CHPTER 9 Cmrssibl Flw 9 Bu f-lb lbm f-lb c 778 6 lbm- R Bu slug slug- R f-lb cv c R 6 76 96 96 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
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
c d / d 6 / 5 m / s d 99 Fr war c d 6 5 m / s L vlciy im 5 6 87 m 9 Sinc c 5 m/s fr h small wav, h im incrmn is d 69 scnds c 5 9 a) c 588 87 88 b) 6/ 76 66 567 c) / 87 668 d) 6/ 76 9 68 ) / 87 8 67 9 c RT 87 6 56 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 87 6 56 m / s sin α sinα 56 anα 68 L 776 m L 776 776 s m L
95 Us Eq 9: a) c sin α r 87 88 sin 98 m / s b) c sinα r 76 59 sin 98 fs 96 Eq 9: c RT fs 7 76 59 ( ) Enrgy Eq: + ct + c( T + T) + c T T c / c 76 59( ) /( 778 ) F Bu f - lb lbm f - lb N: c 778 778 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
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 + + ( )
98 a) s am + 699+ 799 Pa abs 699 Pa abs s s Frm s : s s s 96 79 9 / + 997 g / m 69 9 s / 69 9 79 9 + 77 m / s 96 997 b) s 6 + 6 Pa abs 6 Pa abs Frm s : s s s 6 / + 58 g / m 6 s / 6 6 + m /s 58 / / s s 5 99 a) + s 5 g/m s + 5 8 m/s 5 b) 8-8 8 m/s % rrr % 8 9 Is r < 58? 58 5 7 Pa a) < 58 chd flw RT 5 7 Pa r 87T 98 + T T 8 K, 5 8 m / s 5 7 8 g / m m& 8 π 5 8 7 g / s 87 8 b) r > < + 58 98 8 8 787 g / m 57 9 m / s 87 98 m& 787 π 57 9 9 g / s 9 Is r < 58? 58 5 85 sia a) r < 5 85 chd flw and, 5 85 sia RT 76 T 5 + T T 7 R, fs ( 778 ) 5
585 slug/f 76 7 5 m & π 69 slug/sc b) > 585 <, and sia r 75 slug/f 76 5 / 75 556 slug/f 5(778 ) + 556 5 88 9 fs m& 556 π 88 9 67 slug / sc (N: f-lb f-lb c Bu/lbm R 778 778 ) lbm R slug R 9 a) < 58 58 5 7 Pa T 8 98 8 K r 5 7 8 g / m 87 8 5 9 m / s 87 8 m& 8 π 5 9 7 g / s b) r > 58 Pa, 65 8, T 88T 79 g / m, 8 87 6 6 5 m / s 87 6 m& 79 π 6 5 g / s 9 a) < 58 58 5 85 sia r T 8 5 6 R 5 85 slug 76 6 76 6 fs f m & 5 π 69 slug / sc b) r > 58 sia 6667 785 T 89T 56 785 76 7 86 fs 76 7 5 m& 56π 86 66 slug /sc 6
9 58 Pa abs T 8 5 5 K m 87 5 5 8 5 m / s & π 5 8 5 7 9 g /s 87 5 5 95 58 Pa 9 Pa abs T 8 8 5 8 K m 87 5 8 7 8 m / s & π 7 8 g /s 87 5 8 9 8 Pa abs 58 Pa abs T 5 8 K 7 8 m / s sinc m& π 7 8 6 g / s 87 5 8 96 58 7 sia 7 8 sia T 8 5 6 6 R 76 66 fs g/m and 99 Pa abs 78 58 9 sia, T 66 R, fs m& slug / sc 97 Tra h ilin as a rsrvir Thn, 58 6 5 Pa abs and 87( 8 8) 7 8 m /s 6 5 m & 7 8 6 g / s 87 ( 8 8) m& 6 6 6 m 6 5 / ( 87 8 8) 667 77 T 98 59 + 59 5 5 T T K 975 Pa Nx, T 5 K, 97 5 Pa; 667 77 5 88 6 m /s 975 85 g/m 85 886 75 π π 77 5 667 667 59 + 667 77 Pa / 667 + 9 5 6 667 r 6 + 6 Trial - and - rrr: 9 8 m / s 667 g/m and 99 Pa abs 667 667 7
