16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 1: Introduction. - Depending on gas acceleration mechanism/force on vehicle mechanism.

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1 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur : Introduton Typs of Rokts (Engns) - Dpndng on gas alraton mhansm/for on vhl mhansm. Thrmal Gas pushs drtly on walls by P (prssur) fors Nozzl alrats gas by P fors (most larg rokts, hm, nular, som ltr ) Eltrostat Ions alratd by E fld (a) Eltrostat for (push) on ltrods (Ion ngns) (b) For (push) on magnt ols through gas j (Hall thrustrs) Eltromagnt Gas alratd by j B fors For (push) on ols or ondutors (MPD thrustrs, PPT s) 6.5 onntrats on Thrmal - Dpndng on nrgy sour: Sold Propllant Chmal (always thrmal ) Lqud Propllant Monopropllant Bpropllant Hybrd Nular (Thrmal) Nular (Eltr) Solar (Thrmal) Solar (Eltr) an b Thrmal, ES or EM an b Thrmal, ES or EM 6.5 dals mostly wth Chmal. - Dpndng on Thrust lvl (pr unt mass) - Hgh thrust ( g) for launh, fast spa manuvrng (6.5) - Low thrust ( g) for ffnt n-spa manuvrs (6.5) 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 3

2 Prforman Masurs Spf Impuls F F Isp or = mg m (s) (m/s) = ( ) Domnant for hm. Rokts, rang s Trad-off vs. mass for EP, rang s Thrmal Effny (Thrmal Rokts) Also for ltral thrustrs η = th Jt knt powr Thrmal nput powr η th η= Powr to jt Input ltral powr ~ % Vry los to 00% n hm. (non-ssu) mportant n solar thrmal (60-80%) ltrothrmal, t. Thrust/wght F/W Vry larg ~ (0-00) for Chm. Mdum (5-0) for Nular Vry low (~0-3 ) for (Solar, EP, powr lmtd) Othrs (dsgn slton fators) - Lf, most manngful n total mpuls apaty - R-start apablty - Throttlablty - Dsprson - Cost Rokt Slton Gud (by msson) 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 3

3 ) Non-Spa mssons Rokt Typ Atmosphr/Ionosphr Soundng Sold Propllant, -4 stags Tatal Mssl Sold Prop., - stags Mdum-Long Rang Mssls Sold or Lqud Prop., -3 stags (vry hgh alraton) ) Launh to spa Sold, lqud or ombnatons, -4 stags (-4g) Possbl: hybrd, -4 stags 3) Impulsv V n spa Small Sold Prop. (Apog (tm-rtal manuvrs, kk, t) nrgy hang from llpt orbts, B-propllant (storabl) plan hang from llpt orbts, lquds, Monopropllant non-ful-lmtd stuatons...) (storabl) lquds, V 000 m/s Futur: Nular thrmal 4) Low-Thrust V n spa (Mass-lmtd mssons V 000 m/s non tm-rtal mssons, small, ontnuous orbt orrtons nar-rular orbts...) Solar-ltr systms: Arjts (a bt fastr, lss Isp) Hall, Ion (slowr, hghr Isp) PPT (prson manuvrs) Nular-ltr systms Drt solar-thrmal 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 3 of 3

4 6.5, Rokt Propulson, Prof. Manul Martnz-Sanhz Ltur : Rokt Nozzls and Thrust Rokt Thrust (Thrmal rokts) m = A ρ u da n ( ) P ds P da = u ρ u da x x x n Sold nt. A A surfas (Tanks nludd) dm Not: s.,nt PdS PdA = 0, a x a x A so subtrat, ( a) x = ( a) + x P P ds P P da ρ u u da Sold nt. A A x n Thrust F 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 7

5 In gnral thn, dfn u = A A ρuuda x n n ρu da and P = A PdA A x x ( ) F = mu + P P A a x If thngs ar narly onstant on sphral aps, modfy ontrol volum to sphral wdg: m = A ρu da r x nt. A A solds. ( a) x ( a) = ( ρ r) x P P ds P P da u u da da = da osθ u = u x x r osθ 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 7

6 Dfn u A = ρuuda r m x ; P PdA A = A x x and us da = π r snθ rdθ For dal onal flow, ρ, ur, P ar onstant ovr A. Thn ρ θ α u u u osα or α u os r da sn os sn πr θ θdθ A 0 = = = r α r ρu r da πr snθdθ A 0 u = u r + osα Also, sn P = onst on th xt surfa, P = P 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 3 of 7

7 ss.. ( ) + ( ) = ( ρ ) P P da P P da u da u F a x a x n x A ( a) x F = mu + P P A A m = A ρu da n u = A A ρuuda n n x ρu da P = A A PdA da x x A x = A da x At dsgn, P = (and paralll flow byond). Also P a Thn unform F = mu x u x 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 4 of 7

8 Enrgy Consdratons So, momntum balan gvs th Thrust Equaton. What dos an Enrgy Balan gv? Start wth a nar-stagnant flow n th upstram plnum ( ombuston hambr, or nular hatr or ar hatd plnum ). Th total spf nthalpy ht = h + υ h may b dffrnt for dffrnt stramlns, du to ombuston straks:, ar onstrton, t., But along th flow xpanson n th nozzl, h t s onsrvd for ah stramln. At th xt, h + υ = h t (ah stramtub) o or υ = ( ) ( ) h h h h t For a wll-xpandd nozzl, wth larg ara rato, h o by adabat xpanson, and υ tnd to a max. υ = h. In any ral, fnt xpanson, h o, so som of h t MAX t s wastd as thrmal nrgy n th xhaust. Dfn a nozzl ffny. η N ht h h h = = h h t t h For dal gas, h T P = = h T P γ γ. But, n any as, υ υ = υ ηn = ηn ht (.., ηn = MAX h ) t Sn P unform, so s (n proporton to h t ). η, vn whn h s not. Also, υ s non-unform f s N t h t Th Jt Powr s th knt nrgy flow out of th nozzl P ( ) η = = m h h h m Jt t N t 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 5 of 7

9 Efft of Stagnaton Enthalpy Non-unformts Consdr a as whr h vars from stramtub ( dm ) to stramtub (but P =onst., so η N = onst.). Thn t ( ) F = υ dm + P P a A For P a = o (vauum opraton) and P a A << F (larg xpanson), (or f P = P a ) F υ dm = η h dm N t () and th nput powr s P = h dm () t Psmnmum(Foragvn F, m) It an b shown that or F s maxmum ( gvn P, m ) h t ( =onst.). If t wr, w would hav f th flow s unform F = η = UNIF. N m h t ; P UNIF mh t Elmnatng h t, P UNIF F = UNIF F UNIF F m = = ηn m η N m ηn m PUNIF Dfn an ffny η UNIF = (for a gvn thrust) P ACTUAL Now, xprss n gnral F by () and P by () η UNIF ( η N ht dm) ( ) ( N dm ht dm) = η Dfn gnralzd vtors u = υ = h t n th spa of th dm valus. Thn η UNIF ( u υ ) ( θu υ ) = = os u υ. 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 6 of 7

10 Equalty appls only whn υ s a onstant,.., Exampl: 50% of flow has ht = % of flow has ht =.5 h t h t h t =onst. Ths provs th ansatz η UNIF = = (6.7% nrgy loss du to nonunf. ( ) Important n arjts, lss n flm-oold hmal rokts. 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 7 of 7

11 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 3: Idal Nozzl Flud Mhans Idal Nozzl Flow wth No Sparaton (-D) - Quas -D (slndr) approxmaton - Idal gas assumd ( ) F = mu + P Pa A C F F PA t Optmum xpanson: P = P t a A - For lss, P > Pa, ould drv mor forward push by addtonal A xpanson 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of 0

12 A - For mor, P < Pa, and th xtra prssur fors ar a suton, A bakwards t Comput m = ρua at son throat: γ + γ m = ρ γrgt At = g γ + γ + γ + ( γ ) PA g t RT ; R g = R M all Γ 3 all RT * g = ( haratrst vloty ) Γ ( γ ) PA m = t * Can xprss u, P, A, t n trms of thr M or P P or A A : t P P = γ + M γ γ ; γ + γ A + Pt ut Pt T M t = = = A γ + t P u P M T M ( γ ) + γ P P T = T γ γ, T and =, T γ + M Baus p p g g u γ M γ T + = T RT + γ RT = RT γ γ 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of 0

13 C F m P Pa A u P Pa A = u + = + * PA t P At P P At u * T M γ Rg γ γ + + M ( γ ) = M = γ RT γ + g γ + M Γ In vauum, ( = 0) C P a ( γ γ γ + ) ( γ ) γ + M u P A M = + = + * γ + γ + P A γ M + M γ + ( γ ) M F v t γ + γ γ ( C ) F v γ + γ M + ( γ ) M = γ + γ + M and othrws, 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 3 of 0

14 A C = C A P a F F v t P M Not: For P = P, a ( C ) F Mathd γ + u ( γ ) γ M = = * γ + γ + M For P = P = 0 ( C ) a F Max,Va = γ γ γ + γ + ( γ ) Cho of Optmum Expanson For a Rokt Flyng Through an Atmosphr ( varyng) P a Th thrust offnt C = F F PA t was drvd n lass n th form C Pa A = C P A F F va t () and w also found C A A F va t γ + γ M + ( γ ) M = γ + γ + M γ + M = M γ + γ + ( γ ) () (3) Th thrust-drvd vloty nrmnt s tb F tb CF Δ VF = dt = PAt 0 dt (4) m 0 m 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 4 of 0

15 ( ) whr C = C t du only to th varaton of a n (), whl m = m t baus of F F mass burnout. Th quantts C Fva and ar tm-nvarant. Substtutng (), () and (3) nto (4), A A t P ( ) dpnd on (or nozzl gomtry), but M tb dt A t b Pa dt Δ VF = PAt CF va 0 0 m At P m or PA ΔV t F tb t b P a dt 0 P m A = CF dt va tb dt At m m 0 0 (5) W now mak th approxmaton that th trajtory wll hang lttl whn w vary A M (and hn CF va, ). W an thn rgard th tm ntgrals n (5) as fxd A quantts whl w optmz t. Dfn th non-dmnsonal varabls M v = PA ΔV t F t b dt m ; p = 0 0 t b 0 t b P a dt P m dt m (6) so that (5) boms A v = CF ( M) p ( M va ) A (7) t and w an now dffrntat v w.r.t M (holdng p=onst.) A C v F A va t = p = 0 (8) M M M From () and (3), th fator gnord. W thn hav γ + γ + ( γ ) appars n both trms of (8) and an b 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 5 of 0

16 γ + γ ( γ ) γ M + + M M = p M γ M M + M γ ( γ ) γ γ + γ M M M M γ + γ + = γ M p 3 ( γ ) γ M γ + M M + M M + M γ + Multply tms 3 γ + M, and not that γ + γ + = γ ( γ ) γ γ γ γ γ + M γ γ + ( γ + M ) = ( γ + ) + M p M M M M Expand & smplfy γ + γ ( γ ) γ γ ( γ ) γ M γ + = M M p M M = M M M γ γ ( M ) Canl th fator M M ( M = s larly not an optmum!) γ = p + M or γ γ 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 6 of 0

17 γ + M = p γ γ M OPT γ γ = γ p (9) Not that th xt prssur s gvn by P P = γ + M γ γ (0) and so th optmum xt prssur turn out to b P P OPT = p () P Howvr, f p < 0.4 P a o sparaton at th hghst must b amndd to, ths would mply P < 0.4P P a ao, and thr would b flow (on th ground). To avod ths, th optmalty ondton P P ao = Gratr of ρ, 0.4 () P OPT P wth a smlar xprsson for M : M OPT γ γ γ γ P = Last of γ,.5 p γ P a o (3) Th lmtng ondton n whh th whol burn ours at P W thn obtan a o s smpl. p t b 0 = 0 P ao dt P m dt m t b = P ao P (4) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 7 of 0

18 and th optmalty ondton () ylds ( P ) = prssur-mathd, as xptd. OPT P a o,.., th nozzl should b As mor and mor of th burn shfts to hghr alttuds, p drass from P long as t stll rmans abov 0.4 P P optmum dsgn, and f p drops blow 0.4 P on th vrg of sparaton on th ground. a o, quaton () gvs som ntrmdat a o P a o P. As, th nozzl should b dsgnd to b Nozzl Flow Sparaton Effts Rul of thumb (to b xplord latr): Flow sparats at th pont n th nozzl whr P 0.4P (Summrfld rtron) a So, f P 0.4P (vn f P < P ), no sparaton > a a Aftr sparaton, roughly paralll flow, at watr rgon to turn flow). P=P a (no strong p gradnts n dad So zro thrust ontrbuton wth xt prssur P ' = 0.4P a Prforman wth sparaton at that of a nozzl So, (a) P a P (full nozzl) <, 0.4 C F P = a A CF va P A o t ( f M ) g ( M ) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 8 of 0

19 (b) P a P (full nozzl) >, 0.4 alulat ' = = ' ( 0.4 a) M M P P ' ' A A ' = ( M ) At At P = a A va P A ' thn CF CF ( M) o ' t 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 9 of 0

20 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 0 of 0

21 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 4-5: Nozzl Dsgn: Mthod of Charatrsts Th Mthod of Charatrsts (Idal Gas) (Rf. Phllp Thompson Comprssv Flud Dynams, MGraw Hll, 97, Ch. 9) -D or axsymmtr Homntrop as wll as sntrop u ur = o ur plus ( ρ u ) = 0 and 0 ur ur u ur ur u u + p = o + ω u + p = 0 ρ ρ Us ntrns o-ordnats Eq. of moton along s r u : p u + = o () s ρ s Eq. of moton along n : r u p u = = R ρ n n Now ϑ ϑ u u = u R s s n (an also gt ths from u ϑ + u = o n s (good no p n t) ur u = o ) () s ρ Contnuty: ( u r n) π δ = o 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag of 0

22 ρ u r δn = o ρ s u s r s n s r δ ϑ = sn ϑ s r ( s) sn ( s) ϑ n δn ϑ ds = δ n δ s s n ρ u ϑ snϑ + = ρ s u s n r Homntrop: p dp = dρ = ρ s 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag of 0

23 so p u ϑ snϑ + = ρ s u s n r p u and, from s q. of moton, = u ρ s s so you an lmnat p s : u u u ϑ snϑ + = s u s n r u u ϑ nϑ r M - s + + = u s n M u ϑ snϑ + = u s n r (4) Introdu th Mah angl μ = sn = tan M M tan μ = M Thn () tan μ M u ϑ + = o u n s And (4) M u ϑ tanμ snϑ + tan μ = u s n r Introdu th Prandtl-Myr funton ω ( M) by u d ω= M (to b ntgratd latr) u d 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 3 of 0

24 thn ω ϑ tan μ + = o n s ω ϑ tan μ sn ϑ + tan μ = s n r add and subtrat to obtan th Charatrsts form (sngl dffrntal oprator pr quaton) tan μsn ϑ sn μ sn ϑ + tan μ ( ω+ ϑ ) = os μ + sn μ ( ω+ ϑ ) = s n r s n r tan μ sn ϑ sn μ sn ϑ tan μ ( ϑ ω) os μ sn μ ( ϑ ω ) = s n r s n r r+ os μ r os μ In (s, n) oordnats, = = sn μ sn μ so r+ os μ + sn μ = = s n m r os μ sn μ = = s n m + (m +, m - ar lngths along haratrsts) m m + nlnd ϑ+ μ nlnd ϑ μ m m + sn μ sn ϑ ϑ+ω =+ r sn μ sn ϑ ϑ ω = r ( ) ( ) F or -D, r ϑ +ω=onst. along m + I + (nlnd ϑ μ ) ϑ ω=onst. along m - I - (nlnd ϑ+μ ) 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 4 of 0

25 -D Smpl Rgons Consdr a unform rgon; th flow from t ntrs som dsturbd rgon, lk a wall turnng. On of th m famls orgnat n th unform rgon (m + n xampl) and arrs a onstant nvarant,.g. ϑ +ω=ϑ +ω vrywhr downstram. Along on o o of th othr haratrsts (m - hr), w arry a onstant (whh vars from h. To h. of that famly); w an valuat t at th wall, for nstan ϑ ω=ϑ ω along ah m - ln and ω w w w o o w w o ( ) =ϑ +ω ϑ so ϑ ω= ϑ ϑ +ω o thn along ah m ϑ=ϑ o +ω o +ϑw ω w = ϑw ϑ=ϑw ( ) ω = ϑ + ω ϑ ω along ah m ω = ϑ + ω ϑ = ω o o w w o o w w s o ϑ and ω (and hn M, μ) ar onstant along ah m, and ths m - lns ar stra ght (onst. ϑ+μ ). - Ths s a Smpl Rgon (on of th nvarants s onstant). Th turnng of th flow s dtatd by how ϑ hangs on a m + ln, as dffrnt m - lns ar rossd. Sn on a m + w hav ϑ+ω=ϑ o +ω o, hangs of ϑ ar qual and oppost to thos n ω (ω nrass as M dos n th xpanson, so ϑ drass (nras ngatvly) at th sam rat.) So, w an ntrprt ω as th magntud of th sntrop flow turnng n a smpl rgon,.., whn nothng vars along on haratrst famly. Calulaton of ω (M) du dm dt dω= M = M + u M T but 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 5 of 0

26 γ M M M dm dω= dm = M γ γ + M M + M In tgrats (wth ω =0 at M=) to M ω= K tan tan M K K = γ+ γ π π π π γ + ω = = γ For M K ( K ) γ /3 ω( M ) 08 o 80 o 30.5 o 90 o (Max. turnng from M=) S o rokt xhaust ( γ..3 ) an turn bakwards at a son nozzl xt to vauum) (but vry long dnsty, long mfp, so ontnuum approah fals at som pont n th xpanson, moluls thn ontnu n straght ln). A tually, on starts from som hgh M >, so th atual turnng s through ω ω ( M ), not ω, vn n a vauum: 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 6 of 0

27 Exampl of Applaton: Idal -D plug Nozzl at Dsgn Condton Along m, ϑ ω=ϑa ω a Smpl rgon (Prandtl-Myr fan ntrd at lp L) In partular, at nlt M=, so ϑ = ω throat a r dϕ tan μ = = r d M From gomtry, μ ϑ+ϕ= π ϑ t π Sub. ϑ=ω ω a μ ω+ω a +ϕ= +ωa ϑ = ω t a ϕ = π + ω μ π M γ + ϕ= + K tan tan M tan K = K γ M π 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 7 of 0

28 ϕ= K tan M K ϕ M tan = = K K K tan μ ϕ ϕ sn d os r dϕ dr d K = = K K ϕ = K dr ϕ r ϕ ϕ K tan os os K K K r = onst. ϕ os K K For ht ϕ= 0,r = h t (throat hght) r = K ϕ os K In partular, at nd of xpanson M = M ϕ = K a a M tan a K, thn h t r a = and K os ϕ a K ra ha = ra sn μ a = M x = r os μ = r a a a a Ma a 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 8 of 0

29 Numral Applaton γ γ P o P γ γ γ o = 00 = Ma M a P + a = γ P a Tak γ=.3, Ma = 00 = Thn M γ+.3 ω a = Ktan tan M K = = =.769 K γ 0.3 ω = ϑ = a o o t Also M ρ a = K tan = 4.0 K o and so r = = a ht K.769 ρa 4.0 os K os.769 r a h = t 34.4 ra From th gomtry, h a =r a sn μ a = M a ha = h t 34.4 h a h =9.684 t 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 9 of 0

30 xa and = ra osμ a = 34.4 h t x a h =33.0 t Vry long and ponty, should b trunatd. Non-Smpl Rgons. Whn th haratrsts of both famls ntrst som upstram dsturban, thy afft ah othr s nvarant, and haratrsts ar no longr straght, and no longr arry onstant flow proprts (xpt for thr own nvarant) 0--4 s a smpl rgon s unform -4-0 s non-smpl s a smpl rgon Thn w nd to alulat n a stp-by-stp mannr, arryng to ah pont th two nvarants I +, I - from nghborng upstram ponts, along th m +, m - lns from thm to us. Aftr ths s don, w know th nw sgmnts of m +, m - from our pont (slops ϑ + μ, ϑ μ ), so w an xtnd th grd as w go. Not th flowfld proprts an b found frst vrywhr and only thn w nd to om bak and pla th ponts gomtrally. Exampl: Dsgn a -D dal nozzl to xpand from nar son ondtons (M 0 =.) to M = 3. Us only 4 haratrsts. Us a ornr xpanson as a startr, γ =.5 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 0 of 0

31 K.5 = = ω. 3 o o = 3 tan tan. =.435 ; ϑo = 0 ω 9 3 o = 3 tan tan 9 = ; ϑ = 0 At nlt: I = ϑ ω = ϑ ω =.435 (also at 4) o At xt: + o I = ϑ + ω = ϑ + ω = (also at 4, 7, 9, 0) o o ϑ = = ϑ = At 4, thn, ω4 = = ω4 = o o 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag of 0

32 + o + o + o + Slt: I =.44 I = 0.76 I3 = I4 = 59.4 o Pont M + I =ϑ+ω( o - ) I =ϑ ω( o ) ω ( o ) ϑ ( o ) μ ( o ) ϑ+μ( o ) ϑ μ( o ) o (and 4w) (and 7w) (and 9w) (and ) Nots Ponts -4: sam ϑ ω Pont 5: Hr ϑ= 0 (a boundary ondton) and ϑ +ω= 0.76 Pont 6-7: sam ϑ ω Pont 8: Hr ϑ= 0 and ϑ+ω= Pont 9: sam ϑ ω Pont 0: Hr ϑ= 0 and ϑ ω= 59.4 Not th vry shallow angls of th m + lns (from 4, 7, 9, 0) whh wll put pont 0 far to th rght, and pont (at slop (m + ) of 9.47 o vn farthr. Loatng th ponts gomtrally ( ) ( ) x xa tanα + x xb tan β = ya yb x = y y + x tanα + x tan β a b a b tanα + tan β 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag of 0

33 ( ) + ( ) ya yb xa xb tanα y yb = ( x xb) tanβ = tanβ tanα + tan β y = ( ) y tanα + y tan β + x x tanα tan β b a a b tanα + tan β put x a, y a nto, tanα nto 3 x b, y b nto 4.5 tan β nto 6 Run Run 3 x (st. n 7) y (st. n 7) Fndng th (x,y) s s vry laborous. Auray an b nrasd by avragng togthr th angls at th nds of ah sgmnt, whh an b don baus thos αa + α βb + β angls om from th frst pass. For nstan, α =, β =. 6.5, Rokt Propulson Ltur 8 Prof. M. Martnz-Sanhz Pag 3 of 0

34 Boundary wth Prsrbd Prssur P = P r + M r r, ( ) and M = M w, ( ) so P P w (gvn P ) = o or w = w( P) So, f P s fxd on a boundary (ontat surfa), w an assgn w thr (just as w assgn θ on a sold boundary) From known pont a, θ w =θ w a a So θ= w( P) +θa wa and ths dtrmns th slop of th boundary, and that of th rfltd m + θ+ w = w( P) +θa wa Mor Gnral Contat Surfa Condton If th outsd flud s also suprson, w must solv on both sds of th ontat surfa, makng sur P and θ ar ommon at ah boundary pont 6.5, Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 4 of 0

35 P ( w ) = P ( w ) θ = θ From m know I =θ+ w ( P) From m know I =θ w ( P) ( ) ( ) + + So, I I = w P + w P Solv for P w, w and ( ) ( ) + + w + I P + w P I =θ+ known now Modfatons for Axsymmtr Condtons sn μsn θ ( θ+ w) = + m r sn μsn θ ( θ w) = m r 6.5, Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 5 of 0

36 () Calulat ( x, r ) from th angl ( μ θ), ( μ+θ ) a xa tan α + xb tanβ + ra r x xa tanα + x xb tanβ = ra rb x tan α+ tanβ ( ) ( ) and ( ) r r = x x tanβ b b b b () Δ m = x x a os α + Δ m = x x b os β (3) Advan nvarants θ+ w, θ w basd on μ, θ at a, b: ( ) sn μ sn θ a a + θ+ w = θ+ w + Δm a ra sn μb sn θb θ w = ( θ w) Δm b rb (r at, omputd n ()) (4) M = M( w ), μ = μ( M) (5) Itrat from () wth αβ, avragd btwn (a, b) and : α β ( μ θ ) + ( μ θ) a ( μ+θ ) + ( μ+θ) b 6.5, Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 6 of 0

37 ra + rb r μ a + μ b μ bttr a b θ θ + θ (6) Contnu to nw pont. So, omputaton of (,M) s now θ oupld to that of ( x, r ), whras n -D ( θ,m) an b found frst. But th atual amount of omputaton s not muh mor (only th traton stops). On th axs, r 0, but θ = 0, so. x = xa + ra tanα. + Δ m = r a sn α a a 3. θ = 0; w = ( θ + w) + a ( ) ( ) 4. M = M w, μ = μ M sn μ r sn θ a r,θ, a a 5. α ( μ θ ) +μ a, go to () (on) 6. Contnu Extnson to Cass wth Wak Shoks Sn th ntropy jump n a shok nrass only as th ub of ΔP ρ shok u0, th sntrop assumpton an b approxmatly xtndd whn haratrsts of our famly show som mld onvrgn (n prnpl, that s always ndatv of shok formaton, baus thy arry onfltng nformaton). Whn s th onvrgn too strong? Sn haratrsts ar dsrtzd, for wak onvrgn th ons of our famly wll onvrg, but not ross, and as long as thy don t, t should b OK. Of ours, wth fnr rsoluton thy wll ross, but th loss of auray n gnorng that s of th sam ordr as that nrasd by th oars dsrtzaton n th frst pla. Ths allows us to alulat off-dsgn nozzl prforman, lk an ovrxpandd plug nozzl. 6.5, Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 7 of 0

38 -D Spk Nozzl wth P < P a a dsgn Prssur fors from hot-gas bathd surfas ar th sam as at dsgn. Nt thrust s nrasd baus th ontrbuton PA s rdud: Pa a ( ) F = F + P P A ds. ads. a and so F va = F ds. + P a A ds. (just as for a bll) Pratal Nozzl Dsgns Idal nozzl ar too long, last porton has small wall angl, so small thrust ontrbuton. Wth small?, mayb ngatv ontrbuton. So, optons: (a) Constran lngth, ask for ontour that gvs hghst thrust/gvn L. Mthods of alulus of varatons (Raw nozzl, Rf : Exhaust Nozzl Contour from Optmum Thrust, Jt Propulson 8 (Jun 958): Ext flow non-paralll, non-unform, omputatonally hgh. (b) Ad-ho mthod (wdly usd) s to trunat ontrat an dal nozzl. () Dsgn wall ontour for dsrd P P () If longr than dsrd L trunat t to som ntrmdat ara rato. (3) Contrat ths trunatd nozzl to dsrd lngth ' ' x ' L dsrd xa = xb = xb xd L dal (4) Translat profl to rght mtal knk at P smooths out. 6.5, Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 8 of 0

39 Gvs adquat prforman but lss than? nozzl. Not rally justfd. (Hoffman, J. D. J. of Propulson 3 (Marh-Aprl 987): , Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 9 of 0

40 XRS-00 Engn Data Thrust, lbf At Sa Lvl 06,500 In Vauum 68,000 Spf Impuls, s. At Sa Lvl 339 In Vauum 439 Propllants O, H Mxtur Rato (O/F) 5.5 Chambr Prssur, psa 857 Cyl Gas Gnrator Ara Rato 58 Throttlng, Prnt Thrust 40-9 Dffrntal Throttlng +/- 5% Dmnsons, Inhs Forward End Aft End 33 hgh x 88 wd 46 hgh x 88 wd Forward to Aft , Rokt Propulson Ltur 4-5 Prof. Manul Martnz-Sanhz Pag 0 of 0

41 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 6: Hat Conduton: Thrmal Strsss Efft of Sold or Lqud Partls n Nozzl Flow An ssu n hghly alumnzd sold rokt motors. 3 Al + O Al O 3 m.p. 07 C, b.p. 980 C In modrn formulatons, wth 0% Al by mass, th AlO3 mass fraton of th xhaust an b 35-40%. Ths matral dos not xpand, so thr must b a loss n xt vloty, hn n I sp. m g Assum mass flows (gas) m (solds), non-onvrtng. Th momntum quaton s s m du + m du + Adp = 0 g g s s Call ρ s th (mass of solds)/(volum) (not th dnsty of th sold, thory) ρ u du + ρ u du + dp = 0 g g g s s s Dfn a mass flux funton m x = = m + g ρ u s s s ρgug + ρsus ms x ρ gug dug + dus + dp 0 x = udu dp = g g g s ρg x x udu Th nrgy quaton s smlarly, ( )( ) ( ) x dt + u du + x dt + u du = 0 pg g g g s s s s 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag of 0

42 Substtut hr udu g g from abov: dp x x dt u du + dt + u du = 0 ( ) pg g g s s s s s ρg x x dp ρ g ( ) x = pgdtg + sdts us ug dus x + wth no partls (x=0), ths gvs dp P T RgT = pdt = P P 0 T0 γ γ Wth partls, w nd to know th hstory of th vloty slp u s u g and of th tmpratur slp Ts T g. Ths s a dffult problm, rqurng dtald modlng of th moton and hatng/oolng of th partl. But w an look at th xtrm ass asly. (a) Vry Small Partls good ontat. For sub-mro partls (not a bad rprsntaton of ralty), w an say that u u = u, s s g T T = T. Thn g dp ρ g x = pg + s x dt Not that th man spf hat ( and ar pr unt mass) s ( ) ( ) x x pg s p = pg + s Rg = p v = ( x)( pg vg) = ( x) Rg v = vg + s and also x x So that dp ρ g p = dt x dp p dp p dt RgT = dt = P x P R T g and dfnng an fftv γ by th usual γ= p, v P P γ γ T = T , Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag of 0

43 Th quaton of moton s now ( ρ g + ρ s) udu + dp = 0 Or ρg udu + dp = 0 x ρg P P x = R T x = R T g ( ) g P udu + dp = 0 R T g From th two boxd quatons w s that vrythng from hr an prod as f th gas wr smpl, but wth molular mass M g M = x (or ( ) R = x R ), g and wth p g ( ) pg s = x + x. For xampl, γ P u = RgT γ P γ γ T,P n hambr t. For snstvty analyss t may b of ntrst to lnarz ths for algbra s tdous, but on gts, x. Th u + u 0 ( η ) ln( η ) η 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 3 of 0

44 ps P wth =, η 0 = pg P and, of ours, γ γ γ P u = R 0 gt γ P γ γ W s from ths that f < ( ps < pg, whh s ommon), thn u < u (and v-vrsa). For a numral xampl, look at Problm (attahd) 0 (b) Vry Larg Partls Hard to quantfy, but probably for damtr > 00 µ m or so, th partls hav too muh nrta (and thrmal nrta) to follow th gas alraton and oolng. W thn hav du du ; T T ( at hambr) s g s Tg du 0; dt 0 or s s Rturnng to th dp ρ g = pg dt g dp ρ g quaton, t now looks as f thr wr no partls: (.., partls just do not partpat n th dynams or n th thrmal γ γ P T balans). So, w stll hav =. Ths dos not man zro P0 T0 prforman fft, though. W do not gt th full gas xt vloty γ γ γ P u = RgT γ P but th partulats do not ontrbut to thrust, baus thy xt at u u : s mgu + msus gis = = ( x) gi m + m p sp0 g s 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 4 of 0

45 Ths s atually mor loss than n th small partl as (about tw as muh, dpndng on ). From th xampl, ths s a srous loss n sold rokts. Crtron for Slp 4 3 Rp s 3 π ρ p pρs p ρs p = πµ g p ( g p) = g p τ R = µ g µ g du R du R mp 6 R u u u u dt 9 dt 9 dup ug u = dt τ R p all u u = s u = u s g p p g dug ds s ds s dug = + = dt dt τ dt τ dt R R Say τ R du and dt g t τ R = a g ar onstant s a τ + C g R s( 0) = 0 C = agτ R ε ε +... ε = = +... ε s = agτ R t τ R and u = a g g t s u g = t τ R τ R t t τ t τ R R t τ... t τ R R 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 5 of 0

46 So, small slp for t τ R L u τ R L sr ρ u 9 µ g g p R p 9 µ gl ρ u s g Say g 5 µ 3 0 Kg/m/s L 0.3m p 3 3 ρs 3 0 Kg/m u.5 0 m/s g 3 6 R 4.5 = = 3 0 m = 3µ m So, R 3µ m no slp p R 3µ m full lag p 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 6 of 0

