16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 7: Convective Heat Transfer: Reynolds Analogy

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1 6.5, ok Propulsion Prof. Manul Marinz-Sanhz Lur 7: Conviv Ha Transfr: ynolds Analogy Ha Transfr in ok Nozzls Gnral Ha ransfr o alls an aff a rok in a las o ays: (a) duing h prforman. This nds o b a -3% ff on only, and is hrfor sondary. (b) Craing gra diffiulis in h dsign of ho-sid sruurs ha hav o 7 8 surviv ha fluxs in h 0 0 / m rang. Th prinipal mods of ha ransfr o nozzl and ombusor alls ar onvion and radiaion. Of hs, onvion dominas, and radiaion nds o b imporan only for paril-ladn flos from solid propllan roks. I sp Conviv Ha Transfr W ill rvi hr h omprssibl D boundary layr quaions in ordr o xra informaion on all ha ransfr. Th govrning quaions ar (in h B.L. approximaion) Coninuiy ( ρu) ( ρv) x + y 0 () X-Momnum u u p τxy u ρu + ρv + µ () x y x y y y Y-Momnum P y 0 (3) 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag of 6

2 h h T ρu + ρv uτ + k (4) x y y y y Toal nhalpy ( xy ) u hr h h+ is h spifi oal nhalpy, and µ is h visosiy. For a µµ is a fluid propry. ok boundary layrs ar almos alays laminar flo, ( T) urbuln, and µ is hn h urbuln visosiy, hr momnum ranspor is ffd by h random moion of urbuln ddis. If hs ddis hav a vloiy sal u' and a lngh sal l', hav, in ordr-of-magniud. µ ρ u'l' (5) urb. hr u' is som fraion of h loal u, and l'nds o b of h ordr of h all disan y. Th imporan poins abou (5) ar (a) (b) µ µ, mosly baus l' man fr pah and urb. µ is proporional o dnsiy (hras µ is no, baus h m.f.p. is invrsly urb. proporional o ρ ). Similarly, h las rm on h righ in h nrgy balan, rprsning h onvrgn of ha flux, onains h urbuln hrmal onduiviy K ρ u'l'. On again, noi ha K is hr proporional o dnsiy. W also no ha h urbuln Prandl numbr p P r µ p (from h ordrs of magniud) k I is of som inrs o no h origin and omposiion of h visous rm in quaion (4). If oll h do produs u i τ around a fluid lmn as shon (in B.L. approximaion), 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag of 6

3 obain h rm ( uτxy ) as rin in (4). This 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 s rm in (6) is jus h vloiy ims h visous n for pr uni volum, so i is h par of h oal visous ork ha gos o alra h loal flo. Th sond rm in (6) is posiiv dfini, and i is h ra of dissipaion of nrgy ino ha du o visous ffs. W ill rurn lar o his haing ff. Approxima Analysis L us manipula h righ hand sid of quaion (4): u T u K T uµ + K µ u + y y y y y y µ y and, sin h y p T, his yilds y u h µ p µ + Pr y y Pr y k W no hr ha, boh for laminar and urbuln flos, P r is a onsan, indpndn of P and T o a good approximaion. In fa, as nod bfor, i is also of ordr uniy ( 0.9 for urbuln flos). So, h HS of h nrgy quaion boms h u µ + y y Pr (7) If mad h approximaion P r, hn his ould rdu h furhr o µ y y ih h fla pla approximaion (), (4) ould bom u h h +. If in addiion, mad approximaions P 0, hn h pair of quaions x u u u ρu + ρv µ x y y y h h h ρu + ρv µ x y y y (8) 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 3 of 6

4 Ths ar idnial quaions for u and linarly ransformd variabls h. Th sam quaion ould also govrn h u h u ; h h u h h (9) hr h ( ) subsrip dnos h valu of a variabl in h loal xrnal flo (jus ousid h boundary layr). Boh u and h saisfy idnial boundary ondiions: u h 0; u h (0) and, as nod, idnial govrning quaions. W onlud ha, undr h assumpion P Pr, 0 x, h h u h h u () u hr also noid h h + h. This similariy rlaion bn vloiy and oal nhalpy profils is knon as Croo s analogy. Approxima ha flux a h all W ar inrsd in h magniud of h all ha flux T q K y () q K h K h µ y µ p y p hr usd h u h h h+ + u y y y y 0, sin u 0 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 4 of 6

5 K Th group µ P approximaions. Thus p r should b s qual o uniy, for onsisny ih h sad q h µ y Us no quaion (): u h h u q µ h + ( h h ) µ y u u y and noi ha u µ y is h all shar srss, τ. So q h h u τ (3) hih is also alld ynolds analogy. A mor ompa form of his an b rin in rms of h Friion Coffiin f τ ρ u (4) and h Sanon numbr S q ( ) ρ u h h (5) ih h rsul (from (3)) S f (6) On imporan poin an b mad abou h rsul (3): Th ha flux o h all is drivn by h nhalpy (or mpraur) diffrn bn Toal xrnal and Wall valus, no bn sai valus. This an b noninuiiv. Considr h siuaion nar h xi of a highly xpandd spa nozzl, hr h bulk mpraur may hav droppd o, say, 300K du o h srong T xpansion from a hambr mpraur of, say, 3000K. Th all ould b mad of Tungsn so as o b abl o susain rlaivly high mpraur and ool islf by radiaion o spa, so T ould b, say, 500K. Is h nozzl all bing had or oold by h 300K gas? Th ansr is ha i is bing had, baus 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 5 of 6

