrplay o at tranr to t ra and diion o l and idizr into t la zon. In t rrnt odl, witot o a pyial ai at ti point, t ollowing rlationip will d (not tat t l ponnt n = + ): Ti prion wa dtrind y looing at n, propllant oination. Ea = RT (23) E a and T trnd in t ard data o dirnt Svral point an ad onrning qation 9 and 22. Mot iportantly i tat rgrion rat dra wit al. For intan, inraing t diatr y a ator o tn wil aintaining ontant and / rlt in a rgrion rat dra o 50%. A iilar t not a pronond al t wa otaind optationally in rrn. It old notd tat radiation (not inldd in t prnt odl) old ot t rgrion rat al t to o dgr. A ond point to orv aot qation 9 and 22 i tat wn qation 23 i iplntd, t idizr l ponnt i arond 0.7. Ti lvl i in lin wit tat otaind printally or vral oon propllant oination (and i l tan t laial ponnt o 0.8 tat Maran otaind). A inal point i tat ot t intantano and avrag rgrion rat law dpnd on a priori nowldg o t O / F at t la t, (or altrnatly, φ ). Ti i t only qantity in t odl tat i not radily otainal. It i ptd tat at o point, a tory will dvlopd tat prdit t loal O / F (and tror, φ ) at t la t loation or a diion la in a ondary layr. Until ti or pirii t rlid on to piy ti qantity. It old ritratd tat t odl dvlopd witin ti papr i ot applial to l tat do not or igniiant lt or ar layr. A ar layr at li an inlating lant on t l ra tat altr t at tranr rat to t l. It ay poil to inld t t o ar via a anton nr ratio ad on ar nr iilar to t approa d to aont or t lowing t. T lt layr tat or on t ra o o ig rgrion rat l (i.. n-alan a parain) rlt in a l ntrainnt ani into t la zon tat altr t ai l ra nrgy alan o qation 3. Additional ator (aid ro radiation t) nown to ipat rgrion rat tat ar not inorporatd in t odl inld t t o l ra rogn, otion ar prr (or o l) and otion iiny. Frtror, init rat ial inti, additional apt o l pyrolyi, and la t rvatr ay iportant or o l yt. T loal l tpratr T i rqird in ordr to piy. It an n in igr 3 tat t tpratr i igly varial trogot t doain wit t tpratr along t ytry ai varying twn T at t inlt and T. T l tpratr i optd appriatly a T ( T + T )/ 2 wr T i t adiaati la tpratr optd y an qiliri itry od. An additional opliation i tat t ga i a itr o vral pi. ring t l pyrolyi pro, t polyr ra down into low ollar wigt volatil l pi nar t l ra. For HTPB rnd wit ygn, t ot prvalnt pi ond twn t ra and t la zon ar C 4 H 6, C 2 H 4, CH 4 and 4 Arian Intitt o Aronati and Atronati

15 CO. Aov t la, t ain pi ar O 2, CO 2 and H 2 O. Tror dtrining t vioity o t port ga i diilt. T approa tan r i to t appriat orla 22 (ro t Kinti Tory o a): = MT 0 2 σ 7 g (24) Wr M i t ollar wigt, and σ i t ard-pr diatr in Angtro and T i piid in K. A a rog appriation, it i ad tat on avrag, M = ( M + M = ) 2 and σ = 5. 0 Angtro (an avrag val). Fortnatly, t vioity in A o t rgrion rat prion i raid to appriatly t powr o 0.2, o a ator o two ang in T rlt in only an 8% dirn in avragd rgrion rat allation, = /. T i tat ond at 2. For T tiv at o gaiiation at aint tpratr and prr, T o and prodt at T and P : g = g i t dirn twn t ntalpy o t olid in t initial tat T T0 p P o and t ntalpy o t volatil trally dopod dt + + d + v (25) Wr i t ntalpy o ion, d i t ntalpy aoiatd wit l tral dgradation and v i t ntalpy o vaporization o t dopod prodt. T tiv at o gaiiation i g i a qantity tat an ard in a ontant at l gaiiation dvi or a laing aloritr. T arnt old don ndr ondition tat at t l ra rgrion rat. In t odl, t qantity i t ntalpy ang o t ga twn t la and t l ra. Ti qantity i dpndnt on t loal O F in addition to tpratr. Ti qantity an allatd ing: p ( T T ) ( T T ) = = (26) o p o C. Fl Rgrion Rat ata T avrag rgrion rat or vario oonly d yrid rot l rnd wit ygn i own in Figr 7 and data rlatd to t rning tt tat t rgrion rat rv wr drivd ro ar own in Tal 2. T rv own wr otaind y applying a nonlinar rgrion to ard data ro any or in t opn litratr ovr t idizr a l rang nopaing t tt. Only data ro ylindrial otion ar onigration ar own and no additional orrtion yond to o t original or av n applid. 5 Arian Intitt o Aronati and Atronati