47 Problms Problm As notd n lass, th fft of arryng a mass fraton x of fn sold partls n th xpandng gas n a rokt nozzl an b aountd for by usng an avrag spf hat rato γ = ( ) ( ) x + x pg x + x vg s s and an avrag molular mass M M g = x For Al O th hgh tmpratur spf hat s = 60 J/Kg/K. 3 Consdr a sold rokt wth γ ( ) s =.7.5, P P = 0.0, M = 8 g / mol. For a 0% alumnum loadng n th propllant, x s of th ordr of 37%. Calulat th mathd spf mpuls of th rokt and ompar to what t would b for th sam T = 3300 K, but wth no partls. g 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 7 of 0

48 Problm Spf hat of lan gas.5 r R pg = = = 380 J/Kg /K r M vg 309 pg = = = 78 J / Kg / K r Th spf hat of th sold (or lqud) AlO3 s s = 60 J / Kg / K. Th avrag spf hat rato s thn 309 ( x) pg + xs ( 0.37) r = = =.336 ( x) vg + x s ( 0.37) And th avrag molular mass (M s ) s Mg 8 M = 8.57 g /mol x = 0.37 = P Th xt spd for 0.0 P = and T = 3300 K s thn γ γ Q P γ u = T = 63 m / s γ M P 5 NOTE: Altrnatvly, and asr to do, you an us P u = Cp T P γ γ wth ( ) C = = 469 J/Kg/K p (so M s not rally ndd) 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 8 of 0

49 As a hk, γ R = = 469 γ M J/Kg/K as t should. Undr prssur-mathd ondtons, thr s no xt prssur ontrbuton to thrust or, and hn I sp 63 Isp = = 66.3 s s Wthout partulat but wth th sam P,P and T, w would obtan γ γ Q P γ u = T = 399 m/s 0 γ Mg P and Isp 0 = = 36. s Thr s thrfor a loss of % 00 = 8.3% n I gp It s ntrstng to tst th auray of th lnar approxmaton gvn n lass for small x: ( η ) ln( η ) u x P η = ; η 0 u 0 pg 0 P γ γ u W fnd η 0 = , and thn u 0 = (6.3% loss) (not too dffrnt, dspt larg x) 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 9 of 0

50 NOTE: Altrnatvly, and asr to do, you an us P u = Cp T P γ γ wth ( ) C = = 469 J/Kg/K p (So M s not rally ndd) As a hk, γ R = γ M = 469 J/Kg/K as t should. 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 0 of 0

51 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 7: Convtv Hat Transfr: Rynolds Analogy Hat Transfr n Rokt Nozzls Gnral Hat transfr to walls an afft a rokt n at last two ways: (a) Rdung th prforman. Ths tnds to b a -3% fft on only, and s thrfor sondary. (b) Cratng grat dffults n th dsgn of hot-sd struturs that hav to 7 8 survv hat fluxs n th 0 0 w / m rang. Th prnpal mods of hat transfr to nozzl and ombustor walls ar onvton and radaton. Of ths, onvton domnats, and radaton tnds to b mportant only for partl-ladn flows from sold propllant rokts. I sp Convtv Hat Transfr W wll rvw hr th omprssbl D boundary layr quatons n ordr to xtrat nformaton on wall hat transfr. Th govrnng quatons ar (n th B.L. approxmaton) Contnuty ( ρu) ( ρv) x + y = 0 () X-Momntum u u p τxy u ρu + ρv + = = µ () x y x y y y Y-Momntum P y = 0 (3) 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag of 6

52 h h T ρu + ρv = uτ + k (4) x y y y y t t Total nthalpy ( xy ) u whr ht = h+ s th spf total nthalpy, and µ s th vsosty. For a µ=µ s a flud proprty. Rokt boundary layrs ar almost always lamnar flow, ( T) turbulnt, and µ s thn th turbulnt vsosty, whr momntum transport s fftd by th random moton of turbulnt dds. If ths dds hav a vloty sal u' and a lngth sal l', w hav, n ordr-of-magntud. µ ρ u'l' (5) turb. whr u' s som fraton of th loal u, and l'tnds to b of th ordr of th wall dstan y. Th mportant ponts about (5) ar (a) (b) µ µ, mostly baus l' man fr path and turb. µ s proportonal to dnsty (whras µ s not, baus th m.f.p. s nvrsly turb. proportonal to ρ ). Smlarly, th last trm on th rght n th nrgy balan, rprsntng th onvrgn of hat flux, ontans th turbulnt thrmal ondutvty K ρ u'l'. On agan, w not that K s hr proportonal to dnsty. W also not that th turbulnt Prandtl numbr p P r µ tp = (from th ordrs of magntud) k t It s of som ntrst to not th orgn and omposton of th vsous trm n quaton (4). If w ollt th dot produts u τ around a flud lmnt as shown (n B.L. approxmaton), t 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag of 6

53 w obtan th trm ( uτxy ) as wrttn n (4). Ths an b xpandd as y τxy u u u ( uτ xy ) = u + τ xy = u µ + µ y y y y y y (6) Th st trm n (6) s just th vloty tms th vsous nt for pr unt volum, so t s th part of th total vsous work that gos to alrat th loal flow. Th sond trm n (6) s postv dfnt, and t s th rat of dsspaton of nrgy nto hat du to vsous ffts. W wll rturn latr to ths hatng fft. Approxmat Analyss Lt us manpulat th rght hand sd of quaton (4): u T u K T uµ + K = µ u + y y y y y y µ y and, sn h y = p T, ths ylds y u h µ p µ + Pr y y Pr y k W not hr that, both for lamnar and turbulnt flows, P r s a onstant, ndpndnt of P and T to a good approxmaton. In fat, as w notd bfor, t s also of ordr unty ( 0.9 for turbulnt flows). So, th RHS of th nrgy quaton boms h u µ + y y Pr (7) If w mad th approxmaton P r =, thn ths would rdu h furthr to t µ y y wth th flat plat approxmaton (), (4) would bom u ht = h +. If n addton, w mad approxmatons P 0, thn th par of quatons x u u u ρu + ρv = µ x y y y ht ht ht ρu + ρv = µ x y y y (8) 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 3 of 6

54 Ths ar dntal quatons for u and lnarly transformd varabls h t. Th sam quaton would also govrn th u h u = ; t htw h = u h h t tw (9) whr th ( ) subsrpt dnots th valu of a varabl n th loal xtrnal flow (just outsd th boundary layr). Both u and h satsfy dntal boundary ondtons: u = h w = 0; u = h = (0) w and, as notd, dntal govrnng quatons. W onlud that, undr th assumpton P = = Pr, 0 x, ht hw u = h h u t w () w u whr w also notd h tw = h w + = h w. Ths smlarty rlaton btwn vloty and total nthalpy profls s known as Croo s analogy. Approxmat hat flux at th wall W ar ntrstd n th magntud of th wall hat flux T = qw K y w () q w K h K h t = = µ y µ p y p w w whr w usd ht u h h = h+ = + u y y y y w w w w 0, sn uw = 0 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 4 of 6

55 K Th group = µ P approxmatons. Thus p r should b st qual to unty, for onsstny wth th statd q w ht = µ y w Us now quaton (): u ht h w u qw = µ hw + ( ht h ) w = µ y u u y w w and not that u µ y w s th wall shar strss, τ w. So q w h h t w = w u τ (3) whh s also alld Rynolds analogy. A mor ompat form of ths an b wrttn n trms of th Frton Coffnt f τ w ρ u (4) and th Stanton numbr S t = q w ( ) ρ u h h t w (5) wth th rsult (from (3)) S t f = (6) On mportant pont an b mad about th rsult (3): Th hat flux to th wall s drvn by th nthalpy (or tmpratur) dffrn btwn Total xtrnal and Wall valus, not btwn stat valus. Ths an b nonntutv. Consdr th stuaton nar th xt of a hghly xpandd spa nozzl, whr th bulk tmpratur may hav droppd to, say, 300K du to th strong T xpanson from a hambr tmpratur of, say, 3000K. Th wall ould b mad of Tungstn so as to b abl to sustan rlatvly hgh tmpratur and ool tslf by radaton to spa, so T ould b, say, 500K. Is th nozzl wall bng hatd or w oold by th 300K gas? Th answr s that t s bng hatd, baus 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 5 of 6

56 T t = T =3000K, whl T. Ar w volatng th nd w = 500 < T t Prnpl of Thrmodynams. Rad on. Smplfd Profls, Aross th Boundary Layr To bttr undrstand ths stuaton, lt us rturn to Croo s analogy (quaton ) and wrt h t = h+ u, and solv for h: ( t w) u u h = h h (7) u Ths s a quadrat rlatonshp btwn h and u. For low subson flows h h t, so th last trm s not strong, and th rlatonshp boms lnar n th lmt. Th rlatonshp btwn slops at th wall flows from (7): 0 dh du u = ( ht h ) w u dy u dy y w w w or dh h = du t h u w w (8) W an us (7) and (8) to skth h vs. u aross th boundary layr. For a as wth h >, ths looks lk h w 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 6 of 6

57 Not that whnvr u > h h thr dvlops an ntrmdat tmpratur t w from h. Th as whn h < s mor rvalng vn: maxmum. But n any as, th wall slop s as f th ln wr omng from h h w t, not Now th wall slop s sn to b postv (hat nto th wall), dspt h < h (as long as h > h ) t w So, th quadrat porton of th Croo rlatonshp s rsponsbl for th xtra wall hat; ths an n turn b trad to vsous dsspaton, whh aumulats n th boundary layr and lvats ts tmpratur, so that th wall s hatd vn whn th outsd tmpratur s low (as long as th flow has hgh spd). Modfaton for Pr W lav for now th ssu of th non zro prssur gradnt, xpt to not that t ntrodus small modfatons down to th throat. Th dvatons of P from unty ar small, and, for gass P < (~ 0.9 for turbulnt flow). Ths braks th prft r balan btwn dsspaton and onduton rsponsbl for Croo s analogy, n th sns of favorng onduton of th dsspatd hat. As a sond onsqun, th tmpratur ovrshoot s rdud, and so s th wall slop of T and th hat flux to th wall. Th drt fft of hghr onduton (P ) s aountd for approxmatly by modfyng Rynolds analogy to r < r w S t f = (9) P 0.6 r Th sondary fft (rdud ovrshoot) s aountd for by rplang th drvng nthalpy dffrn h h by h h, whr h s th Adabat-wall nthalpy, dfnd as t w aw w aw 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 7 of 6

58 u haw = h + r ; r 0. 9 (0) (turbulnt) and r s th Rovry fator. Wth ths hangs, th hat flux s now f qw = ρ u ( haw hw) () P 0.6 r Th Bartz hat flux formula A vry rud, but surprsngly fftv rprsntaton for th frton fator suppld by th wll-studd as of fully dvlopd turbulnt flow n a pp ; ρ ud f = R 0. = R µ f s that () whr R s th Rynolds numbr basd on damtr D, and vsosty. Puttng also h = p T + onstant, quaton () now gvs 0. µ s th lamnar 0.03 µ qw = ρu p ( Taw Tw) ( u) p ( T 0. aw Tw) P0.6 = µ r ud ρ (3) ρ D 0.06 It s ommon prat to dfn a hat transfr gas-sd flm offnt, h (not an nthalpy!) by g h g T aw q w T w (4) And, so far, w hav h 0.06 ( ) = ρ u µ p (5) D g 0. At ths pont w not that th formulaton so far has gnord th strong varatons of ρ and µ aross th boundary layr sn ths quantts dpnd on tmpratur as T w ρ (at P=onstant) ; T ( w 0.6) µ (6) A ommonly usd approah to nludng ths varatons s to rpla ρ and quaton (5) by thr valus at som ntrmdat tmpratur <T>: µ n 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 8 of 6

59 T < T > ρ ρ ; < T > µ µ T w (7) and <T> an b valuatd by svral mpral ruls. For Mah numbrs not muh hghr than, w an smply us T + Tw <T> (8) Makng th rplamnts of (7) n quaton (5), w obtan h w T 0. g = ( ρu) 0. p D < T > µ (9) whh s on form of Bartz formula. A mor usful form follows from th ontnuty quaton: ρ u m P At = = A * A, wth RgT, Γ( γ) and whr A s th loal ross-ston, and n (9), and usng A t A A t Dt =, th fnal form s D th throat ross-ston. Substtutng h w 0.06 P Dt 0. T g = 0. pµ D * D < T > t (30) Svral mportant trnds and obsrvatons an b mad now: (a) Smallr throat damtr lads to largr hat flux straght from th Rynolds no. dpndn of. f D 0. t. Ths oms 0.8 (b) Hat flux s almost lnar n hambr prssur ( P ). Ths lmts th fasblty of hgh hambr prssurs, whh ar othrws vry dsrabl..8 D () Maxmum hat flux ours at th throat t. On rtal dsgn D onsdraton s thrfor th thrmal ntgrty of th throat strutur. 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 9 of 6

60 (d) Lghtr gasss lad to hghr hat fluxs, through th ombnd ffts of and * hg 0.6 () Th fator M w 0.68 T T < T > s gratr than unty. Ths < T > nhanmnt of hat flux follows manly from th fat that th gas n th boundary layr s mostly oolr than n th or, hn dnsr. W showd bfor that th turbulnt hat ondutvty s proportonal to dnsty. p Exampl Consdr th Spa Shuttl Man Engn (SSME), whh s a Hydrogn-Oxygn rokt wth (roughly) ths haratrsts: P = 0atm. 0 P 7 a T = 3600 K M = 5g/mol r.5 RgT * = 600m / s Γ γ ( ) p γ R = 800 J / Kg / K γ M 5 µ 3 0 Kg /m/s T = T 300K = T γ+ throat Tw = 000 K W alulat thn T + Tw < T >= = = 00K (at th throat) w 0.6 T < T > , Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 0 of 6

61 and so, usng quaton (30), hg 60,000 w / m / K and qw 60,000 ( T 000) aw t γ Taw = Tt + r µ = K t 0.9 ( ) (slghtly lss than T ) w 8 q 60, = W / m Ths s a vry hgh lvl of hat flux. To vsualz th mplatons, suppos ths q w had to b transmttd through a thn mtal plat (thknss δ, thrmal ondutvty k). On would hav qw = K T δ whr T s th tmpratur drop through th mtal. As an ntal guss, suppos th mtal wr stanlss stl ( K 0 W /m /K) δ =mm. Thn, and 8 3 qwδ T = = = 9,000K!! K 0 Obvously, ths s unaptabl. Try usng Coppr nstad, wth K 400 W / m / K (twnty tms bttr). Ths gvs T =950K, stll not aptabl (oppr would b vry soft thn). Th plat would hav to b thnnr and mad of oppr. Not an asy problm. 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag of 6

62 Mor ratonally Th S or h should dpnd on x, dstan from start of nozzl, sn th B.L. s stll t g dvlopng (not fully dvlopd). In addton, thr should b som aountng for alraton proprty varaton through B.L. ylndral gomtry Th artl by Rubsn and Inony (h. 8 n Rosnhow and Hartntt s Handbook of Hat Transfr, MGraw-Hll, 973) gvs a gnral formula for turbulnt B.L. In an ylndr, wth alraton: S t ( x) = A ρux ff n n s F FRθ µ n (and h = ρ u S t ) g p f s = found walls. h A = onstants, dpndng on Rynolds no. basd on mom. th. n R > 4000, A = 0.03, n = θ 7 R < 4000, A = 0.093, n = θ 5 F FR θ = Fators for proprty varablty. Can tak svral narly quvalnt forms. A smpl on from Ekrt, s F ρ < Τ > = = ρ < > T < Τ > Tw = + + ( Τ ) w µ T FR θ = ( w 0.6) µ ( < Τ > ) < Τ > T T T T aw u γ Taw = T + r = T + r M 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag of 6

63 r 0.9 (rovry fator) Th fftv dstan x ff (aountng for mmory of past alraton) s rlatd to th atual dstan x through an ntgral ( ) ( ) x f x' xff ( x) = dx' 0 f x whr f = ρ u ( ) n zrµ F F n n Rθ n haw h w u z = haw = h +r ht h w R=R(x)= body radus at x. For a quk stmat of R θ, w an smplfy furthr to th flat-plat as, n whh f dθ =, dx wth ( θ ) ( > ) R R < 4000 f θ = R R 4000 θ θ, and wth dθ = dx dr dr θ x 0.08 < R R 54 x R 0.08R θ R = x θ θ θ = R R x.99 0 θ < x 76 6 = x 67 R 7 x > θ θ dr 5 ( ) dr R R 0.05R R 4000 or R R θ θ R x = = x 0.05 x 7 R x 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 3 of 6

64 For larg rokts, tnds to b 0 0, so th hgh formulas should R xthroat b bttr, dspt th ommon us of Bartz s formula, whh ar basd on th low formulaton. Fortunatly, dffrns tnd to b small, and ar markd oftn by R othr unrtants (surfa flms, flud proprts). Exampl and Comparsons: 7 8 R Consdr nozzl R R R R t ± t R = xtanx + x = 4tanα x tanα wth orgn at t R R R x = x = tanα and R.5, 5 R = α = t o and gong through throat at x = x = t R t tanα Usng γ=.5 and th R > 4000 θ opton, w fnd x t Rt ff = M t throat 0.5M M x + + R t x x d R Rt R x R R R x R tan5 Rt o whr = tan5 + 4 ( x) t t o t M R.5 R M t and = M( x) Th ntgraton gvs x R ff t throat =.089 Compard to xt x x =.53 and 0.73 R = t R t 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 4 of 6

65 Sn xff appars to th throat) s nsgnfant. 7 powr, th mmory/alraton fft (up to th Th throat S t s thn (0.9) ( T = 363) aw T (usng r=, Tw = 000K, T = 3300K, = 3300 = 933K ) t.5 ( S ) = 0.03 t throat 0.77 ρ u 7 ( xt x) < T > µ T throat throat (363) (0.695) < T > = = T Tak 7 P = 0 N/m, M= 5 g/mol * 8.34 RgT = = = 59m / s Γ P = = t * ( ρ u) 560 Kg / s / m x R t t x.5.5 = =.866 = = 0.78 o R o tan5 tan5 t and, wth Rt = 0.3m, T 5 throat µ µ = Kg ms , Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 5 of 6

66 On gts, ( ) S = (0.004 usng nstad of t throat Usng th R < 4000 θ opton (small rokts) xt xt x ) ( S ) = t throat ρ uxff < T > µ T throat throat and usng agan x = x x ff t, t, w gt ( ) S = ( usng ) t throat x t For omparson, th fully dvlopd pp flows formulaton would gv S t w 0.9 R * g T At = = µ 0. ρ u p D P t < T> A atthroat ( ) S = t throat Ths s los to th Rθ < rsults abov (and, ndd, th offnts ar for R 4000). But ths appars ondntal, basd on th fat that for most nozzls, θ < x R t. 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 6 of 6

67 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 8: Convtv Hat Transfr: Othr Effts Ovrall Hat Loss and Prforman Effts of Hat Loss () Ovrall Hat Loss Th loal hat loss pr unt ara s q = ρu ( T T ) ntgratd hat loss s w p aw w S, and usng t m = ρuπr, th L dr Qw qwπ R ds ; ds = + dx dx dx (small angls) () x= 0 L m dx Q ( T T ) S π Rdx = m ( T T ) S R L w p aw w t p aw w t πr 0 0 () For an approxmat valuaton, assum th quantty ( ) funton of x, and trat t as a onstant. W thn obtan T T S s a wak p aw w t Q w m T p T aw T T w L L dx T w t R x 0 0 dx S St (3) ( ) T R( x) For many rokts, rato Q w m T p L L dx R R x s of th ordr of 6-0, and Tw, so th T 4 3 ff 0 ( ) (hat loss dvdd by total nthalpy flux) s of th ordr of 8-6 tms 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag of 6

68 th Stanton numbr. As w found bfor, s tslf ~ 0.00, ladng to fraton hat losss of th ordr of -%. Whl ths s a small fraton, ts absolut valu may b larg, baus th total thrmal powr s normous. As an xampl, for th SSME ngn S t 6 F 0 N J m pt = pt K, C 4500 m s KgK or m T = W p 9 (th output powr of four larg powr statons). A.5% fraton of ths mans 66 MW lost to th walls (som 80,000 HP). () Efft on Prforman As a startng guss, w ould magn that all of th losss ( Q w ) ar rfltd n an qual amount of knt nrgy loss n th xhaust. If no losss, u 0 s th xst vloty wth u u m 0 Q w (4) But a lttl rflton shows that th knt nrgy loss must b lss than Q w. Indd, hat losss that our nar th nozzl xt plan ar almost rrlvant for prforman, baus th thrmodynam ffny of th rmanng xpanson from th pont of loss to th xhaust s vry small, so vry lttl of that loss s rfltd n a knt nrgy dras. So, for th tm bng, w smply aknowldg ths by wrtng u u m 0 < Q w (5) 0 Qw or u > u m u Q or > u mu w 0 0 (6) P u = p T T = pt 0 P Rmmbrng that ( ) γ, 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag of 6

69 u u 0 Q > w p P P m T γ γ (7) If th fratonal loss Q w m T p s of ordr.5% and th xpanson ffny γ γ P u 0.05 η= s of ordr 75%, thn > = 0.0 (.., a loss of lss P u than % n spf mpuls, gnorng th xt prssur ontrbuton). Th alulaton an b mad mor prs by trakng th voluton of th gas tmpratur. Th total nrgy quaton, aountng for th losss, s dh dt du = S R dx + π = π dx dx t m ρua p u qw R ρup ( T aw Tw) t T dt du St or p + u = p ( T Tw ) dx dx R (8) But th momntum quaton (gnorng, somwhat nonsstntly, th ffts of du dp du dp frton), gvs ρ u + = 0, or u =. Substtutng n (8), dx dx dx ρ dx p dt dp St = p dx ρ dx R Dvd by pt and not that ( T T ) w dp γ dp = ρ T dx γ p dx p dt γ dp S T T = Tdx γ pdx R T t w (9) Wthout th hat loss trm, ths would ntgrat to (sntrop) rlaton. Mor gnrally now, T T P = P γ γ, th dal flow 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag 3 of 6

70 γ γ L T P S = t T Tw xp dx (0) T P R T 0 and sn th xponnt s a small numbr, γ γ L T P S = t T Tw dx () T P R T 0 To valuat th orrton trm, w us for T th undsturbd T, as f no hat loss had happnd. Ths gvs T Tw T Tw T T w P = T T T T P γ γ and w also assum that T w St T s narly onstant: γ γ L T P γ T γ w P dx St T P T 0 P R () and, n partular, at th xt plan, γ γ L T γ γ P Tw P dx St T P T 0 P R (3) W now xprss th xt knt nrgy as ( p t ) u = T T (4) whr both T and T ar afftd by th losss. For th total nrgy loss, w hav and so t ( ) m T T = Q p t w 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag 4 of 6

71 T Q T dx T R t w w = = St T mpt L 0 (5) whr w hav usd th rsult n quaton (3). For th loss of stat nrgy, w hav th rsult n (3). Usng both n (4), γ γ L L γ T γ w dx P Tw P dx u = pt St + St T R 0 P T 0 P R or γ γ L γ P γ Tw P = p t P T 0 P u T S dx R (6) P γ γ W s now that th fator ourrng n th ntgral of (6) s just th P thrmodynam rlf w had mntond arlr, whh maks th loss of knt nrgy b lss than th hat loss. Indd, ths fator boms zro as P P, so, as antpatd, hat losss nar th xt ar rrlvant. To smplfy th wrtng, us P u = pt 0 P γ γ and dfn γ P γ P η 0, = and η x, = ( ) P P x γ γ : u η Tw x, dx = St u η T 0, R 0 0 L (7) ηx, and, agan, th rght-hand-sd mnus th fator would b th rlatv hat η 0, loss (quaton 3). Numral valuaton shows that th modfd ntgral n (7) s 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag 5 of 6

72 about 3 of th orgnal ntgral L rlatv I sp 0 dx R. Rmmbrng our arlr stmat of th loss ( < %), w onlud that a bttr stmat s about 0.67%. Ths amounts to 3 s. out of I 400s. sp 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag 6 of 6

73 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 9: Lqud Coolng Coolng of Lqud Propllant Rokts W onsdr only b-propllant lqud rokts, sn monopropllants tnd to b small and oprat at lowr tmpraturs. In a b-propllant rokt, both th oxdzr and th ful strams ar n prnpl avalabl for oolng th most xposd parts of th hambr and nozzl pror to bng njtd. Ths s alld rgnratv oolng, baus th hat loss from th gas s rovrd ( rgnratd ) nto th lqud, so no hat saps. Ths s not to say no thrmodynam loss s nurrd, though (hat s transfrrd from vry hot gas to ool lqud, whh mpls rrvrsblty and loss of work potntal). Of th two strams, th ful s normally usd for oolng. Ths s for two rasons: (a) Fuls tnd to hav hghr spf hats, so mor hat s rmovd for a gvn T of th oolant, and (b) Lakag from an oxdzr stram nto th normally ful-rh ombuston gas an produ a loal flam that an b atastroph, whras lakag from a ful ln nto th sam ful-rh gas s nrt. In addton, xposng hot mtal to oxygn or strong oxdants always arrs som rsk of alratd hmal attak, or vn gnton. Som xptons do xst whr oxdzrs ar usd for oolng, though. A typal arrangmnt s as shown blow (Fgur ). Th ful at hgh prssur from th ful pump (FP) s snt through a srs of narrow passags arvd nto th nozzl and hambr walls, pks up th wall hat flux from th gas, and s dlvrd vntually to th njtor manfold. Sn th nozzl rgon 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of

74 s th most thrmally loadd on, oftn th oolant flow s splt, wth on part ntrng at th nozzl xt and anothr provdng xtra ool flud by ntrng just downstram of th throat. A typal onstruton for th oolng hannls s shown n Fg.. Th load-barng part of th strutur s mlld longtudnally wth hannls of varyng dpth and wdth (to obtan varyng lqud vloty), and a hgh thrmal ondutvty thn layr of a Coppr alloy s thn brazd on th nsd. Dsgn Consdratons Two aspts nd to b vrfd n th dsgn of th oolng systm: (a) Th oolant should hav suffnt thrmal apaty to absorb th hat load wthout xdng som rtal tmpratur, whh may b a hmal domposton lmt (thrmal rakng for hydroarbons) or th bolng pont (although, wth ar, bolng an b somtms tolratd or xplotd for ts strong hat absorpton proprts). Suppos Q LOSS s th alulatd total hat loss from th gas. As sn n a prvous ltur, ths amounts to -3% of m T, mor for th smallr ngns. Suppos also th ful only s usd as oolant, wth a flow rat. m m F m Th OF rato s dfnd as OF= mox mf =, and so mf =. If O m + F F th lqud ful has a spf hat, ts tmpratur rs T from nlt to ool xt of th oolng rut wll b gvn by p m F Q LOSS = m T () F ool QLOSS mf mpt = ool T Q O TOT + F 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of

75 Q LOSS p O T = + T () Q C F TOT ool whh must b kpt wthn lmts. For xampl, say O F =4, Q Q LOSS TOT =0.0, p ool =, T =3000K; w obtan T = = 300 K. Ths may or may not b aptabl; for a ryogn oolant t would most lkly b, but a hydroarbon ful, xtng th pump at 300K wll thn lav th oolng rut at 600K, probably too hgh for hmal stablty. (b) Th loal oolng rat at th most xposd loaton (th throat) must b suffnt to avod domposton or bolng vn at th ontat pont of th lqud wth th wall. Th thrmal stuaton n a ut through th front wall of th oolng passags s as shmatzs n Fg. 3. Th T aw tmpratur s shown dashd baus, as w know t s not th atual gas tmpratur outsd th gas boundary layr, but s th on drvng hat. Th lqud bulk s at a tmpratur T l, whh s blow that of th wttd wall ( T w ), baus hat has to b drvn through aordng to l ( w l ) q = h T T (3) whr h s th lqud-sd flm offnt, that an b alulatd, for nstan, l from Bartz formula usng lqud proprts. Th sam hat flux s suppld from th gas through th gas-sd flm offnt: 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 3 of

76 ( ) q = h T T (4) g aw wh and yt th sam flux must ross th wall by onduton: T q = k wh T δ w (5) whr k s th wall thrmal ondutvty, and δ ts thknss. R-wrtng (3)-(5) as T T T aw wh w q Twh = h g δ Tw = q k q Tl = h l and addng, w obtan T aw δ Tl = + + q; hg k h l q = Taw Tl δ + + h h k g l (6) whh w an thn us to alulat ntrmdat tmpraturs from (3), (4), (5). Clarly, what w hav don s addng th srs thrmal mpdans h, δ and of th gas boundary layr, th mtal, and th lqud. k g h l 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 4 of

77 Strsss n Coold Nozzl Walls Rbs arry wak nplan strss. Wall must arry th larg hoop strss du to,. In addton, hot sd wll xpand mor, forng bak (old) sd to hghr tnson. So, us hgh σu stl Fg. l for bak () layr. Front () layr nds to b good thrmal ondutor; us Cu or W-Cu alloy (hghr strngth). Cu has hghr xpanson offnt α >α, whh adds to th fft of hghr T, and nds up puttng ths layr n omprsson. Ths an b rlvd by hot assmbly, so that th Cu s pr-strthd whn old. Cu stl P l P g Plan stran At any z, wthn on of th matrals. whr ( z) σ ε= ( ν ) +α T( z) 0 E T () T 0 an b ntrprtd as th tmpratur at whh th stran ε s dfnd to b zro, wth zro strss. Sn th shap rmans planar, wthn th layr). ε = onstant (at last Wrt () for both layrs. W now assmbl thm wth a tght ft, but zro strsss, at T, whh from now on mans th assmbly tmpratur. Upon hatng or oolng, 0 thrmal strsss wll arrv, vn wth no loadng or T gradnts. Both layrs now hav th sam (onstant) ε ( ) ( z) ( ) 0 E ε =0 by dfnton at assmbly σ ν +α T z T =ε () ( ) For mtals, ( z) σ ν +α T ( z) T =ε (3) 0 E ν vars lttl, so tak ν =ν =ν. 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 5 of

78 T ( z ) ( ) Th tmpratur wll b narly onstant (adabat outr ondton). T vary lnarly wth z, aordng to dt k = q (4) dz z wll W wrt (3) at z = 0, z = t and subtrat: σ σ ν =α Twh Tw E (5) ( ) h and ntgrat (4) to Twh Tw q = k (6) t so that σ σ = h α E t ν k q (7) Also () rads to σ ε= ( ν ) +α T T0 E l, whh an b ombnd wth σ ε= ( ν ) +α Tw 0 E T σ σ T w T l T 0 (8) E E gv ( ν) =α α ( α α ) Hat transfr from wall to lqud gvs q = hl ( T w T l ) (9a) or Tw q = Tl + h l (9b) Substtut nto (8) σ σ q ν = α α l 0 +α E E hl ( ) ( )( T T ) 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 6 of

79 ( α α )( T T ) σ σ α q l 0 or = + (0) E E ν h ν l In all of ths, q s takn as a gvn. It an b alulatd from th gvn T,, plus h, and, k : g h l t aw T l q = Taw Tl () t + + h h k g l Equatons (7) and (0) rlat σh, σ, σ. W nd on mor quaton F or balan Sn T ( z ) s lnar n z, so wll σ ( z ) man valu. For for alulatons, thn, w an us th σ = σ + σ h () Th nt balan thn s σ t + σ t = P D + Plt (3) l g 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 7 of

80 rd whh s our 3 quaton, togthr wth (7), (0). Not that, sn t D, has only a mnor fft on strsss (xpt n th rbs). From hr, w thr solv for σh, σ, σ (gvn gomtry, P, P l, q) or, for dsgn, go th rvrs rout and dd on th gomtry for assgnd strsss. W wll pursu hr th sond approah. Dsgn As notd, σ wll b postv and hgh, whras σ h (and lss so σ ngatv, and probably hgh too. W thn tak th vw that g l ) wll b σ = σ S (4) ult,tns. (S=safty fator ~.5) and P l σ S ult,omp. σ h = last of σ S ( ) buklng (5a) (5b) and us ths ondtons to dtrmn t, t. Th slton of t s omplatd by th fat that ( σ h ) wll dras wth t, but so wll (quadratally) σ buklng : For buklng, us a smpl lampd-bam formulaton: 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 8 of

81 π π = = F EI 4 EI ( l ) l whr I = 3 H t. Dvdng by A = Ht, 3 4π E Ht π t σ = σ = l 3 l ( Ht ) buklng buklng To prod, start by lmnatng σ btwn (7) and (0): ( α α ) ( T T ) σ σh α t α q q = + E E ν k ν h ν l l 0 E (6) ( ) ( α α ) ( T T ) E αe t E σ h = σ + + q + E ν hl k ν l 0 (7) or, rallng (), t + E α E h k σ h = σ + aw l + E t ν ν + + h h k l ( ) ( T T ) g l ( α α ) E ( T T ) l 0 (8) 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 9 of

