16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 10: Ablative Cooling, Film Cooling

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1 6.5, Roet Propulso Prof. Mauel Martez-Sahez Leture 0: Ablatve Coolg, l Coolg Traset Heatg of a Slab Typal proble: Uooled throat of a sold propellat roet Ier layer retards heat flux to the heat s. Heat s s T gradually rses durg frg (60-00 se). Pea T of heat s to rea below atl. lt. Ba T of heat s to rea below weaeg pot for struture. Prototype -D proble: Ca be solved exatly, or a do traset -D ueral oputato. But t s useful to loo at bas ssues frst. Theral odutae of B.L.hg Theral odutae of frot layer Theral odutae of layer ( thess, theral odutvty) 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page of

2 Wat layer to have hg to protet the rest. W//K W (Say, porous, Oreted graphyte, K W hg 50,000, so OK here). K opared to Also, fro goverg equato we see that T T T T α t t x x ( α, theral dffusvty, / s ) x x αt, or x αt, or t. α So the layer wll adapt to ts boudary odtos a te t. α Say, J 70 ad KgK 00 Kg ( 3 sold graphyte), so 6 α.3 0 / s The layer adapts 3 ( 3 0 ) t 7.0se (ore le α.8 se ). Treat frot layer quas-statally,.e., respodg statly to hages heat flux: T T ( t) ( t) wh w q ( t ) Ths also eas we a lup the theral resstaes of BL ad st layer seres: h ( hg ) eff + g 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page of

3 ad se hg, ( hg) eff h g or layer (the heat s), layer) so, very lely, s hgh (etal) ad ( h ) s ow sall (thas to st g eff ( h ) g eff (or stae, say Copper, W 360, wth. We ow have K ( h ) W 350, but K g eff W 36,000 K, so deed, ( h ) ). g eff Uder these odtos, the heat s s beg trle harged through the hgh theral resstae of layer. Most lely, heat has te to redstrbute terally, so that T s early ufor aross the layer. We a the wrte a luped equato. ( ) ( ) ( ) dt T q hg T eff aw T Taw dt dt τ τ + dt ( T ) Defe T T T ( 0) aw 0 ( ) T T T T e τ aw aw 0 t or our exaple, say 3 J (Copper), 8900Kg / 430, KgK τ 30 se 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 3 of

4 Ths s ofortable. Suppose Taw 3300 K, T0 300K, ad we fre for 0 se: (60) ( ) (60) 0 (989) T e 30 50K May eed 4 whh s stll (OK) for Copper (elts at 360K, but o stress bearg, so a go to ~900. Also OK for steel o Carbo str eber). NOTE: ( 0.0).se, so, deed, layer adapts quly to B.C. s 4α ufor / s A More Exat Soluto Cosder T tured o at t0. The B.L. has a fl oeffet h, ad the frst aw ( h ) layer has,, so that h g g eff + h g,, σ, α. The ba s sulated.. Layer has thess, ad has The oe a prove that layer has a teperature dstrbuto g 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 4 of

5 ( ) αt λ aw ae os Taw T 0 T T x,t x λ where a sλ λ + s λ os λ ad λ (,, ) are the roots of λ ta λ ( h ) g eff Graphally, or sall λ λ, sall, so ta λ, so λ λ ad also a α / λ τ fro before So, leadg ter s the ( ) t Taw T x,t x e τ os λ Taw T0 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 5 of

6 whh s what we foud before. The other ters are uh saller, exept at very sall te. or theral proteto of sold roet ozzles read se. 4. (pp ) of Sutto-Bblarz, 7 th ed., espeally, pp A ey oept s ablatve aterals. They ota a C-based hoogeeous atl. ebedded reforg fbres of strog (asotrop) C. Best s C/C, strog expesve fbre se ozzle a get to 3600 K, a be D or 3D. Also good s C or Kelv (Arad) fbres +pheol plast ress (for large ozzles) or the shuttle RSRM, the throat sert (C loth pheol) regresses ~ h/0 se, ad the har depth s ~ 0.5 h/0 s. 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 6 of

7 l Coolg of Roets or applato of data o slot-jeted fls, we eed to defe the tal fl thess s, veloty u, desty, or at least ass flux u. Assue we ow the flow rates ad, where s the ore flow ad the fl flow. We also ow the fully-burt teperatures ad oleular weghts ( T, T ; M, M ). The areas ouped at the fully burt seto are ot ow; let the be A, A. ro otuty, R T P M ua P P s oo to both () 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 7 of

8 R T P M ua () ad the total ross-seto s ow: A + A A (3) We eed soe addtoal forato to fd (egletg frto): u. The two oetu equatos are du dp u + 0 dx dx du dp u + 0 dx dx du u u dx du dx udu udu (4) Both, ad, have bee evolvg as drops evaporate ad bur. We ae ow the approxato of assug ther rato to rea ostat (equal to the fullyburt value). The (4) tegrates to u u u u (5) Substtute to the rato ()/() ua A ua A or A A (6) ad also u u (7) Ths last rato u u s alled the fl oolg paraeter, M : 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 8 of

9 M M M T T The fl thess s (at oplete bur up) follows fro (8) π ( ) A D 4s ( f s D ) A D s D π( ) A D D s A D s s D A 4A D 4 (9) ro Rosehow & Hartett, Chapter 7-B, we haraterze fl oolg by the T for heat flow. I the absee of a hage t dues to the drvg teperature ( ) aw 0 γ fl, Taw T + r M 0 T aw T aw 0, ad we alulate ( qw) hg ( Taw T w ) No l hages to (lower, presuably). The lowest we ould we defe a fl oolg effey aw. The fl T T to get s, so η T 0 aw Taw T aw T (0) Lts: 0 η 0 f Taw T aw (o effet) η f Taw T (ax u effet) If we a predt η, the η( ) 0 0 aw aw aw T T T T () ad the w g ( aw w ) q h T T () where hg paraeter s oputed as f there were o fl. To predt η, we frst oputes the 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 9 of

10 x µ ζ Re Ms µ 4 (3) where x s the dstae dowstrea of the fl jeto (here we assue ths s fro the bur-out seto), ad Re us µ (4) ad u M ( u ), fro before ro ζ, there are several se-epral orrelatos for η. A reoedato fro R & H s 3 r.9p η 0.39 p ζ p (5) (or η f ths gves >) whh s supported by ar data of Seba. Exaple T M 0.8 ; M.6.65 T M 0.5 Say 0. (0.0) 9 (0.00) Say D0.5 x x 0.5 t opl.ob 6 P70 at N / T 300K 5.33K g / ; 8.53Kg / M 0g/ol; γ. 3 3 M , Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page 0 of

11 8.34 u / s 0.0 u / s s 7 Say µ 0 Kg//s Re s 5 0 D 0.5 s OR 0.00 Re ( ) 0.6 µ T 0.6 µ T ( ) 5 ( ) ζ (5.74) e p p µ 0.8 (say, r r), µ Pr η.4 η ( 5.74 ) 0.8 So, ths offers perfet fl oolg, eag T Taw T 600K ( ( )96 K) If the wall s ade of Cu, ad s at T w 700K, the reduto heat flow s w 0 w q q whh a be desve. 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page of

12 (Ths exaple shows oe ould get good fl oolg wth uh less tha 0% flow the fl, aybe wth oly %). 6.5, Roet Propulso Leture 0 Prof. Mauel Martez-Sahez Page of