ANALYTICAL PARAMETRIC CYCLE ANALYSIS OF CONTINUOUS ROTATING DETONATION EJECTOR-AUGMENTED ROCKET ENGINE HUAN V. CAO

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1 ANALYTICAL ARAMETRIC CYCLE ANALYSIS F CNTINUUS RTATING DETNATIN EJECTR-AUGMENTED RCKET ENGINE by HUAN V. CA rsntd to th Faculty of th Graduat School of Th Univrsity of Txas at Arlington in artial Fulfillmnt of th Rquirmnts for th Dgr of MASTER F SCIENCE IN AERSACE ENGINEERING THE UNIVERSITY F TEXAS AT ARLINGTN May 011

2 ACKNWLEDGEMENTS I would lik to thank Dr. Donald Wilson for showing m how intrsting th topic of continuous dtonation is for th application of high spd propulsion. Th mannr in which h taught graduat propulsion courss, spcially hyprsonic propulsion, inspird m to furthr njoy rsarching into this topic as a promising concpt for propulsion. H was fairly hlpful whnvr I hav qustions popping out of my had and would mak himslf availabl in prson for thorough discussions on crtain aspcts of this thsis projct. I would also lik to thank Eric Braun for providing som insights and advic on th physics of continuous dtonation. His ddication and nthusiasm for his rlatd fild of study brings out a sns of admiration from m for this kind of work. Finally, I would lik to ow my dbt of apprciation to my brothr and my parnts for inspiring m to kp up th dtrmination on pursuing highr ducation in my fild of intrst no mattr how rough it may b. April 18, 011 ii

3 ABSTRACT ANALYTICAL ARAMETRIC CYCLE ANALYSIS F CNTINUUS RTATING DETNATIN EJECTR-AUGMENTED RCKET ENGINE Huan Cao, M.S. Th Univrsity of Txas at Arlington, 011 Suprvising rofssor: Dr. Donald Wilson An analytical paramtric cycl analysis modl for th continuous rotating dtonation wav jctor-augmntd rockt was dvlopd to stimat and valuat th maximum potntial prformanc that th continuous rotating dtonation wav rockt (CRDWR) itslf can provid in an jctor ramjt as wll as th two-stag rockt for low arth orbit (LE). This was don by intgrating Bykovskii s modl for CRDWR with Hisr & ratt s modifid jctor ramjt and thir transatmosphric prformanc modl. Th prformanc rsults of this uniqu ngin in comparison to a rgular rockt countrpart wr valuatd primarily in trms of spcific thrust and spcific impuls with rspct to flight Mach numbr in a constant dynamic prssur trajctory as wll as in trms of initial payload mass ratio for a transatmosphric trajctory to LE. iii

4 TABLE F CNTENTS ACKNWLEDGEMENTS... ii ABSTRACT... iii LIST F ILLUSTRATINS... v LIST F TABLES...vi Chaptr ag AENDIX 1. INTRDUCTIN Bykovskii s Continuous Dtonation Modl rojct bjctiv ANALYTICAL METHDLGY FR ERFRMANCE ANALYSIS aramtric Cycl Analysis of Bykovskii s CRDWR aramtric Cycl Analysis of Ejctor CRDWR Transatmosphric rformanc Analysis for LE RESULTS Spcific Thrust and Spcific Impuls rformanc Rsults Two-Stag Transatmosphric rformanc Rsults CRDWR rformanc at Highr Chambr rssur CNCLUSINS & RECMMENDATINS A. MATLAB RGRAMS USED FR TRANSATMSHERIC ERFRMANCE ANALYSIS REFERENCES BIGRAHICAL INFRMATIN iv

5 LIST F ILLUSTRATINS Figur ag 1.1 Diagram of TDW in annular cylindrical chambr....1 Schmatic diagram of combustion annular chambr Schmatic diagram of jctor ramjt lot of various launch trajctoris in trms of altitud vs. vlocity Spcific thrust comparison btwn CRDWR and its rgular H - rockt countrpart Spcific impuls comparison btwn CRDWR and its rgular H - rockt countrpart Spcific thrust comparison btwn CRDWR jctor ramjt and th rgular H - rockt jctor ramjt Spcific impuls comparison btwn CRDWR jctor ramjt and th rgular H - rockt jctor ramjt Spcific thrust comparison btwn all four ngins in transatmosphric launch trajctory Spcific impuls comparison btwn all four ngins in transatmosphric launch trajctory Comparison of initial payload mass ratio btwn all four launch vhicls in transatmosphric trajctory to LE v

6 LIST F TABLES Tabl ag 3.1 aramtric cycl analysis spradsht for Bykovskii s CRDWR modl Spradsht to gt input paramtrs of th jctor ramjt s primary flow Spradsht for input paramtrs of th scondary flow Spradsht for othr input paramtrs aramtric cycl analysis spradsht for th jctor ramjt aramtric cycl analysis spradsht for th CRDWR with a nozzl Rcordd prformanc data of CRDWR with nozzl for constant q trajctory Rcordd prformanc data of CRDWR jctor ramjt for constant q trajctory Input spradsht for paramtric cycl analysis of rgular H - rockt jctor ramjt Input spradsht for paramtric cycl analysis of rgular H - rockt with a nozzl Rcordd prformanc data of CRDWR with nozzl for transatmosphric launch trajctory Rcordd prformanc data of CRDWR jctor ramjt for transatmosphric launch trajctory rformanc analysis spradsht for a -stag CRDWR launch vhicl Rcordd prformanc data of a -stag CRDWR launch vhicl rformanc analysis spradsht for CRDWR-ER 1 st stag, CRDWR nd stag launch vhicl Rcordd prformanc data of CRDWR-ER 1 st stag, CRDWR nd stag launch vhicl vi

7 3.17 aramtric cycl analysis spradsht for Bykovskii s CRDWR modl (1 atm - input)... 4 vii

