CHAPTER 9. Compressible Flow. Btu ft-lb lbm ft-lb c p = = ft-lb slug- R. slug- R. 1 k. p p. p v p v. = ρ ρ

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1 CHPTER 9 Cmrssibl Flw 9 Bu f-lb lbm f-lb c lbm- R Bu slug slug- R f-lb cv c R slug- R Bu 7 lbm R f-lb slug- R Bu 778 f - lb slug lbm c 9 c cv + R c cv c + R r c R c R / ( ) 9 If s, Eq 99 can b wrin as c n T R n n T n l l r l l T T I fllws ha, using c c + R and c / c, R/ c T T Using Eq 97, v v T T r Finally, his can b wrin as 9 Subsiu Eq 58 in Eq 57 and nglc nial nrgy chang: & & Q WS + + ~ u ~ u m& c / R

2 Enhaly is dfind in Thrmdynamics as h u~ + v u~ + / Thrfr, & & Q WS + h h m& ssum h fluid is an idal gas wih cnsan scific ha s ha h c T Thn & & Q WS + c ( T T ) m& Nx, l c c + R and c / c s ha c / R ( ) Thn, wih h idal gas v v law T / R, h firs law as h frm & & Q W S + m& 95 Diffrnia c using d( xy) ydx + xdy: Rwri: d d d d 96 Th sd f sund is givn by c d / d Fr an ishrmal rcss TR / K, whr K is a cnsan This can b diffrniad: d Kd RTd Hnc, h sd f sund is c RT 97 Eq 9 wih Q & W & S is: + ct cns' ( + ) + + ( ) + c T + c ( T + T ) + c T + c T ( ) + + c T c T h W nglcd ( ) Th vlciy f a small wav is c h c 98 Fr war d d 6 Pa Sinc g / m, w s ha

3 c d / d 6 / 5 m / s d 99 Fr war c d 6 5 m / s L vlciy im m 9 Sinc c 5 m/s fr h small wav, h im incrmn is d 69 scnds c 5 9 a) c b) 6/ c) / d) 6/ ) / c RT m / s d c 56 9 m 9 a) ssum T C: c RT 87 9 m /s d c 686 m b) ssum T 7 F: c RT 76 5 fs d c 6 f Fr vry scnd ha asss, h lighning flashd abu f away Cun 5 scnds and i is arximaly n mil away c 9 c m / s sin α sinα 56 anα 68 L 776 m L s m L

4 95 Us Eq 9: a) c sin α r sin 98 m / s b) c sinα r sin 98 fs 96 Eq 9: c RT fs ( ) Enrgy Eq: + ct + c( T + T) + c T T c / c 76 59( ) /( 778 ) F Bu f - lb lbm f - lb N: c lbm - F Bu slug slug - F f / sc f lb - sc F Thn F (unis can b a ain!) f - lb / (slug - F) sc f - lb - f 97 a) + d + d + dd + d + dd + dd + ddd K nly h firs rdr rms (h highr rdr rms hs wih mr han n diffrnial quaniy will b ngligibl): d + d + d Divid by : d d d + + b) Exand h rhs f Eq 95 ( nly firs rdr rms): + d + d d Hnc, d + d + + d + d d d + + d d d d + whr w nglcd d cmard Fr an isnric rcss Eq 98 givs d d, s h abv bcms

5 d d d + ( ) d d + d + d Bu d / d / d / s ha h abv quain is d + d d which can b wrin as d d Sinc c /, and /c, his is u in h frm d c d d r ( ) c) Subsiuing in c, c RT, and R / c ( ) /, w find T T c RT c T c T c T ( ) + + /( ) + d) m& / TR R T ( ) RT h criical ara, Hnc, + ( ) m& RT + ) Sinc &m is cnsan hrughu h nzzl, w can qua Eq 97 Eq 98: r RT + + ( ) + + ( ) + ( ) RT d + + ( )