16 Fig. 7. Mard avrag l rgrion rat wit ygn. No. Fl a n No. o Tt Tal 2. ata ro rgrion rat tt. Rlt ary o t avrag rgrion rat wit ygn or vario l Car Avrag ata Prr O/F Rdtion Rang Ratio Tniq (MPa) Rang Oidizr Ma Fl Rang (g/ 2 -) Parain, SPA A HTPB, (Tiool) HTPB+9.7%A OA HTPB OA HTPB+20%AT PMMA HPE A PE Wa, Mar A PE Wa, Polylo A HTPB OA HTPB+3% nano OA Al 2 Parain, FR OA % nano Al 3 Parain, FR OA Parain, FR ?.3-.8 A Rgrion rat qation: n & wit 0 r = a o = For wit o wit nit o g/ 2 -, prod an avrag rgrion rat in /. A: iatr Avragd, FA: Fl Avragd, AA: Ara Avrag, OA: Otr avraging tniq applid R. 6 Arian Intitt o Aronati and Atronati

17 A opl o dilair ar in ordr onrning t data prntd in Fig. 7. Many ator inlding al, O/F ratio, otion ar onigration, idizr injtor dign, l opoition (i.. tra additiv) and proing, ignition and trt trination tranint, data rdtion and printal tniq gratly ipat t aray o rportd rgrion rat data. Ti i on raon or t diparity twn rportd rgrion rat data or ingly iilar propllant oination. An attpt wa ad to oo data ro rlial or, nvrtl, t data old d or dign prpo only and t atal rgrion rat o a l and idizr oination or any givn appliation old indpndntly vriid. Fl opoition and proing an gratly at t rgrion rat. Ti i partilarly tr or polyr-ad l a HTPB. Typial HTPB l ar long-ain ydroaron tat rlt ro t iing o a rin wit a ardnr and otn a platiizr, anti-idant, diprant and an opaiir. T atal rip and l proing tniq d vari widly and ti inoration i ally loly ld y opani a trad rt.. Coparion o Modl wit Mard ata T pa-ti avrag l rgrion rat prditd y t odl dvlopd in t prding tion i own in igr 8 or tr propllant oination. T odl wa rid ing l grain dinion qivalnt to tat o t tt artil. T Prandtl nr in t odl a n ad to qal to on nrtainty in t la O F ratio and port ga vioity a it diilt to dtrin a val o gratr aray. ata pplid to t odl i litd in Tal 3 wit t data or rrnd wr applial. Svral o t paratr d in t odl wr allatd ing t tro-itry od CEA tat wa dvlopd at t NASA lnn Rar Cntr. It wa ond tat t odl i vry nitiv to piiation o t l ra tpratr T. Sall adjtnt in ti tpratr an aily rlt in a rgrion rat ang o a ator o two. T ra tpratr a an ipat on t l tpratr tat i d in proprty dtrination and alo on t ponnt, o it i ndrtandal wy ti qantity i iportant. Sin T i an inpt to t odl and rlial arnt o T ar diilt to otain, (ty t prord ndr qivalnt at tranr rat in a pially dignd tt rig) o lirty wa tan in t T val oi to prod a avoral rgrion rat at. T lop o t odld rlt in igr 8 do not at t ard data vry wll. Sin t lop i dirtly rlatd to t idizr a l ponnt n, t poor agrnt i an indiation tat qation 23 old iprovd. Finally, t port ga vioity a a igniiant ipat on t odld rlt and ti val a n dtrind ing a vry ipl rlationip (i.. qation 24) tat i proaly not a arat a dird. 7 Arian Intitt o Aronati and Atronati