8 CHATER 1 INTRDUCTIN For yars, various attmpts wr mad to improv th prformanc of air-brathing and rockt ngins. n improvmnt was undrtakn by rvolutionizing th way fuls ar burnd, which is by using dtonation wavs. For dcads, traditional burning of ful lik dflagration has bn usd in ngins in which th flam front travls at low subsonic spd through th fuloxidizr mixtur. But dtonation itslf can combust th mixtur mor fficintly than dflagration and in many cass burn th mixtur mor intnsly. Whil dflagration producs mostly hat, dtonation in combustion producs both hat and high prssur almost simultanously, both of which is ndd to produc high spcific thrust in an ngin. It is known that prssur in th rockt combustion chambr nds to b vry high to produc high thrust with high spcific impuls. With dtonation, th stablishd prssur in th chambr can b lowr sinc th dtonation wav will incras th prssur significantly as it passs through th ful-oxidizr mixtur. Undr idntical initial conditions in th chambr, combustion through dtonation producs a lowr ntropy ris as compard to dflagration [1]. 1.1 Bykovskii s Continuous Dtonation Modl Thr ar various typs of dtonation wav ngin concpts. n of thm is continuous dtonation wav ngin whr th dtonation procss continus as long as th propllant ractants ar fd into th chambr whil th combustion products ar rmovd from th chambr [1]. A spcific concpt of this catgory is th continuous rotating (spin) dtonation wav in an annular cylindrical chambr proposd by Voitskhovskii whr combustion of th mixtur is achivd by a transvrs dtonation wav (TDW) moving prpndicularly to th main axial dirction of th mixtur and raction products [1]. Th TDW travls in a circumfrntial 1

9 trajctory in th chambr whil th propllant mixtur is rnwd bhind ach passing TDW front as shown in Figur 1.1 blow [1]. Figur 1.1 Diagram of TDW in annular cylindrical chambr [Rf 1]. In Figur 1.1, th TDW, rprsntd by numbr 4, is travling countr-clockwis with an adjacnt trailing obliqu shock wav rprsntd by numbr 7 [1]. A frsh propllant mixtur is formd bfor ach TDW as rprsntd by numbr 3 [1]. Th TDW thn propagats through rgion 3 to combust that mixtur [1]. Compard to th pulsd dtonation ngin (DE), this dos not rquir a complt purging of th combustion products from th ntir chambr nor filling th ntir chambr with th propllant mixtur bfor ach dtonation procss. Whil th DE oprats in a frquncy of tns of hrtz, th continuous rotating dtonation wav ngin can oprat in a rang of thousands of hrtz, making th opration a mor stady-stat procss than that of a DE. To hlp gain a bttr undrstanding of this concpt, th focus of this study

10 will b aimd particularly at th rockt propulsion modl of this concpt, which was dvlopd xtnsivly by Bykovskii and othrs at th Institut of Hydrodynamics (LIH) [1]. 1. rojct bjctiv Th purpos of this projct study was to dvlop an analytical paramtric cycl analysis modl to stimat th maximum potntial prformanc that th continuous rotating dtonation wav rockt (CRDWR) can provid in an jctor ramjt as wll as a two-stag rockt for low arth orbit (LE). This was don by intgrating Bykovskii s rockt modl for continuous rotating dtonation with Hisr & ratt s jctor ramjt and thir transatmosphric prformanc modl []. Th propllants usd for such cass wr diatomic hydrogn and oxygn for ful and oxidizr sinc thy ar commonly usd in today s rockts. For a basic lvl of prformanc comparison with a rgular H - rockt, th initial prssur and tmpratur of th chambr of th continuous rotating dtonation wav rockt wr 1 atm and 300 K bfor combustion, which ar ssntially sa-lvl atmosphric conditions. This was don to kp th focus mor on th ffct of prformanc du to th diffrnt combustion mchanisms of ths two ngins rathr than th high total prssurs in thir rockt chambrs. 3

11 CHATER ANALYTICAL METHDLGY FR ERFRMANCE ANALYSIS Th first thing to do is to dvlop a paramtric cycl analysis on Bykovskii s CRDWR modl basd on his thortical quations [1]. Th xhaust xit paramtrs from that analysis is thn fd as inputs into Hisr & ratt s jctor ramjt modl to calculat th ngin prformanc for a constant dynamic prssur, q, trajctory from takoff to hyprsonic spds. Thos paramtrs ar also put into Hisr & ratt s two-stag transatmosphric modl to calculat ngin prformanc in trms of initial payload mass ratio for LE..1 aramtric Cycl Analysis of Bykovskii s CRDWR Th aim is to put togthr a paramtric cycl analysis to calculat th gas proprtis from th xhaust xit of th annular chambr of th CRDWR, utilizing Bykovskii s thortical quations for his CRDWR modl. This analysis in particular dos not includ any nozzl sinc th goal is to calculat total gas proprtis spcifically, such as total tmpratur, T t, and total prssur, t, as wll as gamma,, and th gas constant, R, which ar input paramtrs for th jctor ramjt cycl analysis. Anothr paramtric cycl analysis will b built upon this analysis latr on whr a nozzl is addd to this modl for a two-stag launch vhicl. With that in mind, th first stp is to slct a ful and oxidizr and thir corrsponding prssur and tmpratur bfor combustion. In this cas, it is diatomic hydrogn and oxygn with both having a prssur of 1 atm and a tmpratur of 300 K in th annular rockt chambr. Axially, thy both hav a crtain injction vlocity, but in a transvrs dirction with rspct to th rotating dtonation wav, th initial vlocity is approximatly zro. With that information, th gas proprtis across th dtonation wav is thn calculatd using NASA s CEA cod [3]. Th 4

12 information that is xtractd from that program is th molcular wight of th combustd gas bhind th dtonation wav, dt, its tmpratur, T dt, and its spd of sound, a dt. From that, th gas constant bhind th dtonation wav is calculatd by this rlation shown blow [4]: R dt R (1) M dt whr R is th univrsal gas constant and M dt is th molcular wight of th combustd gas [4]. Th spcific hat at constant prssur of th combustd gas, c,dt this rlation (4):, is thn dtrmind by c,dt R dt dt dt 1 () With c,dt rlation:, th total nthalpy of th combustd gas bhind th wav can b calculatd by this V h c,dttdt dt (3) whr th absolut total vlocity bhind th wav, V dt, which includs th vlocity componnts in both th axial and transvrs dirction, is qual to a dt according to th Bykovskii modl [1]. Whil th absolut transvrs componnt of this vlocity is qual to th TDW s Chapman- Jougut (CJ) vlocity minus th sonic vlocity bhind th wav, th axial componnt of this vlocity is du to th nw propllant mixtur that is pushing th old combustion products forward in th annular chambr for th nxt coming dtonation wav. Now th nxt thing is to slct a valu for input paramtrs for th paramtric cycl analysis such as th outr diamtr of th annular chambr, d C, th ratio of frsh combustibl mixtur layr hight to th distanc btwn succding transvrs dtonation wavs, h 1 l, and 5

