16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 3: Ideal Nozzle Fluid Mechanics
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1 6.5, Rok ropulsion rof. nul rinz-snhz Lur 3: Idl Nozzl luid hnis Idl Nozzl low wih No Sprion (-D) - Qusi -D (slndr) pproximion - Idl gs ssumd ( ) mu + Opimum xpnsion: - or lss, >, ould driv mor forwrd push by ddiionl xpnsion 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g of 0
2 - or mor, <, nd h xr prssur fors r suion, bkwrds ompu m ρu soni hro: + m ρ RgT g ( ) g RT ; R g R ll Γ 3 ll RT * g ( hrrisi vloiy ) Γ ( ) m * n xprss u,,, in rms of ihr or or : + ; + + u T + u T ( ) + T T, T nd, T + Bus p p g g u T + T RT + RT RT 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g of 0
3 m u u + + * u * T Rg + + ( ) RT + g + Γ In vuum, ( 0) ( + ) ( ) + u + + * ( ) v + ( ) v + + ( ) + + nd ohrwis, 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 3 of 0
4 v No: or, ( ) hd + u ( ) * + + or 0 ( ) x,v + + ( ) hoi of Opimum Expnsion or Rok lying Through n mosphr ( vrying) Th hrus offiin ws drivd in lss in h form v () nd w lso found v + + ( ) ( ) () (3) Th hrus-drivd vloiy inrmn is b b Δ V d 0 d (4) m 0 m 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 4 of 0
5 ( ) whr du only o h vriion of in (), whil m m bus of mss burnou. Th quniis v nd r im-invrin. Subsiuing (), () nd (3) ino (4), ( ) dpnd on (or nozzl gomry), bu b d b d Δ V v 0 0 m m or ΔV b b d 0 m d v b d m m 0 0 (5) W now mk h pproximion h h rjory will hng lil whn w vry (nd hn v, ). W n hn rgrd h im ingrls in (5) s fixd quniis whil w opimiz. Dfin h non-dimnsionl vribls v ΔV b d m ; p 0 0 b 0 b d m d m (6) so h (5) boms v ( ) p ( v ) (7) nd w n now diffrni v w.r. (holding pons.) v v p 0 (8) rom () nd (3), h for ignord. W hn hv + + ( ) pprs in boh rms of (8) nd n b 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 5 of 0
6 + ( ) + + p + ( ) p 3 ( ) uliply ims 3 +, nd no h + + ( ) + + ( + ) ( + ) + p Expnd & simplify + ( ) ( ) + p ( ) nl h for ( is lrly no n opimum!) p+ or 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 6 of 0
7 + p OT p (9) Noi h h xi prssur is givn by + (0) nd so h opimum xi prssur urn ou o b OT p () Howvr, if p < 0.4 o sprion h highs mus b mndd o, his would imply < 0.4 o, nd hr would b flow (on h ground). To void his, h opimliy ondiion o Grr of ρ, 0.4 () OT wih similr xprssion for : OT Ls of,.5 p o (3) Th limiing ondiion in whih h whol burn ours W hn obin o is simpl. p b 0 0 o d m d m b o (4) 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 7 of 0
8 nd h opimliy ondiion () yilds ( ) prssur-mhd, s xpd. OT o, i.., h nozzl should b s mor nd mor of h burn shifs o highr liuds, p drss from long s i sill rmins bov 0.4 opimum dsign, nd if p drops blow 0.4 on h vrg of sprion on h ground. o, quion () givs som inrmdi o o. s, h nozzl should b dsignd o b Nozzl low Sprion Effs Rul of humb (o b xplord lr): low sprs h poin in h nozzl whr 0.4 (Summrfild ririon) So, if 0.4 (vn if < ), no sprion > fr sprion, roughly prlll flow, wr rgion o urn flow). (no srong p grdins in dd So zro hrus onribuion wih xi prssur ' 0.4 rformn wih sprion h of nozzl So, () (full nozzl) <, 0.4 v o ( f ) g ( ) 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 8 of 0
9 (b) (full nozzl) >, 0.4 lul ' ' ( 0.4 ) ' ' ' ( ) v ' hn ( ) o ' 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 9 of 0
10 6.5, Rok ropulsion Lur 3 rof. nul rinz-snhz g 0 of 0
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