82 Ths dsplays svral ffts: σ ult. (a) On σ = s fxd, σ h wll nras wth t, although wakly, sn S t, (th Cu wall offrs lttl thrmal mpdan, ompard to th k h h g two boundary layrs). l (b) Ths ( σ h ) (mddl trm n (8)) arss from th hatng of layr rlatv to, du to hat flowng n. () Th postv strss σ (to ountr P mostly) rlvs ths tndny to omprss layr, and mght vn rvrs t. g (d) Th last trm n (8) arsng from dffrntal xpanson offnts α α, ould b usd as a dsgn ad. If =Cu, =stl, α =.3 0 K, 5 α =.4 0 K, so α α > 0. 5 Thn, f to rdu σh s stll too hgh dspt σ, w ould nras T 0, f possbl abov T l, σ h. Ths mpls assmbly at hgh tmpratur. ( 8 ) W an us ( 6) s don, quaton (7) gvs to onstrut a plot lk Fgur 3, and slt a vabl σ, quaton () gvs t. On ths σ, and quaton (3) gvs t. Som data: Matral E ( P a ) (at 500K) Cu St. Stl T Alloy Stl (SAE x430) d ( k ) σ ult ( P a ) (at 500K) ν ( ) Z = ν σ Eα ult. o ( K) Th last olumn s a fgur of mrt xtratd from quaton (8) to gv a prlmnary rough da of matrals xpanson strss. Th hghr Z, th hghr th T to rah σ n a doubl strp of ths matral subjt to dffrntal hatng T. ult. 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 0 of

83 7 Exampl P = 00atm 0 P (nglt P fft) g D=0.3m; l=4mm h = 4000 W / m / K ; g a 5 h =.76 0 W/m /K; k l = 360 W / m / K T = 300K, T = 400K aw Substtutng nto (8), l P a ( alloy stl ) ; E.09 0 P a ;.4 0 K σ = = α = P a ( Cu ) ; E P a ;.3 0 K σ = = α = ult,omp t 6 h ( 400 T t σ = + + ) (9) and, from (6) t buklng σ = (t n m.) Followng ar som alulatd rsults: ( ) ( ) ( P ) 0 h a t mm 0 0. T = 97K σ P σ buklng a COMMENTS. 0 Sn σ ult,om =, assmblng at.5 room T 0 s not aptabl for any t. 8 ( ) ( ) ( P ) 0 h a t mm 0 0. T = 500K σ P σ,bukl. a Closr, but stll no soluton 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of

84 ( ) ( ) h ( a ) ( ) ( Pa ) ( mm) t mm σ ,bukl. a P T = 700 K σ P q w / m = σ t Wth assmbly at T =700K, a vry thn t 0 0.mm s aptabl, but may b qustonabl on robustnss. Also, σ s too hgh ( ) h ( a ) ( ) ( Pa ) ( mm) t mm σ ,bukl. a P T = 800K σ P q = σ t If T 0 =800K, s fasbl, thn t = 0.3mm s aptabl. σ OK (n tnson) also Wth th assumd l=4mm, buklng s not a problm n any as, but omprssv falur s hard to avod. It may b possbl to xd th last lmt and go nto plast omprssv yld f dutlty s hgh nough to nsur no ruptur. But ths mans no rusablty. 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of

85 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 0: Ablatv Coolng, Flm Coolng Transnt Hatng of a Slab Typal problm: Unoold throat of a sold propllant rokt Innr layr rtards hat flux to th hat snk. Hat snk s T gradually rss durng frng (60-00 s). Pak T of hat snk to rman blow matl. lmt. Bak T of hat snk to rman blow waknng pont for strutur. Prototyp -D problm: Can b solvd xatly, or an do transnt -D numral omputaton. But t s usful to look at bas ssus frst. Thrmal ondutan of B.L.=hg k Thrmal ondutan of front layr = δ Thrmal ondutan of layr k = ( δ = thknss, δ = thrmal ondutvty) k 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of

86 k Want layr to hav hg to prott th rst. δ k W/m/K k W (Say, porous, Orntd graphyt, = 330 3mm δ = δ mk W hg 50,000, so OK hr). mk ompard to Also, from govrnng quaton w s that T T T T ρ = k = α t t x x k ( α=, thrmal dffusvty, m / s ) ρ x x αt, or x αt, or t. α So th layr wll adapt to ts boundary ondtons n a tm δ t. α Say, J 70 and KgK 00 Kg m ρ ( 3 sold graphyt), so 6 α= =.3 0 m / s Th layr adapts n 3 ( 3 0 ) t = 7.0s (mor lk δ 4α =.8 s ). Trat front layr quas-statally,.., rspondng nstantly to hangs n hat flux: k T T ( t) ( t) wh w δ q ( t ) Ths also mans w an lump th thrmal rsstans of BL and st layr n srs: h ( hg ) ff δ + k g 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of

87 k and sn hg δ, k ( hg) ff δ h g For layr (th hat snk), layr) so, vry lkly, k s hgh (mtal) and ( h ) s now small (thanks to st g ff k δ ( h ) g ff (For nstan, say Coppr, k W 360, wth δ = m. W now hav mk k ( h ) W k = 350, but mk δ g ff δ = W 36,000 mk k, so ndd, ( h ) δ ). g ff Undr ths ondtons, th hat snk s bng trkl hargd through th hgh thrmal rsstan of layr. Most lkly, hat has tm to rdstrbut ntrnally, so that T s narly unform aross th layr. W an thn wrt a lumpd quaton. k ( ) ( ) ( ) dt ρ T δ = q = hg T ff aw T Taw dt δ ρ δδ dt τ= τ + = = k dt ( T ) Dfn T T T ( 0) aw 0 ( ) T = T T T τ aw aw 0 t For our xampl, say 3 J ρ = (Coppr), 8900Kg /m = 430, δ = m KgK τ= =30 s 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 3 of

88 Ths s omfortabl. Suppos Taw = 3300 K, T0 = 300K, and w fr for 0 s: (60) ( ) (60) 0 (989) T 0 = = 50K May nd 4 m whh s stll (OK) for Coppr (mlts at 360K, but no strss barng, so an go to ~900. Also OK for stl on Carbon str mmbr). NOTE: ( 0.0) δ = =.s, so, ndd, layr adapts qukly to B.C. s 4α unform k = = m / s. ρ A Mor Exat Soluton Consdr T turnd on at t=0. Th B.L. has a flm offnt h, and th frst aw δ k ( h ) layr has,, so that h g g = ff δ δ + h g k k, ρ, σ, α. Th bak s nsulatd. k. Layr has thknss, and has Thn on an prov that layr has a tmpratur dstrbuton g δ 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 4 of

89 ( ) αt λn aw a δ δ n os Taw T 0 δ n= T T x,t x = λn whr a n = snλn λ + sn λ os λ n n n and λ (n=,, ) ar th roots of n λ tan λ = n n ( h ) k k g ff k δ δ δ Graphally, k δ For small k δ λ λ, small, so tan λ, so λ λ = k k δ δ and also a k α k δ / ρ k λ = τ k δ δ δ ρ δδ from bfor So, ladng trm s thn ( ) t Taw T x,t x τ δ os λ Taw T0 δ 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 5 of

90 whh s what w found bfor. Th othr trms ar muh smallr, xpt at vry small tm. For thrmal protton of sold rokt nozzls rad s. 4. (pp ) of Sutton-Bblarz, 7 th d., spally, pp A ky onpt s ablatv matrals. Thy ontan a C-basd homognous matl. mbddd n rnforng fbrs of strong (ansotrop) C. Bst s C/C, strong xpnsv fbr sn nozzl an gt to 3600 K, an b D or 3D. Also good s C or Klvn (Aramd) fbrs +phnol plast rsns (for larg nozzls) For th shuttl RSRM, th throat nsrt (C loth phnol) rgrsss ~ nh/0 s, and th har dpth s ~ 0.5 nh/0 s. 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 6 of

91 Flm Coolng of Rokts For applaton of data on slot-njtd flms, w nd to dfn th ntal flm thknss s, vloty uf, dnsty ρ F, or at last mass flux u F ρ F. m m Assum w know th flow rats and F, whr s th or flow and F th flm flow. W also know th fully-burnt tmpraturs and molular wghts ( T, T F ; M, M F ). Th aras oupd at th fully burnt ston ar not known; lt thm b A, A. From ontnuty, m m F m m R = = T ρ P M ua P = P s ommon to both () 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 7 of

92 m m R F F F F = = TF ρf P MF ua () and th total ross-ston s known: A + A = A (3) F W nd som addtonal nformaton to fnd (ngltng frton): u F. Th two momntum quatons ar du dp ρu + = 0 dx dx duf dp ρfuf + = 0 dx dx du u = ρfuf ρ dx du F dx udu F udu F = ρ ρ F (4) Both, ρ F and ρ, hav bn volvng as drops vaporat and burn. W mak now th approxmaton of assumng thr rato to rman onstant (qual to th fullyburnt valu). Thn (4) ntgrats to uf uf u = ρ ρ = ρ F u ρf (5) Substtut nto th rato ()/() ρ ua m ρ ρ A m = = ρ ρ ρ m F F F F F F ua F A m F or A A F m = m F ρ ρ F (6) ρ and also F u F F u = ρ ρ ρ (7) Ths last rato ρfu ρu F s alld th flm oolng paramtr, M F : 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 8 of

93 M F ρf = = ρ M M F T T F Th flm thknss s (at omplt burn up) follows from (8) F =π ( ) AF D 4s = ( f s D ) A D s D =π( ) A D D s A D s s D A 4A F D m = 4 m F ρ ρ F (9) From Rosnhow & Hartntt, Chaptr 7-B, w haratrz flm oolng by th T for hat flow. In th absn of a hang t ndus to th drvng tmpratur ( ) aw 0 γ flm, Taw = T + r M 0 T aw F T aw 0, and w alulat ( qw) = hg ( Taw T w ) No Flm hangs to (lowr, prsumably). Th lowst w ould w dfn a flm oolng ffny F aw. Th flm T F T to gt s, so η= T 0 F aw Taw T aw T F (0) Lmts: F 0 η = 0 f Taw = T aw (no fft) F η= f Taw = T F (max mum fft) If w an prdt η, thn = η( ) F 0 0 aw aw aw F T T T T () and thn F w g ( aw w ) q = h T T () whr hg paramtr s omputd as f thr wr no flm. To prdt η, w frst omputs th 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 9 of

94 x µ F ζ= RF Ms F µ 4 (3) whr x s th dstan downstram of th flm njton (hr w assum ths s from th burn-out ston), and R = F ρ us F µ F F (4) and ρ u = M ( ρ u ), from bfor F F F From ζ, thr ar svral sm-mpral orrlatons for η. A rommndaton from R & H s 3 r.9p η= 0.39 p ζ p F (5) (or η = f ths gvs >) whh s supportd by ar data of Sban. Exampl T M 0.8 ; MF.6.65 T = M = ρ ρ = 0.5 = = = Say F F F m F m = mf 0. = (0.0) 9 m (0.00) Say D=0.5m x x = 0.5m t ompl.omb 6 P=70 atm= N / m T = 300K ρ = = 5.33K g /m ; ρf = 8.53Kg /m M = 0g/mol; γ =. 3 3 M = , Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 0 of

95 8.34 u = = 53m / s 0.0 uf = 53 = 00m / s s 7 Say µ F = 0 Kg/m/s RF = = s 5 0 DmF 0.5 s = ρ 0.00m 4 = F = ρ m OR 0.00 R = F 5 ( ) 0.6 µ F T = F 0.6 = = µ T ( ) 5 ( ) ζ= = (5.74) p pf µ F = 0.8 (say, rf r), µ Pr = η= =.4 η= ( 5.74 ) 0.8 So, ths offrs prft flm oolng, manng F T Taw = TF = = 600K ( ( )=96 K) If th wall s mad of Cu, and s at T w = 700K, th rduton n hat flow s F w 0 w q = = q = whh an b dsv. 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of

96 (Ths xampl shows on ould gt good flm oolng wth muh lss than 0% flow n th flm, mayb wth only %). 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of

97 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur : Radaton Hat Transfr and Coolng Radatv Losss At throat of a RP-LOX rokt, valuat radaton hat flux P = 70atm D = 0.m T = 3500 K O/F =. γ=.5 M=5 g/mol x 0.38 o x = 0.3 HO x 0.4 H x 0. o P P P γ γ = γ+ = atm throat o P HO P H Po = 4.8 atm =.0 atm = 5.4atm = 4.3atm Tthroat = T = 3K = 5600R γ+ Assum slab f thknss L=0.9 ( PL) o ( PL) HO ( PL) H ( PL) o = 9. ft atm = 7.5 ft atm = 3.4 ft atm =.7 ft atm Dt = 0.9m = 0.63 ft 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 4

98 CO Fg 4- for CO gas only to 400 R ft atm So, xtrapolatd to 9. ft atm, ε ( 400R) 0. at 400 R. But ε falls rapdly wth T. If w onsrvatvly xtrapolat lnarly n Log ε (T). ε o would appar to go to ~ or so. Hn, vn though th gas s CO-rh, radaton by CO s nglgbl. HO At P = atm, PL 7.5 ft atm, Fg 4.5 gvs T a bt to T=5600R, εho 0.5. = ( ) ε 5000R = 0.8, and xtrapolatng Fg 4.5 gvs for P 0,P = atm. To orrt for fnt P and hghr P, us εw w T P 4.6. Hr, for PL 7.5ftatm, thr s som sgnfant fft of w Ho = w T Pw + PT = = 5.5 atm way byond th graph. w + P T. W hav Pw + PT β= w 0.5 () (.4).3 (.)..8 (.8) , Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 4

99 w ( ) n = P + B n n B = B +. 8 = ln ln n B = ( B + 0.5) n = = n 0.5. n B +..8 ln + ln +.8 = ( B +.) = B 0.57 B B. ln B B = 0.05 n = = =.439 = n = 0.3 ln + ln ln B + + B B w ( ) 0.3 =.75 P Thn, for P = 5.5, w = 3.33 ε w = = Suspt! too muh CO For PL=.7 ft atm, T=5600 R εco From fg 4.4, Corrton. ε 0.06 CO For ntrfrn, us Fg 4.7 Pw = = P + P w ( ) P + P L = = 0. ft atm r w ε 0.06 So ε gas Ths s lkly to b an ovr stmat, baus 0.3 T grow as P. ε must saturat as P nrass, not HO T 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 3 of 4

100 4 8 4 Wth ths ε σ T = =.66 0 W / m g t 6 Compar to Convton: Say T = 000K * = w R T δ s ( u) p µ <> <> g Dt Pr ( ) h = ρ * = = 640 m s < T >= = 056 ( ρ ) 5 P 70 0 u = = = 469Kg /m /s * u = Kg /m /s <> 856 = ( ρ ) µ <> 6 0 = Kg /m/s 3000 Sp = 663 J / Kg / K Pr = 0.8 ( ) hg = = 345, = 8960 W / m / K ( q ) 8960 ( ) =.4 0 W / m w onv So q q rad onv = , Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 4 of 4

101 As PT nrass, ah ndvdual msson ln s broadnd by ollsons, and nrass. Howvr, whn PL s rlatvly larg ( >.5 ft atm ), Fgur 4.4 shows th fft s small; ths s baus at that PL th bands ar largly ovrlappd alrady and only th broadnng of thr dgs mattrs anymor. So, w gnor th fft. ε P T Efft of Partulats R For π R, gomtral opts λ For π R, Raylgh rgm, partl appar to b smallr by λ 4 πr λ. For 3000 K, pak of sptrum at λ. µ m λ Rross ovr = 0 µ π.4 m Partls tnd to b nar (somwhat blow) ths valu. For onsrvatsm, assum gomtral oultaton. L mfp nql p P α = Prob. of absorpton = -Prob. of transmsson= = n P x ρ = = x 4π 3 gas Qp π R p 3 Rpρs εp ρ x g 3 L x ρs 4Rp 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 5 of 4

102 6 Say L = R = 0.3m R = µ m = 0 m t p ρ g = = 3.75Kg /m ρ s = 3000Kg /m x= εp = = 4 4 πrp 0. Now, suppos R nstad. Exponnt has a fator p = 0.µm = = 0.4, λ 56 and has a, whh s anothr 4.69 = smallr = 0.99 stll R 0. In ths as p (a) In flam looks sold (blak body radator) (b) Radatv losss doubl ( ε = nstad of 0.5), to ~0% of loss 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 6 of 4

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110 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur : Rvw of Equlbrum Thrmohmstry Thrmohmstry of Combuston for Propulson Dvs.. Introduton Th gnral prnpls that govrn hmal ratons (sps onsrvaton, nthalpy balan and hmal qulbrum) hav bn ntrodud n prvous subjts. In th nxt thr Stons w wll brfly xamn th applaton of ths prnpls to Propulson. Th skth n Fg. ndats th man rgons. Fg... Man rgons n a lqud-ful ombustor In a ombuston hambr whh ould blong to a ramjt or rokt, or, wth mnor varatons, to many othr thnal dvs whh utlz hot gas flows. It s frqunt to supply ful (and somtms oxdzr as wll) n lqud form, and so th frst rgon nvolvs a srs of olldng lqud jts or sprays, dsgnd to brak up ths lquds nto fn droplts. Ths now fnd thmslvs mmrsd n th hot ombuston gass, and vaporat rapdly, a stp whh must nssarly prd atual hmal raton. Mxng of th oxdzr and ful vapors typally prods va th strong turbuln xstng n th ombustor, and on ths mxng has atually 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 0

111 ahvd ntmat molular lvl ntrdffuson wthn th turbulnt dds, raton ours usually on a tm sal whh s short ompard to thos for vaporzaton and mxng. Th ombuston hambr dsgn must tak nto aount th knts of ths stps to arrv at a dmnson whh guarants omplton of th ombuston ratons wthout wast of volum. If ths s so, th last porton of th ombustor ontans at any tm a mxtur of ombuston produts whh hav rahd a stat of thrmodynam qulbrum, and whh ar about to flow nto thr workng dvs, b t a smpl nozzl, a turbn (also prdd by ts nozzls) or othr onfguratons. Equlbrum dos not nssarly man omplt ombuston sn, at hgh tmpraturs, varous domposton ratons ar thrmodynamally favord. In a hgh spd Ramjt wth a subson ombuston hambr th nompltnss of th ombuston s so xtrm du to th hgh tmpratur prvalng- that, byond a rtan flght spd, no nt hat rlas an b obtand. It s mportant to dstngush btwn ths knd of ombuston nompltnss, whh s unavodabl at hgh tmpraturs, and s du to th atvaton of rvrs ratons, and a mor ommon but avodabl nompltnss, whh ours whn th ombuston hambr s nadquatly szd and th vaporzaton-mxng-raton prosss ar not ompltd n th rsdn tm allowd to ah flud partl. A frst objtv of our analyss s to dtrmn th atual gas ondtons (tmpratur, omposton) at th xt of an qulbratd ombustor, for a st of prsrbd ratant flow rats. Ths flow rats, togthr wth th dsharg nozzl gomtry, also dtrmn a ombustor prssur, whh w wll hr assum as an ndpndntly prsrbd quantty. Byond ths pont th flow typally xpands and ools mor or lss adabatally. As t dos so, thr s at last a tndny to shft ts hmal omposton, usually n th sns of ompltng th nomplt ombuston, sn th lowr tmpratur now nhbts domposton ratons. Thus, thr may b a sort of aftrburnng fft along th nozzl xpanson, and ths somtms ontrbuts sgnfantly to nras th prforman of rokts or ramjts. It s worth notng hr that, vn f all th ratons that wr not ompltd n th hambr wr ompltd n th nozzl, thus stll rlasng all th hat of ombuston, th prforman lvl ahvd would b lowr than f no domposton had vr ourrd n th frst pla; ths s baus n any thrmodynam yl lss work an b xtratd from hat addd at lowr than at hghr tmpraturs. In any vnt, t oftn happns that th nozzl rombnaton ratons ar too slow (du to thr ourrng at a rdud tmpratur, whh slows down raton rats) to kp pa wth th rapdty of th nozzl xpanson pross. In that as th flow s sad to b hmally frozn, and t oms out of th nozzl xt n a stat of substantal nonqulbrum, mantanng a omposton los to what t had n th hambr. Ths s th oppost xtrm to th as whn all th rombnaton ratons atually do our along th flow (qulbrum xpanson). Atual prforman always falls btwn ths two xtrms (mor prforman than that for frozn xpanson, lss than that for qulbrum xpanson), and sn th sprad btwn thm may not b too larg, t s usful to brakt th atual rsult by prformng both frozn and qulbrum flow alulatons, both of whh, as w wll s, nvolv only onsdraton of th nd ponts of th xpanson. Anythng n btwn ntals muh mor omplx 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 0

112 alulatons, du to th nd to follow n dtal th knts of a varty of hmal ratons... Combustors Far from Stohomtry Stohomtr proportons of ful and oxdzr ar thos for whh, f th raton wr omplt, no xss of thr would nsu. It s lar that ths would thn nsur maxmum hat rlas pr unt mass, and hn hghst adabat flam tmpratur. Evn undr ondtons of nomplt raton, maxmum tmpratur usually dos our nar stohomtr ondtons, and falls off for thr ful-rh or ful-lan ondtons. For som mportant typs of ombustors th ful/ar rato s purposly sltd to b vry far from stohomtr, so as to kp th fnal tmpraturs blow aptabl matrals lmts. Exampls ar: (a) Jt ngn ombustors (opratd ful/lan) (b) Gas gnrators for rokt turbopumps (opratd typally vry ful-rh). In ass suh as ths th fnal tmpratur and othr proprts of th ratd gass an b obtand from smpl mass and nrgy balans, sn thr omposton an usually b obtand by nspton by assumng omplt ombuston wth xss oxdzr or ful, as th as may b. Ths s baus domposton ratons rsponsbl for ombuston nompltnss ar not vry atv at ths modrat tmpraturs. Exampl Th lqud hydrogn ful n th Spa Shuttl man ngn s prssurzd to th vry hgh ombuston hambr prssur by a turbopump whh s drvn by a turbn. Th gas drvng ths turbn s gnratd n a prburnr, nto whh th flow rats of oxygn and hydrogn ar (at maxmum powr) m L = 43 Kg/s OX m LH = 40. Kg/s Sn ths s vry ful-rh, w an safly assum all of th hav th raton xh + O HO + ( x) H whr th x s gvn by th oxdzr/ful rato: O s onsumd, and w 3 43 = 8 = 40. x. x x = 7.5 So that 7.5 H + O HO + 6.5H 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 3 of 0

113 Th adabat flam tmpratur orrspondng to ths raton s obtand by statng that th ombuston pross s adabat: Enthalpy bfor raton = Enthalpy aftr raton Bfor raton, assumng th and O ntr th prburnr vry old (say, for an H approxmaton, at O K), w rad n Rf.., h o h H = -868 J/mol, = J/mol, for a total nthalpy h bfor = 7.5 x (-8468) + (-868) = -67,850 J Th tmpratur aftr raton s suh as to gv ths sam nthalpy. Wth som hndsght (w know th turbns ar unlkly to b dsgnd for mor than 00-00K), w try 000, 00 K, and usng th tabls n VW-S (Rf..), App. A. w obtan T (K) h HO -4,87 + 5,978-4, h H 0,678 3,73 h = h + 6.5h -8,390-57,46 HO H Intrpolatng lnarly gvs T = ,540 3,939 = 057 K Th man molular mass of th gas s smply M = 8 x + x = 4.3 g/mol W an also alulat asly th man spf hats p and v and thr rato, γ. For ah of th omponnts, Ho and H, w an approxmat = p h T as p ( ) ( ) h 00 h 000. Ths gvs ( p ) 00 = 4.89 J/mol K, HO ( p ) H = J/mol K, so that p 4.89 x x 6.5 = 7.5 = 3.9 J/mol K 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 4 of 0

114 Sn for ah onsttunt = R, w also hav v p v = p R = = 3.60 J/mol K, gvng p 3.9 γ = = =.35 v 3.60 (Not γ s not th avrag of th ndvdual γ s)..3 Ratons Clos to Stohomtry (or wth Prhatd Ratants) In othr applatons, whr matrals lmts ar thr absnt or an b ovrom by approprat oolng of walls, th ratants may b n proportons los to stohomtr, so as to gnrat vry hgh gas tmpraturs. Exampls ar: (a) Rokt ombustors (opratd typally on th ful-rh sd, but only modratly so) (b) Ramjt ombustors (whr, n addton, th ar s strongly hatd by th ram fft) () Pston ngns (ful-lan for gasoln ngns, but nar-stohomtr n Dsl ngns) (d) Wldng torhs In ths ass on annot asly prdt th fnal omposton of th burnt gass, sn th rlatvly omplx moluls lk Ho, CO (or vn H, O ) tnd to brak down nto smplr ons whh ar mor stabl at hgh tmpraturs (suh as OH, H, O, t.). Thus, n addton to mass and nrgy onsrvaton, on nds to us also th laws of hmal qulbrum to dtrmn th tmpratur and othr proprts of th gas produts. In most ral stuatons, many molular sps may b prsnt at th hgh tmpraturs of th flam, and so mor than on qulbra ar smultanously to b onsdrd. In ths ass, traton s rqurd to obtan onsstnt solutons. A produr whh, wth som judgmnt, usually works, s as follows: () Idntfy th possbl molular sps n th produts, lassfy thm nto major and mnor (mnor sps would b thos that ar xptd to b prsnt n small onntraton). () Nglt at frst th mnor sps, us produrs smlar to thos n th prvous xampl to obtan th major sps onntraton and an approxmat tmpratur. (3) Wrt hmal ratons whh would produ on mnor sps at a tm from only major ons. Calulat thr qulbrum onstants at th approxmat tmpratur of stp (), and us th qulbrum laws to alulat th onntratons of th mnors. (4) Corrt th majors onntraton, and do a nw nthalpy balan to obtan an mprovd tmpratur. 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 5 of 0

115 (5) Itrat bak to stp (3) untl onvrgn s obtand. W llustrat ths prnpls blow. Rf... G. Van Wyln and R. E. Sonntag, Fundamntals of Classal Thrmodynam, Appndx A., 3 rd Ed., SI Vrson, J. Wly & Sons, Exampl Calulaton of Thrmohmal Equlbrum n a Rokt Chambr.. Introduton In th Spa Shuttl s Man Engns (SSME S), th rokt ombuston hambr oprats at p = 0 atm and an oxdzr-to-ful mass rato O/F = 6, th ratants bng ryogn and O. Th stohomtr rato for full ombuston H H + O HO would b O/F = 8, so ths s now only slghtly ful-rh, and w xpt a vry hgh ombuston tmpratur. In th nxt fw stons w wll analyz ths as stp-by-stp, n ordr to brng out th mportant ponts of any suh alulaton. Pratal problms of ths natur ar routnly solvd by mans of omplx omputr programs basd on th prsnt mthods or on mor gnralzd traton shms, but, of ours, ntalng th sam bas physs. On suh standard od s th so-alld CEC, Chmal Equlbrum Cod, dvlopd and mantand by NASA (Rf..), whh n ts normal onfguraton, an handl about 00 sps smultanously... Atom Sps Conntraton In ordr to mor asly kp trak of mass onsrvaton, t s usful to work wth th numbrs of mols, of th varous hmal produt sps n an arbtrary total n mass of gas. Ths sam mass of gas orgnats from a st of numbrs of mols ratants, and w must mpos that, no mattr what raton shfts our, th numbr of atom mols for ah knd of atom (O, H, N, C, ) s th sam bfor and aftr raton. As an xampl, suppos w hav ombuston of hydrogn n oxygn, wth unknown amounts of watr, molular and atom oxygn, molular and atom hydrogn and hydroxyl (OH) radals formd. Th oxdzr/ful rato s prsrbd to b, say, R, so that, f w arbtrarly slt = /, thn N O n j of 3 x / = R N H ; N H 8 = (.-) R Th raton s 8 H + O nh OHO no O no O nh H nhh no OH H R (.-) Not that th lst of anddat raton produts s arrvd at from our bakground knowldg of what sps atually do our n sgnfant quantts. For nstan, w mght not nlud or O f R wr low nough to mak th mxtur O ful-rh, suh that undr thos ondtons, oxygn sps would b manly dpltd. 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 6 of 0

116 Howvr, f w stll dd nlud thm, wth th propr qulbrum onstants, w would stll fnd som small onntratons of ths moluls prsnt. W would smply gnor thm for pratal purposs, as w gnor othr xot, but potntally ral sps, lk HO, H3, t. Th atom sps ar H and O and n ordr to onsrv atoms, w must mpos: H-onsrvaton: 6 nho n H n H noh R = (.-3) O-onsrvaton: = n + n + n + n (.-4) HO O O OH..3 Equlbrum Rlatons Ths ar two quatons nvolvng th sx unknown th prsn of th domposton produts O, H and OH, (plus H s. Clarly baus of f ful-rh or f lan), onsrvaton alon annot tll us th omposton of th burnt gas. W nd four xtra rlatonshps, and ths ar th qulbrum ondtons (laws of mass-aton) for four lmntary ratons sltd so as to nvolv ths domposton produts, as wll as som of th major produts, HO, CO, H. Ths ould b, for xampl, HO H + OH (.-5) n O H H HO H + O O O (.-6) (.-7) (.-8) or any lnar ombnaton of thm. For ah of th lmntary ratons abov, an qulbrum rlatonshp must b satsfd. Aordng to Ch. 6, ths tak th form P / H P P HO OH = K ( T) (.-9) P P H H = K ( T) (.-0) P / H O P P HO = K 3 ( T) (.-) 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 7 of 0

117 P P O O = K 4 ( T) (.-) whr ah P rprsnts a partal prssur. Ths an b xprssd n trms of th gvn total prssur P and th mol numbrs n : n P P ( n= n + n + n + n + n ) n HO HO= HO O H H OH P O no = P n noh P = P (.-3) n OH... And thrfor th qulbrum rlatons bom / nh n OH n = K T n p HO ( ) (.-4) nh n = K T n p H ( ) (.-5) / nh n O n = K 3 T n p HO ( ) (.-6) no n = K 4 T n p O ( ) (.-7) Th qulbrum onstants,,, K stll dpnd on tmpratur. If K K K3 4 tmpratur s known a-pror (or assumd at som stp n an traton pross), ar an thr look thr valus up n som omplaton of hmal proprts (suh as Tabl A. of Rf..), or mor ratonally, gvn th many possbl ombnatons of 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 8 of 0

118 lmntary ratons, thy an b bult up from tabulatd valus of th standard hmal potntals (or Gbbs fr nrgs) of th ntrvnng sps. Th qulbrum onstants ar gvn n our xampl by K µ + µ µ = xp RT H HO HO (.-8) K µ H µ = xp RT 0 0 H (.-9) K 3 µ + µ µ = xp RT H O HO (.-0) K 4 µ O µ = xp RT 0 0 H (.-) whr ah µ 0 ( T) s th hmal potntal of sps at atm. prssur (standard hmal potntal), at th gvn tmpratur T, and R s th unvrsal gas onstant n onsstnt unts...4 Consrvaton of Enrgy Sn th ratons our at onstant prssur, f w assum a wll-nsulatd h ombustor, th nthalpy of th produts must qual that of th ratants. If ( ) th molar nthalpy of th xampl: th sps at tmpratur T, w must hav n our T s 8 hh T ho T nh OhHO T no h O T n OhO T R ( ) + ( ) = ( ) + ( ) + ( ) ( ) ( ) ( ) + n h T + n h T + n h T (.-) H H H H OH OH Not th lft hand sd of ths quaton rprsnts th nthalpy of th ratants ( 8 R mols of H, mol of O ) at thr njton tmpratur, T, whras th rght 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 9 of 0

119 hand sd rprsnts that of th produts, wth ah valuatd at T, th fnal qulbraton tmpratur, to b found. Ths s what s oftn alld th adabat flam tmpratur for th gvn O/F and prssur. h 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 0 of 0