13 th numbr of transvrs dtonation wavs, n [1]. For simplicity for this analytical analysis, it shall b assumd that no fraction of combustion products will pass through anothr TDW from th prcding TDW. Thrfor, th valu of k, which is th fraction of th total mass flux from th prcding TDW that passs through th nxt TDW, is zro and th hight of th frsh mixtur layr in front of ach TDW, h 1, is qual to th hight of th TDW, h, as shown in this rlation [1]: h1 (1 k) h (4) Sinc th aim is to dtrmin th maximum potntial prformanc that th CRDWR is capabl of, th rlationship btwn th optimal lngth of th annular cylindrical chambr, L opt, and h was usd to calculat blow [1]: h 1 l for th bst continuous dtonation procss for th CRDWR as shown L 4h 0.7l (5) opt In this cas, h1 h, which would quat to h Th distanc btwn th TDW s is l thn calculatd with this rlation [1]: d l C (6) n which is thn usd to calculat h 1 dirctly with this rlation [1]: h h l 1 1 l (7) With that, th distanc btwn th annular walls,, for a chambr of possibl minimum siz is dtrmind by this rlation [1]: * 0.h (8) Th propllant flow rat across ach TDW, G 1, is thn calculatd with this rlation [1]: 6

14 G h q 1 1 (9) whr is th dnsity of th combustd gas bhind th TDW and q is th total vlocity of that gas, which is qual to th spd of sound bhind th TDW, c, according to Bykovskii [1]. Th spcific flow rat of th propllant in th axial dirction, g, is thn dtrmind by this rlation [1]: G 1 g l (10) Figur.1 Schmatic diagram of combustion annular chambr [1]. From that point, th xit paramtrs for th constant annular cylindrical chambr configuration of th CRDWR as shown in Figur.1 abov, can finally b calculatd, which ar xit vlocity, V, xit prssur,, and xit dnsity,. Thy ar dtrmind by ths rlations blow [1]: V ( dt 1) h ( 1) dt (11) 7

15 gv dt (1) g V With thos paramtrs, th spcific impuls, (13) I sp, of th CRDWR can finally b calculatd, which is th maximum valu for an annular chambr at vacuum ambint prssur with no nozzl. Th I sp is dtrmind with th assumption that th axial xit vlocity of th chambr is th sonic vlocity and is found by this rlation [1]: I sp ( dt 1) h V (14) g dt. aramtric Cycl Analysis of Ejctor CRDWR Basd on th xit paramtrs of th CRDWR, th nxt stp is to dtrmin th input paramtrs of th primary flow for th jctor ramjt whr th CRDWR acts as th primary cor ngin. Th aim is to calculat th prformanc of th jctor ramjt in a constant dynamic prssur, q, trajctory with th CRDWR utilizd. This is don by using th paramtric cycl analysis of Hisr & ratt s idal jctor ramjt modl as illustratd in Figur. blow []. Figur. Schmatic diagram of jctor ramjt []. 8

16 Th input paramtrs of th primary flow for th jctor ramjt ar th ratio of total xit prssur of th CRDWR to ambint prssur, tp, ratio of total xit tmpratur of th CRDWR to ambint tmpratur, Ttp T, p, and th gas constant of th primary flow, R p. To calculat ths paramtrs, th spd of sound at th xit, xit Mach numbr, a, th static xit tmpratur, T, and th M, from th xhaust of th CRDWR ngin in th primary flow must first b dtrmind, which ar found by ths rlations blow: a dt (15) M V a (16) T a R dt dt (17) From thr, th total tmpratur and total prssur from th primary ngin can finally b calculatd by ths rlations: T T M dt 1 t 1 (18) dt 1 t 1 M dt dt 1 (19) Thy ar thn dividd by th ambint paramtrs to gt T T tp and tp as shown blow: T T tp T T t amb (0) 9

17 tp t amb (1) Th ambint prssur and tmpratur would corrspond to flight conditions along th constant q trajctory. Th dynamic prssur itslf can b a function of flight Mach numbr and ambint prssur as shown blow []: q 1 M () Th ambint prssur,, in that abov quation would corrspond to a crtain altitud, from which th valu of ambint tmpratur can b dtrmind. Th gas constant and gamma valu (spcific hat ratio) of th primary flow ar th xit paramtrs from th CRDWR as shown in ths rlations blow: Rp R dt (3) p dt (4) Th rason bhind thos rlations is bcaus th xhaust flow of th CRDWR is th xpansion of combustion products from bhind th TDW [1]. Thrfor, th xhaust gas only consists of combustion products from bhind ach TDW. For th input paramtrs of th scondary flow of th jctor ramjt, thy ar M, ts, Tts T, s, and R s. Th flight Mach numbr, M, is st as an input control variabl for th constant q trajctory, which for this study is q qual to 47,880 N/m (1000 lbf/ft ). Th total prssur ratio, th total tmpratur ratio, s, and th gas constant of th scondary flow can b dtrmind by ths rlations blow [5]: s air (5) 10

18 ts d t (6) T T R ts s s 1 1 M R air (7) (8) whr t s 1 1 M s s 1 and [5]. In this study, th valu of d,max is d d,max r 0.96 using lvl 4 tchnology for suprsonic aircraft with th ngin in th airfram [5]. Th total prssur rcovry of th suprsonic inlt, r, is stimatd by this rlation blow for military spcification MIL-E-5008B [5]: 1 M r M 1 M M 4 M 935 (9) With th input paramtrs dtrmind, th paramtric cycl analysis of th jctor ramjt can b carrid out by first calculating th Mach numbr of th primary and scondary flow right bfor thy start mixing in th shroud, which ar dtrmind by ths rlations blow [, 6]: M pi tp p 1 i p 1 p 1 (30) M si ts s 1 i s 1 s 1 (31) 11

19 whr i is th static prssur ratio usd as an itration paramtr for th paramtric cycl analysis of th jctor ramjt []. Th ratio of ara of th primary flow bfor mixing ovr th throat ara of th primary flow, A A pi * p, th ratio of ara of th primary flow bfor mixing ovr th shroud ara of th jctor ramjt, A pi A, and th ratio of ara of th scondary flow bfor mixing ovr th shroud ara, A si A, ar thn calculatd in th following ordr [, 6]: A A 1 p 1 1 M 1 pi * p M pi p A A A * pi pi p * A Ap A pi p 1 p 1 (3) (33) A si 1 A pi A A (34) whr A A * p is th input paramtr for th cross-sctional siz of th jctor ramjt []. Th bypass ratio of th jctor ramjt,, which is th ratio of th scondary mass flow rat to th primary mass flow rat is givn by this rlation [, 6]: p 1 1 M pi tp s p ts Asi A M T T R si tp Api A M pi Tts T pr s s 1 1 M si p 1 p 1 s 1 s 1 (35) Th spcific hat at constant prssur for th primary, scondary, and fully mixd flow ar thn dtrmind by ths rlations blow [, 6]: 1