6 T T 3000K, hil T. Ar violaing h nd 500 < T Prinipl of Thrmodynamis. ad on. Simplifid Profils, Aross h Boundary Layr To br undrsand his siuaion, l us rurn o Croo s analogy (quaion ) and ri h h+ u, and solv for h: ( ) u u h h h (7) u This is a quadrai rlaionship bn h and u. For lo subsoni flos h h, so h las rm is no srong, and h rlaionship boms linar in h limi. Th rlaionship bn slops a h all flos from (7): 0 dh du u ( h h ) u dy u dy y or dh h du h u (8) W an us (7) and (8) o skh h vs. u aross h boundary layr. For a as ih h >, his looks lik h 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 6 of 6

7 No ha hnvr u > h h hr dvlops an inrmdia mpraur from h. Th as hn h < is mor rvaling vn: maximum. Bu in any as, h all slop is as if h lin r oming from h h, no No h all slop is sn o b posiiv (ha ino h all), dspi h < h (as long as h > h ) So, h quadrai porion of h Croo rlaionship is rsponsibl for h xra all ha; his an in urn b rad o visous dissipaion, hih aumulas in h boundary layr and lvas is mpraur, so ha h all is had vn hn h ousid mpraur is lo (as long as h flo has high spd). Modifiaion for Pr W lav for no h issu of h non zro prssur gradin, xp o no ha i inrodus small modifiaions don o h hroa. Th dviaions of P from uniy ar small, and, for gass P < (~ 0.9 for urbuln flo). This braks h prf r balan bn dissipaion and onduion rsponsibl for Croo s analogy, in h sns of favoring onduion of h dissipad ha. As a sond onsqun, h mpraur ovrshoo is rdud, and so is h all slop of T and h ha flux o h all. Th dir ff of highr onduion (P ) is aound for approximaly by modifying ynolds analogy o r < r S f (9) P 0.6 r Th sondary ff (rdud ovrshoo) is aound for by rplaing h driving nhalpy diffrn h h by h h, hr h is h Adiabai-all nhalpy, dfind as a a 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 7 of 6

8 u ha h + r ; r 0. 9 (0) (urbuln) and r is h ovry faor. Wih hs hangs, h ha flux is no f q ρ u ( ha h) () P 0.6 r Th Barz ha flux formula A vry rud, bu surprisingly ffiv rprsnaion for h friion faor supplid by h ll-sudid as of fully dvlopd urbuln flo in a pip ; ρ ud f 0. µ f is ha () hr is h ynolds numbr basd on diamr D, and visosiy. Puing also h p T + onsan, quaion () no givs 0. µ is h laminar 0.03 µ q ρu p ( Ta T) ( u) p ( T 0. a T) P0.6 µ r ud ρ (3) ρ D 0.06 I is ommon prai o dfin a ha ransfr gas-sid film offiin, h (no an nhalpy!) by g h g T a q T (4) And, so far, hav h 0.06 ( ) ρ u µ p (5) D g 0. A his poin no ha h formulaion so far has ignord h srong variaions of ρ and µ aross h boundary layr sin hs quaniis dpnd on mpraur as T ρ (a Ponsan) ; T ( 0.6) µ (6) A ommonly usd approah o inluding hs variaions is o rpla ρ and quaion (5) by hir valus a som inrmdia mpraur <T>: µ in 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 8 of 6

9 T < T > ρ ρ ; < T > µ µ T (7) and <T> an b valuad by svral mpirial ruls. For Mah numbrs no muh highr han, an simply us T + T <T> (8) Making h rplamns of (7) in quaion (5), obain h T 0. g ( ρu) 0. p D < T > µ (9) hih is on form of Barz formula. A mor usful form follos from h oninuiy quaion: ρ u i m P A A * A, ih gt, Γ( γ) and hr A is h loal ross-sion, and in (9), and using A A A D, h final form is D h hroa ross-sion. Subsiuing h P D 0. T g 0. pµ D * D < T > (30) Svral imporan rnds and obsrvaions an b mad no: (a) Smallr hroa diamr lads o largr ha flux sraigh from h ynolds no. dpndn of. f D 0.. This oms 0.8 (b) Ha flux is almos linar in hambr prssur ( P ). This limis h fasibiliy of high hambr prssurs, hih ar ohris vry dsirabl..8 D () Maximum ha flux ours a h hroa. On riial dsign D onsidraion is hrfor h hrmal ingriy of h hroa sruur. 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 9 of 6