120 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 3: Exampls of Chmal Equlbrum 3. Numral Itraton Produr For th Shuttl man ngn, w tak R = O/F = 6, and P = 0 atm. To gt startd, w nglt n, n, n and n, so that Eqs. (.-3) and (.-4) rdu to n O O H 8 + n H = 3 HO nho = OH whh gv nho =, n = /3 and (sn all othr n s ar takn to b zro, n = 4/3). H 0 Th nthalpy bfor raton was (assumng vry old ratants,.. T = 0 K), 4 H= hh ( O) + h ( ) O O = -5,63 J (for th mols of H, mol of O ). For th produts to hav th sam nthalpy, th tmpratur must b vry hgh. Usng th tabl n Rf.. (pp ) w hav th tral valus tabulatd blow: T(K) h HO (J/mol) -9,973-58,547-46,94 h HO (J/mol) 03,738 6,846 34, ,65 -,04 hho + h H 3 Sn w must obtan an nthalpy of -5,63 J, ths valus ndat a tmpratur vry los to 4000 K. Lnar ntrpolaton btwn 4000 and 400 K gvs T = 4009 K W an now us ths tmpratur to alulat th four qulbrum onstants through K. Ths ar gvn n Tabl of Rf... Wth som ntrpolaton, w K 4 obtan: / K = ( atm ) K =.6 ( atm ) / K = ( atm ) K =.86 ( atm ) , Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of 5

121 At ths pont w an obtan our frst nonzro stmat of th mnor sps onntratons. From Eqs. (.-4) through (.-7), n HO n 4/3 noh = K = = 0.38 / n p /3 00 H nnh = 4/3 x /3 nh K = x.6 = P 00 n HO n 4/3 no = K = ( ) = nh p / 3 00 nno = 4 / 3 x 0.0 no K4 = x.86 = 0.08 P 00 W ar now at th nd of th frst loop of our traton produr. W an nxt r-alulat and from th atom onsrvaton quatons, nludng th n HO n H nw mnor n s w just omputd. Wth th nw omplt st of n s, w an ralulat th nthalpy at a fw tmpraturs and ntrpolat for a nw st of t. W wll do on mor yl n som dtal to llustrat th natur of th typal rsults and th way thy may tnd to osllat or dvrg. Byond that, a tabulatd summary of th sudng rsults wll suff. K j s, n HO n H Corrtd,, n: From Eqs. (.-3) and (.-4), nho + n = = H.6 3 n = x = 0.80 HO Hn nh = = 0.44 and n = = , Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of 5

122 Corrtd tmpratur: Sn th partal domposton of HO and H nto th mnor sps s an ndothrm pross, th nw tmpratur wll b lowr. Usng th tabls of nthalps and th omputd mol numbrs w alulat h(800 K) = 0.80 x (-6,533) x 8, x, x 69, x 90, x 30,587 = -,338 J h(3000 K) = 0.80 x (-5,466) x 88, , x 74, x 98, x 305,77 = -8,769 J Sn w stll want h = -5,63 J, lnar ntrpolaton gvs a tmpratur 00 T = ( 5,63 +,338) = 899K 8,769 +,338 It s lar that at ths nw, muh lowr tmpratur, thr wll b muh lss of th mnor sps, sn th nw qulbrum onstants, ( through K ) wll b smallr. K 4 Sn t was th prsn of ths mnor sps that ford a rduton from 4009 K to 889 K, w would nxt obtan a rfnd T agan rlatvly hgh, and th pross s lkly to ovrshoot at ah traton stp. Ths suggsts that w an alrat onvrgn by adoptng as a nw tral tmpratur th avrag of th last two: T = = 3454K Corrtd mnor sps: Th nw qulbrum onstants (at 3480 K) ar: K K3 = 0.5( atm) / K = 0.36 ( atm) = 0.84 ( atm) / K = ( atm) 4 and, prodng as bfor, noh = nh = no = no = Tabl 3. summarzs ths two tratons, and shows th nxt fw tratons as wll, ladng to th pratally onvrgd valus of th last ln. Th mol numbrs ar thn onvrtd to mol fratons by smply dvdng ah by n (Tabl 3.). Not that th numbrs of mol gvn n Tabl 3. hav bn onvrtd to mol/kg, by dvdng th n numbrs usd so far (whh orrspond to 8 x x 0.06 = Kg of ratants) by ths mass Kg , Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 3 of 5

123 ITERATION N HO (mol/kg) N O N O N H N H N OH T (K) T + TOLD K K / / ( atm ) ( atm ) ( atm ) ( atm) K 3 K Rvsd Tabl 3. Th man molular wght of th gas s thn obtand from Tabl 3. by: M = x M = 3.48kg /Kmol An approxmat molar, spf hat for ah sps an also b obtand as h (3800) h (3400) / 400, and ths an also b avragd to obtan p = = J x 50.6 =. p p and thn th avrag spf hat rato s al mol K ( mol) K ( ) p 50.6 γ= = =.96 p R Fnally, for purposs of alulatng th flow n th rokt nozzl, t s usful to dtrmn th ntropy of th ratd gasous mxtur. W an rfr ths to a unt of mass, or, altrnatvly, to th sam arbtrary amount of mass w hav so far workd wth,.., 4 mols of H 3 and mol of O (w should not alulat t pr mol, sn th numbr of mols n ths muh mass may hang from th prsnt valu n =.384, du to ratons ourrng n th nozzl). Tabl A. of Rf.. gvs 0 valus of th spf ntrops s of th varous sps at atm, as a funton of tmpratur. W an asly orrt ths to th propr prssurs by usng ( ) 0 ( ) S T,P = S T R ln P (3.) whr P = Px s th orrspondng partal prssur. Th rsults ar summarzd n Tabl , Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 4 of 5

124 Sps TABLE 3.3 Entrops at 3630 K Entropy at atm KJ / (Kmol)K Partal Prssur atm HO H Entropy at own prssur KJ / (Kmol)K OH H O O Th total ntropy n our ontrol mass (4/3 mols of plus / mol of O, n th form of th N s lstd n th last ln of Tabl..) s thn S = s N = 7.050KJ / Kg / K H Rf.. Computr Program for Calulaton of Complx Chmal Equlbrum Compostons, Rokt Prforman, Indnt and Rfltd Shoks and Chapman Jongnt Dtoratons, by S. Gordon, NASA Asson No. M84-06, NASA Lws Rsarh Cntr. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 5 of 5

125 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 4: Non-Equlbrum Flows Ratng Nozzl Flow 4. Introduton As ndatd n th ntrodutory dsusson of ston., th atual xpanson pross n a rokt or ramjt nozzl s ntrmdat btwn th xtrms of frozn and qulbrum flow, wth th lattr produng hghr prforman du to rovry of som of th hmal nrgy td up n th domposton of omplx molular sps n th hambr - a knd of aftrburnng fft. Th two lmts, frozn and qulbrum flow shar an mportant proprty: both ar sntrop flows (f w gnor frton or hat losss). Ths s baus, n th frozn as no hmal hang durng xpanson), thr ar no rat prosss at all ourrng, th moluls prsrvng thr dntty all th way, whl n th qulbrum as, n whh ratons do our, thr rat s so hgh (ompard to th xpanson rat) that ondtons adjust ontnuously to mantan qulbrum at th loal prssur and nthalpy lvl, wth th rsult that th whol pross an b rgardd as rvrsbl (and hn sntrop). For any mor ralst ntrmdat ondtons, n whh ratons may prod at rats omparabl to that of th xpanson, ths fnt rat ratons produ an rrvrsblty, and a onsqunt ntropy nras. In ths Ston w wll dsuss n dtal an xampl of ah, frozn and qulbrum xpanson. W wll us for ths purpos th rsults of th qulbrum hambr alulatons of S. 3., rlatv to th spa Shuttl Man Engn. 4. Frozn Flow Calulaton Th smpl dal gas modl w hav usd throughout most of ths ours (onstant molular mass, onstant spf hats) s an xampl of a frozn flow modl, sn t s mpld by ths assumptons that no hmal hang taks pla. Thus all of our onstantγ rsults blong n ths atgory, and an b usd as a frst approxmaton for nozzl flow alulatons (usng for nstan th valu of γ and M omputd for th ombustor). Howvr, vn wth no hmal hangs, th spf hats of th varous moluls do hang wth tmpratur, gnrally drasng as tmpratur drass n th rang nountrd n nozzls. Ths mans that γ= C C = C C R s not a onstant, sn, th gas onstant, dos not vary ( ) p v p p g du to th onstany of th molular mass. For a mor prs alulaton, w mak us of th onstant-ntropy ondton for frozn flow. Suppos, for nstan that th nozzl xt prssur s spfd. Ths prssur, togthr wth th hambr ntropy ar nough to dtrmn all othr thrmodynam varabls of th frozn gas at th xt plan, n partular ts nthalpy h pr unt mass. Sn th hambr nthalpy h o s alrady known, w an fnd th xt vloty by th stady-stat nrgy quaton: R g 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag of 7

126 h0 = h + u ( ) u = h0 h (4.) Th pross by whh h s found on P and S ar gvn dpnds on th data avalabl. Tabls or graphs (Mollr harts) ar th smplst mthod, but ths ar unlkly to xst for th partular omposton of ntrst. Mor fundamntally, on an rpat th stps at th nd of S. 3., whn S was alulatd aftr th gas omposton was onvrgd upon. In ths as, th omposton (.., x, x, x, t) H H OH s fxd throughout ("frozn"), and for ah P, w wll try varous tmpraturs untl S, th ntropy, quals th hambr valu. In addton to th xt ondtons, t s usually of ntrst to alulat th throat ondtons, sn t s th throat ara that dtrmns th mass flow rat. Two altrnatv produrs an b followd for ths: (a) Try a rang of prssurs about / th hambr prssur, alulat th orrspondng u = h h and dnsty ρ, and look for a maxmum of ρu. ( o Ths ours at th throat. ) (b) For th sam P rang, alulat th loal spd of sound, fnd whr t s qual to th loal vloty u. Th loal spd of sound s gvn n gnral by a = P ( ρ ) s (4.) and for frozn flow ths an b shown to hav th famlar form a = γr T (4.3) g Condtonal Throat : = ρ u A, u = ( h h) da dρ du = = 0 A ρ u at throat S = onst. d ρ du dh ρ dρ t = = = ρ t t t t t u h h h h u 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag of 7

127 So, dρ du dρ = = ρ u ρ u t t t t dp ut = u = t dρ P ρ s M t = (both, qul. and frozn) Frozn and Equl. Spd of Sound a ρ = ρdt = ρ s Tds RT + d ρ = s d ρ ρ ρ and P = R T g ρ Frozn: R = onst g dp dρ dt = p ρ T T C ρ dp d P ρ ρ = R T g P dp dp ( Rg + Cp) = Cp p dρ ρ C v dp Cp P ρ = = γrt a = = γ T RT ( ) g g dρ C ρ s v ρ s γ ( T) Equl. : R g = R M dp dρ dμ dt + = p ρ Μ T C P T dp dρ dμ + = p ρ Μ R g T p dp C v dp ρ Μ = C d d P ρ Μ p 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 3 of 7

128 dp P ρ dμ =γ( T) dρ ρ Μ dρ s In qul., durng xpanson Μ (rombnaton) and ρ (xpanson) dμ P < 0, a > γ dρ ρ whr th only novlty s that γ. γ tslf s varabl, and must b takn to b th loal A gnral alulaton produr an b as follows: () Gvn P 0, T 0, S 0, all x s, R () Spfy an xt prssur P 0 (3) Calulat T 0 =T(P 0, s 0 ; x ) (4) Calulat h, ρ, u = h ( h) 0 (5) Calulat th spf mpuls. For a rokt, gi sp ( ) F mu + Pg Pa A Pg Pa = = = u + ρ u A ρu m (4.4) whr P g s th xtrnal (ambnt) prssur? All th abov s ndpndnt F of sz. If a partular thrust F s dsrd (or an m = ), w must also gi fnd th rqurd throat ara A * : (6) Loat P x usng on of th produr (a) or (b) abov. * * ρ u = ρu,.., from mthod (a) abov. (7) Calulat ( ) (8) * m A = ; ( ρ ) µ MAX A g MAX * = ( ρ u) F (9) Thrust offnt = o = P A A MAX F * 0. ρ u sp 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 4 of 7

129 4.3 Frozn Flow xampl (Shuttl Man Engn) Usng a smpl omputr program, and startng from th hambr ondtons stablshd n S 3., th produrs abov lad to th rsults shown n Tabl 4.. W hav assumd mathd xt ondtons throughout (P g = P ) P /P P (atm) T (K) u (m/s) I sp (s) A /A * P * (atm) T * (K) TABLE 4. Frozn flow prforman of nozzl, from P 0 = 0 atm, T 0 = 3640K, (LOX LH, O/F = 6). * For rfrn, th Shuttl nozzl has an ara rato of A A = 76.5, whh would gv a (mathd) frozn spf mpuls of about 435 s, aordng to Tabl 4.. γ If w had smply usd th onstant approxmaton, usng (from S. 3.) =.9 and M = 3.48 g/mol, th xt vloty would hav bn smply: γ γ γ γ R = P u T0 γ M P0 For th as of P / P 0 =0.0005, ths would hav gvn an xt vloty = 436 m/s, whh s qut los to th 430 m/s shown n Tabl 4.. Thus, n ths as, onstant s a good modl, but ths may not b tru n vn hghr γ tmpratur ass, lk n ltrally hatd gass (ar-jt rokts). 4.4 Equlbrum Flow Calulaton Hr w must mpos at ah prssur blow P 0 th sam qulbrum ondtons that wr usd n S. 3., wth th dffrn that th ntropy, not th nthalpy, s now prsrbd. Th mol fratons ( x, x, x, t.) ar now H H OH varabl along th xpanson, and so ar thrfor th molular mass, th spf hat and γ. Th alulatonal dffulty rsds prsly n th nd to prform ths rpatd qulbrum alulatons, but, on agan, omputr programs ar avalabl to as th burdn. Othrws, th produr s ntrly analogous to that outlnd n S. 4. (stps () through (9)). Th on notworthy dffrn s that th spd of sound, f 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 5 of 7

130 rqurd, annot b alulatd as th loal valu of γ RT g, but must b found from th bas Eq. (4.). Also, of ours, th x valus ar undrstood to b unknown n prnpl, and ar, n fat rsults of ah loal qulbrum alulaton. 4.5 Equlbrum Flow Exampl (Shuttl Man Engn) For th sam hambr ondtons as n S. 4.3 and on agan assumng mathd xt ondtons throughout, th rsults of a alulaton usng a smpl omputr program ar as shown n Tabl 4.: P /P P (atm) T (K) X ( H ) ( x HO) ( x O ) ( 0 ) ( ) x x H ( ) H x U g (m/s) I sp (s) A / A * P * (atm) T * (K) x * ( H ) ( x HO) * ( x O ) * ( x ) * 0 ( x ) * 0H ( x ) * H TABLE 4. Equlbrum flow nozzl prforman LOX-LH, O/F = 6, P 0 = 0 atm, T 0 = 3640 K. 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 6 of 7

131 On agan, for th Shuttl, (A /A * = 76.5), w would obtan I sp 45 s n ths as, ompard wth 435 s for th frozn flow as. If w want to fnd th spf mpuls n vauum rathr than at th mathd prssur pont, w would smply add P g g ρ u g g (s Eq. (4.4)) and w would obtan ( I SP ) = 465 s. Ths s ndd va, qul qut los to th atual prforman of th Shuttl man ngns, ndatng that qulbrum dos prval durng xpanson. Ths s a rsult of th hgh prssur and larg sz of ths ngns, and would most lkly not b th as n a smallr, lowr prssur ngn. In addton to th hghr xt vloty n th qulbrum as, othr ntrstng dffrns btwn th rsults of Tabls 4. and 4. ar: (a) Th xt tmpraturs ar hghr n th qulbrum as by about 00 K (a rsult of th aftrburnng). (b) Th xt ara ratos ar largr n th qulbrum as (for a gvn prssur rato). Ths s also du to th sam rhatng fft, whh produs mor volum nras. Not, fnally, that th onstant γ approxmaton would stll gv u g = 436 m/s for th as of P /P 0 = ; ths s now wll blow th 45 m/s shown n Tabl 4. for ths as. Thus, onstant γ may srously undrstmat prforman n larg, hgh prssur rokts. 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 7 of 7

132 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 5: Slton of Propllant Mxturs Sold Propllant Rokt Fundamntals Sold Propllants Rokts Rad Sutton h. bas prforman Surfa rgrsson rat r, mprally orrlatd to gas prssur as = n t m =ρ p a P Ab = * n r ap (n < ) PA n * Ab P = ρp a A d n< for stablty Boostr motor Spa Engn (IUS) Tatal motor 6.5, Rokt Propulson Ltur 5 Prof. Manul Martnz-Sanhz Pag of

133 B. Doubl bor propllants: Homognous, Ntroglrn/Ntrollulos + addtv C. Compost propllants: Ammonum Pr hlorat (AP) + Rubbr bndr (ful) + Alumnum Ex. Fg.7 Sutton. AP-CMDB 30% AP (50 µ m ) n ~ 0.4 r (00 atm) m/s 0.4 P -5 7 a=.58 x 0 r C* = 600 m/s ρ = lb/m = 760 Kg/m p Want P = 50 atm 5 x 0 6 N/m A ( b A = ) t = 35 annot us nd burn Gran onfguraton For Sold Rokt Componnts and Motor Dsgn Rad Sutton, Chaptr 4 6.5, Rokt Propulson Ltur 5 Prof. Manul Martnz-Sanhz Pag of

134 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 6: Sold Propllants: Dsgn Goals and Constrants Sold Propllants Rad Sutton s, Chaptr Doubl Bas (DB) Ntrollulos + Ntroglyrn + Addtvs (for opaty, plastty, ). Both NC and NG ar xplosvs, dangrous somtms JPN NC 5.5%, NG 43%, Dthyl phthalat 3.%, Ethyl ntralt %, H SO 4.% + arbon blak + andllla wax Compost Modfd Doubl bas (CMBD) DB + Ammonum prhlorat (AP) or Alumnum (Al) Compost (C) AP (somtms A Ntr) + Syntht Rubbr bndr (ful) + Al. Safr than DB Othr Compost ontan ntramn xplosv (RDX, HMX), rplang som AP Typ I sp (s) at 000/4.7 ps T (K) ρ p (g/m 3 ) Al % r ps η Fabraton DB Extrudd DB-AP-Al Extrudd CTPB/AP/Al Cast HTPB/AP/Al Cast Th addton of Alumnum s not nssarly bnfal, as th followng xampl shows: Problm 3. Addng Alumnum to th formulaton of a sold rokt propllant nrass th gas tmpratur, but nurs prforman pnalts rlatd to th sold partls that ar gnratd. Consdr a smpl modl for th fft of addng a mass fraton x Al of Alumnum, of th form T T o x Al = + rx ; x=.85 () whr r.4 s a sparatly alulatd offnt, T o s th flam tmpratur wthout alumnum (~500K), and x s th solds fraton n th gas (th.85 fator aounts for th oxygn n th Al O 3 partls). 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag of 3

135 Consdr also a lnarzd modl for th fft of th partulats, of th form u fx u = (at fxd T ) () o whr f s as drvd n lass: f = γ s ( η) ln( η) P γ + ; η = (small partls) pg η P (larg partls) (a) Show that th optmum loadng s gvn by r f x x ; ( x ) 3rf.85 OPT OPT = Al = (3) OPT (b) For P P J = 0.0, γ g =.5, Mg = 8g / mol, s = 60, alulat ( xal) for KgK OPT both small and larg partulats. Commnt on rsults. Problm 3 Soluton Ignorng th xt prssur fft (or at mathd ondtons), g I sp = v = P CpT P γ γ whh s proportonal to T. Th rst of th dpndn (,) γ p ar afftd by partulats, but that s ountd sparatly n th loss analyss. So w hav (ountng both ffts) (a) ( ) v fx + rx To optmz, tak th logarthm drvatv and quat to zro 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag of 3

136 f r + = 0 fx + rx f ( + rx) = r ( fx) r - f = 3rfx x = OP T r-f 3rf and thn ( ) OPT x Al OPT x =.85 (b) For small partulat f = valus, s + pg ( η) ln( η) η, and usng th gvn η = - (0.0) 0.5/.5 = 0.60 ; Cpg = = 309 J / Kg /K ln(0.3988) f = + = x OP T = xop T = and thn th Alumnum fraton should b XAl = = % Al loadng OP T.85 ( ) For th as of largr partl, th lass drvaton showd f=, and so.4 xopt = < Ths nonsnsal rsult smply mans thr s no good Al loadng n ths as. Th losss du to th partls ar strongr than th gans du to nrasd tmpratur, so no Alumnum should b addd. Fortunatly, th partls ar small, not larg. 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 3 of 3

137 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 7-8: Sold Propllants: Othr Tops Combuston of Sold Propllants For a gnral dsusson, rad Sutton, Chaptr 3. A dtald modl of ombuston of ompost propllants s prsntd nxt. Combuston of Compost Propllant (Rf: Guy Lngllé, Jan-Robrt Dutrqu, Jan-Claud Godon, Jan-Franos Trubrt, ONERA, Sold Propllant Stady Combuston Physal Aspts. In AGARD-LS-80-Combuston of Sold Proplllants,99 (TL507.N867, no. 80)) Compost propllants ar htrognous mxturs of oxdzr grans and powdrd alumnum ful, both mbddd n a rubbr-lk bndr, whh s also a ful. Th most ommon oxdzr by far s Ammonum Prhlorat (AP), (ClO 4 NH 4 ), a rystalln substan wth ρ =.95 g/m 3, p = 0.3 al/g/k, thrmal dffusvty = =.5x0 4.55x0 T C m /s, and an stmatd m.p of 835K. AP s ground d p ( ) to szs from a fw to around 00 µ m. Th fnr grads ar dangrous, so grndng s don just pror to fabraton. AP has M = 6.5 g/mol and 55% by mass s oxygn. Th alumnum s also ground to smlar szs at th last mnut. Al s a vry xothrm ful, produng Al O 3 whh s lqud at th flam tmpratur ( 3500K), and ondnss latr to a sold. Th bndr s oftn polybutadn (syntht rubbr), thr Carboxyl Trmnatd (CTPB) or Hydroxyl Trmnatd (HTPB). Th omposton of CTPB s 4 al / s 3 C 7 H.4 O 0., wth thrmal ondutvty λ = 3.6x0, = 0.97 g / mpρ, ( m)( Kp) = 0.39 al / g / K p Bst prforman s obtand wth vry hgh prntag of AP, although mhanal proprts rqur a mnmum of bndr, and AP onntraton rangs from 70% by mass whn thr s Al ( 6%), th balan bng bndr (4%), to about 80%-85%, wth no Al (as n smok-lss ompostons), th balan thn bng all bndr. Ovrvw of Combuston Mhansm Th burnng of AP-bndr propllants (no Al) s a omplx srs of phnomna, and th dtald gomtry of th grans dos mattr (sz, partularly). At P 0 atm, AP tslf an dflagrat xothrmally, and t domposs partly n a thn lqud layr on th surfa of a gran, partly n an AP flam a fw µ m abov t. Th hat of domposton rass T to 05K by tslf; hat from th outr flam (mor blow) an ras th AP flam tmpratur wll abov ths, howvr. Around th AP grans, th hat from th man flam domposs th bndr, whh gnrats a mxtur of short-han hydroarbons, whl absorbng about 360 al/g, plus th nrgy to hat t to th surfa tmpratur T s 000K-00K. 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag of 5

138 Th O -rh gas gnratd n th AP flam o-flows outwards wth th bndr domposton produts, wth ntrdffuson along th way. Ths s a dffuson flam, and th fnal ombuston taks pla n t, rasng T to about 3540K. AP flam ( k) lqud sold AP Dffuson flam (Man flam ~ 3450 k) bndr Whn thr s Alumnum, th Al Partls ar jtd whn th bndr holdng thm rds; thy thn burn at svral hundrd µ m from t; durng ths burnng thy agglomrat to svral tns of µ m (thy ar lqud, m.p=930k, but rman nasd n Al O 3 untl ths shll braks at 300K, thn th Al spws µ m mropartls whh burn qukly) t = 0 t = x0-4 s t = 4x0-4 s t = 6x0-4 s t = 8x0-4 s 0-3 s From th surfa rgrsson pont of vw, Al burns nstantly, as ts partls ar jtd. Ovrall Burn Rat from Burn Rats of Consttunts Lt υ p b th man surfa rgrsson spd (m/s), and υ AP, υ b th orrspondng rats for th AP and th bndr ndvdually and n solaton. Although th gomtry s mor omplx, w an dalz th propllant as a layrd mdum, wth altrnatv thknsss δ AP, δ b. Th tm to burn through both δ AP and δ b s t δ + δ δ δ = = t + t = + AP b AP b AP b υp υap υb δ AP δb Callng ξap =, ξb = ξap = δ + δ δ + δ AP b AP b ξ ξ = AP + b υ υ υ p AP b th volum fratons of th onsttunts () 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag of 5

139 Th mass flux (g/m /s) burnt s whr ρ p s th man dnsty m= ρ υ () p p p ρ = p m m ρ AP AP AP + mb mb + ρ b α α = AP + b ρp ρap ρb (3) whr α AP s th mass fraton of AP, and αb = α AP s that of th bndr. Not: ρ AP =.95 g/m 3, ρ b = 0.9 g/m Th mass fluxs of th ndvdual onsttunts ar, smlarly, mm= ρ υ ; = ρ υ (4) AP AP AP b b b From () and (), ρap ρb ξap ξb ρp ρp = m= + ρ υ ρ υ ρ υ p p AP AP b b p ρ V M M M and sn ξ = / = ρp V V V M = α m= + p α m AP AP αb mb (5) AP AP AP Al b b For alumnum loadd propllants, w notd that th Al partls ar jtd whn th bndr holdng thm burns through. Lookng at th smplfd gomtry blow, t an b sn that th man burnng spd υ p would b th sam f th bndr wr rally fllng n for th Al. 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 3 of 5

140 W thn gt th approxmat xprsson ξap ξb + ξ = + υ υ υ p AP b A, dntal to th Eq. (), vn wth Al partls. Eq. (5) also follows, wth α AP n m pla of α b. Th only dffrn s that th man dnsty ρ p n p = ρpυ p s modfd by th Al α AP αb αa = + + ρp ρap ρb ρai Sparat Burnng of AP Hat pntrats nto th rdng AP partl to a small dpth only. Ths an b sn from th hat balan wrttn n th rdng fram: x Surfa fxd T s AP movng υ AP T = T o dt d dt ρap υap AP λap = 0 Dfn d = dx dx dx λ ρ (hat dff., m /s) ( T ) dt d T υ T d = υ T ; d = υ T T dx dx ( ) 0 AP AP AP 0 AP AP 0 υap x d T T AP 0 T T0 = = T T s 0 dap x / υap Not: 3 d. x0 m / s, 0.3 al / g / K. p 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 4 of 5

141 Ths shows an xponntal tmpratur day nto th sold, wth a * d haratrst thrmal thknss x =. Assumng υ = m / s, υ 3 d. 0 m / s, ths gvs AP AP * 3 x =. 0 m = m µ. Th top fw mrons of th AP ar moltn whn ts tmpratur xds T 835 K (whh may not happn f th prssur s so low, undr 0atm, that th AP flam-surfa dstan s too larg to provd suffnt hatng to t. Not ths rat s dt * q = λ = ( λ / x )( T T ) = ρ υ ( T T ) s AP dx AP s 0 AP AP AP s 0, so o q s MIN = ( ) ( ρυ ) typally. AP In ths moltn layr, about 70% of th AP undrgos omplt domposton to fnal (onydzng) gasous produts, aordng to NH 4 ClO N + 0.N O+ 0.3NO +.6H O HCl + 0.Cl +.05O (7) whras th rmanng 30% sublmats as a mxtur of ammona, NH 3, and prhlor ad, HClO 4 ; ths mxtur thn omplts th domposton to th fnal produts of (7) n a prmxd (AP) flam about mµ from th surfa. Lt us look at th nrgts of ths ffts: (a) Enthalpy pr gram to brng AP to ts surfa (moltn) tmpratur (nludng som ntrmdat phas transtons): h,ap= H835) (al/g) (8) ( T s, AP (b) Hat of th sublmaton nto NH 3 + HClO 4 =sh,ap476 to 50 al/(g/sublmd) (493 avrag) (9) () Hat of ombuston of NH 3 wth HClO 4 h,ap= 850 to C885 al/(g.ratd) (hat rlasd) (0) (d) Hat rlasd pr gram of AP drtly dgradd n th lqud phas: h,ap= D375 al / g () 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 5 of 5

142 APhhNot: W must hav + h = sh ) D W an ombn ths valus to obtan th Adabat Flam Tmpratur AP flam : h D,AP ad T f, AP of th hhh,ap+ 0.7 D,AP T =H0 () s,apc,apad ( ) g ( f, APTs, AP) whh, usng for th gas a spf hat = 0.3 / g / K hal g ad gvs T, = 05 (3) f AP K Rats and AP flam strutur. Th rat of pyrolzaton s found xprmntally to dpnd on th surfa tmpratur aordng to an Arrhnus-typ xprsson map Es, AP/ RTs = S, AP A (4) wth = 0 Kal/mol = g/m /s E S, AP A S, AP Th raton rat for th prmxd AP flam obys a smlar law, xpt that, bng prmarly a bmolular raton, ts rat s proportonal to p : ω pa (5) Eg, AP / RTf, AP = g, AP wth p n atm, n g/m PA 5Kal / mol; ωe,=gg AP 3 /s, and wth A, = 650g / m / s / atm 3 If th vloty gυ of th gas normal to th surfa wr known, Eq. (5) would allow xalulaton of th flam standoff dstan,. Th tm to burn th gas s ω= ρg /, and thn τ f,x,paf g, AP Eg, AP + RTg gρg = υ τ = υ ρ / ω = (6) AP g h g g υ (and not ρgυg = map ) 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 6 of 5

143 A sparat xprsson for x f an b obtand from th fat that th AP flam has to supply to th vaporzng surfa th rqurd hat of raton, so f th flam movs too far, th raton s too slow, and v vrsa. Th nt (onvton + onduton) hat flux n th gas btwn surfa and flam s onstant (no hat volvs thr) dt d dt 4 = dx dx λ dx ( g 0.3 al / g / K, λ g =.9 x0 al / s / m/ K ) (7) map g g 0 wth th boundary ondtons T=T S,AP at x=0 and dt dx λ g = map Q = map ( hh, A + Q ) x= 0 P S (8) Hr Q s s th nt hat rqurd to tak th AP from lqud at T s to gasous produts (bfor th AP flam ) Q s = 0.3 x (-375) = -5 al/g (9) Intgratng (7) wth (8), dt map gt λg = map gts,ap map Q dx dt map map g ( S, AP) T T = Q dx λ λ g and mposng T= T S, AP at x = 0 agan, Q =, so g T-T S, AP map g Q x λg = g (0) Th AP flam s at x = x f, AP, whr T rahs T fap (adabat T f for AP alon, but mayb hghr f thr s hat supply from th man flam). Solvng for x f,ap thn, x f, AP = λ g map g (,, ) T T n + Q g f AP s AP () Ths must b th sam as (6). Equatng thm, 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 7 of 5

144 Eg, AP AP RT λ f, AP g g, AP AP m = n pa m g or ( ) Eg, AP λg A g AP RTf, AD g Tf AP T,, s, AP map p n = + g Q () So, for AP alon, whr T, 05 K s known, and Ts,AP an b stmatd (not too f AP muh hghr than 835 K), th rgrsson rat s proportonal to prssur ( n ). Pyrolyss of Inrt Bndr. A smlar formulaton an b usd for th alulaton of th hatng rat du to th man flam, whh srvs to pyrolyz th surfa of th bndr (and also to lvat th tmpratur of th AP flam). AP W f W f, AP T s, AP X F, AP AP T F,AP Bndr X f Tf T s,b Th papr by Lngllè t al. Smplfs th modl by tratng for ths purpos th flow of gas as -D (vn though t rally s -D or 3- D du to th htrognty of th surfa. It also assgns a unform mass flux m p abov both, AP and bndr. Ths s qustonabl, but w ll prss on. Smlar to Eq. (7), w hav now dt d dt dx dx λ = dx mp g g 0 and, dfnng dt q = λ g th magntud of th (surfa-drtd) onduton hat flux, dx dq dx mp g λ g q = 0 (3) W ntgrat th ondton mp Qf = q( xf ) (4) 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 8 of 5

145 whr x f s th loaton of th man flam, at whh th nrgy of ombuston Qf 700 al / g s rlasd (q=0 abov t, q = mp Q blow t): f mp g x f x g q = mp Qf ( ) λ (5) At x=0, on th bndr surfa, th hat flux q(0) s usd to pyrolyz th bndr at th rat or m b (p.u. ara), and f Q,b s th rqurd hat (al/g), w obtan q(0) = mb Q b,, Qf mb = mp Q b, mp g x f λg (6) Th pyrolsaton hat Q,b s omposd of that rqurd to hat th bndr form T 0 to T s,b, plus th hat of domposton Q s =360 al/g: ( ) Qb, = b Tsb, T ( b 0.39 al / g / k ) (7) For T s,b =00K, Q,b =675 al/g. Eq. (5) s also usful to stmat th rat of arrval of hat from th man flam at th loaton of th AP flam (x=x f,ap ): q f, AP = p g ( x f x f, AP ) mg p Q (8) f mλ and, from ths, th AP flam tmpratur, usng a modfaton of Eq. (): ( ) map hh, AP + hd, AP + g Tf, AP T s, AP = qf, AP (9) or ad ( ) map g Tf,AP Tf,AP = qf,ap (9b) whr ad T s th valu of Tf,AP wth no q f,ap prsnt ( 05K, as w found). ad f,ap T f,ap 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 9 of 5