20 C C C pp ps p prp 1 p srs 1 s C pp C 1 ps (36) (37) (38) Not that th fully mixd flow is at station of th jctor ramjt, which is th rason for th subscript notation at this point of th paramtric cycl analysis. With that, th gas constant and of th fully mixd flow in th jctor ramjt can b calculatd by ths rlations blow [, 6]: R Rp R 1 s (39) C p C R p Aftrwards, th ratio of total tmpratur of th fully mixd flow ovr ambint tmpratur, (40) Tt T, and th ratio of its total prssur ovr ambint prssur, rlations blow [, 6]: T 1 C pp T t tp C ps T T 1 C T 1 C T p p ts t, can thn b dtrmind by ths (41) t tp Api R Tt T p 1 M pi A R p Ttp T p 1 1 M pi p 1 p (4) 13

21 Th ratio of static prssur of th fully mixd flow ovr ambint prssur, calculatd by this rlation [, 6]: t t, can b (43) whr t 1 1 for a fully mixd flow at sonic Mach numbr. Th total prssur and static prssur of that flow can b individually found by ths rlations blow: t t (44) (45) Th solution for th paramtric cycl analysis of th jctor ramjt convrgs whn th itratd valu of i is such that th following rlation blow basd on th consrvation of mass and momntum is satisfid [, 6]. 1 A i pi A A A si 1 1 pm pi sm si 1.0 (46) Th maximum attainabl Mach numbr for th primary flow whn xpandd to ambint prssur, M p, and th rsultant xit Mach numbr of th jctor ramjt, 10 M, can b found by ths rlations blow [, 6]: M p tp p 1 p 1 p 1 (47) 14

22 M 10 t (48) Th total tmpratur of th primary and fully mixd flow, and th static tmpratur of that mixd flow ar dtrmind by ths rlations blow: T T T tp t T T T T T T tp t tp T T T tp (49) (50) (51) whr T T C T T 1 C C T T 1 C 1 tp p ps ts pp pp tp []. According to Hisr & ratt s idal jctor ramjt modl, th total tmpratur at th xhaust xit, T t10, is approximatly th sam as th total tmpratur of th primary flow, T tp, sinc th total tmpratur of th fully mixd flow is incrasd by th ramjt s main burnr downstram of th shroud []. For such an idal modl that can b considrd a vry bold assumption. With that in mind, th static tmpratur at th xit of th jctor ramjt, T 10, and th static tmpratur of th primary flow at its maximum attainabl Mach numbr, ths rlations blow: T p, can b found by T 10 Tt M 10 (5) 15

23 T p Ttp p 1 1 M p (53) From thr, th vlocity at th xhaust xit, V 10, th maximum attainabl vlocity of th primary flow, V p, and th flight vlocity, V, can b obtaind by th following rlations blow [, 6]: V M R T (54) V M R T (55) p p p p p V M R T s s With that, th thrust augmntation ratio, p, spcific thrust, F m p (56), and spcific impuls, I sp, can finally b calculatd by th following rlations blow [, 6]: V V 10 p 1 (57) Vp Vp F m p V p p (58) I sp F m g p (59) whr g is th acclration du to gravity, which has a valu of 9.81 m/s for Earth. Thos prformanc paramtrs abov ar calculatd with th assumption of an idal nozzl whr xit prssur is prfctly xpandd to ambint prssur, which is This ntir paramtric cycl analysis of th jctor ramjt can b rpatd for various flight conditions in trms of flight Mach numbr, ambint prssur, and ambint tmpratur. 16

24 .3 Transatmosphric rformanc Analysis for LE Anothr aspct in valuating th maximum potntial prformanc of th CRDWR is by calculating th minimum initial payload mass ratio,, which is th ratio of th total initial mass of th launch vhicl ovr its payload mass. Th smallr th valu for is, th mor payload mass that could b liftd into orbit. In this study, that paramtr will b dtrmind for a twostag launch vhicl to LE in which th optimum valu for is found with a particular valu of stag-sparation Mach numbr. To do so, th first thing to dtrmin is th rfrnc trajctory through th atmosphr to LE, which would b paramtrizd by altitud with corrsponding vlocity of th launch vhicl. Altitud vs. Vlocity for Transatmosphric Trajctory y = 7.653x R² = Altitud [m] S Kosmos (Russia) 4-S Constoga (USA) 4-S Minotaur (USA) 4-S Taurus (USA) Linar (-S Kosmos (Russia)) V [m/s] Figur.3 lot of various launch trajctoris in trms of altitud vs. vlocity. Figur.3 abov provids a st of various trajctoris of diffrnt rockts, of which th twostag Kosmos rockt from Russia has th most appropriat trajctory for this study [7, 8]. Its launch trajctory is thn mathmatically quantifid with a curv-fit trnd lin in th figur, which is rprsntd by a black solid lin with a corrsponding quation of h for V altitud in mtrs. For a particular flight vlocity, an altitud is givn by that quation, from 17

25 which th ambint prssur and tmpratur can b dtrmind with th standard atmosphric tabls. Th corrsponding flight Mach numbr at that vlocity is thn calculatd by this rlation blow: M V R T air air (60) Th flight Mach numbr, ambint prssur, and ambint tmpratur can thn b fd as inputs into th paramtric cycl analysis of th jctor-augmntd CRDWR or CRDWR jctor ramjt for th first stag of th rockt. As for th CRDWR itslf, only th ambint prssur from that st of paramtrs is ndd for th paramtric cycl analysis. For ithr th first stag or scond stag of th launch vhicl, a paramtric cycl analysis for th CRDWR with an attachd spik nozzl was dvlopd. Just lik in th jctor ramjt cycl analysis, th spik nozzl is approximatd as an idal nozzl whr th xhaust xit prssur is always prfctly xpandd to ambint prssur. With th ambint prssur givn along a crtain trajctory, th first thing to calculat is th throat ara of th CRDWR, which is givn by this rlation: 1 1 A throat dc dc (61) 4 4 Basd on that, th total mass flow rat from th CRDWR is dtrmind by this rlation blow: m g A throat throat whr th spcific flow rat at th throat, (6) g throat, is th sam as g that was dtrmind prviously from Bykovskii s CRDWR modl. Th Mach numbr of th xpandd xhaust flow of th CRDWR and its corrsponding xit ara ar thn calculatd by ths rlations blow [4]: M t, throat throat throat 1 1 throat 1 (63) 18