+ 99 757 757 g / m 6 g / m RT 87 9 757 6 5 9 / 9 + + + + 757 6 m 7 5 s m& 757 π 5 7 5 95 g / s ( 5 + 7) slug 9 96 RT 76 5 f / slug / f 96 5 7 857 59 7 96 857 95 59 7 95 5 7 + + 9 fs 96 857 m& 96π ( / ) 9 5 slug / sc 9 Enrgy : + T RT 67 5 Pa l 79 59 59 Pa 7 g/m 87 79 Enrgy : + 8 5 87 m / s, T K 5 5 5 Pa 55 g / m 87 5 d Cninuiy: 55π 5 8 7π 87 7 9 d 65 m 9 RT 9 87 T + T T K m / s 5 6 5 6 5 Pa abs 76 g / m 9 87 8
π π 76 6 6 5 9 + 76 5 75 9 + Trial-and-rrr: m/s, 659 m/s 5 897, 987 g / m 9 Pa, 9 Pa abs 9 9 997 frm Tabl D 5 997 985 Pa and 855 frm Tabl D 8 Pa abs 9 58 6 sia, T 8 5 R 8 slug π d m& 8 76 d f 9 f 5 5, T 55 5 87 R, 76 87 68 fs π d π 9 78 78 d 7 f 95 7, 6586 7 Pa, 8 9 6976 K T Fr 7, 58 9976 9976 995 Pa abs 5 96 L Nglc viscus ffcs 87 π 5 π d 57 86 m r 86 cm d 57 57 97 58 Pa abs T 8 5 5 s T 96 T 55 K 87 55 98 m/s 55 m& π 5 98 77 g/s 87 55 9
98 Isnric flw Sinc ~ fr nirgn, h isnric abl i < may b usd > : 5 i 97 7 8 m/s i 97 g/m 97 7 m& 98 i 98 m m i i 97 8 5, T 57 T, 7 7 T T K r 77 C 67 Pa 57 7 99 Isnric flw Sinc fr nirgn, h isnric abl i < may b usd > : 5 ~ i 776 66 8 fs 5 slug i 8 776 66 f 8 i 8 f 667 f 8 8 5, T 57 T, 7 66 5 T T 88 R r 88 F 55 sia 57 7 9 ssum Pa Thn 98 g / m 89 7 8 98 mnum: F m & 98 π 5 6 6 m/s 9 F m & g / m (ssum gass ar air ) 87 87 98 9 m/s 9 ; 9, 98 665 T 665 995 K, T 98 56 Pa abs F B
9 87 9 95 68 m / s F B π 5 68 + 56 π N 87 9 95 9 ssum an isnric flw; Eq 9 rvids + Using his givs r 6 Fr sandard cndiins c 6 87 88 7 m / s 9 a) 985 8 985 ( ) 8 + 87 8 985 g /m 87 8 + ( 985 + 65 ) 8 985 78 + 78 6 m / s 77 g / m Subsiu in and find 88 Pa 88 966 T 76 K r 7 C 87 8 87 77 6 77 87 76 b) / 87 8 97 77 89 6 Pa 89 6 K r 75 C 77 g / m 87 78 T 6 8 78 slug 95 a) 76 5 f mnum: ( ) + 76 5 6 + 7 9 85 6 6 6 (, ) 6 6 slug 6, + 5 8 fs 75 f 9 sia 9 7 T 9 R r 7 F 76 5 76 75
8 9 76 9 b) / 76 5 7 9 859 sia R r 7 F 75 slug / f 76 9 T 86 5 9 96 T + ( + ) ( + ) + [ + T + ( ) + ] + + (This is Eq 9) Subsiu in abv: ( + ) ( + ) + ( ) + ( ) ( + ) + ( ) + ( + ) / + + ( ) / Fr a srng schc in which ( + ) ( + ) + ( + ) + ( )( + ) >>, + 97 ssum sandard cndiins: T 5 C, Pa 87 88 68 m / s 577 T 688 88 86 K 5 5 Pa 577 87 86 55 m/s inducd 68 55 5 m/s sainary shc Th high rssur and high inducd vlciy caus xrm damag 98 If 5, hn 65 65 87 9 98 m /s 6 Pa abs 8 g / m 87 ( 85 9) 8 6 99 If 5, hn 65 65 76 5 8 fs sia 695 slug / f 76 ( 85 5) 8