18 Fig. 8. Modld avrag l rgrion rat wit ygn. Tal 3. ata d to dtrin t odld l rgrion rat (wit ygn). PMMA HPE HTPB Avrag Forla (C 5 H 8 O 2 ) n (C 2 H 4 ) n (C H O ) n MW o rpat nit, (g/ol) , Hat o Foration (al/ol) O/F oi O/F opt (at optial Ip va ) T ( at O/F, K) T (K) * M ( at O/F, g/ol) p (ga at ra, J/g K ) p (ga at la, J/g K ) (q. 26, J/g) g (J/g) (olid, g/ 3 ) Arian Intitt o Aronati and Atronati

19 (ga, q. 24, g/ ) at T φ (allatd ing dinition) E * a (J/ol) ^ () ^ ^ Val dtrind ro grain onigration o tt otor d in oparion Callatd ing CEA tro-itry od, NASA lnn. * ro ard data ndr iilar avrag rgrion rat ondition III. Conlion Ti papr prnt a rgrion rat odl tat a n dvlopd ad on t rlt o vral prvio tdi. T odl i applial to vaporizing l in a ylindrial grain onigration tat do not or igniiant ar or lt layr. It aont or t prn o a pr-otion ar ptra o t l grain and alo varial ga proprti (to a liitd dgr). T odl i opard wit iting plid rgrion rat data and t oparion i raonal givn t lvl o appriation in t odl. T odld idizr a l ponnt i too ig in oparion to tat otaind y rv it o ard data t it i lor to ard val tan t ponnt prditd y Maran laial rgrion rat tory. T odl i vry nitiv to vral paratr inlding t avrag l ra tpratr and port ga aolt vioity. T odld rgrion rat an o y a ator o two or or dpnding on t aray o t data d in t odl. A ajor ort oing o t odl i t nity o piying t idizr-to-l ratio a priori (piid a toiiotri vn tog t atal val ay l or o propllant oination). Nvrtl, t odl rv a a good tarting point in aing t ator tat inln yrid l rgrion rat. Anowldgnt Ti papr ontain data and analyi originatd ndr NASA ooprativ agrnt NA3-265 wit t NASA lnn Rar Cntr and agrnt NCC2-72 and NCC2-300 wit t NASA A Rar Cntr. Rrn Maran,. and ilrt, M., Trlnt Bondary ayr Cotion in t Hyrid Rot, Nint Sypoi (Intrnational) on Cotion, Aadi Pr, 963, pp Maran. A., C. E. Wooldridg and R. J. Mzzy, Fndantal o Hyrid Bondary ayr Cotion, AIAA Htrogno Cotion Conrn, Prprint No , Pal Ba, Fl. -3, Karayogl, A., Cantwll, B.J. and Zillia,., vlopnt o Salal Spa-Ti Avragd Rgrion Rat Eprion or Hyrid Rot, AIAA/ASME/SAE/ASEE 4 t Joint Proplion Conrn, Ton AZ. AIAA , Srin, N and og, Y.A., Navir-o Invtigation on Rating Flow Fild o HTPB/O2 Hyrid Motor and Rgrion Rat Evalation AIAA , Hntvill, Al, Arian Intitt o Aronati and Atronati