26 A A M throat throat M throat 1 throat 1 throat 1 (64) whr and t, throat ar qual to th xit paramtrs of th CRDWR without th nozzl as throat dtrmind prviously from th paramtric cycl analysis of Bykovskii s CRDWR modl. Th xit static tmpratur and xit vlocity of th xpandd flow from th CRDWR ar thn found by ths rlations blow [4]: T 1 throat Tt 1 M (65) V M R T (66) throat throat From thr, th thrust, spcific thrust, and spcific impuls of th CRDWR basd on th flight conditions can finally b dtrmind by ths rlations blow (5): F m V A (67) F spcific _ thrust (68) m I sp F m g (69) With th prformanc of th CRDWR rpatdly calculatd by this paramtric cycl analysis along th launch trajctory, th two-stag transatmosphric prformanc can thn b carrid out by first slcting th stag-sparation Mach numbr, M 1. This corrsponds to a sparation flight vlocity and altitud of th launch vhicl by th following rlations: V M R T (70) 1 1 air air, avg h V (71) 19

27 whr T, avg is th avrag ambint tmpratur for th ntir atmosphr []. Basd on how th spcific impuls of th CRDWR or CRDWR jctor ramjt varis along th trajctory, th diffrntial chang in th mass of th launch vhicl can b dtrmind by this rlation []: V d dm m gispv gdr D D F 1 (7) whr r is th distanc of th launch vhicl from th cntr of th Earth, and 1 D D F is th installd thrust paramtr. For LE, th ratio r r, whr r is th radius of th Earth, is approximatly clos to 1 and thrfor, th trm gdr is omittd in quation (7) abov. Sinc it is convnint to hav that ntir diffrntial quation in trms of on variabl on th right-hand sid, V is thn st as variabl b with th ntir quation simplifid into this rlation blow: dm m g I sp db D D b 1 F (73) D D whr 1 has a constant avrag valu for ach rockt stag and I sp is st as a F function of variabl V or b. Equation (73) is solvd by using th classical 4 th -ordr Rung- Kutta mthod in a MATLAB program basd on ths fundamntal rlations blow [9]: ' y f x, y y x y (74) k hf x y 1 n, h k1 k hf xn, yn n (75) (76) 0

28 h k k3 hf xn, yn (77) 4 n n 3 k hf x h, y k (78) y k k k k n1 yn (79) For this study, th variabl y rprsnts th mass m whil variabl x rprsnts th vlocity paramtr b of th launch vhicl. As for th spcific impuls function in quation (73), it is ssntially th curv-fit trnd lin quation that approximats th variation of spcific impuls of th propulsion systm along th transatmosphric launch trajctory with rspct to flight vlocity. Aftr solving quation (73) along th launch trajctory, an array of valus for m with corrsponding valus for b is producd that is usd to plot th variation of mass with rspct to th vlocity of th launch vhicl for th 1 st stag. From that plot, a curv-fit trnd lin quation is cratd to approximat that variation. That quation is usd to dtrmin th final mass of stag 1 at th instant of sparation, rprsntd as m f V, whr V 1 is th flight 1, final 1 vlocity at stag sparation. Th ful mass fraction of stag 1, f 1, and th initial payload mass ratio of stag 1, 1, ar thn calculatd by ths rlations blow []: m f 1 1 m 1, final 1 i f 1 (80) (81) Aftr stag sparation, th initial mass of stag is dtrmind by this rlation: m m m (8), initial 1, final 1 i1 At th nd of stag whn th launch vhicl finally rachs LE, th flight vlocity and Mach numbr at that particular instant ar calculatd by ths rlations blow: 1

29 hle Vfinal (83) M final V final (84) R T air air, avg whr h LE is th altitud of LE. To find th final mass of th launch vhicl at stag aftr stag sparation, quation (73) must b solvd again for stag whr m m and,initial V1 b as th initial conditions. Aftr solving, it again producs an array of valus for m with corrsponding valus for b from stag sparation all th way to LE whr V V final. Thy ar thn usd to plot th variation of mass with rspct to th vlocity of th launch vhicl for th nd stag, from which a curv-fit trnd lin quation is again cratd. That quation is thn usd to dtrmin th final mass of stag whn th launch vhicl rachs LE, rprsntd as m, final f V final. From thr, th ful mass fraction of stag, f, and th initial payload mass ratio of stag,, can thn b calculatd by ths rlations blow []: m f 1 m, final, initial 1 1 f (85) (86) With th paramtrs for th 1 st and nd stag calculatd, th ovrall initial payload mass ratio,, for th ntir launch vhicl can finally b dtrmind by this rlation []: 1 (87) This ntir two-stag transatmosphric prformanc analysis is thn rpatd for diffrnt stag-sparation Mach numbrs until th minimum valu of is obtaind.

30 CHATER 3 RESULTS With th analytical mthodology dvlopd to stimat th prformanc of th CRDWR in an jctor ramjt and a two-stag launch vhicl, calculations wr carrid out using a combination of Excl spradshts and MATLAB programs basd on that mthodology. 3.1 Spcific Thrust and Spcific Impuls rformanc Rsults In Excl, th calculations for th paramtric cycl analysis of Bykovskii s CRDWR modl is don in th spradsht shown in Tabl 3.1 blow. As mntiond bfor, th pr-dtonatd prssur and tmpratur in th chambr is 1 atm and 300 K, using H - mixtur. 3

31 Initial conditions of th rockt chambr: [atm] T [K] Tabl 3.1 aramtric cycl analysis spradsht for Bykovskii s CRDWR modl. Total ntalpy of th mixtur (bhind th dtonation wav) input: output: R-univrsal [J/(kmol-K)] Molcular-wight [kg/kmol] R-dt [J/(kg-K)] gamma-dt R-dt [J/(kg-K)] Cp-dt [J/(kg-K)] Cp-dt [J/(kg-K)] T-dt [K] a-dt [m/s] h [J/kg] h [kj/kg] input control variabls: d c [in] corrsponding d c [m] no. of transvrs dt. wavs h 1 /l (optimum) Bykovskii approach in calculating xit paramtrs for multipl TDW: (rf #1) (for constant ara annular chambr - Modl A) input: output: n wavs d c [m] distanc l btwn wavs [m] h 1 /l distanc l [m] h 1 [m] h 1 [m] k h [m] h [m] [m] (rf #1 qn. 9) h 1 [m] [m] rho [kg/m 3 ] q [m/s] G 1 [kg/s] G 1 [kg/s] [m] distanc l [m] g (spcific flow rat) [kg/(m -s)] gamma-dt h [J/kg] V [m/s] g [kg/(m -s)] V [m/s] gamma-dt [N/m ] g [kg/(m -s)] V [m/s] rho- [kg/m 3 ] [N/m ] rho- [kg/m 3 ] V [m/s] g [kg/(m -s)] I sp [m/s] I sp [sc] gamma-dt h [J/kg] I sp [m/s] (using nd qn to vrify) I sp [sc]