146 Estmatng th Man Flam Dstan. Lt D ox b th damtr of th oxdzr partl. Th surfa of th partl volvs oxdzng gas (aftr AP flam), whl th bndr around t gnrats ful gas. ful gas oxdzr gas AP X f Ths ntr-dffus to form a dffuson-flam, smlar to that from a Bunsn burnr (but nsdout). Th radal dstan ovrd by a substan dffusng wth a dffusvty D, n a tm t, s of th ordr of Dt. W say th flam s nd s at th pont whn ths quals D ox / (tms som fator of ordr, to aount for ral gomtry): D ox D ox / d ( ) A Dt A d Equatng th tm t to hr), ρg x f / υ g = xf (not w agan us th ovrall mass flux mp m p D ox / A d Dρgx f / mp or x f = 8A mp DOX d ( ρgd) (3) Th dffusvty D s nvrsly proportonal to th gas dnsty, so that ρ D s g ndpndnt of P at a gvn T (and wakly dpndnt on T). Thus, (3) gvs a man flam dstan whh s ndpndnt of prssur, and sals wth th squar of AP partl damtr. To a good approxmaton, th mass dffusvty D s qual to th hat dffusvty: 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 0 of 5

147 λg λ g ρg gg g D dg = D (3) ρ whh an b substtutd nto (3): x f mp g D OX (3) 8A λ d g Th fnal gas tmpratur (aftr th man flam) an b stmatd now from α T Q + α T + = T (34) f AP f,ap b s,b f g ad Usng α AP = 0.8, α b = 0., T = 05 K, T f,ap s,b = 00 K, g = 0.3al/g/K, Q f = 700 al/g. W alulat T f = 350 K. Atually, Eq.(33) s approprat only whn th dffuson tm (dstan) s muh mor than th raton tm (dstan), as at hgh P (short raton tms), and/or larg AP partl damtrs (long dffuson tms). In th oppost lmt, x f s rally dtatd by th hmal raton tm (dstan); smlar to Eq. (6), ths dstan an b wrttn as x f,r Eg,f m + p RTf = pa (35) g,f Th valus of A g,f and E g,f ar not gvn by Langllé t al. Ths s spally rgrttabl for E g,f whh s vry snstv. W hr tak tntatvly A g,f = 650g/m 3 /s/atm, as for AP, and dtrmnng E g,f by mathng approxmatly on of th small-damtr data ponts quotd n th papr. Ths lads to E g,f 9. 4Kal / mol (36) For th gnral as, thn, w tak x f to b th sum of th raton and dffuson dstans: xf = m D + Eg,f + g RTf p ox 8Adλ g pag,f (37) Soluton Produr. Gvn P,T 0,D ox, α AP, w want to alulat p, as wll as svral of th ntrmdat varabls. Th quatons ar farly omplx, so som traton must b dvsd. Frst, from th avragng law (Eq. (5)) wth α = α, m b AP 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag of 5

148 mb mp = α α AP (38) mp AP map But mb s also alulabl from Eq. (6); ths nvolvs x f, whh s gvn by Eq. (37): mp mb = m p Q Q f,b xp g ( + E /QT ) D ox 8A λ p A (39) mp g g,f s mp + λg g g g,f Equat (38) and (39) and solv for m p : mp = g Q λg n Q D 8A,b AP m p/ map α α xp + AP ( E g,f / RTf ) g ox dλg p Ag,f (40) Ths s smlar to th xprsson Eq. () for provds a rlatonshp btwn mp map m AP. In fat, th rato of both quatons, and T f,ap wth D ox and p as paramtrs: mp Qf AP m p/ map n α E g,ap / RT f,ap Q,b α AP = A T T E / RT ( ) xp ( + ) AP g,ap g f,ap s,ap g g,f f n + pdox + Q 8Adλ g Ag,f m (4) Th AP flam tmpratur T f,ap dpnds, n turn, on ombnng Eqs. (8) and (9): mp map, as an b sn by mp g ( x f x f, AP ) λg AP H, AP + D, AP + g f, AP s, AP = p f (4) ( ) m h h T T m Q Hr, w not that th fator mp g x f λg s 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag of 5

149 mp g x f λg Q,b mb Q,b α AP = = Qf Q mp f mp α AP m AP (43) Also, th fator mp g x f,ap + λg an b xprssd, usng () as mp map gxf,ap m p / map λ g g ( Tf,AP Ts,AP ) map = + (44) Q Combnng (4), (43) and (44) ( ) α g Tf,AP T map AP s,ap + + ( ) = + mp hh,ap hd,ap g Tf,AP Ts,AP Q,b Q mp map α AP map mp (45) Hr w rall that Q dpnds on T s,ap (Eqs. (8), (9) and (8)) ( ) ( ) Q T = 0. 38T 3 al / g s,ap s,ap (46) For th prsnt purposs, t s suffnt to assum a valu for T s,ap somwhat abov 835K; ths ould b rfnd latr by mathng hat fluxs at th lqud surfa. W adopt for most of th followng T s,ap =95K. Gvn T s,ap, w an s that Eq. (45) unquly rlats to T f,ap to followng alulatonal produr an than b followd: m p/ map. Th (a) Tak a valu of p AP (typally n th rang ) m / m (b) Solv (45) for T f,ap (non-lnar quaton, rqurs som ntrnal traton) () Solv (4) for th group ψ + 8A + E g,f / RTf g ( pdox ) dλg Ag,f and, hn, for th produt p Dox. Gvn p Dox, Ths gvs p. (47) (d) Calulat m AP from Eq. (). () mp p map m = map 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 3 of 5

150 Ths produs a urv of D ox, partl damtr). m mp p vs. p, wth m AP as th runnng paramtr (for a fxd Not how T f,ap and m AP dpnd on th ombnaton pdox rathr than on th sparat varabls. Howvr, sn burnng rat of m p s of th form m AP ontans th fator p drtly, th fnal m p = p f(p Dox) (48) whr f s som omplatd funton. Ths s an mportant salng law (not pontd out n Lngll t al s papr). To vrfy t, w an us th data rportd n Fg. 3 of th papr. 00 Burnng Rat (mm/s) 0 80% 5µm AP-CTPB 80% 90µm AP-CTPB Computd burnng rat Fg.3 Prssur (atm) For Dox=5 µm, w hav P(m) υ (mm/s) p PD ox (atm µm ) p ( ) υ / p mm / s / atm For Dox=90 µ, m P(m) υ (mm/s) p PD ox (atm µm ) ,000 p ( ) υ / p mm / s / atm , Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 4 of 5

151 Plottng now both urvs as ( υ p /p) vs. (pd ox ), thy do ond: υ p /P (mm/s/atm) 0.5 D ox = 5 µm D ox = 90µm ,000 pd ox (atm x µm) 6.5, Rokt Propulson Ltur 7-8 Prof. Manul Martnz-Sanhz Pag 5 of 5

152 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 9: Lqud Propllants Propllant Slton Rad Sutton, Ch. 7 Fuls Hydrazn LH MMH RP- UDMH N H 4 H N H 3 -CH 3 CH.97 N H -(CH 3 ) m.p. (K) b.p. (K) Kal Sp. ht Kg K ρ (g/m 3 ) H@5C f (Kal/mol) +.0 (l) -.4 (@0K) (l) +.7 (l) (l) Unstabl Cryogn Unstabl Unstabl Oxdzrs NA NTO LOX Ntr Ad Ntrogn Lqud Oxygn Ttroxd Hydrogn Proxd HNO 3 N O 4 O H O m.p. (K) b.p. (K) Kal Sp. ht Kg K ρ (g/m 3 ) H@5C f (Kal/mol) Kal mol (@90K) (l) Corrosv Tox Ratv, Cyrogn Unstabl 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of 3

153 Dsrabl: Low m.p., hgh ρ,, b.p. (for hgh H n oolng) Stablty (ngatv H f ) Low vapor prssur (hgh b.p.) Hyprgolty Non-toxty Storablty Non-orrosv Exampls of us: Hydrazn LH MMH RP- UDMH Monoprops. J- Spa bprops Atlas Ttan II (50% N H 4 ) SSME Shuttl OMS Thor Lunar Landr Vulan (Arant) Dlta (St.) Proton? J7 (Japan) RL-0 (Cntans) RS-68(EELV) Dlta IV Ttan I Saturn RD-70 RD-80 Fastra (X-34) RS-76 NA NTO L ox H O Ttan II Atlas X-5 Shuttl OMS Spa bprops Proton? Wth RP- Juptr RD-70 RD-80 Dlta (St.) Fastra(X 34) RS-76 Wth LH SSME Vulan RS-68 RL-0 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of 3

154 NA NTO Lox H O Ttan II Atlas X-5 Shuttl OMS Spa bprops Proton? Wth RP- Wth LH Juptr RD-70 RD-80 Dlta (St.) Fastra(X 34) RS-76 SSME Vulan RS-68 RL-0 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 3 of 3

155 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 0: Combuston of Lqud Propllants Smplfd Drop-Ws Lqud Combuston Thory. Introduton In ths ltur w xamn n mor dpth th flud phnomna whh domnat th opraton, and hn th dsgn, of a lqud rokt. W start by onsdrng th two phas aspts of ombuston, whh usually dtrmn th ombustor sz. Nxt w xamn th stablty of ombustor opraton and th dsgn modfaton rqurd to ahv t. Hat transfr to walls, wall ompatblty and wall oolng form th nxt top. W onlud wth som notons n nozzl arodynam dsgn and nozzl optmzaton.. Drop Vaporzaton and Spray Combuston Rturnng to Fg.. (ltur ), w not agan th xstn of a zon AC whr th propllants go from a srs of lqud jts ssung through a multplty of small njtor hols, through brakup of ths jts nto droplts, mpngmnt (n som dsgns) of jts or droplt strams on ah othr, dsprson of th droplts nto a rrulatng mass of ombuston produts, vaporaton of th droplts, ntrdffuson of th vapors and kntally ontrolld ombuston. Ths ar obvously omplatd prosss, and a omprhnsv analyss good nough for frst prnpls dsgn rqurs larg-sal omputaton [8, 9, 0]. In fat, th largst numbr of xstng lqud rokt ombustors, thos datng from bfor 970, wr dvlopd manly through mpral mthods, supplmntd by vry xtnsv tstng. Improvd modlng and omputatonal apablts hav mor rntly prmttd a mor drt approah, wth fwr hardwar tratons, but thory s stll far from ompltly dvlopd n ths ara, and srvs at ths pont manly to asrtan trnds and vrfy mhansms. For an n-dpth dsusson of lqud propllant ombuston, s Rfs. and. Hr w wll only rvw th fundamntal onpts whh undrl urrnt spray ombuston modls.. Sngl-Drop Combuston Evn whn lquds ar orgnally njtd nto th hambr, atual ombuston taks pla n th gas phas, followng vaporzaton of both, oxdzr and ful. In fat, vaporzaton s usually th rat ontrollng stp n th whol pross, although ts rat s tslf afftd by th ratons ourrng nar ah droplt. In a fw nstans, both propllants ntr th hambr as gass or asly vaporzd lquds, and thn gas phas mxng s th lmtng stp. (An xampl s th Shuttl SSME, whr th hydrogn s vaporzd n th oolng rut, and th oxygn s partally vaporzd by th hydrogn n a o-flowng hat xhangr arrangmnt ahad of th njtors, th rst bng atomzd and vaporatd shortly aftr njton). 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of 6

156 In a ful- rh nvronmnt, vapor from a ful droplt dffuss wthout ratng nto th surroundng gas, whl hat dffuss nwards to supply th latnt hat of vaporaton. In th sam nvronmnt, vapor from an oxdzr drop oms n ontat, and rats wth th ful-rh gas at som dstan from th droplt, formng a sphral flam, from whh hat dffuss both ways, whl raton produts dffus outwards and also stagnat n layr around th droplt. A smlar dsrpton appls to a loally oxydzng nvronmnt. It wll b suffnt to analyz, for xampl th as of a ful drop n an oxydzng nvronmnt, from whh w an asly drv othr ass by sutabl rntrprtatons. As a prlmnary, som rlatonshps wll b drvd btwn th ovrall stohomtry (oxdzr/ful rato, or OF, by mass), and th mass fratons of produts and xss oxdzr or ful to b found n th burnt gas far from th droplts. Lt th atual mass rato b OF, and th stohomtr mass fraton b OF s (latr alld for short), Assumng xss oxdzr, and omplt ombuston (a rough assumpton), th raton an b mass baland as (Ful) + OF (Oxdzr) (+OF s ) (Produts) + (OF-OF s ) (Oxdzr) and so th mass fraton of xss oxdzaton at nfnty s Y ox, OF OF OF OF s = = ( + OF ) + (OF OF ) + OF s s s Smlarly, for ful-rh opraton, OF + Produts + (Ful) OF OF OF (Ful) + I (oxdzr) ( ) s s and th mass fraton of xss ful at nfnty s Y F, = OF OF + + OF s OF OF s s = OF s OF ( + ) OF OF s whh ould b also formally obtand from and oxdzr (nludng OF ) OF YOX, by rvrsng th rols of ful 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of 6

157 Consdr a sphral ful droplt of radus R and dnsty ρ l. Its rat of vaporzaton s dr Γ= 4πR ρ ρ dt and th man purpos of th analyss s th alulaton of Γ from gvn R and gas proprts. Lt Y r, Y r by th mass fratons of ful and of ombuston f ( ) ( ) pr produts n th rgon R < r < r Fl btwn th drop and th flam loaton (no oxdzr s to b found thr, so Y f + Y pr = ). Ths two onsttunts ar both onvtd by th man outwards flow, and dffus aordng to thr onntraton gradnts. Sn th produts do not pntrat th drop surfa, th nt outflow of ful at radus r s qual to th total mass flow, and s gvn by dy Γ π ρ = Γ f Yf 4 r Df dr whr D f s th dffusvty n th produts. Th nt flux of produts rman zro n ths rgon (but not thr onntraton Y = Y ). Imposng that approahs zro at th flam radus, th abov quaton ntgrats to pr f Y f f λf r r Fl Y = (R < r < r Fl ) whr λ = f Γ 4π ρd f ( ) and th produt ρ Df s takn to b onstant. Whl ths gvs nformaton about th omposton of th gas nar th droplt, th mor mportant rsults om from a smlar mass balan outsd th flam radus (r > r F ). 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 3 of 6

158 + Assum, k k, p p p f OX prods. k ρg D k P For r > r F, th oxdzr mass balan s dyox Γ Yox = 4πr ρd =-Γ () dr or dy Γ ox Γ + Yox = dr 4πρD 4π ρd ( ) () Dfn λ= C p Γ Γ (3) 4πρD 4πk Yox r Yox, and mpos ( ) ox = = (4) λ r ( ox, ) Y = + Y (5) 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 4 of 6

159 Impos also ( ) Y r = r = 0, solv for ox F λ : r F λ = ln + rf Y ox, (6) whh s a rlatonshp btwn Γ and r F (r F turns out to b proportonal to Γ,.., a blowng fft) Th dstrbuton of both, mass fluxs and mass fratons ar dsplayd blow: W now nd th hat flux balan, both nsd and outsd th flam: R < r <r F - Takng T = T v as th (tmporary) nthalpy zro, th onvtd hat Γ T T. Th ondutv flow nsd th flam, whr only ful vapor movs, s ( ) flow s dt 4π r k, Th total, flowng nward (ngatv) s usd at r = R to vaporz dr lqud, at L (latnt hat) pr unt mass: dt Γp ( T Tr) 4π r k = Γ L (7) dr p v dt Γ Cp Γ L + ( T Tv ) = dr 4πk 4πk ( ) (8) λ agan 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 5 of 6

160 Imposng T(R) = T v, ths ntgrats to T - T v = L p λ R r (9) and, n partular, at r = r F T F - T v = L λ R r f p (0) whh also rlats Γ (n λ) to r F, but also ntrodus T F, th flam tmpratur, as a thrd unknown. W wll nd th hat flux balan for r > r F to omplt th formulaton. For ths purpos, w now mov th nthalpy zro to T=T F, th flam tmpratur. Th total hat flow (onvton plus dffusv) s now ( ) dt p T TF 4 r k dr Γ π and ths must b th hat rlasd at th flam front, mnus th hat snt nwards from th flam to th lqud. Th formr s Γ Q (Q= hat of ombuston pr unt ful mass). Th lattr would b Γ L, xpt for th xtra Γ T T du to th hang of nthalpy rfrns. Altogthr, ( ) p F v dt Γp ( T TF) 4π r k = Γ Q L p ( TF Tv) dr () or dt Γ p Γ + F = p F dr 4πk 4πk ( ) ( T T ) Q L ( T T ) v () Intgratng wth T( ) = T as a boundary ondton, ( ) Q L T T T TF = + T T λ λ p F v r r ( F ) p (3) and partularzng at r = r F (T=T F ), 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 6 of 6

161 λ ( ) F Q L p TF T v r TF T = p (4) whh s th ndd xtra rlatonshp to omplmnt (6) and (0) Eqs. (0) and (4) an b smplfd f λ s substtutd from (6): r F λ R L TF Tv = (0 ) Y p ox, + T F p ( ) Q L p TF Tv Yox, T = (4 ) Elmnat T F btwn ths two quatons: T + T = v λ R L Q L Y p ox, p + L p λ R Y + ox, Y ox, L p λ R Y + ox, + Y ox, Q Y L ox, = + + T Tv (5) p p Solv for R λ, and us λ= Γ p π : 4 k 4π k QYox, + p(t T v) Γ= Rln + (6) p L 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 7 of 6

162 In th ltratur, ths s wrttn as 4πk Γ= Rln + B (7) p wth B ( ) QY + T T ox, p v L (8) (th Spauldng paramtr). W hav solvd for Γ, and an go bak to (6) to gt th flam radus r F = R ( + B) ln Y ln + ox, (9) and to (0 ) to gt T F. From (5), (8), λ R l = + B, and thn, from (0 ), T F L B Yox, = Tv + + Y p ox, (0) 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 8 of 6

163 Ordr of Magntud: Assum an oxdzr rh LOX-krosn burnr, wth (O/F) s ==.4, and O/F = Ths gvs Y ox, = = 0.5. Th hat of raton s roughly Q=4.3 x J/Kg/ful and th ful latnt hat s L.5 0 J/Kg. Assum th surroundng gas s at T T = v 000K, and tak Cp 000J / Kg /K. W alulat Yox, 0.5 = = B = = 6.8 In vw of ths numbrs, an smplfy furthr to Y + Y ox, ox, ln, but B, so (9) and (0) rf R ln( + B) (9 ) Y ox, L QY TF Tv + B = T + p ox, p (0 ) Vaporzaton tm - Th droplt s atually bng onsumd, so ts radus vars (on a slowr tm than th gas transt tm), aordng to dr ρ π = Γ () l 4 R dt and sn Γ s gvn by (7) ρ 4 π R dr 4 dt = π k R p ( + B) ln R0 R k l ln ( B ) t p ρ = + ( + B) k ln R = R t 0 ρl p R 0 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 9 of 6

164 or R t R = τ 0 v ; ρl p R0 τ v = () k ln ( + B) τ v s th dp vaporzaton tm 5 3 For an ordr of magntud, assum R = 50 µ m = 5 0 m, ρ = 800 Kg /m, 0 l K 0. W/m/k, plus th numbrs usd n th prvous xampls. W alulat 5 ( ) Γ v = = = 0. ln ( ) s 6 ms. On th othr hand, sn τ R 0, f R 0 wr nstad 0 µ m, τ v would b 5 v tms shortr, or To s th sgnfan of ths, assum th gas movs n th ombuston hambr at M 0., or u 40m / s. A droplt of radus 50 µ m would thn rqur a ombustor lngth no lss than =.44 m R0 = 0µ m,, whh s xssv from th mass pont of vw. A drop of on th othr hand, rqur only 6 m to vaporat. Ths brngs out larly th mportan of good atomzaton. g 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 0 of 6

165 Othr ass For a ful drop n a ful-rh bakground, vrythng s th sam f w mak Q=0 (no flam), and, of ours, gnor rf and T F. In fat, th only quaton ndd s th r < r F hat flow balan (Eq. 9), whh, usng T( ) = T, gvs L λ R T Tv = p and so, fnally ( ) λ p T T = ln + R L v wth 4πk Γ= Rln + B (as n (7)), or ( ) p B = (T T) p L v (F / F Cas) (3) Agan, ths s as n (8) f Q=0. So, th ful drop vaporats fastr f t s n an oxdzng nvronmnt (largr B). Th rason, larly, s th xtra hatng from th narly flam n that as. For an oxdzr drop n an oxdzng bakground (0/0), w hav a vry smlar stuaton (no flam), so, on agan, Γ s gvn by (7), wth B gvn by (3), but now T v ox and L must b rntrprtd to b thos for th oxdzr lqud. Fnally, for an oxdzr drop n a ful-rh bakground, w rturn to th frst as (ful drop n oxdzng gas), and swth th rols of ful and oxdzr throughout. For, xampl, Q must b th hat of raton pr unt oxdzr mass, Y must b rplad by, = (O/F) boms, and (T, L) rfr to th oxdzng lqud. Yf, st. v ox, 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of 6

166 CALCULATION OF CHAMBER LENGTH (Gnralzng Spauldng s ) (000 vrson) Th drop vaporzaton tm T v ρ R = kln l p 0 ( + B) dpnd on B. For a ful-rh bakground, as normally sn n rokt, w hav two typs of droplts: Ful drop n ful-rh mdum: B F,F = p ( v,f) T T L f (3) Oxdzr drop n ful-rh mdum: ( ) QYf, + p T Tv,ox B (4) O,F = L ox (not Q s stll th hat of raton pr unt ful mass). In many ass L (spally for Lox), and also Q s a larg quantty apparng n B < L, but not n. So, B B, manng th oxdzr drops ar lkly to vaporat muh BF,F O,F F,F arlr than th ful drops. For hambr lngth alulatons, thn, w fous on th ful drops mostly. Assum ful drops ar njtd nto th hambr at a vloty u Do. Oxdzr drops ar assumd to vaporz vry fast (or to b fully vaporatd at njton, as whn Lox s usd). At any ston pror to ful vaporzaton, th ful lqud fraton (by flux) s O,F ox f 4 R 3 l n D u D γ= π ρ 3 A mf (5) 3 Th numbr-flux of drops nu A s onsrvd, so γ R and from (), sn th ntal γ s unty, D D t γ= τv 3 ( τ v = ful drop vap. tm) (6) Th vaporzd ful fraton s - γ, so th gas man vloty s u g = ( γ ) mf+ mo ρ g A x (7) 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag of 6

167 and, at th nd of th vaporzaton, ths boms u m = (8) ρ A g If th molular mass dffrn btwn ful and oxdzr ful and oxdzr vapors s gnord (onsstnt wth arlr approxmatons), and ρ,t g g ar narly unform, thn w trat as a onstant, and so ρ g ug mf mox γ+ O ( γ ) + = u + m m /F O/F (9) or 3 u t + (30) + O/F τv ug O/F Not that (9) mpls an ntal gas man vloty ( ) du to th vaporzd oxdzr flow. O/F ug 0 u, whh s + O/F Th partl vloty, u s dffrnt from u n gnral, so thr s slp, and D thrfor drag for. Assumng small Rynolds numbr around ah drop, Stoks flow ours, so g 4 du R 6 R u 3 dt 3 D π ρ l = π µ g ( g D) u (3) or, dfnng a vloty rlaxaton tm R0 ρl τ Rl = (3) 9 µ g R dud τ Rl = ug ud (33) R0 dt W now ntrodu non dmnsonal varabl τ τ v t = b ; = θ ; τ Ru v u u D = υ (34) 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 3 of 6

168 and wrt (33) as 3 θ dυ + O/F θ = +υ b dθ + O/F (35) Ths s a lnar, frst-ordr quaton for υ(), θ although wth non-onstant offnts. It an b ntgratd by standard mthods, lk paramtr varaton. ud 0 Imposng ud ( t = 0) = u D, or υ( θ = ) =, th soluton s found to b o u udo b b ν= + (3 b)( O/F) θ + θ + (3 b)(+ O/F) b 3 (36) On mor ntgraton wll yld th vaporzaton lngth: τv D v ( ) (37) L = u dt = u τ υ θ dθ 0 0 Aftr smplfaton, ths gvs th rsult L udo 5 = + u τ v + b u + O/F b (38) Not that, from () and (3), th paramtr b an b wrttn as b τv l p 0 = = = τ Rl ( + ) ρlr 0 g ( µ g p /k) ρ R 9 k ln B 4 ln B 9 µ ( + ) or b = P r 9 4ln(+ B) (39) For B=30 and P r =0.8, ths gvs b=0.5. Spauldng dfnd a rlatd paramtr ξ = b and prsntd rsults for th monopropllant as. W an rdu to that as by sttng O/F=0. Rsult ar ud 0 shown n th nxt fgur for ξ = 0.5 (b = 0.5). Th paramtr χ 0 s u, and 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 4 of 6

169 x ξ=. u τ v As an xampl, for th mddl graph χ 0 = 0.5, so usng O/F = 0 n Eq. (38) L w alulat ξ = = ( ) u τ.5 v wth th rsult n th fgur. or ξ = Ths onds Not how th droplts ar ntally slowd down by gas drag, but latr, as th gas volvs and alrat, thy ar pulld along. Nar th nd of vaporzaton, th rats alrat, baus th droplts ar so lght. Evoluton of vaporatng droplt sz and vloty. Hr th rfrn vloty s that of th gas aftr omplt vaporaton, and for lngth, l v rf for = v τ rf rf v 6.5, Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 5 of 6

170 Rfrns-Cont. 8. R.D. Sutton, W.S. Hns and L.P. Combs, Dvlopmnt and Applaton of a Comprhnsv Analyss of Lqud Rokt Combuston, AIAA JI., Vol. 0, no., Fb. 97, pp P-Y Lang, S. Fshr and Y.M. Chang, Comprhnsv Modllng of Lqud Rokt Combuston Chambr. J. of Propulson, Vol., no., Marh-Aprl 986, pp M.S. Raju and W. A Srgnano, Multomponnt Spray Computatons n a Modfd Cntrbody Combustor, J. Propulson Vol. 6, No., pp 97-05, Marh-Aprl F.A. Wllams, Combuston Thory, Bnjamn/Cummngs Publshng Co., In., nd Ed., W.A. Srgnano, Ful Droplt Vaporzaton and Spray Combuston Thory, Progrss n Enrgy and Combuston Sn, Vol. 9, 983, pp D.B. Spauldng, Aro. Quartrly, 0,. (959). S also Rf., S H.H. Chu, H.Y. Km and E.J. Crok, Intrnal Group Combuston of lqud Droplts, Pro. 9th Intrnatonal Symposum on Combuston, P.Y. Lang, R.J. Jnsn and Y.M. Chang, Numral Analyss of SSME Prburnr Injtor Atomzaton and Combuston Pross. J. of Propulson, Vol. 3, No., Nov-D.987, pp D.T. Harrj and F.H. Rardon (Edtors). Lqud Propllant Rokt Combuston Instablty, NASA SP-94, R. Bhata and W.A. Srgnano, On dmnsonal Analyss of Lqud-Fuld Combuston Instablty, J. of Propulson, Vol 7, No. 6, Nov-D. 99, pp G.A. Flandro, Vortx Drvng Mhansm n Osllatory Rokt Flows, J. of Propulson, Vol., No., May-Jun 986, pp R.J. Prm and D.C. Guntrt, Combuston Instablty Lmts Dtrmnd by a Non lnar Thory and a On-dmnsonal Modl. NASA TND-409, Ot M. Habballa, D. Lourm and F. Pt. PHEDRE-Numral Modl for Combuston Stablty Studs Appld to th Aran Vkng Engn. J. Propulson, Vol. 7, no. 3, May-Jun 99, T. Ponsot, F. Bournn, S. Candl and E. Esposto, Supprsson of Combuston Instablts by Atv Control. J. Propulson, Vol.5, No., Jan- Fb. 989, pp NASA SP-83, Lqud Rokt Combuston Stablzaton Dvs, Nov , Rokt Propulson Ltur 0 Prof. Manul Martnz-Sanhz Pag 6 of 6

171 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur : Lqud Motors: Injton and Mxng Ths s a vry omplx ssu, whh ontnus to rv a grat dal to rsarh attnton, du to ts mportan for: (a) Prforman, and (b) Stablty of ombuston. Th sngl-droplt modls of th prvous ltur only bgn to srath th surfa of ths problm. A rnt rvw by Srgnano s nludd and rommndd for radng n ordr to apprat th urrnt stat of ths top. 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of

172 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur : Lqud Motors: Combuston Instablty (Low Frquny) Combuston Stablty. Gnral Dsusson Elmnaton of nstablts has bn hstorally on of th largst omponnts of all nw lqud rokt dvlopmnt programs. Ths s baus thr has bn lttl rlabl mthodology to nsur stablty through dsgn, and also baus of th potntally atastroph onsquns of nstablty. Th stuaton has mprovd to som xtnt baus of th vastly nlargd smulaton apablts xstng now. As w wll s, ths ara s ntmatly rlatd to that of spray ombuston, whr a smlar stuaton has prvald. As n th ombuston ara, th advans n rnt yars ar promsng, but not yt suffnt to provd rlabl tools for a pror dsgn, at last aganst th hgh-frquny nstablty problm. In a systm wth th vry larg nrgy dnsty of a rokt ombustor, thr ar bound to b many mhansms by whh a small fraton of ths nrgy an b hannld nto undsrabl osllatons. Th rsultng nstablts ar usually atgorzd nto low frquny and hgh frquny typs. Th formr nvolv prssur osllatons whh ar slow nough ompar to th aoust passag tm that th whol hambr partpats n phas, whl th lattr xhbts aoust bhavor, wth dffrnt parts of th ombustor osllatng wth dffrnt phas or ampltud. In thr as, th nstablty arss whn nrgy (or somtms mass) an b addd to th gas at, or nar, paks n ts prssur osllaton (ths s th lassal Raylgh rtron for nstablty, as xpland) for xampl n Rfs. () and (6)*. Ths mpls a synhronzaton of two mhansms, on whh gnrats th nrgy or mass n som unstady mannr, and anothr on whh allows th gas prssur and othr thrmodynam quantts to osllat. Ths two mhansms must hav omparabl tm onstants for th mutual fdbak to dvlop. Th aoust mods of typal rokt ombustors hav wavlngths whh ar som fraton of thr lnar sz, and a wav spd (th spd of sound) of th ordr of 000 m/s. Thus thr frquns ar n th KHz rang, and hn th osllaton tm onstant s 0-4 to 0-3 s, and potntal ouplngs ar to b sought to nrgy or mass rlas mhansms wth tm onstants of that ordr. To b mor prs, t must b notd that a tm lag n th rlas of ombuston nrgy (suh as, for xampl atomzaton, vaporaton, mxng or raton dlays) wll not pr s provd ouplng wth th aoust fld. Thr must also b a snstvty of th hat rlas rat, and hn of th tm dlay, to th prssur or othr wav quantty. Sn, n gnral, only som of th addtv tm lags show suh a snstvty, and thn mayb only for a fraton of th fftv lag, t s found that th xtd aoust mods hav tm onstants whh ar a fraton of th tm lags thmslvs. As an xampl, Eq. ( of ltur 9) gvs for a droplt wth =000 kg/m 3, ln (l + B)=5 and ρ D 3 x 0-5 g kg/m/s a vaporzaton tm of 8. ms. Yt, ths sz droplts ould d-stablz aoust mods of about KHz frquny, as rportd, for nstan n Rf. 7. ρ * S Rf. Lst and nd of Ltur 9 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 6