20 5 Wooldridg, C. E. and Mzzy, R.J. Marnt in a Trlnt Bondary ayr wit Poro Wall Injtion and Cotion. 0t Sypoi (Intrnational) on Cotion, Aadi Pr, 965, pp art, B. Hat Tranr, 2 nd Ed., Mraw Hill, Nw Yor, Krall, K.M. and Sparrow, E.M. Trlnt Hat Tranr in t Sparatd, Rattad, and Rdvlopnt Rgion o a Cirlar T, J. o Hat Tranr, Vol 88, No. 2, 966. p Maran,.A., Cotion in t Trlnt Bondary ayr on a Vaporizing Sra, 0t Sypoi (Intrnational) on Cotion, Aadi Pr, 965, pp Ciavrini, M. J., Srin, N., Jonon,.,, Y. C., Ko, K. K., and Ria,. A., "Rgrion Rat Bavior o Hyrid Rot Solid Fl," Jornal o Proplion and Powr, Vol. 6, No., Janary-Frary 2000, pp Vnatwaran, S. and Mrl, C.., Siz Sal-p in Hyrid Rot Motor AIAA , Aropa Sin Mting and Eiit, 34t, Rno, NV, Jan. 5-8, 996. Karayogl, M.A., Altan,. and Bradr,., Tranint Cotion in Hyrid Rot AIAA , 3 t AIAA Joint Proplion Conrn, San igo, CA yon, R.E. and Jann, M.., Polyr Flaaility, OT/FAA/AR=05/4 May, Wild, J.P. d, T Hat o aiiation o PE and PMMA, Morand M-593, lt Univrity o Tnology, lt, T Ntrland, Rainovit, B., Rgrion Rat and Kinti o Polyr gradation, Tnt Sypoi (Intrnational) on Cotion, Aadi Pr, 965, pp Karayogl, A., Zillia,. Cantwll, B., Zilwa S. and Pal Catlli, Sal-Up Tt o Hig Rgrion Rat Parain-Bad Hyrid Rot Fl, Jornal o Proplion and Powr, Vol 20., No. 6, Nov-, ton,.p. and Bilarz, O., Rot Proplion Elnt, 7 t d. Wily and Son, Nw Yor, org, P. Krinan, S., Vary, P., Ravindran, M. and Raaandran,. Fl Rgrion Rat in Hydryl-Trinatd-Polytadin/ao-Oygn Hyrid Rot Motor, Jornal o Proplion and Powr, Vol. 7, No., 200, pp Hdon, M. K., Wrigt, A. M., ini, C., Wynn, P., and Roo, S., anidini Azo Ttrazolat (AT) a a Hig Proran Hyrid Rot Fl Additiv, J. Pyrotni, No. 9, 2004, pp rinr, B., Fdri, R, Rlt o aal Hyrid Rot Motor Invtigation, AIAA/ASME/SAE/ASEE 28 t Joint Proplion Conrn & Eiit, Navill, TN, AIAA, Papr , Karayogl, M, Cantwll, B. and vn, J. Evalation o Hoologo Sri o Noral-Alan a Hyrid Rot Fl, AIAA/ASME/SAE/ASEE 4 t Joint Proplion Conrn & Eiit, Ton, AZ, AIAA, Papr , Evan, B. Favorito, A. and Ko, K. dy o Solid Fl Brning-Rat Enannt Bavior in an X-ray Tranlnt Hyrid Rot Motor AIAA/ASME/SAE/ASEE 4 t Joint Proplion Conrn & Eiit, Ton, AZ, AIAA, Papr , Arian Intitt o Aronati and Atronati

21 22 Rid, R.C., Pranitz, J.M. and Srwood, T.K. T Proprti o a and iqid, 3 rd d., Mraw-Hill, Nw Yor, NY Krir, H., and Krznr, H., An Analyi o t Cially Rating Bondary ayr ring Hyrid Cotion, AIAA 72-44, AIAA Joint Proplion Conrn, Nw Orlan, A, Arian Intitt o Aronati and Atronati