6 98 a) s am Pa abs 699 Pa abs s s Frm s : s s s / g / m 69 9 s / m / s b) s Pa abs 6 Pa abs Frm s : s s s 6 / + 58 g / m 6 s / m /s 58 / / s s 5 99 a) + s 5 g/m s m/s 5 b) m/s % rrr % 8 9 Is r < 58? Pa a) < 58 chd flw RT 5 7 Pa r 87T 98 + T T 8 K, 5 8 m / s g / m m& 8 π g / s 87 8 b) r > < g / m 57 9 m / s m& 787 π g / s 9 Is r < 58? sia a) r < 5 85 chd flw and, 5 85 sia RT 76 T 5 + T T 7 R, fs ( 778 ) 5

7 585 slug/f m & π 69 slug/sc b) > 585 <, and sia r 75 slug/f 76 5 / slug/f 5(778 ) fs m& 556 π slug / sc (N: f-lb f-lb c Bu/lbm R ) lbm R slug R 9 a) < Pa T K r g / m m / s 87 8 m& 8 π g / s b) r > 58 Pa, 65 8, T 88T 79 g / m, m / s 87 6 m& 79 π 6 5 g / s 9 a) < sia r T R 5 85 slug fs f m & 5 π 69 slug / sc b) r > 58 sia T 89T fs m& 56π slug /sc 6

8 9 58 Pa abs T K m m / s & π g /s Pa 9 Pa abs T K m m / s & π 7 8 g /s Pa abs 58 Pa abs T 5 8 K 7 8 m / s sinc m& π g / s sia 7 8 sia T R fs g/m and 99 Pa abs sia, T 66 R, fs m& slug / sc 97 Tra h ilin as a rsrvir Thn, Pa abs and 87( 8 8) 7 8 m /s 6 5 m & g / s 87 ( 8 8) m& m 6 5 / ( ) T T T K 975 Pa Nx, T 5 K, 97 5 Pa; m /s g/m π π Pa / r Trial - and - rrr: 9 8 m / s 667 g/m and 99 Pa abs

9 g / m 6 g / m RT / m 7 5 s m& 757 π g / s ( 5 + 7) slug 9 96 RT 76 5 f / slug / f fs m& 96π ( / ) 9 5 slug / sc 9 Enrgy : + T RT 67 5 Pa l Pa 7 g/m Enrgy : m / s, T K Pa 55 g / m 87 5 d Cninuiy: 55π 5 8 7π d 65 m 9 RT 9 87 T + T T K m / s Pa abs 76 g / m

10 π π Trial-and-rrr: m/s, 659 m/s 5 897, 987 g / m 9 Pa, 9 Pa abs frm Tabl D Pa and 855 frm Tabl D 8 Pa abs sia, T 8 5 R 8 slug π d m& 8 76 d f 9 f 5 5, T R, fs π d π d 7 f 95 7, Pa, K T Fr 7, Pa abs 5 96 L Nglc viscus ffcs 87 π 5 π d m r 86 cm d Pa abs T s T 96 T 55 K m/s 55 m& π g/s

11 98 Isnric flw Sinc ~ fr nirgn, h isnric abl i < may b usd > : 5 i m/s i 97 g/m 97 7 m& 98 i 98 m m i i , T 57 T, 7 7 T T K r 77 C 67 Pa Isnric flw Sinc fr nirgn, h isnric abl i < may b usd > : 5 ~ i fs 5 slug i f 8 i 8 f 667 f 8 8 5, T 57 T, T T 88 R r 88 F 55 sia ssum Pa Thn 98 g / m mnum: F m & 98 π m/s 9 F m & g / m (ssum gass ar air ) m/s 9 ; 9, T K, T Pa abs F B

12 m / s F B π π N ssum an isnric flw; Eq 9 rvids + Using his givs r 6 Fr sandard cndiins c m / s 9 a) ( ) g /m ( ) m / s 77 g / m Subsiu in and find 88 Pa T 76 K r 7 C b) / Pa 89 6 K r 75 C 77 g / m T slug 95 a) 76 5 f mnum: ( ) (, ) 6 6 slug 6, fs 75 f 9 sia 9 7 T 9 R r 7 F