173 Othr xtaton mhansms for hgh frquny wavs hav bn rportd. Gas-phas propllant mxng dlays n rapdly vaporatng ryogns may b of th rqurd ordr and show snstvty to aoust flds. Convrson of vortx nrgy to aoust dpol radaton upon mpngmnt of a vortx strt or a turbulnt shar layr on an obstal an also b a dstablzng fator [8], although ths mhansm may also work as a dampr, f th phas s arrangd appropratly. On th othr hand aoust frquns ar normally too hgh to allow nvolvmnt of lmnts outsd th hambr tslf, suh as th lqud-handlng qupmnt or th ovrall rokt strutur. Also, hmal raton tms n wll-mxd gass tnd to b shortr than 0.- ms, and ar not usually nvolvd thr. Th low frquny typs of nstablty, by ontrast, usually rsults from ouplng of osllatons of prssur n th whol hambr to lmnts outsd of t. Ths an b th njtors, th lqud lns (or th lqud n thm), th turbopumps, and, n th nstablty alld POGO, th whol rokt strutur. Thr s substantally mor undrstandng of ths than of th fast nstablts, and analytal thnqus wr dvlopd arly on [6] to hlp avod thm n th dsgn pross. Many of th low-frquny nstablts an b dsrbd n trms of a sngl, non-snstv, ombuston dlay; as dsussd, ths dlay an b svral tms longr than ts snstv fraton. In othrs ass, th mhansm may b mor omplx, prhaps nvolvng dffrnt dlays for th two propllant strams. Th ltratur also mntons an ntrmdat and bngn typ of nstablty, whh s a hybrd somtms dsrbd as ntropy wav. Aoust osllatons may modulat dffrntly th oxydzr and ful njton rats, ratng paths of varyng stohomtry and tmpratur. Ths onvt wth th man flow, and, upon mpngmnt on th onvrgng nozzl, may synhronz wth and rnfor th aoust wav, whh thn propagats bak to th njtor rgon. Sn part of th mhansm s aoust and part onvtv, th frquny falls btwn th frst aoust longtudnal mod and th flow passag tm. Th ffts of ths nstablts rang from smpl vbratory and aoust dsturbans to vry rapd burnout of walls. Th most svr ffts ar assoatd wth hgh-frquny nstablts of th tangntal hambr mods. Ths rat sloshng gas motons whh ompltly dsrupt th boundary layr nar th njtor nd of th hambr, and an lad to falur n a mattr n sonds. Thus, most ffort has gon nto undrstandng and lmnatng tangntal mod nstablts. Low frquny nstablts, f allowd to prod, an also b struturally damagng, but, as notd, thy an oftn b dsgnd out, and f not, thr ar rlabl ountrmasurs aganst thm.. Mthods of Analyss for Low Frquny Instablts Th omplaton by Harrj and Rardon (Rf. 6)* rmans th bst sour of nformaton on ths top. Th analyss typally sparats nto a hambr porton, balanng th gas gnraton, storag and dsposabl, plus a srs of submodls dtalng th rspons of varous othr parts of th systm, suh as njtors, pumps, t. For prlmnary analyss, lnar prturbaton modls ar usd, and Rf. (6)* shows a varty of subsystm rspons funtons adquat for ths purpos. * - D.T. Harrj and T.H. Rardon (Edtors) Lqud Propllant Rokt Combuston Instablty, NASA SP-94, , Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag of 6

174 As a smpl prototyp, w dsuss hr th ouplng of hambr and njtor alon (sngl stram). It s assumd that vaporzaton of th lqud happns a tm τ v aftr njton, whh may b a rflton of th tm dlay for atomzng nto droplts, plus th droplt vaporaton tm gvn by Eq. (Ltur 9), although ths part would b bttr modld as a dstrbutd, rathr than a lumpd lag. Lt P C and T C b th hambr prssur and tmpratur, both of whh ar takn to b unform, and V C th hambr volum. Th vaporzaton rat s: thn hav ( ) = ( τ ) m t m t () v v whl th gas dsposal rat through th nozzl s as gvn by m = P A t /. W P V = m ( t τ v) g d P A dt R T t () Th njton rat through an fftv njtor orf ara A, wth a prssur P 0 bhnd t, s m = ρl ( P0 P) A (3) whr ρ l s th gas lqud dstny. Lnarzaton of ths quatons, assumng onstant R g, T, P 0 and *, gvs ( t τ ) ' ' ' V dp P v P = m RT g C dt Po P P (4) P At whr l ( 0 ) hambr prssur, and tm m = = ρ P P A s th man mass flow rat, P s th man ' P ts prturbaton. W an dfn a hambr vauaton P V τ = RT (5) g m and an njtor ovrprssur paramtr P P = 0 (6) P 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 3 of 6

175 and assum prssur varatons of th form P P ' ωt = ĉ (7) Substtutng nto (.5) ylds an quaton for th omplx ω : ωτ v ωτ = (8) and an nstablty wll rsult whn th magnary part of ω as gvn by ths quaton s ngatv. Th stablty thrshold ours whn ω s ral. Imposng ths and lmnatng ω btwn th ral and magnary part of Eq. (8) ylds th thrshold ondton τ τ 4 ( ) v = π os (9) at whh th osllatons our wth a frquny gvn by ( ) ωτ v = π os (0) Not that ths nstablty thrshold annot b rahd f >. Ths s baus of th strong nrgy-dsspatng rol of th njtor, whh ats as a dampr for th osllatons. Rlatvty larg njtor prssur drops ar routnly usd, of th ordr of 0.. Valus abov = would lad to xssvly and lossy prssurzaton and njton systms. NOTE: (at thrshold) ' ωt v m ' = P ' P L ωt ' ' ωτ ( π ωτ ) = P = P os P P ( ) Hn, mass addton ours at an advan phas angl os - ( ) w.r.t. prssur paks. If s small, narly on top. 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 4 of 6

176 Allowng a non zro (postv or ngatv) magnary part of ω, w an obtan th rsults shown n Fg.. In gnral, stablty s nhand by nrasng njtor drop or hambr rsdn tm (n rlaton to th vaporaton lag). As shown n Fg..3b, th stabl or nutral osllatons hav frquns of th ordr of f / 4τ. As ( ) an xampl of th knds of dlays nountrd n LOX-LH rokts, w rprodu Fg. (from Rf. 6). R v 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 5 of 6

177 So, thr ass:. mv P' = 0.5 = P m Gas mass produton n phas wth gas xpulson (as wth P' ).. 3. > 0.5 m' v P' = P m m' lags v ( ) m', hambr bgns mptyng bfor nw gas s ntrodud Stabl. ont mv P' + < 0.5 = P m ( ) m' v lads an b m' ont, Nw gas starts aumulatng bfor t an b xplld. Ths unstabl, f th τ s small nough ompard to τ v. Not that th snstv (to prssur) part of ths tm lag s 0. to 0. of th total lag. Aordng to th rsults n Fg., th margnally stabl osllatons wth th total dlay of Fg., (and wth 0. ) would b at approxmatly a frquny.7 f= 70-3 π 0 Hz Fg. Exprmntal valus of ombuston tm lag for suprrtal hambr prssur 6.5, Rokt Propulson Ltur Prof. Manul Martnz-Sanhz Pag 6 of 6

178 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 3: Lqud Motors: Stablty (Hgh Frquny); Aousts Combuston Instablty: Hgh Frquny Mthods of Analyss for Hgh Frquny Instablts Pror to th advnt of larg-sal omputatons, th most sussful thortal dvlopmnt n ths ara was th snstv tm lag thory of L. Croo [6]. Mor than a dtald physal thory, ths was a modl n whh a fw bas paramtrs wr ntrodud from ntutv onsdratons, and thn usd to orrlat xprmntal obsrvatons on stablty thrsholds. Th prnpal paramtr was th snstv tm lag, durng whh th varous rats whh vntually rsultd n vaporzaton at th total tm lag τ T aftr njton wr assumd to vary wth prssur, vloty, stohomtry, t. Ths varaton was haratrzd by mans of othr mportant paramtr, th snstvty ndx. For prssur snstvty, ths s n = ln ( Rats) ln P () and th dfnton of τ s suh that th varatons n gas gnraton rat du to ths snstvty ar gvn loally by m m τ P'( t) P'( t τ ) = = n t P m () Smlar snstvty nds an b ntrodud for vloty, t. On ths paramtrzaton s aptd, t s only a mattr of mathmatal modlng to obtan th stablty lmts of a gvn aoust wav or avty. Ths modlng ould b lnar or vn allow for nonlnarts n th gas dynams. It s on of th strngths of ths thory that th aoust part of th problm, namly, ombustor gomtry, stady stat ombuston and hat rlas, t. ar sparatd from th unstady ombuston ffts, whh allows for gnralzaton of tst rsults and aumulaton of manngful stablty data. Th rsults of alulatons usng th lnar snstv tm lag thory ar dsplayd as shown n Fg. (Rf. 6). 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of

179 Fg. An n-τ dagram, showng nstablty zons for varous mods. Hr ar shown th lo of margnal stablty for svr mods of a ombustor, on a map of ntraton ndx n vs. snstv lag τ. Eah pont on on of th lns orrsponds to a partular osllaton frquny, and ths frquns ar found to b wthn ± 0% of th undsturbd aoust frquny of th mod. Th goal of th dsgnr s to manpulat th fators nflunng n and τ n ordr to pla th opratng pont outsd all th stabl rgons of th varous mods. Th paramtrs τ and n (nto whh s lumpd th modfyng ffts of vloty or othr snstvts) ar basally mpral, and a larg data bas has bn laborously aumulatd on thr dpndns upon many dsgn fators. As an xampl, Fgs. (a) and (b) (Rf. 6) show data on τ for oaxal njtors, for whh n 0.5 throughout. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of

180 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 3 of

181 Startng wth th arly of Prm and Guntrt [9], ths mthods hav bn progrssvly rplad by dtald numral tm-dpndnt smulatons of th ombuston pross. Th omplxty of ths prosss s suh that, vn now ths smulatons stll ontan larg lmnts of modlng and approxmaton, and ar gnrally lmtd to on or two sltd spatal dmnson plus tm. A rnt xampl of ths approah s dsrbd n Rf. 30. Hr th mphass s on th tangntal aoust mods. A dtald -D flud mhanal modl (n th transvrs plan) s oupld to a srs of mprally drvd droplt vaporzaton laws for UDMH and N O 4. Th flow s turbulnt, modld usng a K- ε approxmaton, and th drops ar allowd to slp, xrtng drag fors whh ar omputd from mpral drag offnts, and also modfyng th vaporzaton rats du to onvtv hat transfr. Th drop-hatng transnts ar gnord. Th omputatons yld dtald tm hstors of all th flud paramtrs, and omparsons to lmtd tst data on paramtr ffts of prssur and njtor typ ar found to b favorabl. A smlar omputaton, but for longtudnal mods only, s dsrbd n Rf. 7. Hr th spatal dpndn s on on dmnson only (axal), but th droplt ntratons ar alulatd n somwhat mor dtal, nludng drop thrmal nrta. Ths alulatons show strongst nstablty whn th rato of th aoust prod to th droplt vaporzaton tm s 0.5, whh, as notd bfor, an b ntrprtd as ndatng a snstv tm lag whh s a fraton (0.-0.) of th total vaporzaton tm. Th alulatons also show ass of ntropy wav xtaton, for whh th frquny orrsponds losly to th onvtv tm n th hambr. In what follows, w provd a smplfd analyss of th aoust ffts of svral ombuston-rlatd phnomna (hat rlas, mass addton, t.), and thn us ths n onjunton wth Croo s thory for an assssmnt of stablty n a smpl -D stuaton. Som gnral onlusons about stablty and dstablty ffts ar also drawn from th aoust analyss. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 4 of

182 Aoust quatons wth hat and mass addton, flud fors, and molular mass hangs. z = = 0 y ( ρ u) ρ + = m (mass addton p.u. volum, p.u. tm) t x ρ u + ρ u u + p = f (for p.u. volum) t x x ( /p) ( /p) T T q v + u = p + u t x p t x p ρ = R T M (q=hatng rat p.u. volum) Assum small prturbatons about a unform stady bakground, (wth µ, m,f, q): ρ = ρ + ρ', u = u + u', p = p + p', T = T + T ', µ = µ + µ ' (5) u = 0 (6) Baus of (6), Lnarzng thn, 0 D() () + u () = ()' + (u + u') ()' ()' Dt t x t x t nd ordr (7) ρ ' t u' + ρ = m (8) x ρ u' p' + = f t x (9) ρ u' p' + = f x t x x () ρ v T' p p' = q + t ρ t p' ρ ' T ' M' = + p ρ T M (0) () T' p' ρ ' µ ' = T + p ρ µ (3) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 5 of

183 (3) nto (0): p' ρ ' M' p ρ ' ρ v T + = + ρ q p t t M t ρ t p' R = T = R g T p M ( M'/ M) R ρ ' v p' v + = q + + p M t Rg t Rg t T v p ( M'/ M) ρ ' = p' p q + t pt γ t γ t (4) Sub. Into (8) ( M'/ M) p' p u' q + + ρ = m pt γ t γ t x (5) Dffrntat w.r.t. t: ( M'/ M) p' q p u' m + + ρ = pt γ t t γ t x t t (6) but, from, ρ u' p' f = + x t x x substtut n (6) ( M'/M) p' q p p' f m + + = pt γ t t γ t x x t (7) Now ( ) γ pt =γ Rg T = ( =spd of sound) (8) ( µ '/ µ ) p' p' = + ( γ ) f m q p t x t t x (9) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 6 of

184 ( ) ( 5) ( µ ' µ ) u' p' q m γ C pt ρ + = ( γ ) + p (0) x x x t x x t x (9) u' p' f ρ + = t t x t t () Subtrat, dvd by ρ : ( M' M) u' u' f = m + ( γ ) q p () t x ρ t ρ x t Not th ombnaton ( ) m + γ q p ( M' M) t Addng hat, addng mass, or havng a dras rat of molular mass, all ar (up to fators) quvalnt aoust dsturbans. To los th problm, on nds to rlat th prturbatons (m, q, f, th stat varabls ( p', ρ ',u',t ' ), by lookng at th partular mhansms µ ' ) to t (vaporzaton, ombuston, t), and how thy dpnd on prssur, vloty, and so on. Gnral Condtons for Instablty Lookng at Eq. (9), w s that th ffts of gas mass gnraton m, hat µ '/ µ gnraton q and molular mass hang ar smlar. All of thm hav th t fft of nrasng th loal volum, and w suspt thrfor that whn ths quantts (atng togthr) pak whn th prssur also paks, w wll hav unstabl ondton. To xamn ths, dfn th quantty ( M' M) Q = m + ( γ ) q p (3) t and gnor for now th loal dsturbng fors f. Thn p' p' = Q t x t (4) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 7 of

185 For a snusodal wav of th typ ( t kx) p'(t) R p ω = (5) wth ω omplx and k ral, w thn hav p k p Q ω + = ω (6) Dfn ω Q ν ; h = (7) k k p and r-wrt (6) as ν + hν = 0 (8) whh has th omplx soluton ν = h ± + h (9) W now ask what form h should hav for stablty. Frst, w not that at th h = ν / ν must also b ral. From (7), stablty thrshold, ν s ral, and so ( ) ths mans that Q must b n phas (or ountr-phas) wth addton rat must b 90 o ahad of or bhnd prssur osllatons: p, namly, volum Mor gnrally, r-wrt (8) as ν + h = 0, and put xpltly h = h, R + hi ν ν = ν + ν : R I 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 8 of

186 ν ν νr + ν I + hr + hi = 0 (30) ν R I R + ν I Sparat ral and magnary parts: ν ν R + hr = 0 ν R R + ν I ν ν I + hi + = 0 ν I R + ν I (3) From (3b), hi = ν I + (3) ν R + ν I ωt Ths shows that whnvr h, I > 0 ν I < 0. Sn = ωit ωrt p', ν I < 0 mpls nstablty. Also, from (7b), h mpls Q I > 0 > 0, whh mans that th p R volum addton rat Q(t) must hav a postv projton on prssur p (t), namly, a part n-phas wth t. Ths s a onfrmaton of th physal ntuton that rlasng volum whn prssur s hgh must b d-stablzng. W rpat that ths may man hat addton, gas addton (vaporzaton) or molular mass rduton rat (domposton of omplx moluls). Lt us now onsdr brfly th fft of body fors f. Rturnng to (9) and dfnng th omplx quantty. f ρ = (33) kp w an s that (8) s xpandd to th form ( ) ν + hν ρ = 0 (34) For nutral ondtons (stablty thrshold), w must hav ral ν ( ν I = 0), so, takng th magnary part of (34), ν + = 0 h I ρ I (35) In th absn of fors, w found that h would lad to nstablty. W s now I > 0 that f ρ = ν h, whh s ngatv, thn th pross stablzs at last to th pont I I of nutral stablty. From (33), th onluson s that f < 0 s stablzng,.., th p R body fors should b n th bakwards drton (aganst H vloty) whn prssur s hgh, and v-vrsa. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 9 of

187 Lookng at on nstant of tm vs. dstan: So, th body fors ar n quadratur wth th prssur gradnt for ( P ), and ar takng nrgy away from th wav moton. Th Snstv tm lag thory (Harrj and Rardon, p.) Bas snaro: - Drops njtd at t τ omplt vaporaton and ombuston at T - Howvr, from t τ to t τ, no omplt vaporaton ours, only T prursor prosss. τ snstv lag. - Th rat of vaporaton + ombuston durng τ s snstv to prssur and/or ( ) ln rats vloty. For p, ln P (n= snstvty ndx ). n - Th duraton τ of atual vap. + omb. hangs n tm, n rspons to ths rat hangs. Howvr, th total mass burnt s that of th drop, and t s assumd thr mass (and thr numbr) ar ndpndnt of P, υ n th hambr. () Say, R s th rlvant rat. Undr qut or man ondtons, R = R. Whn P P, 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 0 of

188 R R + n p p p () Exprss that th total mass burnt n ( t τ,t) that burnt undr man ondton s always th sam, qual to t t τ t p p R + n dt = R dt (3) p t τ t τ R dt t τ + R n t t τ P P dt P = 0 hr rplang τ by τ s ok. ( nd ordr rror) t P P τ τ = n dt (4) P t τ Lt now, spfally, m b th rat of gas gnraton from lqud (loal, pr unt volum). Th lqud njton rat s onstant, qual to m. Th gas gnratd n (t, t+td) s mdt. Ths gas orgnats from lqud that rahd ts maturty for vaporzaton btwn t τ and ( t τ )+d ( t τ ), and sn lqud arrvs at m, m dt = m d ( t τ ) m m dτ = (5) m dt From (4) dτ = n dt ( ) P ( t τ ) P t P ( ) P ( t τ ) m m P t and so = n (6) m P ( ) P' ( t τ ) m' P' t or, n trms of prturbatons, = n (6 ) m P NOTE: Eq. (3) s wrong n Harrj and Rardon. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of

189 Control of Instablts Dspt all fforts at avodan through dsgn, th nd rmans for dvs that wll hlp damp th many potntal mods of nstablty n any gvn rokt ombustor. A nw thnology for atv ontrol s now volvng [3], n whh fdbak ontrolld aoust gnrators ar usd to anl unstabl wavs. Sn th growth an b dttd at small ampltud, t may not b nssary to njt vry larg aoust powrs for ths purpos. Nvrthlss, th manstay of urrnt prat s basd on passv dampng mthods. A good dsrpton of ths mthods, wth dsgn gudlns, s gvn n Rf. 3. Th most mportant hgh frquny stablzaton dvs ar njtor had baffls and aoust absorbrs. Baffls ar radal or rumfrntal barrrs attahd to th njtor had and xtndng damtrs n th axal drton. An xampl of a baffld njtor s that of th SSME (Fg. 3, from Rf. ). Th xat mhansm by whh baffls nhan stablty s not wll undrstood, whh has ld to som dvrgn n dsgn. It appars that th fft s rlatd to th snstvty of th droplt vloty ross-ovr pont, whh ours qut los to th njtor fa, and t may nvolv dsrupton of th tangntal gas moton assoatd wth tangntal mods, or shftng of th loal aoust frquns to valus abov th haratrst drop vaporzaton frquns. Aoust absorbrs ar avts on th hambr walls wth rlatvly narrow onntng hannls to th hambr, so as to dsspat powr durng prssur osllatons n thr vnty. Thr aton s muh bttr undrstood than that of baffls, and dsgnrs an prod wth som onfdn, usng mthods dsrbd, for xampl n Rf. 3. Absorbrs ar oftn loatd on th ylndral walls, nar th njtor, or as ornr slots btwn njtor and ylndr. Thy an also tak th form of a ontnuous doubl wall wth an array of hols prodally arrangd to onnt to th hambr. Th absorpton offnt of a wll-dsgnd absorbr an b hgh ovr a rlatvly wd frquny band, so as to ontrbut dampng to th most prvalnt mods. Somtms svral dffrnt absorbrs ar usd, ah tund to a dffrnt frquny. Fg. 4 (Rf. 3) shows a baffld njtor wth ornr absorbrs, and Fg. 5 (Rf. 3) shows an xtndd aoust lnr. A smplfd analyss of an aoust absorbr s dsrbd nxt. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of

190 Aoust Absorbrs: Rsonators Th shmat shows a dv whh s a varaton on th Hlmholtz Rsonator onpt. Consdr frst th halfyl durng whh flow s lavng th rsonator ( m >0). Th prssur nsd th rsonator s P R, whl at th ntran to th narrow nlt dut, t s ' PR = PR ρ v, du to th (subson) alraton towards th nlt. Th dnsty hangs δρ an b quatd to δρ / ( = spd of sound n th rsonator). Th mass balan s thn V R dp dt R = m () and th momntum balan n th dut s or, usng m ρ A v, = d dt ρ ' ( A Lv) = A ( PR P) dm A m = PR P () C dt L ρ A Consdr nxt th othr half-yl, whn th avty s fllng ( m <0). Th prssur at th hambr-sd of th dut, whh s now th flow nlt s ' P = P ρ ( v ), whras that at th xt of th dut s now just P R. Eq. () stll holds, whras th momntum balan s now dm A m = PR P + (3) C dt L ρ A 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 3 of

191 W now dffrntat () and (3), and unfy th two half-yl momntum quatons as dm dp = L dt dt dt m d m A dp R dt ρ A Substtutng Eq. () hr, m d m dm A A dp + + = m dt ρ LA dt LVR L dt (4) Ths quaton s non-lnar n stat opraton f w rpla m, but w an obtan rasonabl rsults for stady m by ts tm avrag ovr on yl. Th analyss rdus thm to that of a ford lnar osllator, xpt that m nds to b alulatd slf-onsstntly rathr than bng a prsrbd quantty. Assum solutons of th form 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 4 of

192 t m = R m ω, ωt δ PR = R δ PR,t. (5) Eq. (4) boms m A + + = A ω ω m ω δ p (6) ρ LA LVR L whl Eq. () gvs V ω R δp R = m (7) and solvng (6) and (7) togthr, δ P R δ P = A LV R A LV R m + ω ω ρ L A (8) A From ths, th natural frquny of th osllatons s sn to b ω n = (9) LV and th damagng fator ζ s gvn by R ζω n = m ρ LA, or ζ = m ρ LA LV A R (0) For th rtfd sn wav m (t), w an s that m = m, and from (7) and (8), π V V LV π π m A + ω ω LV ρ L A R R R = ω δp = R ω δp m R A () 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 5 of

193 m m A Usng A ω + ω = ω + ω, ths ylds a b-quadrat LVR ρ L A LVR ρ L A quaton for m n trms of th ampltud δ P of th prssur flutuatons n th hambr. Th algbra s smplr n non-dmnsonal trms. Dfn ω ω ν = ω n LV A R () VR ρ π LA δ P ρ (3) W thn obtan m ( ν ) ( ν ) ρ A 4 4 = + + p ν, and substtutng n ω VR (0), th dampng rato s p ν / ζ = (4) 4 4 ( ) ( ) ν + ν + p ν Svral ponts an now b mad: (a) For any frquny, th dampng nrass wth prssur flutuaton ntnsty. Ths s a favorabl rumstan baus w nd th dampng most whn ombuston s rough. Mathmatally, ths s a onsqun of th nonlnarty of th quaton. Physally, nrgy s dsspatd both durng aspraton and durng xpulson of gas from th avty, (by th mxng out of th jt knt nrgy), and t s lar that mor nrgy s dsspatd whn th drvng prssur dffrns ar strongr. (b) Although th algbra s stll tdous, dffrntaton of (4) shows that ζ s maxmum at ν =,.., whn th avty s tund n rsonan wth th prssur flutuatons (by sltng paramtrs so that ω = ω ). Puttng ν = n (4) gvs ζ MAX = p/ = V R π LA δ P ρ whh agan shows ζ nrasng wth flutuaton ntnsty. n (5) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 6 of

194 () Sn w should dsgn for rsonan for th gvn ω, th dmnsons should A ω satsfy =. Eq. (5) an thn b put n th two quvalnt forms LV R ζ MAX δp δp VR = = ω ωl π ρ A π ρ (6) Ths ndats w should us short nlts, larg avty volums and A ω small nlt aras (subjt to = LV ). R As an ndaton of what s possbl, onsdr % flutuatons on a P prssur P suh that RT g ρ = s P = 3000 =. 5 0 m / s. Th ρ 00. aoust frquny to b dampd s at f = 000 Hz, or ω = π f =, 5000 m/s. Thn from th frst of Eqs. (6), w an obtan rtal dampng ( ζ MAX hoosng an nlt lngth = ) by δ P L = = (. ) =. = ωζ π ρ, 500 π m 7mm MAX. whh appars rasonabl. To nsur rsonan, thn, w nd to slt. 6 VR γ R T = = = 34m. A 3 ω L ω L, ( ) 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 7 of

195 A possbl ho of gomtry s thn to tak an nlt damtr of, say, 5mm, and a avty volum π m 6 3 m V R =.. =. =. 4 If ths s shapd as a ub avty, ts sd s about 3 m. Ths dmnsons ar skthd (roughly to sal) blow. As a fnal not, rmmbr that th lumpd paramtr dalzaton usd n th thory may not b vry prs. Only sm-quanttatv auray s to b xptd, but th trnds should b orrt and th analyss an b usd for prlmnary dsgn, to b rfnd through numral smulaton or physal tstng. How many aoust damprs: δξr = δ mv + ρ δ PRVR = d ( δξ ) dt R dδ PR = δ mvδ v+ V dt R v = m ρ A δv = m δ ρ A Wth no dsspaton dm dt A = ( PR P L C ) dδ dt m = A δ PR L L m A δ V dp = R R m dt dδ PR = dt V R δ m R V δ P R 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 8 of

196 d L d L δ m V δ m V P = dt A dt A δ P R + R = 0 + R ( δ R) 0 ζ R L δ m + V R ( δ P R) A L VR δ m ρ ρ A ρ = = + ( δ P ) R Wth dsspaton m δ m dδ m A = δ PR dt L ρ A L m A δ ( δ ) d PR = dt V R m δ R R V δ P d L δ m + V R ( δ P R) dt A m = δ m ρ ρ A (nrgy dsspaton rat) ζ R Avrag 3 m ε R = ρ A m π m = At rsonan, A = δ P m L ρ LA ρ A = m m δ P ρ A = δ P m π m π = ρ A δp m 3 / 3 3 / A R = = ρ A δp δp ρ A π π ρ ε 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 9 of

197 Avrag hambr aoust nrgy ξ = V δ P ρ ontrbuton of ω I of hambr, du to N rsonators ( ω ) 3 ζ R N A ρ NA ρ I = N = = h. 3 / ξ π π V ρ V P δ P δ Also, ( ω ) ( ω ) n = = n R A LV R ζ ( ωi ) NA = = ω n 3 π V ρ δ P LVR N ρ = LVRA 3 A π V δ P Exampl ω = π 000Hz δ P. ρ = = L =. m V = 63 0 R. m A π = 5 0 m and say V = 0 m. ζ = π 3 5 π =. 7 0 N N 0. 4 So,for ζ = 0,. nd 3730 damprs. 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag 0 of

198 Say l = 4 m D = 40 m N = π Dl T l l T Nl = = = 475 m π D π 40 Not nough room Can gt ζ 00. n = m l T 6.5, Rokt Propulson Ltur 3 Prof. Manul Martnz-Sanhz Pag of

199 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 4: Prssurzaton and Pump Cyls Thrust Chambr Prssurzaton. Introduton Th most thnally dmandng part of a modrn hgh prssur lqud propllant rokt s th hambr prssurzaton systm. Consdr for xampl th lustr of four RD-70 ngns powrng th Enrga frst stag. Thr shard oxygn pump must dlvr 79 Kg/s (.63 m 3 ) of LOX at 64 atm prssur to th ombustors, for a powr of 76,000 HP. Th whol ngn wghs 9500 Kg. Smlarly, th SSME LH pump (s Fg. 3 n ltur 5) dlvrs 73 Kg/s of LH (.06 m 3 /s) at 470 atm prssur, for a powr of 76,000 HP. Th SSME mass s 870 Kg, of whh th LH turbopump s 344 Kg. And ths flow rats ar both dwarfd by thos of th old F Saturn ngn, whh swallowd a small rvr (3.5 m 3 / s) of LOX, albt at a mor modst prssur. Baus of th mportan of ths systm, w wll dvot ths ltur to ngn prssurzaton yls and omponnts.. Gas Prssurzaton Systms Th smplst way to ahv th rqurd thrust hambr prssur s to provd a small, hgh prssur gas rsrvor, whh, at frng tm, prssurzs th propllant tanks. Th tanks must thn b thk nough to support th thrust hambr prssur, plus th njtor drop. In addton, th gas rsrvors ar also rlatvly havy. Thus, th prforman of a gas-prssurzd rokt lags that of a pump-prssurzd rokt (whr th tanks an b muh lghtr), as shown n Fg. (from Rf. 40). W not hr that th rlatv sz of th gans du to a turbo pump systm bom ovrpowrng for hgh V rokts, but may not b worth th xtra omplxty for small V 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag of 5

200 Prnt Improvmnt wth Pump-Fd Engn Payload Wght/Gross Wght Prssur Fd Pump Fd ,000 0,000 30,000 40,000 Msson Vloty (fps) Fg.. Payload rato for typal gas prssurzaton and pump prssurzaton Thus, gas prssurzaton ontnus to play a rol, partularly n small spabasd ngns, suh as monopropllant manuvrng and atttud ontrol thrustrs. A rlatvly larg and sophstatd xampl s th Shuttl OMS/RCS systm. If a gas prssurzaton systm s adoptd, a ho must b mad btwn rgulatd or blow-down gas dlvry. Th rgulatd typ s usd most oftn, baus t avods hambr opraton ovr an xtndd prssur rang, whh may b lad to stablty problms. On th othr hand, blow-down opraton dos rsult n a smplr and lghtr systm, wth lss gas nvntory, and s standard n hydrazn monopropllant applatons. Consdr a rgulatd gas fd systm, shmatzd n Fg.. Lt PG t b th drasng prssur n th gas rsrvor, of onstant volum VG and Pp th rgulatd tank prssur, whr Vp ( t ) s th nrasng ullag volum n both U+propllant tanks. Th ntrnal nrgy of th gas at som tms t s GPand th work of gas xpanson n dt s P dv p. Thrfor, ntgratng n (0,t), Up (assumng adabat xpanson) ( ) UG + Up + PpVp = UG o () 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag of 5

201 NOTE: If sothrmal xp., UG UP UG o whh an b r-wrttn for an dal gas as + = and thn ( ) ( ) PG t VG + Pp VP t = PGoVG Usng U = P v γ ( ) ( ) P t V +γ P V t = P VG () G G p P Go Ths rlats P(t) to V(t) at any tm durng opraton. If th fnal gas prssur s PG G P (larg nough to stll ovrom th rgulator and njtor drops), and f rprsnts th whol tankag volum, w obtan for th gas rsrvor volum V TK V G = P Go γ P p P G V TK (3) (no γ for sothrmal as) (mor gnrally n btwn) Th mnmum mass of ths gas rsrvor, assumd sphral of radus R G, an b stmatd by notng that ts wall thknss t G must b t G Pgo = R G (4) σ GW whr σ GW matral s s th workng strss of th wall matral. If th dnsty of ths wall ρ GW, w obtan for th gas tank mass M GTK 3 ρ γ P GW p VTK = σgw P G PGo (5) A smlar alulaton an b mad for th mass of ah of th propllant tanks. If, for smplty, both, oxdzr and ful tanks ar assumd to b gomtrally smlar and mad of th sam matral (a ylndral body of lngth L p, appd by hmsphrs of radus R p, wth qual L p /R p for both ), thn th propllant tank mass s M 3 ρ + L /R PW p p PTK = Pp V TK σ 3 PW + L p /Rp 4 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 3 of 5