32 Th xit paramtrs from th abov spradsht ar thn fd into anothr spradsht that calculats th input paramtrs for th primary flow of th jctor ramjt, which is shown blow in Tabl 3.. Tabl 3. Spradsht to gt input paramtrs of th jctor ramjt s primary flow. aramtrs of primary flow for th Ejctor-Augmntd Rockt modl: (th xit of th annular dtonation rockt chambr) input: output: V [m/s] gamma-dt [N/m ] rho- [kg/m 3 ] a [m/s] M gamma-dt M R-dt [J/(kg-K)] a [m/s] T [K] T t [K] gamma-dt M [N/m ] t [N/m ] Corrsponding ambint frstram conditions: [N/m ] T [K] t [N/m ] T t [K] tp / T tp /T M [N/m ] T [K] gamma-dt R-dt [J/(kg-K)] gamma-p R p [J/(kg-K)] Th spradsht for calculating th input paramtrs for th scondary flow of th jctor ramjt is shown in Tabl 3.3 blow. 5 Tabl 3.3 Spradsht for input paramtrs of th scondary flow. aramtrs of scondary flow: input: gamma-air output: gamma-s gamma-s M T ts /T t / π d,max ηr t / π d ts / R air [J/(kg-K)] R s [J/(kg-K)] 87 87

33 Th spradsht that contains othr input paramtrs for th jctor ramjt, including th paramtr for itration, is shown in Tabl 3.4 blow. Tabl 3.4 Spradsht for othr input paramtrs. Itrativ paramtr: i / : 61.7 thr input paramtr: A/A p *: 1 Th paramtrs from Tabls 3. to 3.4 ar thn fd into th paramtric cycl analysis spradsht of th jctor ramjt or jctor-augmntd rockt as shown blow in Tabl

34 7 Ejctor-Augmntd Rockt paramtric cycl analysis: input: Tabl 3.5 aramtric cycl analysis spradsht for th jctor ramjt. gamma-p tp/ i/ : Mpi gamma-s ts/ i/ : M si M i gamma-p Api/Ap* Api/Ap* A/Ap*: Api/A A pi/a: Asi/A ts/ tp/ Asi/A Api/A M si M i T tp/t T ts/t gamma-p gamma-s R p [J/(kg-K)] R s [J/(kg-K)] α (bypass ratio) gamma-p gamma-s R p [J/(kg-K)] R s [J/(kg-K)] C pp [J/(kg-K)] C ps [J/(kg-K)] α C pp [J/(kg-K)] C ps [J/(kg-K)] C p [J/(kg-K)] R p [J/(kg-K)] R s [J/(kg-K)] α R [J/(kg-K)] C p [J/(kg-K)] R [J/(kg-K)] gamma α C pp [J/(kg-K)] C p [J/(kg-K)] C ps [J/(kg-K)] T tp/t T ts/t T t/t gamma- / t gamma- gamma-p R [J/(kg-K)] R p [J/(kg-K)] M i tp/ T tp/t Api/A T t/t : α t/ / t t/ : / / [N/m ] t/ [N/m ] t [N/m ] numrator: ( / )*(1+γ ) dnominator( i/ )*[(A pi/a)*(1+γ p*m pi^)+(a si/a)*(1+γ s*m si^)] ratio: (itrativ procdur) /(gamma_+1) Cpp/Cp Cps/Cpp Tts/T T/Ttp α T/Ttp output: gamma-p tp/ gamma- t/ : M p M T t/t T tp/t T [K] T t [K] T tp [K] T /T tp T tp [K] T [K] T tp [K] T t10 [K] (incrasd by downstram burnr) T t10 [K] T tp [K] M 10 M p gamma- gamma-p T 10 [K] T p [K] M gamma-s R s [J/(kg-K)] T [K] V [m/s] M p gamma-p R p [J/(kg-K)] T p [K] M 10 gamma- R [J/(kg-K)] T 10 [K] V 10 [m/s] V p [m/s] α V 10 [m/s] V p [m/s] V [m/s] φ p (thrust augmntation ratio) φ p V p [m/s] F/m p-dot [(N*s)/kg] F/m p-dot [N/(kg/s)] g [m/s ] I sp [sc]

35 From th abov spradsht in Tabl 3.5, th spcific thrust and spcific impuls of th jctor-augmntd CRDWR is finally calculatd. As for th CRDWR with an attachd spik nozzl, a spradsht to calculat its spcific thrust and spcific impuls is shown in Tabl 3.6 blow. input control variabl: amb or [N/m ] Tabl 3.6 aramtric cycl analysis spradsht for th CRDWR with a nozzl. Sizing of Cylindrical Annular Chambr: input: output: d c [m] [m] A throat [m ] (at th xhaust nd of chambr) Continuous Dtonation Annular Rockt Chambr (Modl A with nozzl xpansion) paramtric cycl analysis: input: output: [N/m ] t,throat [N/m ] gamma-throat A throat [m ] M (whr = ) Corrsponding A [m ] g throat [kg/(m -s)] A throat [m ] m _dot [kg/s] T t [K] M gamma-throat R throat [J/(kg-K)] T [K] V [m/s] m _dot [kg/s] V [m/s] A [m ] [N/m ] [N/m ] F [N] F [kn] F [N] m _dot [kg/s] g [m/s ] I sp [sc] F/m -dot [N/(kg/s)] With thos spradshts abov, th prformanc calculations wr carrid out and tabulatd for th constant CRDWR and th CRDWR jctor ramjt (jctor-augmntd CRDWR) as shown in Tabls 3.7 and 3.8 blow. q trajctory for both th

36 Tabl 3.7 Rcordd prformanc data of CRDWR with nozzl for constant q trajctory. Constant q trajctory of 47,880 N/m (1000 lbf/ft ): (for const annular chambr w Expansion Nozzl, idal cas whr = amb ) M (no ffct on rockt) altitud [m] gamma amb. prssur [N/m ] amb. Tmpratur [K] q [N/m ] F/m -dot [N/(kg/s)] Isp [sc]

37 Constant q trajctory of 47,880 N/m (1000 lbf/ft ): Tabl 3.8 Rcordd prformanc data of CRDWR jctor ramjt for constant q trajctory. M altitud [m] gamma amb. prssur [N/m ] amb. Tmpratur [K] q [N/m ] thrust augmntation ratio F/m p -dot [N/(kg/s)] Isp [sc] α (bypass ratio) no solution no solution no solution no solution 30 For th rgular H - rockt countrpart in th jctor ramjt and its pur rockt mod, th following input spradshts wr usd as shown blow in Tabls 3.9 and 3.10, using th sam total prssur of th chambr lik in th CRDWR.