13 b) / sia R r 7 F 75 slug / f 76 9 T T + ( + ) ( + ) + [ + T + ( ) + ] + + (This is Eq 9) Subsiu in abv: ( + ) ( + ) + ( ) + ( ) ( + ) + ( ) + ( + ) / + + ( ) / Fr a srng schc in which ( + ) ( + ) + ( + ) + ( )( + ) >>, + 97 ssum sandard cndiins: T 5 C, Pa m / s 577 T K 5 5 Pa m/s inducd m/s sainary shc Th high rssur and high inducd vlciy caus xrm damag 98 If 5, hn m /s 6 Pa abs 8 g / m 87 ( 85 9) If 5, hn fs sia 695 slug / f 76 ( 85 5) 8

14 Pa T K / Pa T 69 5 K Fr isnric flw frm : Fr 58, 866 and T 96 T 9/ Pa abs T 69 5/ 96 7 K r 8 C 95 fr h shc 75, 8 86 Pa abs Fr isnric flw frm : Fr 75, / Pa abs / Pa abs Pa T K g / m m / s m& 7599 π g / s If hra ara is rducd, rmains a, 7599 g / m and m& 7599 π 5 9 g / s 95 Pa 9, and / 9 98 / Pa 9, / 98 8/ 98 Pa abs, 58 8 Pa abs T 8 9 K 87 m / s 9, 8 Pa abs T K m / s 788, Pa T T K m / s 95 7 sia 9, and / / sia 9, / 98 8/ sia, sia T 8 5 R 76 fs 9, 8 sia T R fs

15 788, 7 sia T T R fs Pa T K , T 5 98 K m / s 56, Pa T 8 8 K fr h shc i s isnric flw 56, π Pa 85 m π Pa abs r T K m / s / T Pa 585 K 655 g / m m /s ( ) π d m& 59 d 6 m r 6 cm T 67 / 8 K 575 g / m (Enrgy frm ) (c 87 J / g K) πd 5 m / s d 9 m r 9 cm Pa T g 5 K m πd m / s d m r cm

16 958 sia T 9 R slug fs f πd 5 9 d 99 f r 9" / Pa T 585 K m / s g / m m& π g / s r nzzl / T K Frm Fig 95, β 6, 79 a) n β 6 9sin sin(6 ) n 9 Pa abs T K m / s b) β 79 n 9 sin 79 5 n 5 sin( 79 ) Pa abs T K m / s c) a dachd shc 5 5

17 96 β θ sin 9 79 sin( ) 58 n n hn, wih β n n β β 5 If θ 58, 5 58sin 5 n sin( ) 96 n 5sin 5 n 576 T K θ θ β 7 sin( 5 ) 6sin sin( 7 ) n n T 5 7 K RT m /s 96 n 5sin 5 n 576 T R θ θ β 7 sin( 5 ) 6sin sin( 7 ) n n T 8 8 R RT 76 8 fs 96, θ β 8 n sin 8 n Pa Pa sin( 8 ) ( ) 965, θ 9 8, µ 9 7 θ nrmal Pa (S Fig 98) θ Frm isnric flw abl: 5 8 Pa 7 T T T T K r 6 C µ 8 T T m / s α θ 6 Fr, θ 65 8 (S Fig 98) θ T T T T K T T m / s T 56 C 6

18 967 θ 6 Fr, θ 65 8 θ T T T T 9 8 R r 5 F T T fs 968 a) θ 9 θ u u Pa Fr θ 5 and 5, β 7 n 5sin 7 n 889 l 6 Pa 889 l 7 sin( 7 5 ) b) 7, θ 5 β 5 n 7sin 5 5, n u 56 sin( 5 5 ) Fr 7, θ 6 Fr θ 6 + 5, l 58 c) Frc n la ( 6 ) F F C cs L 9 5 F d) C sin 5 87 D 5 irfil surfac F Drag Lif 969 β 9 n sin Pa n θ 5 6 θ sin( 9 5 ) shc 6 Pa 88 C D 6 sin

19 97 If θ 5 wih, hn Fig 95 β 8 n sin8 n Pa l 88 l 6 sin( 8 5 ) shc l u shc u 88 u 77, θ Pa Lif 5cs 5 / 6 6 / cs CL 85 C Drag 5 sin / sin D 8