202 P P REGULATOR V P (t) P P OXIDIZER Hgh Prssur Gas P G (t) V G FUEL Fg.. Shmat of gas prssurzaton systm W not from (5) that th gas tank mass s narly ndpndnt of ts ntal prssur: a hghr P allows a smallr volum V (Eq. 3), but rqurs thkr Go walls (Eq. 4). Valus of PGo 5-0 tms P ar ommon. To ompar th mass of P/Pth gas tank to that of th man propllant tanks, assum GGo= 0., γ= 5 3 (H gas) and L p R p = 6. For qual matral rato ( ρ σ ), Eqs. (5) and (6) thn gv MGTK =.64, ndatng that th gas rsrvor s lkly to b th havr M PTK omponnt (although addng ant-slosh baffls, nsulaton, t. may modfy ths onluson). G 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 4 of 5

203 =Th mass of th propllant tslf s Mp = VTK ρ p, whr th dnsts of oxdzr and ful hav bn avragd to ρ p. To ompar to th mnmum tankag mass, 4 assum stl onstruton, wth σρ (m / s) and = 0 atm. Eq. (5) 3 3 thn gvs M /V = 84Kg/m, and Eq. (3.6) gvs M / V = Kg / m, for a GTK TK total of 96 Kg / m 3. Ths s to b ompard to th man propllant dnsty; for N O 4 -UDMH ths s of th ordr of 000 Kg/m 3. Thus, not ountng flangs valvs, rgulators, t., th tankag mass s 30% of th propllant mass, a fgur whh s xssv for any ambtous msson. Advand ompost matrals an rdu ths mass sgnfantly, howvr, Ttanum s usd mostly. An altrnatv mthod to rdu th rqurd gas rsrvor mass s to hat th prssurzng gas pror to njton nto th propllant tank. Ths an b aomplshd through hat xhangng, prhaps usng th gas as a nozzl oolant, or by nludng n th gas a vry small amount of oxygn and hydrogn, blow th flammablty lmt, and thn passng th gas through a atalyt bd rator. In thr as, assumng a gas tmpratur rs T s aomplshd, a trm MG v T nds to b addd on th rght hand sd of Eq. (), whr (MG(t)) s th gas mass that has flowd through th hatr. Ths ylds vntually a modfd xprsson for th gas rsrvor volum: GTK P p TK VGγ P ΔV TK(7) +T PGo -PGT Go pas an xampl, % O plus % H (by mol) n H ylds upon raton T 30K, whh narly uts n half th rqurd gas tank volum (and mass). On problm wth hatng mthods s th potntal for rrat varatons n fd prssur f propllant sloshng ools th prssurzng gas n an unstady mannr. 6.5, Rokt Propulson Ltur 4 Prof. Manul Martnz-Sanhz Pag 5 of 5

204 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 5: Bas Turbomahn Prforman Turbopump Prssurzaton Systms. Cyls For hghr prforman, mhanal pumps must b usd to fd th ombuston hambr. In turn, ths pumps rqur drv powr, whh s always provdd by turbns usng xss thrmal nrgy n th propllants (although ltral motors hav bn onsdrd for small rokts). Th mannr n whh hot gas s provdd to drv th turbns srvs to dstngush among svral prssurzaton yls, of whh th most mportant ar summarzd n Fg.. (From Rf. 4). Th most ommon of ths ar th b-propllant gas gnrator (G.G.) yl, th xpandr yl and th stagd ombuston yl. (a) Th Gas Gnrator Cyl Th GG yl was usd n th F ngn, and s also n us n th Dlta II, Atlas and Ttan rokts. In ths yl, a small fraton of th prssurzd oxdzr and ful ar dvrtd to a mdum-tmpratur burnr (Gas Gnrator), whh produs typally vry ful-rh gas to drv th turbn or turbns. Ths ar dsgnd wth a larg prssur rato, and thr xhaust s thr dumpd ovrboard, or njtd at som pont nto th man nozzl to provd som xtra thrust. Nvrthlss, ths yl s nhrntly somwhat lossy, n that th turbn gas s not fully utlzd n th man ombustor. On th othr hand, th powr ontrol s rlatvly straghtforward, and thr s lttl ntraton of th fd systm wth th rst of th rokt. Any propllant ombnaton an b usd, all powr lvls ar sutabl, and any dsrd prssur lvl an b obtand although th I sp loss nras wth prssur (.5-4 s pr 00 atm). Th mhanal powr rqurd to drv th pumps s P OP PFP PP = mox + mf η ρ η ρ op ox FP F () whr P and P OP FP ar th prssur rs n th oxdzr pump (OP) and ful pump (FP), rsptvly whh hav ffns If th gas gnrator mass flow rat s ' ' PT = mgg TPT n TT mgg η OP, η. Also ρ, ρ ar th lqud dnsts. FP ox, th (sngl) turbn powr s η η () n whh η T s th turbn sntrop ffny (60-80%), η ' TT s ts thrmodynam xpanson ffny ( ) ' ' ' η = P / P γ γ (3) TT n whh dpnds on prssur rato P /P n and GG gas spfy hat rato, F γ '. Also, th spf hat of ths gas, and T t ts tmpratur, whh s ontrolld through stohomtry to valus aptabl by unoold turbns (700-00K). Rf. 4: Turbopump Systm for Lqud Rokt Engns. NASA SP-807, Aug ' P s 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag of

205 Equatng P and PT ylds th rqurd mgg. Suppos now th turbn xhaust s xpandd to th sam xt prssur P as th man flow. Th xhaust spd s thn ' ' GG = P Tt.out ( P / Pt ) γ γ (4) whr ' T = T ( η η ) (5) t.out n T TT Ths spd s gnrally lowr than th man flow xt spd, = T η ; η = (P / P ) (6) p γ γ whr p and η blong to th nozzl gas, and P, tmpratur. Th rlatv I sp loss s: T, ar th hambr prssur and 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag of

206 δ Isp mgg GG Isp m (7) and an b alulatd on th turbn xhaust prssur prssur rato) s sltd. A tradoff s nvolvd hr: f mgg P t P t (and hn th turbn s not vry low, too muh mass must b dvrtd (larg n Eq. 7), whr as f s too low, th m xhaust provds almost no addtonal thrust (small GG n Eq. 7). An optmum an thrfor b found. As an xampl, onsdr th LOX-RP F- yl, for whh Also, P t =844 Kg/s, 7 7 m F = 777 Kg / s, P OP =.06 0 N / m, P FP =.4 0 N / m, T t = 06 K ηop = 0.746, ηfp = 0.76, ηt = and 3 3 = 45 Kg / m, ρ = 80 Kg / m. Th GG ' ' s stmatd to hav γ =.35 and = 40 J / Kg / K, whl th man gas has p γ =.5, = 080 J / Kg /K. Th hambr prssur s P = 95 atm, and at th xt, P = 0.68 atm. Us of th quatons abov ylds, aftr som sarhng, an optmum δi turbn prssur rato of 3, for whh sp = Th atual F- ngn had a I turbn prssur rato of 6.4. sp ρ ox F m ox P (b) Th Expandr Cyl For ngns, utlzng hydrogn (and possbl mthan) as ful, th gas gnrator an b lmnatd. Instad, th ful s smply routd from th xt manfold of th nozzl oolng rut to th turbn nlt. Ths s possbl baus hydrogn s suprrtal at th pump xt, and t smply xpands smoothly nto an ordnary gas as t pks up hat. Th rsultng Expandr Cyl s smpl and ffnt (th ful s fully utlzd n th thrust hambr). Ths yl s usd n th RL-0 ngn and n th start-up squn of th Japans LE-5 ngn (whh thn transtons to gasgnratory opraton). Th prnpl lmtaton of ths yl s th rlatvty small amount of hat avalabl from rgnratv oolng, whh lmts applablty to hamb prssurs undr approxmatly 70 atm. A smpl analyss an dmonstrat ths pont. Th pump powr s gvn by Eq. () and th powr drvd from th ful-drvn turbn s γ F P γ F nj P = T mfη TPFTι (8) Pι whr, th njtor prssur, s also th turbn xhaust prssur. Th turbn P nj 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 3 of

207 nlt prssur rut loss, Pn, s rlatd to th ful pump prssur rs by P ool whl th oxdzr pump has PFP and th oolng PFP = Pt + Pool P TK (9) POP = Pnj PTK. Th shaft powr balan thn gvs ( O/F) γ F ρ + F δtk πt δool δtk γ F + = ψ π T ρ ox ηop η FP (0) whr Pnj ρf PF Tt δtk = P TK / P nj, δool = P ool / P nj, πt =, and ψ = η () T P P t nj Assumng δtk = 0., δool = 0., ηop = η NF = 0.7, O / F = 5 and ρ F / ρ ox = 69/40 = , th rlatonshp btwn π and ψ, as gvn by (0) s shown plottd n Fg.. As th T fgur shows, th turbn nlt prssur nrass rapdly whn ψ drops blow about 30 (at whh pont π T =.33). In fat, as also shown n Fg., th quantty ψ has a mnmum valu of approxmatly 7.76 whn π xsts. Th turbn nlt tmpratur t T =.68, blow whh no soluton T s 00K n th RL-0. Usng also 3 ηt =0.7, ρf =69 kg/m and CPF = 4,600 J/Kg/K, w alulat from () ψ 390 / P nj (atm) and so th maxmum s 390/7.76=78 atm. For rfrn, th RL-0 has PC P nj = 3 atm, P 40 atm, whh orrsponds to ψ = 35. Of ours, as () ndats, hghr Pnj nj valus ould b ahvd f T t ould b nrasd furthr. 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 4 of

208 () Stagd Combuston Cyl For rokts whr hgh hambr prssur as wll as hgh ffny s dsrd, th stagd ombuston yl s th prfrrd ho. On ould thnk of ths as a modfd xpandr yl, n whh a small amount of oxdzr s addd to th ful aftr th oolng rut, thus nrasng th avalabl nthalpy for th turbn drv. As n th xpandr yl, all of th propllant s ntrly usd n th ombuston hambr. Unlk th xpandr, through, any oxdzr-ful ombnaton an b usd. Two promnnt xampls of ths yl ar th Spa Shuttl Man Engn (SSME), and th Russan RD-70 boostr ngn (Fgs. 3 and 4 from Rf. ). In th SSME, th pr-burnrs ar norporatd nto sparat ful and oxdzr turbopump assmbls, and pross most of th ful (LH) wth a small fraton of th oxdzr (LOX), produng a lght vtatd hydrogn turbn drvng gas. In th RD-70, th prburnrs pross all of th oxdzr (LOX) and a fraton of th ful (krosn) to produ a ful-lan gas whh drvs th sngl ntral turbn. In both ass, th 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 5 of

209 turbn xhaust s dutd to th man ombustor njtors, togthr wth th rmanng LOX (SSME) or krosn (RD-70). Th ho of ful-rh prburnrs s prludd by arbon dposts on turbn blads and othr surfas whn hydroarbon ful s nvolvd. 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 6 of

210 Th stagd ombuston yl provds th hghst lvls of rokt prforman, but at th ost of gratly nrasd omplxty. Ths s both, baus of th many duts and valvs nvolvd, and baus of th vry hgh pump xt prssurs. A sondary potntal dffulty, whh s shard by th xpandr yl, s that th turbns ar unhokd and thr s a possblty of low-frquny nstablty dvlopng. Consdr a smplfd shmat (Fg. 5) of a ful-rh yl analogous to that of th SSME. Th njtor prssur, P, th turbn nlt tmpratur,, and th ovrall O/F rato, r, ar prsrbd. Th hat nput and prssur drop n th oolng rut (Q ool, P ool ) ar also assumd gvn. W wsh to dtrmn th rqurd prssur rs P FP by th ful pump, as wll as th O/F n th pr-burnrs (r PB ) and th ful splt S F btwn thm. Th prssur rs n th oxdzr pump s smply P = P P. OP TK T t 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 7 of

211 Th analyss s bst don tratvly. If P FP s tmporarly assumd known, thn th turbn nlt prssur s whr P PB Pr = PTK + PFP Pool PPB () s th prssur drop n th pr-burnr (manly assoatd wth njton). Th rqurd rs n th oxdzr boostr pump s thn POBP = Pt + PPB P (3) Th nrgy balan n on of th pr-burnrs s wrttn as P P P r r h h + + Q + h = ( + r ) C ( T T OP OBP FP PB f PB OTK FTK ool PB Pt t rf ρη 0 OP ρη 0 OBP ρf ηfp rst ) (4) whr h OTK and h FTK ar th nthalps of oxdzr and ful n thr tanks (ths gnors th low prssur boostr pumps) r st s th stohomtr O/F rato and hf th ful hat valu at th rfrn tmpratur T rf. Th spf hat of th ful-rh burnt gas s pt. Eq. (4) an b solvd for r PB f P s assumd known. FP 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 8 of

212 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 9 of

213 Th shaft powr balan for th ful turbopump s r ' r ' PFP = ( + ) P S F rpb pttnη FT ρη F FP Pt (5) whr η FT s th turbn ffny and γ ' γ t = blongs to th pr-burnr gas. A smlar balan an b wrttn for th oxdzr turbopump, and, by dvson, w an solv for th ful splt S F. η FT ρf ηfp POP POPB SF = + r + r ηot ρ0 PFP ηop ηopb PB (6) Aftr r PB has bn alulatd from (4), S F s gvn by (6), and thn (5) an b usd as a hk on th assumd PFP. In ralty, an outr traton loop s rqurd by th fat that γ ' and pt thmslvs dpnd snstvly upon th prburnr stohomtry, r PB. Wth th approxmatons ph 7.67 al / mol / K, and 0.63 al / mol / K pho rPB γ ' = (7) r and thn (for H -O ), PB ρt γ ' = γ ' R M ; M = (+r PB ) g / mol (8) As an xampl, Fg. 3.8 shows som rsults n whh w hav usd and also T = 0 K h h 0 rf OTK FTK r =6, r =8, P =atm =atm, st OTK P FTK η op =0.78, η OBP = 0.696, η FT = 0.790, η OT = 0.79, Pool =0.5 P, P PB =0.05 P, 5 = J/kg, P=P-P =0.P Q ool f 8 h =.x0 J/Kg, t =00 K, T η =0.74 As shown n Fg. 6, th prssur rs n th ful pump nrass mor stply than th hambr prssur, just as t dd n th xpandr yl. Howvr, vry hgh prssurs ar now attanabl. Th P for th SSME s shown for rfrn. Not FP how th pr-burnr stohomtry s ssntally fxd by T t, and dos not vary muh ovr th prssur rang. FP 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag 0 of

214 6.5, Rokt Propulson Ltur 5 Prof. M. Martnz-Sanhz Pag of

215 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 6: Turbopumps Turbopump Componnts Th mhanal omponnts of th prssurzaton yl (pumps and turbns) ar nxt to b onsdrd. An xllnt rnt survy of ths ara s gvn n Rf.40. A mor omprhnsv, but oldr survy s ontand n a srs of NASA SP rports [4-43]. Pumps and turbns wll frst b dsussd sparatly, and thr ntgraton wll thn b xamnd. (a) Pumps Almost all xstng rokts hav ntrfugal turbopumps. Ths dlvr mor P pr stag than axal flow pumps, wth only slghtly lss ffny. Only f multstagng boms nssary s thr a possbl nntv to go to axal pumps; ths happns wth LH ful, whr, du to th low dnsty, th P pr stag s lmtd by th attanabl rm spds. In gnral, th dsgn attmpts to maxmz opratng spd, sn ths rdus th pump sz, and hn th wght. Pump spd s lmtd by svral ffts, most mportantly avtaton at nlt. Othrs ar ntrfugal strsss (thr at th mpllr or n th drvng turbn), lmtng prphral spds for barng and sals, and avodan of rtal spds. Had rs s usd ommonly nstad of prssur rs to xprss th prforman of pumps. W an dfn had rs as th hght to whh on ould ras on Kg of flud wth th amount of dal work pr Kg don by th pump: P dp H = h s /g = P () ρg Th rs s drtly rlatd to th pump work, vn f th flud has sgnfant omprssblty: Work/mass= gh h = () η p and ths s on of th advantags of ts us. Obvously, f ρ = onst., P H = Work P, = (3) ρ g mass ρη p 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag of 8

216 Th had rs s drtly to th prphral spd of th mpllr dsk. Th flud ntrs axally nar thy mpllr hub, wth no angular momntum; t lavs th mpllr wth absolut tangntal spd ωr V r tan β, whr β s th bak-lanng blad angl at th rm Fg, and V r s th flud radal xt vloty, rlatd to th volum flow rat as Q = πr b V r (4) Th torqu ndd to drv th mpllr s th nt outflow rat of angular momntum, and th work rat s ths, tms ω. Thus, m R V tan ω β ωr Powr = ( ) r (5) and sn w also hav Powr = gh m, th had rs s η p ( R ) ω V =η anβ (6) r H p t g ωr 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag of 8

217 V r Th quantty ψ = ηp tanβ ωr s somtms alld th had offnt. In trms of ψ, ( ω ) R H =ψ (7) g ψ s typally btwn 0. and 0.8. Valus gratr than unty ould b obtand f th blads wr dsgnd to larn forward ( β < 0 ), but thn P would nras wth Q (through th fft of V r ). Ths postv slop of th P vs. Q haratrst s known to produ nstablts n th pumpng systm [44]. Ths ar gnrally dynam n natur, and dpnd to som xtnt on th haratrsts of th rst of th systm (fr volum, throttlng ffts, t), but t s rlatvly asy to undrstand thr orgn from a quasstat argumnt: f th pump tmporarly dlvrs mor flow than an b dsposd of n stady stat by th downstram omponnts, ( P) and f ts haratrst has a postv slop, th pump prssur rs wll also Q b hghr than normal. Th downstram prssur wll thrfor tnd to nras for both rasons, and a runaway stuaton nsus. In addton to ths systm nstablty, thr s also a tndny for flow maldstrbuton analogous to rotatng stall, sn th flow s thn unstabl wth rspt to mass ntrhang btwn paralll stramtubs [45]. Eq. (6) would prdt a lnar dpndn of had on flow rat. In ralty varyng th flow at a gvn spd wll vary th ntrnal flow angls wth rspt to blads, and wll thrfor rsult n varatons of th slop H Q. Exampls of ths bhavor ar shown n Fg (Rf. 4), whr th flow offnt s dfnd as = R π Q ( ω ) R b (8) 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 3 of 8

218 Th throttlng rang s dfnd by th pont of maxmum had, blow whh opraton s unstabl. Smlar (but strongr) rstrtons apply to axal dsgns, suh as that usd n th J- LH pump, and so ths dsgns tnd to b lmtd to applatons whh rqur vry lmtd throttlng. Th pump s dsgnd to spfd had rs H or ( P) and volumtr flow Q. Ths quantts an b usd to onstrut th non dmnsonal quantts alld spf damtr and spf spd: 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 4 of 8

219 d s 4 D(gH) = (9) Q n ωq = (0) ( gh) s 3 4 In th U.S. ltratur, D s xprssd n ft, Q n gpm and ω n rpm, and g s omttd. Ths produr, of qustonabl pratally, rsult n rlatd paramtrs D s, N s gvn By N s = 78 n s, Ds = d s. Not that nd and hn, from (7), ωd = () ( gh) s s nd = () ψ s s or, n Englsh unts, NsDs = 08.3 ψ. Sn for ntrfugal pumps, w found ψ <, w an s that thr doman s a ( n,d s s) map s nd s s> (or NsDs > 08 ). Th onsdrabl mpral vdn on pump prforman has bn dstlld (Rf.46) by onstrutng ( n,d ) maps on whh favorabl rgons ar shown for s s varous typs of mahn. A vry gnralzd xampl (takn from Rf. 4) s shown n Fg. 3, whr lns of onstant pak ffny ar shown for a wd varty of pumps. Ths us a st of laran, tolran, roughnss, t. fators, and ar to b takn only as ndatv, sn atual dsgn may dpart from adoptd valus. W not n Fg. 3 th ln ND 00, dnotd as th lmt for dynam pumps, n s s aordan to our dsusson abov. Radal and axal pump dsgns narly mrg, although th axal typ s ndatd for th hghst spf spd, whh as Eq. 0 shows, may b smply a rflton of low had rs, as for xampl, n th ndur stags faturd ommonly at th nlt of ntrfugal pumps for avtaton supprsson. Tabl gvs th faturs of th hgh prssur SSME pumps, and th rsultng ( N, D ) ponts ar nludd n Fg 3, whr thy ar sn to l roughly on th η = 0.8 ln. s s 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 5 of 8

220 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 6 of 8

221 Th pumps must b dsgnd so as to avod avtaton, whh both, dgrads prforman, and auss damag to rusabl artls. Cavtaton rsk xsts at th pump nlt, whr th lqud prssur s lowst, and t nrass wth both, th flud vloty and th spd of th pump ndur blads mtng th flud. Th ndur damtr nds to b larg nough to rdu th nlt flud dynam had ρ C m to som fraton of th xss nlt prssur ovr saturaton P -P sat,. Ths last quantty s alld th Nt Postv Suton Prssur (NPSP), and s usually gvn as a suton had NPSH= NPS/( ρg ). Hydrogn Pump Oxygn Pump P (N/m ) (pr stag) H (m) 0,000,930 Q (m 3 /s) (pr sd) ω (rad/s) D (m) n s (N s ) (050) (90) d s (D s ) 6.6 (0.03) 435 (0.0866) Tabl. Charatrsts of th SSME hgh-prssur ful and oxdzr pumps Th rato: ( NPSP ) ( NPSP ) σ= = C / ρ C m m g (3) s alld th Thoma paramtr, and mpral vdn [40,4] ndats that t should b gratr than for LH, for LOX and 3 for watr and storabl propllants. Th mor favorabl stuaton for hydrogn appars to b rlatd to a gratr vapor supprsson fft du to vaporat hllng whn bubbls start to form. Svral othr paramtr rprsntatons of avtaton data xst. Thus, Rfs [40] and [4] us a suton spf spd S s dfnd as n Eq (0), but wth H rplad by th NPSH, and wth orrton for flow blokag by th hub. Ths paramtr an b shown to b rlatd to Thoma s paramtr σ and to = / ω R (R= ndur radus) by.98 S = (4) s 3 σ 4 t t m In Englsh unts, th numral fator s 83. Th data on avtatons onst for a varty of lquds show that σ rmans approxmatly onstant for ah, as notd abov. Indpndntly, Rf. 47 shows avtatons rsults n th form τ= f( Z t ) (5) 6.5, Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 7 of 8

222 whr τ= NPSP ρ ω ( R ) ; Zt sn ϑ = + osϑ t (6) and ϑ s th ndur ladng dg blad angl, whh, at dsgn ondtons, s los τ = 3Z. Wth small 0 to tan t and s typally Th data ls los to th ln angl approxmatons for ϑ and t, ths an b shown to b quvalnt to σ= 3, ntrmdat btwn Stangland s rommndatons [40] for LH and LOX. t Wth th ndur damtr hosn from th abov rtra, th shaft spd s lmtd [40] so as to kp th ndur tp spd blow 70 m/s (LOX) or 340 m/s (LH). Ths s don n ordr to ontrol avtaton n th blad-tp lakag vortx [40]. Ths spd lmtaton may onflt wth th dsr to pla th ( r, d pont on a s s ) favorabl spot n th ffny maps. In that as, th NPSH must b rasd, thr by partal prssurzaton of th tanks, or, as n th SSME, by th us of sparat lowprssur boostr pumps. Ths ar only rough gudlns, and, th prs allowabl lmts dpnd upon dtald dsgn of th ndur. Progrss n ndur dsgn has bn a pang tm n allowng turbopump spd to nras, thus rdung wght (as wll as nrasng lf). Rfrns td: 40. M. L. Jo Stangland. Turbopumps for Lqud Rokts Engns. Nnth Clff Garrtt Turbomahnry Award Ltur, Aprl 7, 99. SAE/SP-9/ Turbopump Systms for Lqud Rokt Engns, NASA SP-807, Aug Lqud Rokt Engn Turbns, NASA SP-80, Jan Lqud Rokt Engns Cntrfugal Flow Turbopumps, NASA SP-809, D E.M. Grtzr, Th Stablty of Pumpng Systms, ASME, Transatons, Jl. of Fluds Engnrng 03 (98), pp J.L.Krrbrok, Arraft Engns and Gas Turbns, MIT prss. 99 (S.5-7). 46. O.E. Balj, Turbomahns: A Gud to Dsgn, Slton and Thory, J. Wly & Sons, Nw York, L.B. Strplng, Cavtaton n Turbopumps, pt., Trans. ASME, Srs D, J. Of Bas Engnrng (96): , Rokt Propulson Ltur 6 Prof. Manul Martnz-Sanhz Pag 8 of 8

223 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 7: Turbns. Rokt turbn dsgn mphaszs powr dnsty, baus of th ovrrdng onrn for mass savng. Effny, whl larly a onsdraton, taks a lss promnnt rol than n arraft turbn dsgn. Ths tnds to favor low raton turbns and whn mult-stagng, vloty ompoundng ovr prssur ompoundng. Th dgr of raton of a turbn stag (stator nozzls plus rotor blads) s th fraton of th flud stat nthalpy drop whh ours n th rotor (s ston. of ths Ltur). In an mpuls turbn, th dgr of raton s zro, manng that th gas xpands and alrats as t turns n th stator passags, and thn s mrly rdrtd at onstant thrmodynam stat by th movng rotor blads. Th stag vloty trangls ar shown n Fg, whh also nluds th as of a 50% raton turbn for omparson. In both ass, th flow lavs axally. Th torqu (and hn th powr) s proportonal to th hang n th tangntal omponnt of th absolut vloty. Fg shows that ths hang s th whl spd ω R for th 50% raton turbn, but s ω R for th mpuls turbn. It s also lar, howvr, that flow vlots ar hghr n th mpuls as whh wll lad to largr vsous losss. Also, th lak of alraton n th rotor passag wll favor flow sparaton on th suton sd. Altogthr, th mor powrful mpuls stag s also lss ffnt. 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag of 7

224 If two stags ar usd, on an ovr-turn th flow n th frst rotor, so that t lavs wth a omponnt of vloty ontrary to rotaton. Th sond stator thn an r-drt ths flow wthout alratng t, and snd t nto yt anothr pur mpuls rotor. It an b shown that ths vloty ompoundng (s Ston 3. of ths ltur) wll yld four tms as muh torqu as th sngl mpuls stag suh as thos n Fg. b so that th flow s r-xpandd n th sond nozzl (prssur ompoundng). Ths only doubls th powr of th sngl stag, but mprovs th ffny. Th SSME ful turbopump turbn s ssntally of ths typ, although thr s som non zro raton (R 0.). For prlmnary dsgn purpos, Rf [46] (Rfr to th lst of th Rfrns n Ltur 5) prsnts ( n, d dagrams smlar to thos n us for pumps. On of s s ) ths s rprodud n Fg.. Th quantts Q and gh n th dfntons of d s and n s (Eqs ) ar now to b spfd mor fully: th volumtr flow rat Q s at 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag of 7

225 turbn xt stat ondtons, and th nrgy gh s th total-to-stat sntrop nthalpy drop: γ γ P t gh =. ptt t Ptt. Not that, as n th as of a pump, th produt nd s s () has a smpl onnotaton: nd s s = 8 ω R = C ψ 0 () whr 0 = gh s somtms alld th spoutng vloty. Th vloty rato ω R 0 = s oftn usd as a dsgn paramtr. For optmum ffny, t should ψ b about 0.4 n an mpuls stag, 0. for a vloty-ompound doubl stag, or 0.7 for a 50% raton stag [40]. Th optmum rotatonal spd of th turbn was n th past hghr than that for th drvn pumps, whh rqurd havy garng. Th stady advanmnts n ndur dsgn hav mor rntly allowd mountng both turbn and pumps on on shaft, wth major mass savngs. Whras pump mpllrs (xpt for hydrogn) ar rarly strss lmtd, th rvrs s tru for turbns, whr th blad root axal strss and/or th ds rm hoop strss tnd to lmt spd. Th radal root strss s approxmatly (s S. 4 of ths Ltur) A σ 6 = ρbω (3) π whr A s th annulus flow ara and ρ b s th dnsty of th blad matral, whh s dtatd by th flow rat and nlt gas dnsty. Thr s thus an nntv to us matrals, suh as drtonally soldfd supralloys, that rtan a hgh strngth at th unoold opratng ondtons. 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 3 of 7

226 Fg.. n s d s dagram for turbns alulatd for mnmum loss offnts.. Som Mor Gnral Prforman Rlatonshps for Turbns Ignor nffns: h = P P Dfn Dgr of Raton R ( h) ( h) ( P) ( P) Rot. = stag Rot. stag (low p.r. pr stag, ρ onst.) 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 4 of 7

227 =absolut vl. ω=rlatv vl. Assum axal xhaust (dsgn ond.) Torqu = α IN sn r m Powr m ( ) = ωr snα = ( ) IN = ωr tan α OUT P stag ρ OUT = IN osα ( ) x =onst. through stag P stator IN OUT OUT OUT = = = tan α ρ os α ( ) ( ) P tanα R = = P ωr stator OUT stag Also, ψ= ( P ) P ( ωr) = tanα stag OUT ωr So, ψ = R ψ= ( ) R Mor work/stag wth lss raton R=0 (mpuls turbn) ψ = R= (50% raton) ψ = Also, φ= x = OUT, so, ωr ωr tan α tanβ ψ = φ = φ For, 50% dsgn, α = β and ωr tan β = For mpuls, ta n α = tanβ OUT 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 5 of 7

228 3. Vloty Compoundng: Follows ths dagram startng from th xhaust (at bottom) St. Rotor Powr = m(4ω r + ωr)rω = 6m( ωr) St. Rotor Powr = m(ωr)rω = m( ωr) Total = 8 m( ωr) = 4 Sngl mpuls st stator must gv 4ω r tangntal vloty. 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 6 of 7

229 4. Blad root strss: Pb ω r A dr ( ) dr d = σa dr σ=σ P b ω (r r ) σ = σ = ω (r ) 0 P b (r r ) A =π(r r ) σ = ρ ω b A π 6.5, Rokt Propulson Ltur 7 Prof. Manul Martnz-Sanhz Pag 7 of 7

230 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 8: Mhanal Dsgn of Turbomahnry Intgraton and Rotordynams of Turbo Pumps. Intgraton and Mhanal Componnts As notd bfor, turbns and pumps ar oftn mountd on a ommon shaft. If th oxdzr and ful hav smlar dnsts, thr rsptv pumps an also b on on shaft. Ths s th as n th MK-3 Atlas and Dlta II boostr turbopump, whh, howvr, has a gard turbn-pump transmsson. Th Russan RD-70 taks a furthr ntgraton stp by havng a sngl turbn drv both, ful and oxdzr pump, all on a sngl shaft. In addton, ths pumps fd not a sngl thrust hambr, but a lustr of four n th as of th Enrga vhl (Fg 4, L. 4). Engns usng LH ful rqur dffrnt spds for th oxdzr and ful pumps. Th frst of ths ngns, th RL-0, had a sngl-stag oxygn pump on on shaft, gar drvn by a sond shaft on whh wr mountd a -stag hydrogn pump and th drv turbn. Mor rnt ngns (LE-7, SSME) fatur sparat shafts for th oxygn and ful, ah arryng ts own drv turbn. Barng dsgn and barng plamnt hav a sgnfant mpat on th ovrall turbopump haratrsts. Exstng ngns us rollr lmnt barngs, and n rnt dsgns, ths ar lubratd and oold by th propllant bng pumpd, whh smplfs th onstruton. On th othr hand, ths dpartur from tradtonal barng prat has nsstatd xtnsv rsarh on ompatbl matrals. Rf [48] dsrbs work on advand ball barngs for th futur Spa Transportaton Man Engns (STME) and th SSME Altrnat Turbopump Dvlopmnt (ATD). Th lubraton onpt rls on sarfal war of th barng ag (bronz-60% Tflon), and ts transfr from th rollng lmnts to th raway surfa. Rollr lmnt barngs an provd som stffnss, through angular ontat dsgn, but th bulk of th axal thrust of th pumps and turbns must b hydraulally baland, thr by bak-to-bak pump arrangmnts wth no fdbak, or, as n th SSME turbopumps, by provdng hydraul fdbak to som surfa atng as th balanng pston [40]. Futur dsgns ar lkly to fatur hydrostat barngs, whh rly on a vry thn flud flm to support th rotor wthout sold ontat wth th asng. Th advantags of ths barngs ar summarzd n Tabl from Rf [40]. Th most mportant ar th rmoval of th surfa spd lmtaton of ball barngs and th muh hghr radal stffnss. Th surfa spd s xprssd by th DN produt n onvntonal barngs, and, as 6 Tabl ndats, s n th rang of 0 (mm) (Rpm). Ths lmtaton fors th dsgnr to sk barng loatons wth th smallst possbl damtr, suh as outboard of th pumps and turbns, but ths barng loatons tnd to lowr th st natural frquny, and to ntrfr wth flow approah to th pumps. 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag of 3