38 ambint frstram conditions: Tabl 3.9 Input spradsht for paramtric cycl analysis of rgular H - rockt jctor ramjt. From CEA Rockt cod: M [N/m ] T [K] T tp [K] tp [N/m ] aramtrs of scondary flow: aramtrs of primary flow: input: output: tp / T tp /T gamma-p R p [J/(kg-K)] gamma-air gamma-s gamma-s M T ts /T t / Itrativ paramtr: i/ : 51 π d,max ηr t / π d ts / thr input paramtr: R air [J/(kg-K)] R s [J/(kg-K)] A/A p *: Tabl 3.10 Input spradsht for paramtric cycl analysis of rgular H - rockt with a nozzl. input control variabl: amb or [N/m ] E Sizing of Exhaust Cylindrical Annular Chambr for Rockt: input: output: d c [m] [m] A throat [m ] CEA: H- 8.9-psi Rockt t [N/m ] gamma R [J/(kg-K)] throat [N/m ] T t [K] With that, th prformanc calculations wr again carrid out and rcordd for th constant q trajctory for both th rgular H - rockt and th rgular H - rockt jctor ramjt. Th prformanc comparison was thn mad btwn th CRDWR and its rgular

39 H - rockt countrpart in trms of spcific thrust and spcific impuls with rspct to flight Mach numbr for th constant q trajctory as shown blow in Figurs 3.1 to 3.4. Spcific Thrust vs. Mo (for const q trajctory) F/m_dot [N/(kg/s)] Mo CDR no nozzl CDR with nozzl Rgular H- Rockt Figur 3.1 Spcific thrust comparison btwn CRDWR and its rgular H - rockt countrpart. Spcific Impuls vs. Mo (for const q trajctory) Isp [sc] CDR no nozzl CDR with nozzl Rgular H- Rockt Mo Figur 3. Spcific impuls comparison btwn CRDWR and its rgular H - rockt countrpart. 3

40 Spcific Thrust vs. Mo (for const q trajctory) F/m_dot [N/(kg/s)] Mo CDR with nozzl CDR Ejctor Ramjt Rgular Rockt Ejctor Ramjt Figur 3.3 Spcific thrust comparison btwn CRDWR jctor ramjt and th rgular H - rockt jctor ramjt. Spcific Impuls vs. Mo (for const q trajctory) Isp [sc] Mo CDR with nozzl CDR Ejctor Ramjt Rgular Rockt Ejctor Ramjt Figur 3.4 Spcific impuls comparison btwn CRDWR jctor ramjt and th rgular H - rockt jctor ramjt. Th spcific thrust and spcific impuls prformanc calculations wr also carrid out for th transatmosphric launch or vrtical launch trajctory for both th CRDWR and th CRDWR jctor ramjt as rcordd in Tabls 3.11 and 3.1 blow. 33

41 Vrtical Launch Trajctory for CDR: Tabl 3.11 Rcordd prformanc data of CRDWR with nozzl for transatmosphric launch trajctory. V (no ffct on rockt) [m/s] altitud [m] altitud [km] gamma amb. prssur [N/m ] amb. Tmpratur [K] M (no ffct on rockt) F/m -dot [N/(kg/s)] Isp [sc] E E E E E Tabl 3.1 Rcordd prformanc data of CRDWR jctor ramjt for transatmosphric launch trajctory. Vrtical Launch Trajctory for CDR Ejctor Ramjt: V [m/s] altitud [m] altitud [km] gamma amb. prssur [N/m ] amb. Tmpratur [K] M thrust augmntation ratio F/m -dot [N/(kg/s)] Isp [sc] no solution no solution no solution no solution no solution no solution no solution no solution no solution no solution no solution no solution no solution no solution no solution E no solution no solution no solution E no solution no solution no solution E no solution no solution no solution E no solution no solution no solution E no solution no solution no solution

42 With that, th prformanc comparison was again mad btwn th CRDWR and its rgular H - rockt countrpart in trms of spcific thrust and spcific impuls for th transatmosphric launch trajctory as shown blow in Figurs 3.5 and 3.6. Spcific Thrust vs. Mo (for vrtical launch trajctory) F/m_dot [N/(kg/s)] H- CDR H- Rgular Rockt Rgular Rockt Ejctor Ramjt CDR Ejctor Ramjt Mo Figur 3.5 Spcific thrust comparison btwn all four ngins in transatmosphric launch trajctory. 35

43 Spcific Impuls vs. Mo (for vrtical launch trajctory) Isp [sc] H- CDR H- Rgular Rockt Rgular Rockt Ejctor Ramjt CDR Ejctor Ramjt Mo Figur 3.6 Spcific impuls comparison btwn all four ngins in transatmosphric launch trajctory. From obsrving Figurs 3.1 to 3.6, th CRDWR has xcdd th spcific thrust and spcific impuls prformanc of its rgular H - rockt countrpart in both th pur rockt mod and th jctor ramjt mod whil using idntical valus of total prssur in th rockt chambr. This is du to th fact that th dtonation procss in th CRDWR burns th propllant mixtur mor intnsly with highr total tmpratur from th combustion than its rgular H - rockt countrpart, thus lading to highr xhaust xit vlocity from th ngin. 3. Two-Stag Transatmosphric rformanc Rsults In Excl, calculations wr also mad for th transatmosphric prformanc of th twostag launch vhicl with th CRDWR for both th 1 st and nd stag by using th spradsht shown in Tabl 3.13 blow. 36

44 input control variabl: Tabl 3.13 rformanc analysis spradsht for a -stag CRDWR launch vhicl. π 1 π Stag CDR: Input: Stag Sparation Mach numbr, M 1 (choos a valu) corrsponding V 1 [m/s] Corrsponding h [km] (altitud) output: V 1 [m/s] (at sparation) (using curv fit qn. from MATLAB) m 1,f inal [kg] m 1,f inal [kg] m i1 [kg] π f π 1 π f 1 Г π 1 m 1,f inal [kg] m i1 [kg] m,initial [kg] final altitud [km] V f inal [m/s] M f inal V f inal [m/s] (using curv fit qn. from MATLAB) m,f inal [kg] m,f inal [kg] m,initial [kg] π f π π f Г Г 1 Г Г For diffrnt stag-sparation Mach numbrs, th prformanc calculations for this launch vhicl ar rcordd blow in Tabl 3.14.