10 (d) Lighr gasss lad o highr ha fluxs, hrough h ombind ffs of and * hg 0.6 () Th faor M T T < T > is grar han uniy. This < T > nhanmn of ha flux follos mainly from h fa ha h gas in h boundary layr is mosly oolr han in h or, hn dnsr. W shod bfor ha h urbuln ha onduiviy is proporional o dnsiy. p Exampl Considr h Spa Shul Main Engin (SSME), hih is a Hydrogn-Oxygn rok ih (roughly) hs hararisis: P 0am. 0 P 7 a T 3600 K M 5g/mol r.5 gt * 600m / s Γ γ ( ) p γ 800 J / Kg / K γ M 5 µ 3 0 Kg /m/s T T 300K T γ+ hroa T 000 K W alula hn T + T < T > 00K (a h hroa) T < T > , ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 0 of 6

11 and so, using quaion (30), hg 60,000 / m / K and q 60,000 ( T 000) a γ Ta T + r µ K 0.9 ( ) (slighly lss han T ) 8 q 60, W / m This is a vry high lvl of ha flux. To visualiz h impliaions, suppos his q had o b ransmid hrough a hin mal pla (hiknss δ, hrmal onduiviy k). On ould hav q K T δ hr T is h mpraur drop hrough h mal. As an iniial guss, suppos h mal r sainlss sl ( K 0 W /m /K) δ mm. Thn, and 8 3 qδ T 9,000K!! K 0 Obviously, his is unapabl. Try using Coppr insad, ih K 400 W / m / K (ny ims br). This givs T 950K, sill no apabl (oppr ould b vry sof hn). Th pla ould hav o b hinnr and mad of oppr. No an asy problm. 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag of 6

12 Mor raionally Th S or h should dpnd on x, disan from sar of nozzl, sin h B.L. is sill g dvloping (no fully dvlopd). In addiion, hr should b som aouning for alraion propry variaion hrough B.L. ylindrial gomry Th aril by ubsin and Inony (h. 8 in osnho and Harn s Handbook of Ha Transfr, MGra-Hill, 973) givs a gnral formula for urbuln B.L. In an ylindr, ih alraion: S ( x) A ρux ff n n s F F µ n (and h ρ u S ) g p f s found alls. h A onsans, dpnding on ynolds no. basd on mom. h. n > 4000, A 0.03, n 7 < 4000, A 0.093, n 5 F F Faors for propry variabiliy. Can ak svral narly quivaln forms. A simpl on from Ekr, is F ρ < Τ > ρ < > T < Τ > T + + ( Τ ) µ T F ( 0.6) µ ( < Τ > ) < Τ > T T T T a u γ Ta T + r T + r M 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag of 6

13 r 0.9 (rovry faor) Th ffiv disan x ff (aouning for mmory of pas alraion) is rlad o h aual disan x hrough an ingral ( ) ( ) x f x' xff ( x) dx' 0 f x hr f ρ u ( ) n zµ F F n n n ha h u z ha h +r h h (x) body radius a x. For a quik sima of, an simplify furhr o h fla-pla as, in hih f d, dx ih ( ) ( > ) < 4000 f , and ih d dx d d x 0.08 < x x x.99 0 < x 76 6 x 67 7 x > d 5 ( ) d or x x 0.05 x 7 x 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 3 of 6

14 For larg roks, nds o b 0 0, so h high formulas should xhroa b br, dspi h ommon us of Barz s formula, hih ar basd on h lo formulaion. Forunaly, diffrns nd o b small, and ar markd ofn by ohr unrainis (surfa films, fluid propris). Exampl and Comparisons: 7 8 Considr nozzl ± xanx + x 4anα x anα ih origin a x x anα and.5, 5 α o and going hrough hroa a x x anα Using γ.5 and h > 4000 opion, find x ff M hroa 0.5M M x + + x x d x x an5 o hr an5 + 4 ( x) o M.5 M and M( x) Th ingraion givs x ff hroa.089 Compard o x x x.53 and , ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 4 of 6

15 Sin xff appars o h hroa) is insignifian. 7 por, h mmory/alraion ff (up o h Th hroa S is hn (0.9) ( T 363) a T (using r, T 000K, T 3300K, K ).5 ( S ) 0.03 hroa 0.77 ρ u 7 ( x x) < T > µ T hroa hroa (363) (0.695) < T > T Tak 7 P 0 N/m, M 5 g/mol * 8.34 gt m / s Γ P * ( ρ u) 560 Kg / s / m x x o o an5 an5 and, ih 0.3m, T 5 hroa µ µ Kg ms , ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 5 of 6

16 On gs, ( ) S (0.004 using insad of hroa Using h < 4000 opion (small roks) x x x ) ( S ) hroa ρ uxff < T > µ T hroa hroa and using again x x x ff,, g ( ) S ( using ) hroa x For omparison, h fully dvlopd pip flos formulaion ould giv S * g T A µ 0. ρ u p D P < T> A ahroa ( ) S hroa This is los o h < rsuls abov (and, indd, h offiins ar for 4000). Bu his appars oinidnal, basd on h fa ha for mos nozzls, < x. 6.5, ok Propulsion Lur 7 Prof. Manul Marinz-Sanhz Pag 6 of 6