231 Itm Hydrostat barng Ball barng Spd lmt Non.0M DN LH.75M DN LO Lf lmt Unlmtd stady stat Transnt rub onrn = h Dsgn onstrants Supply prssur avalablty Shaft damtr for torqu transmsson Drt stffnss to >5M lb/n. 0.5 to M lb/n. for duplx par Dampng Rotor-dynams 50 to >500 lb s n. No onstrants for optmum poston Adjustabl stffnss and dampng to 5 lb s n. Poston onstrants No adjustabl dampng Tabl. Hydrostat barng bnfts As n all turbo mahns, sals ar rqurd to rdu or prvnt lakag of fluds around th shaft from hgh to low prssur aras. Th hgh lnar spds of th rokt turbopump surfas, as wll as, n som ass, th oxdzng natur of th flud, dtats th us of non-ontat typ sals, whh, by thr natur, allow a nonzro lakag rat. Thus, n oxdzr turbopump wth th ful-rh turbn on sam shaft, thr s a nd to ntrodu som hgh-prssur nrt gas nto a rgon sparatng th two fluds, wth sals provdd to mnmz lakag (and hn nvntory) of ths purg gas. Mult-tooth labyrnth sals ar standard n jt ngns, and wr norporatd at varous ponts n th orgnal SSME turbo pumps. Howvr, aftr a squn of rdsgns to orrt vbraton problms most of ths hav bn rplad by vry low laran smooth ylndral sals, whh hav muh hghr radal stffnss and sgnfantly ontrbut to rasng th lowst natural frquns of th rotor (n fat, ths sals an b vwd as a transton to hydrostat barng dsgns). 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag of 3

232 6.5, Rokt Propulson Ltur 8 Prof. Manul Martnz-Sanhz Pag 3 of 3

233 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 9: Rotordynams Problms. Turbopump Rotor Dynams Baus of hgh powr dnsty and low dampng n rokt turbopumps, ths mahns xhbt n thr most xtrm form a varty of vbraton ffts, whh ar thr absnt or maskd by normal dampng mhansms n othr turbo mahns. Th low dampng s spally promnnt n lqud hydrogn pumps, baus of th vry low vsosty and dnsty of ths mdum. Ol squz flm damprs ar prludd n any ryogn mdum. Th gnral fram work of Rotor Dynams s now wll stablshd, through a ombnaton of lassal analyss and dtald numral smulaton [49, 50, 5]. Intnsv fforts on th applaton of ths thortal mthods to a spf rokt turbopump ar dsrbd by Ek[5], and wr nstrumntal n pontng th way to a srs of mprovmnts that rsolvd a srous dvlopmnt problm n th SSME. Th gratst dffulty n ths fld rmans th prs haratrzaton of th flud fors atng on th rotor at omponnts suh as sals, turbns, or mpllrs. On ths ar spfd, numral modls of grat powr and vrsatlty an b brought to bar for analyzng th dynams of a gvn onfguraton. Baus of th rmanng unrtants n th bas fors. Ek s 978 rommndaton [5] rmans vald today: Prdton of stablty n a nw dsgn must b vwd wth skptsm. A prdton of nstablty should, howvr, b takn vry srously.. Ford and Slf- Extd Vbratons Thr ar two typs of rotor dynams problms: (a) Rsonans whh usually our whn th rotatng spd onds wth on of th natural ( rtal ) frquns of th rotor (nludng ts supportng strutur). Ths fall n th atgory of Ford Vbratons, n whh an xtaton for produs dflton rsponss of an ampltud whh nrass as th xtaton frquny approahs a rtal frquny. If th xtaton s at xat rsonan, th ampltud grows lnarly n tm at frst, and thn, f vsous dampng xsts, t approahs a lmt whh s nvrsly proportonal to th dampng fator. Rmoval of th xtaton rmovs th rspons. Th xtng fors ar typally rlatd to rotor mass mbalan or gomtral mprftons. Rsonans rarly pos srous problms, unlss th stady opratng pont ls vry los to on rtal. On th othr hand, sn th strutur s mad as lght as pratal, many natural mods usually xst, svral of thm thr blow or not far abov th opratng rang. Efforts ar mad n th dsgn phas to rat a rlatvly wd rang of rsonan- fr spds around th normal opratng pont. Passag through rtals, f mad rapdly nough, s not a svr ondton. Tabl 3 (Rf [53]) shows th rtal frquns of th SSME ful turbopump. Not that svral of th shaft mods ar splt nto adjant pars of rtal frquns baus of th lak of symmtry of th asng strutural supports, 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of 3

234 vn though th barng strutur tslf s symmtr. Ths asymmtry s n gnral a bnfal fft, and provds a sort of fftv dampng [54]. (b) Slf- Extd Vbratons Ths ar autonomous osllatons, n whh th shaft vbrats at on of ts natural frquns (not qual to th shaft spd), and du to som postv fdbak mhansm, absorbs nrgy from an xtrnal sour (usually th flud) nto th vbratonal mod. Exat balanng dos not rmov ths typ of vbraton. On ntatd, f dampng s nsuffnt, th vbraton wll nras xponntally n ampltud untl som nonlnar mhansm ntrvns, or untl rubbng ours. Slf-xtd vbratons ar also alld rotor-dynam nstablts or sub- synhronous vbratons. Th lattr dsgnaton s du to th fat that thy ar most oftn obsrvd n th lowst shaft mod whn th rotatng spd s wll abov th frquny of ths mod. For som mhansms of xtaton, th rato of th rotaton spd at onst of nstablty to th frquny of th vbraton xtd s a smpl ntgr, COUPLED HPFTP ROTOR AND CASE MODES Mod Frquny(Hz) Dsrpton - Rotor fr spn, X 47.0 Cas rokng, Y Cas rokng, Z Cas rokng + bndng, Y Dffusr torson Cas rokng + bndng, Z Rotor translaton, Y a Rotor translaton, Z Rotor rokng, Z a Rotor rokng, Y Rotor axal, X Cas + rotor rokng, Z Rotor bndng, Y a Rotor bndng, Z Turbn as torson, X Dffusr bndng, Y Dffusr bndng, Z Cas + rotor rokng, Y Cas torson, X Cas axal, X Turbn as bndng, Z 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of 3

235 or an xtaton thrshold Tabl 3 xsts at an ntgr multpl of th xtd frquny [55]. Howvr, thus s not unvrsally th as, and, n fat, no suh smpl rato or thrshold sms to xst for th most mportant mhansms (sal or turbn blad-tp ffts). It s tru, howvr, that th rat of nrgy nput nto th vbraton nrass wth powr lvl of th turbopump, and hn wth ts spd; thus, th mahn dampng may b suffnt to ompnsat for ths fft at low rotaton spds, but, as spd nrass, a thrshold wll appar, byond whh th opraton s unstabl. Th two typs of vbraton dsrbd an b asly dstngushd n tsts by plottng a srs of vbraton sptra at nrasng rotatonal spd (a asad dagram, Fg. ). Hr Fg.. A asad dagram showng on mod only. Ford vbratons rlatd to mbalan ar sn at Ω =ω, wth a rsonan whn ω =Ω r. Also shown s a sub-synhronous, slf-xtd vbraton at Ω =Ω. r ω s th shaft spd, Ω s th angular frquny of th vbraton, and Ω s th rtal or natural frquny (only on shown). As ndatd, no slf-xtd vbraton s vsbl untl ω rahs som thrshold, and thn th nstablty boms mor and mor promnnt. Th ampltud tslf dpnds on ondtons whn th sptrum s takn (varaton rat of ω dwll tm at th gvn ondton, t.), but th frquny nformaton s stll qut usful. r 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 3 of 3

236 3. Sours of Rotor-Dynam Extaton: Cross-Coupld Fors Th most mportant xtaton mhansms ar rlatd to th produton of ross-fors whn th shaft s offst from ts ntral loaton. Bfor dsrbng som spf xampls, th gnr dynam ffts of ths ross-fors wll b dsussd. Qualtatvly, f a ross-for F y rsults from a shaft offst x, ths for xrts a torqu F y x wth rspt to th nomnal shaft ntr. It s wll known that any lnar vbraton n th x-y plan an b rsolvd nto a forward and a bakward rular osllatons. Of ths two, on (th forward omponnt n Fg. s rnford by th rsultng torqu F y x, whh, bng produd by th dsplamnt tslf, wll rman synhronzd to t. Ths s th bas nstablty mhansm. For a smpl lnarzd analyss, suppos th flud ffts ar suh that a gnral transvrs dsplamnt ( x, y ) and dsplamnt rat x, y of th shaft produs fors (Fx,F y ) aordng to F Kxx K x xy Cxx C x xy x = + Fy K K y C C xy xx xy xx y () 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 4 of 3

237 Th offnts Kx x and K x y ar th drt and ross-oupld stffnss, rsptvly. Not that th ylndral symmtry has bn xplotd to rdu to two th numbr of stffnss fators, and that a postv K s rstorng, and wll augmnt th strutural stffnss K 0, although, n gnral, Kxx << K0. Smlar ommnts apply to th dampng offnts C, C, xpt that Cxx may b of th sam ordr as any addtonal dampng C 0. xx xy x x Dfn a omplx dsplamnt x + y = z () thn dx M = ( K + K ) + K ( C + C ) + C dt dy 0 xx x xy y 0 xx x xy y ( K0 + Kxx ) y xy x yy x ( 0 xx ) y M = K C C + C dt (3 a, b) Add x (3b) to (3a): d z ( 0 xx )( x y ) xy ( y x ) M dt = K + K + + K z - z d z = ( ) ( ) xx xy xx xy z M K K K z C C C dt K C C K z + z z 0 M + M = (4) Put now Ωt z = z (unstabl f Ω I < 0) + Ω Ω = C K 0 (5) M M C C K Ω= ± + M M M (6) Call K xx = K K xy 0 K yy K xy 0 = K 0 C + C ζ= xx KM 0 η= C xy KM 0 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 5 of 3

238 Ω Ω = Ω = K M 0 0 C + C + C C + C + C K + K + K ± + KM 0 KM 0 K0 0 xx xy 0 xx xy 0 xx xy Ω K0 M = = ( ζ + η ) ± ( ζ + η ) + + Kxx + Kx y (7) γ ζ, η, Kxy Ω K0 M K K ζ η± + xx + xy + Kxx + η+ ζ + k xy Ω = Ω 0 (8) - Kxx η+ ζ k xy For K xy > 0, f K xy > ζ, unstabl (9) K C + C xy 0 xx > xy >Ω 0( 0 + ) 0 KM 0 K K C C xx (For nstablty) (0) Clarly, th flud dampng C, f postv, rnfors th othr mahn dampng C 0, xx and promots stablty, whras th ross-oupld stffnss K x y s quvalnt to a ngatv dampng K xy Ω0. Th ross-dampng C xy s sn to hav rlatvly mnor fft on th dynams, sn, as K xx, t only affts th natural frquny, and not th growth or day rat. In som nstans, th flud-rlatd stffnss s not nglgbl, and an b xplotd to hlp rloat th rtal frquns away from unfavorabl rangs. Ths was, n fat, th approah takn n th SSME rdsgn [5]. Th two prnpal avnus for mprovng rotordynam stablty-nrasng dampng or rasng th natural frquny ar both xmplfd on th rght-hand sd of Eq. (0). 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 6 of 3

239 4. Som Exampls of Cross-Coupld For Gnraton Among th mhansms whh hav bn dntfd as ontrbutors to th produton of dstablzng ross-fors ar (a) Prssur non-unformty n labyrnth and othr sals; (b) Non-unform drvng for n turbns du to tp lakag ffts [59, 60]; () Non unform prssur and drvng for n pump mpllrs, also rlatd to lakag ffts [55, 56]. Othr ffts ar rvwd n Rf [57]. Rathr than attmptng hr a gnral ovrag of ths rathr rh fld, w wll only xplan n som dtal th labyrnth sal and blad tp ffts, whh, asd from thr mportan n pratal ass, an b sn as prototyps of th rlvant physs. (a) Labyrnth Sals. Th xstn of flow swrl at th ntran to an offst labyrnth sal (and, n modfd form, to othr sals as wll) gvs rs to a rotaton of th prssur pattrn produd by th offst, and hn produs ross-oupld fors. Two prnpal ffts an b dntfd hr [58]. On of ths an b dsrbd as follows: th flud n th gland of th sal (a sngl-avty labyrnth, for smplty) rulats azmuthally n th varyng ara ratd by th shaft offst (Fg. 3). Fg. 3: Knmat quantts assoatd wth a labyrnth sal. Th shaft s spnnng at an angular frquny ω, whl smultanously undrgong a rular prsson of ampltud and frquny Ω. 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 7 of 3

240 If th tangntal vloty V wr onstant (whh s approxmatly tru undr som ondtons), th rulatng flux ρ Vl( h +δ) would nd to b nrasng wth ϑ at ponts whr th dpth ( h +δ) (h= salng strp hght, δ( ϑ ) = loal gap) s nrasng. Ths mpls an xss of lakag from upstram of th sal nto th avty ovr that from th avty to downstram of t, whh s aomplshd by a loally dprssd P( ϑ). Thus, th prssur would b mnmum at pont C n Fg. 3. and maxmum at pont A. Th nt ntgratd prssur for s thn rotatd 90 from th shaft dsplamnt n th drton of th swrl. Sn th prssur pattrn s anhord to th whrlng gap pattrn, t s lar that, for a gvn nlt swrl, th fft wll dpnd on th whrlng spd Ω and wll, n fat, b zro whn Ω R = V, baus th gland flud wll thn not mov tangntally wth rspt to th whrlng gap pattrn. Ths argumnt s modfd by th azmuthal varatons ndud on th vloty V, whh wr so far ngltd. A mor omplt analyss [58] gvs th for omponnts paralll and prpndular to th dsplamnt as π lh δ lh P ρ F = ( l P) ; f = 4 Rδ + f Rδ V ΩR () ( ) π δ F = l ρ P V ΩR + f () whr P = P P, and w hav assumd nomprssbl, nvsd flud and a rular whrlng moton about th asng ntr wth ampltud. Asd from a ngatv (n th notaton of (Eq. ), w s from () that postv Kxy and C xx ar prdtd. Th prsn of hghr ordr trms n Ω n Eq. also ndatd fftv mass and othr ffts, but ths hav lttl dynam mpat. Th man fators ar rlatd n ths as through K xx R K V C xx = xy (3) and th smpl as n whh vloty varatons ar ngltd s rovrd whn lh Rδ n Eq s. () and () Th sond sal mhansm dpnds on th xstn of frton btwn th rulatng flud and th rotor and asng surfas (although th rsultant ross-for s stll a prssur, and not a frtonal for). Baus of frton, th man tangntal vloty V s usually slghtly lss than th nlt tangntal vloty V and lakag flud ntrng th gland from upstram wll ontnuously add tangntal momntum 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 8 of 3

241 to th gland flud as thy mx, n a mannr smlar to th opraton of an jtor pump. Ths fft s strongst at pont D (Fg. 3), whr th gap s wdst, and P wakst at pont B. Thus, a postv prssur gradnt wll xst at D, and a ϑ ngatv on at B, agan ladng to a prssur maxmum at A and to a ross-for. Sn ths fft dos not dpnd on flud vlots along th prpndular axs ( ) rlatv to th whrlng fram, t dos not show whrl vloty dpndn, and dos C. A smplfd analyss, ngltng tangntal vloty not ontrbut to dampng ( ) xx varatons, gvs for ths fft ( V V) δ ρ R F =πr hl + g h (4) ( ) ρ δ =π P V V R F R ; g = V V + g h hl P ρ (5) Baus of th mportan of nlt swrl n promotng sal ross-fors, dswrlng fns an b usd ahad of th sal, f ths s at all pratal. Exprmnts [58] hav valdatd th abov ross-for mhansms, whl also pontng out th mportan of othr sondary ffts, partularly for drt stffnss. Both for omponnts ar gratly magnfd n smooth sals wth vry small laran [5, 6]. Whthr th addd ross-ouplng or th addd stffnss s mor mportant whn on of ths s substtutd for a labyrnth, must b drtly assssd through dynam analyss for ah spf as. (b) Turbn Blad-Tp Effts. It was ndpndntly pontd by Alford [6] and Thomas [63] that th snstvty of blad-tp losss to blad-tp gap n turbns should produ forward-whrlng ross fors. Th bas mhansm s smpl: Whn th turbn rotor s offst from ts ntr, th blads on th sd whr th tp gap s rdud wll gan ffny, and hn produ mor tangntal drvng for than avrag, and th oppost wll happn on th sd whr gap opns up. Intgratng ths fors around th prphry ylds, n addton to th dsrd torqu, a sd for n th forward-whrlng drton (s Fg. 4). It s asy to translat ths η argumnt nto an xplt quaton for th ross-for. Lt β= b th δ / H ( ) snstvty of blad ffny to rlatv tp laran δ / H, whr δ s th laran and H s th blad hght. Ths fator s of th ordr of -5, dpndng on dsgn and opratng pont. 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 9 of 3

242 Th tangntal for f pr unt lngth along th prmtr s thn approxmatd as f δ δ f β (6) H although t must b pontd out that ths quats work loss to ffny loss, and hn t gnors prssur rato varatons also ndud by th offst. Th laran prturbaton δ δ vars n proporton to th offst, and snusodally wth azmuth ϑ, and so, upon projton n th drton prpndular to th offst, and ntgraton, on obtans F Q =πrf β =β H R H (7) whr Q = πrf s th turbn torqu. Th drt for F s prdtd to th zro. Th xstn of ths fors was onfrmd xprmntally n Rf. [64] and, n mor dtal, n Rfs [59,60] whr t was found that, n addton to th abov mhansm, a sond sour of ross-fors s a tangntally rotatd prssur nonunformty s atng on th turbn hub. Th ontrbuton of ths omponnt s addtv to th bas Alford/Thomas fft, and amounts typally to 40% of th total ross-for. In addton, both mhansms also gv stffnng for omponnts (along wth offst drton). 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 0 of 3

243 On mportant quston whh rmans xprmntally unanswrd s whthr or not th Alford/Thomas fors show a sgnfantly snstvty to whrl spd,.., whthr thy provd C omponnt for th stffnss matrx. Th orgnal argumnt xx would prdt no suh snstvty. A farly obvous xtnson, aountng for hang Ω os ϑ to th bas blad spd ωr an b shown to yld a modfd Alford fator β ' = β ψ H R Ω ω (8) whr Q m( R) ψ= ω ω s th turbn work offnt, whh s los to for mpuls turbns. Sn HR 0. and Ω ω (whrl to spn rato) s 0.5, w s that β' β 0.05, whh would not b sgnfant. Othr vloty-dpndnt ffts may ars form tm lags n th azmuthal rdstrbuton of flow approahng a whrlng rotor; ths would also b of ordr HR, but no drt xprmntal vdn xsts for thr magntud. Rf [60] dos provd a thortal modl for ths fft howvr th Alford-Thomas fors an b vry larg, rqurng dampng log drmnts of th ordr of 0. n typal rokt turbopumps. 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of 3

244 Rfrns Ctd 48. F. Frlta t al. Rotor Support for th STME Oxygn Turbopump. Papr AIAA th Jont Propulson Confrn, July J.P. Dn Hartog, Mhanal Vbraton, MGraw-Hll book Company, Chlds, Dara W. Turbomahn Rotor Dynams: Phnomna, Modlng and Analyss, Wly Intrsn, N.York, J.M. Van, Rotordynams of Turbomahnry, Wly, Nw York, M.C. Ek Soluton of th Subsynhronous Whrl Problm n th Hgh-Prssur Hydrogn Turbomahnry of th Spa Shuttl Man Engn. J. of Sparaft, Vol. 7, No G.R. Mullr, Fnt Elmnts Modls of th Spa Shuttl Man Engn NASA TM-7860, Jan F. Ehrh Th Rol of Barng Support Stffnss Anstropy In Supprsson of Rotordynam Instablts. ASME th Bnnal Conf. on Vbraton and Nos. Montral, Canada, Spt Jry, B. Aosta, A. J., Brnnn, C.E. and Caughy, T.K. Hydrodynam Impllr Stffnss, Dampng and Inrta n th Rotordynams of Cntrfugal Flow Pumps. NASA CP-338, 984, pp Cham, D.S., Aosta, A.J., Brnnn, C.E. and Caughy, T.K. Exprmntal Masurmnts of Hydronam Radal Fors Stffnss Matrs for a Cntrfugal Pump Impllr, ASME JI. Of Fluds Engnrng, Vol.07, No. 3, 985, pp F. Ehrh and D.W Chlds, Slf-Extd Vbratons n Hgh Prforman Turbomahnry, Mh. Engnrng, ay Knox T. Mllsaps, Th mpat of Unstady Swrlng Flow n a Sngl-Gland Labyrnth Sal on Rotordynms Stablty: Thory and Exprmnt. Ph.D. Thss, MIT, M. Martnz Sandus, B. Jaroux, S.J. Song and S. Yoo Masurmnt of Turbns Blad-Tp Rotor dynam Extaton Fors J. of Turbomahnry, 7, 3, July 995, pp S.J. Srg and M. Matrnz-Sanhur, Rotordyanam Effts du to Turbn-Tp Lakag: Part I, Blak-Sal Effts. Part II, Radus Sal Effts and Exprmntal Vrfaton. J. of Turbomahnry, 9, 4, Ot 997, pp D.W. Chlds, Dynam Analyss of Turbulnt Annular Slas Basd on Hr s Lubraton Equaton ASME J. of Lubraton Thnology, V.05, No. 3, 983, pp , Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag of 3

245 6. J.S. Alford, Protstng Turbomahnry from Slf-Extd Rotor Whrl, J. of Engnrng for Powr, Ot H.J. Thomas Unstabl Osllatons of turbn Rotors Du to Stam Lakag n th Claran Of th salng Glands and th Bukts. Bull. Sntfqu A.J.M., Vol. 7, 958, pp K. Urlks, Claran Flow Gnratd Transvrs Fors at th Rotors of Thrmal Turbomahns. NASA TM -779, Ot.983. (Translatd from Dotoral Dssrtaton at th Thnal Unv. of Munh, 975). 6.5, Rokt Propulson Ltur 9 Prof. Manul Martnz-Sanhz Pag 3 of 3

246 6.5, Rokt Propulson Prof. Manul Martnz-Sanhz Ltur 30: Dynams of Turbopump Systms: Th Shuttl Engn Dynams of th Spa Shuttl Man Engn Oxdzr Prssurzaton Subsystms Sltd Sub-Modl In th omplt SSME ngn, all varabls afft ah othr n omplx ways. In ordr to tst our fault dtton algorthms, a dynam subsystm s dsrd, wth rdud ordr, but wth th unmodlld stats ntratng as wakly as possbl wth thos modlld. Attnton was fousd on th lqud oxygn subsystm for two man rasons: (a) Th O/F rato at ratd powr s 6, so that th LOX dynams should domnat ovr th LH ffts whrvr thy ntrat, and (b) Th turbopump pr-burnrs ar run vry ful-rh n ordr to lmt th turbn nlt tmpraturs blow th mtallurgal lmts of th unoold blads; small xursons of th LOX flow to th pr-burnrs ar thn mmdatly translatd nto larg and potntally rtal turbn tmpratur xursons. In our submodl w thrfor fous attnton on th LOX turbopump, whh fds both, th man LOX njtors, and (aftr th boost stag) th two turbn prburnrs. Also modld ar th dynams of th LOX fdng ln to th pr-burnrs, as wll as to th man LOX valv and man njtor, plus th LOX pr-burnr tslf and th man hambr prssur. Varatons of th LH-rlatd stats should ndd oupl wakly to th LOX systm, manly through LH flow varatons onto th prburnrs (nsnstv, sn thy ar ful-rh), and nto th man hambr (hokd flow, no fdbak). Th Dynam Equatons Thr ar thr typs of dynam quatons to b onsdrd: () Rotatonal dynams of th LOX turbopumps. () Equatons xprssng th lqud nrta undr prssur dffrn varatons (analogous to ndutan n ltr ruts). (3) Equatons xprssng th ablty of avts to stor flud du to ts omprssblty undr prssur flutuatons (analogous to apatv ffts). () Rotatonal Dynams If I OTP s th momnt of th nrta of th Oxdzr Turbo Pump (OTP) rotor, angular vloty, τ OT th torqu dlvrd by th OTP turbn, τ OP th torqu absorbd by th man oxdzr pump stag, and τ OP 3 th torqu absorbd by th oxdzr boostr pump, thn 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag of Ω 0 ts

247 dω = τ τ τ () dt 0 OTP OT OP OP3 I In a hybrd systm whr Ω 0 s n rad/s, t n sonds, and th torqu n lbn, th onstant I OTP has a valu 0.96 (whh mpls I OTP = 4 lb m n ). 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag of

248 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 3 of

249 () Inrta of LOX n pr-burnr ommon supply ln Ths s a prototypal quaton of typ (), and, n ordr to llustrat th undrlyng physs, w wll gv hr a brf drvaton of t. Consdr a pp of lngth L and ross-ston A, fd by th boostr pump dsharg at a prssur P OD3, and havng a man prssur P POS (Pr-burnr Oxdzr Supply). Frtonal fors along th pp, and at bnds and rstrtons, ontrbut a total prssur drop λ ρυ, whr ρ and υ ar th LOX dnsty and vloty, and λ (of ordr unty) s a prssur loss offnt. Th lqud s thn atd on n th forward drton by a nt for POD3 λ ρυ PPOS A.. Th mass of lqud n th pp s ρ AL, and w must hav dυ λ ρal = P P ρυ POS A dt ( ) OD3 Now, th flow rat n th pp s mop 3 = ρ υ A, so th quaton an b r-wrttn as dmop 3 λ L = P 3 3 OP OP PPOS m dt ρ A A or L dmop 3 A dt OP3 = P P K m POS OP 3 () whr K = λ ρ A (3) In unts of lb m / s for m and lb / n for P, th onstants hav th valus L =, K = A 00 n and λ = 0.06 ( ) A n. A Ths mpls L = 3.86 (3)Flud Capatan n prburnr LOX supply ln blow: Ths s a prototypal quaton of typ (3), and w also provd a drvaton Consdrng agan th POS supply ln, t rvs LOX flow at a rat m OP 3 from th boostr pump, and dshargs m FPO nto th ful prburnr (FP) and m nto OPO th oxdzr pr-burnr (OP), plus a small amount whh s dvrtd to ool th pump. Undr dynam ondtons, thr s a (gnrally non-zro) nt nflow 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 4 of m OP C

250 m m m m 3 m. Lt ρ V b th mass stord n th pp, whr OP FPO OPO OP C V π D = 4 L s th volum avalabl. Thn w must hav d ( ρ V) dt = m (4) Evn though LOX s a lqud, t has fnt omprssblty at th vry hgh prssurs nvolvd hr. Ths s masurd by th thrmodynam paramtr K dρ 0 6 = 5 0 Pa n / lb ρ dp In gnral, th volum V also vars slghtly undr prssur flutuatons, but t an b shown that ths fft s sondary. W thrfor rwrt (4) as d ρ dp dp dt ρ POS ( ρv) = m 3 m m m POS LOX OP FPO OPO OP C (5) In th sam unts as bfor, th fator dρ ( ρv ) has a valu of POS ρ dp LOX. Usng dρ n / lb, ths mpls a ln volum V = 409 n, whh 38,0 ρ dp ombnd wth prvously stmatd L / A = 3.86 n, ylds L 40 n, A 0 n. Ths ar not xptd to b xat dmnsons of th POS ln, baus th modl lumps togthr svral sondary nrtos and apatans, but thy do appar rasonabl. Indntally, from th prvous rsult λ 0.06 A, w now stmat λ.8, agan a rasonabl valu for a prssur loss fator. Th Complt Submodl In addton to th thr quatons drvd abov, thr ar thr othrs of th flud nrta typ and thr othrs of th flud apatan typ. Th omplt submodl, n th sam unt usd so far, s shown n Tabl.on nxt pag. Equatons (6), (7), (8) ar th ons just drvd. Eq. (9) dals wth th nrta of th LOX movng through th valv and th njtors of th Ful Prburnr (FP) undr th flutuatng drv of th prssur dffrn P P, lss th prssur drops n th valv and n th njtors. Ths drops hav th haratrst m form, just as n Eq. (), but, n addton th valv opn ara fraton A/ A FPV appar squard n th dnomnator, as t should aordng to Eq. (3). Ths ara fraton wll at as on of our ontrol varabls. Eq. (0) s dntal n strutur to Eq. (9), but rfrs to th flud nrta n th LOX dom of th Oxdzr Prburnr (OP). On agan, th OP valv ara fraton A/ A OPV appars hr as a ontrol varabl. 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 5 of POS FP ( n)

251 Th rmanng nrta-typ quaton s Eq. (3), whh rfrs to th LOX movng through th Man Oxdzr Valv (MOV) undr th drv of th dffrn btwn th man oxdzr dsharg prssur,, and th man ombustor prssur, P, lss th sum of th prssur drops n th MOV (assumd 00% opn) and th njtors. Th rmanng thr apatan-typ quatons ar Eqs. (), () and (4). Eq. () dsrbs aumulaton of gas n th Oxdzr Prburnr (OP), wth mass flow m OPF (th un-modlld ful flow nto th OP) plus P OD m OPO (th LOX flow nto th OP) ntrng, and almost all of th gas flow nput to th oxdzr turbn, m OT, lavng. Tabl: DYNAMICS OF LOX PRESSURIZATION SUBSYSTEM Equaton No. Equaton Dsrpton Tm Constant (s.) (6) dω Rotatonal O = τot τop τop3 dynams of 0.96 dt OTP (7) dn OP3 LOX nrta n = POD3 PPOS m OP3 prburnr 00 dt supply ln (8) dp POS Mass storag = mop 3 mfpo mopo mop C n prburnr 380 dt supply ln (9) LOX nrta n dm FPO = mfpo njtor dom PPOS PFP m OPO dt A / AFPV of FP (0) dn OPO 0.60 mopo = P P.463 m POS OP dt A / AOPV () () 0, dp dp OP dt F dt = m + m m OPF OPO OT + = m m + M.085 m FT OT FT OT (3) dm MOV = POD PC m MOV 5 dt (4) dp = mf + mmov mcn 4000 dt OPO LOX nrta n njtor dom of OP Mass storag n Oxdzr Prburnr Mass storag n ful duts to njtor LOX nrta n man njtor dom Mass storag n man ombustor , Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 6 of

252 Equaton () dsrbs th gas aumulaton n th two larg duts whh brng th partally oxdzd hydrogn to th man hambr njtor dom. Fdng ths volum ar th (unmodlld) dsharg flows on th man Ful Turbn m and of FT th low prssur Ful Turbn m OT m FT, plus th dsharg of th man Oxdzr Turbn, ; lavng ths volum s basally th man Ful Injtor flow m, plus som FI smallr LH oolng flows, up to.085. m FI Fnally, Eq. (4) govrns th hangs n th man ombustor prssur, P, du to mass aumulaton. Th mass nputs ar th ful and oxdzr njtor flows ( m, ), whl th mass loss s th nozzl flow rat. FI m m MOV CN Charatrst Tms For ah of th dynam quatons (6)-(4), w an stmat th haratrst tm onstant, whh provds sam prlmnary appraton for th dynams of th systm. For ths purpos, w balan th rat trm wth on of th domnant trms on th rght; for nstan, for Eq. (6), th tm onstant s Ω τ = O, wth and valuatd at thr nomnal valus (ratd powr). Ths 0.9τ OT Ω O τ OT tm onstrants ar nludd n Tabl. Th prburnr supply ln flow and th man ombustor prssur ar sn to adjust rapdly (undr ms). Fllng and mptyng of th Oxdzr Prburnr s rlatvly slow (4 ms), and th shaft spd of th OTP s vry slow (58 ms). All othr dynams ar omparabl n spd wth tm onstants of a fw ms. Calulaton of non-stat varabls Th squn of algbra omputatons (no addtonal dynams) rqurd to alulat th rght-hand-sds of Eqs. (6)-(4) s summarzd n Appndx. Th data for ths alulatons ar th valus of th nn stat varabls, th valus of th ontrol varabls (prburnr valv opnngs), and a fw unmodlld varabls arsng from th ful sd of th ovrall systm. Th lattr ar gnrally kpt at thr nomnal valus. 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 7 of

253 Appndx B. Stady Stat Valus from th SSME Thrmodynam Modl 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 8 of

254 Appndx C. Stat Varabls n a smulatd Throttlng Squn 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 9 of

255 Appndx D. Charatrst Tms (approx.) for th SSME Dynam Modl 6.5, Rokt Propulson Ltur 30 Prof. Manul Martnz-Sanhz Pag 0 of