45 Tabl 3.14 Rcordd prformanc data of a -stag CRDWR launch vhicl. sparation Mach numbr π f 1 Г 1 π f Г Г Similar calculations ar also mad for a launch vhicl whr th CRDWR jctor ramjt is usd for th 1 st stag and th pur CRDWR for th nd stag using th transatmosphric prformanc analysis spradsht shown blow in Tabl

46 Tabl 3.15 rformanc analysis spradsht for CRDWR-ER 1 st stag, CRDWR nd stag launch vhicl. CDR Ejctor Ramjt 1st-Stag, CDR nd-stag: Input: output: Stag Sparation Mach numbr, M 1 (choos a valu) corrsponding V 1 [m/s] Corrsponding h [km] (altitud) V 1 [m/s] (at sparation) (using curv fit qn. from MATLAB) m 1,f inal [kg] m 1,f inal [kg] m i1 [kg] π f π 1 π f 1 Г π 1 m 1,f inal [kg] m i1 [kg] m,initial [kg] final altitud [km] V f inal [m/s] M f inal V f inal [m/s] (using curv fit qn. from MATLAB) m,f inal [kg] m,f inal [kg] m,initial [kg] π f π π f Г Г 1 Г Г Although th abov spradsht is similar to th on in Tabl 3.13, th curv-fit trnd lin quations from MATLAB (s Appndix A for th MATLAB programs usd) that rflct th vhicl mass variations for ach stag basd on spcific impuls variations ar diffrnt for ach spradsht. From th spradsht in Tabl 3.15, th prformanc calculations for this launch vhicl ar rcordd blow in Tabl 3.16.

47 Tabl 3.16 Rcordd prformanc data of CRDWR-ER 1 st stag, CRDWR nd stag launch vhicl. sparation Mach numbr π f 1 Г 1 π f Г Г With that, similar prformanc calculations wr again carrid out and rcordd for th rgular H - rockt countrpart. Th prformanc comparison was thn mad btwn th -stag CRDWR, CRDWR-ER 1 st stag & CRDWR nd stag, -stag rgular H - rockt, and rgular H - rockt-er 1 st stag & rgular H - rockt nd stag launch vhicl in trms of ovrall initial payload mass ratio with rspct to stag-sparation Mach numbr for th transatmosphric trajctory to LE as shown blow in Figur 3.7. mi/mp vs. Stag Sparation Mach Numbr mi/mp Stag CDR CDR ER 1st Stag, CDR nd Stag Rgular Rockt ER 1st Stag, Rockt nd Stag -Stag Rgular Rockt M1 Figur 3.7 Comparison of initial payload mass ratio btwn all four launch vhicls in transatmosphric trajctory to LE. In Figur 3.7, th minimum valu of obtaind is significantly lowr for th CRDWR than for its rgular H - rockt countrpart vn though thy hav idntical valus of total prssur in th rockt chambr, which is promising. A highr spcific impuls from th CRDWR will undoubtdly lad to lowr initial payload mass ratio for th -stag launch vhicl. Basd on Figur 3.7, a launch vhicl with th CRDWR jctor ramjt for th 1 st stag and th CRDWR for th nd stag 40

48 would provid th bst solution to rach LE in which th stag sparation occurs at around Mach CRDWR rformanc at Highr Chambr rssur All ths prformanc calculations abov for th CRDWR wr don whr th prdtonatd chambr prssur is 1 atm. n wondrs how much highr lvl of prformanc would th CRDWR obtain if th pr-dtonatd chambr prssur is incras to 1 atm, for xampl, whil th pr-dtonatd chambr tmpratur is kpt th sam at 300 K? To find out, this input valu is fd into th paramtric cycl analysis spradsht of Bykovskii s CRDWR modl as shown in Tabl 3.17 blow. 41

49 Tabl 3.17 aramtric cycl analysis spradsht for Bykovskii s CRDWR modl (1 atm - input). Initial conditions of th rockt chambr: [atm] T [K] Total ntalpy of th mixtur (bhind th dtonation wav) input: output: R-univrsal [J/(kmol-K)] Molcular-wight [kg/kmol] R-dt [J/(kg-K)] gamma-dt R-dt [J/(kg-K)] Cp-dt [J/(kg-K)] Cp-dt [J/(kg-K)] T-dt [K] a-dt [m/s] h [J/kg] h [kj/kg] input control variabls: d c [in] corrsponding d c [m] no. of transvrs dt. wavs h 1 /l (optimum) Bykovskii approach in calculating xit paramtrs for multipl TDW: (rf #1) (for constant ara annular chambr - Modl A) input: output: n wavs d c [m] distanc l btwn wavs [m] h 1 /l distanc l [m] h 1 [m] h 1 [m] k h [m] h [m] [m] (rf #1 qn. 9) h 1 [m] [m] rho [kg/m 3 ] q [m/s] G 1 [kg/s] G 1 [kg/s] [m] distanc l [m] g (spcific flow rat) [kg/(m -s)] gamma-dt h [J/kg] V [m/s] g [kg/(m -s)] V [m/s] gamma-dt [N/m ] g [kg/(m -s)] V [m/s] rho- [kg/m 3 ] [N/m ] rho- [kg/m 3 ] V [m/s] g [kg/(m -s)] I sp [m/s] I sp [sc] gamma-dt h [J/kg] I sp [m/s] (using nd qn to vrify) I sp [sc]

50 From Tabl 3.17, th spcific impuls of th CRDWR turns out to b sc as compard to 96.5 sc prviously for 1 atm in pr-dtonatd chambr prssur. A slight 4.45 % incras in prformanc. With an attachd idal spik nozzl, its spcific impuls was 41.9 sc at sa lvl and 64. sc at th vacuum as compard to 70.6 sc at sa lvl and 639. sc at th vacuum prviously. This corrsponds to a 5.6 % incras in prformanc at sa lvl, but only a 0.47 % incras in prformanc at th vacuum. This mans that th CRDWR would alrady hav clos to maximum prformanc at a much smallr initial chambr prssur as compard to an alrady high chambr prssur of th rgular rockt. 43