95 65 6 Pa T K / 87 578 85 6 9 Pa T 69 5 K Fr isnric flw frm : Fr 58, 866 and T 96 T 9/ 866 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, 857 86/ 857 96 Pa abs 95 7 985 / 985 5 Pa abs 58 5 5 5 Pa T 8 98 8 K 5 5 7599 g / m 87 8 87 8 5 9 m / s m& 7599 π 5 5 9 7 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 / 9 98 8 Pa 9, / 98 8/ 98 Pa abs, 58 8 Pa abs T 8 9 K 87 m / s 9, 8 Pa abs T 665 9 7 K 9 87 7 6 m / s 788, Pa T T 69 7 8 K 788 87 8 6 m / s 95 7 sia 9, and / 9 98 7 / 9 98 8 sia 9, / 98 8/ 98 9 7 sia, 58 9 7 6 sia T 8 5 R 76 fs 9, 8 sia T 665 5 9 6 R 9 76 9 6 989 fs
788, 7 sia T T 69 9 6 97 R 788 76 97 5 fs 955 58 5 6 Pa T 8 98 8 K 8 56 7, 6 5 65 5 T 5 98 K 7 87 57 m / s 56, 6 95 65 Pa T 8 8 K fr h shc i s isnric flw 56, 5 5 55 5 π Pa 85 m π 5 5 9 55 5 Pa abs r 85 98 T 857 8 7 8 K 98 87 7 8 99 m / s / 956 56 56 655 T 67 655 Pa 585 K 655 g / m 6 585 59 m /s ( ) 6 585 π d m& 59 d 6 m r 6 cm T 67 / 8 K 575 g / m 6 8 + 87 8 87 67 (Enrgy frm ) (c 87 J / g K) πd 5 m / s 575 5 d 9 m r 9 cm 957 56 56 56 Pa T 6 56 56 g 5 K 8 6 5 m πd 6 5 57 m / s 5 8 57 d m r cm
958 sia 56 5 8 9 6 8 9 T 9 R 5 8 9 slug 76 9 76 9 9 fs f πd 5 9 d 99 f r 9" / 959 56 655 Pa 67 655 T 585 K 655 6 585 59 m / s g / m 6 585 m& π 75 59 5 g / s r nzzl / T 67 96 K 8 96 9 87 Frm Fig 95, β 6, 79 a) n β 6 9sin6 65 65 sin(6 ) n 9 Pa abs T K 87 9 6 m / s b) β 79 n 9 sin 79 5 n 5 sin( 79 ) 6 5 7 Pa abs T 9 576 K 87 576 6 m / s c) a dachd shc 5 5
96 β θ sin 9 79 sin( ) 58 n n hn, wih β n n 8 5 6 β β 5 If θ 58, 5 58sin 5 n sin( ) 96 n 5sin 5 n 576 T 696 5 K 576 6 θ θ β 7 sin( 5 ) 6sin 7 65 65 sin( 7 ) n n T 5 7 K RT 87 7 78 m /s 96 n 5sin 5 n 576 T 696 9 8 R 576 6 θ θ β 7 sin( 5 ) 6sin 7 65 65 sin( 7 ) n n T 8 8 R RT 76 8 fs 96, θ β 8 n sin 8 n 76 5 86 Pa 76 8 6 86 555 Pa sin( 8 ) ( ) 965, θ 9 8, µ 9 7 θ nrmal Pa (S Fig 98) θ + 9 8 + 5 7 8 78 Frm isnric flw abl: 5 8 Pa 7 T T T T 5 795 7K r 6 C µ 8 T T 57 78 87 7 8 m / s α 9 + 5 7 5 8 966 θ 6 Fr, θ 65 8 (S Fig 98) θ 65 8 6 9 T T T T 7 8 7 K T T 5556 87 7 867 m / s T 56 C 6
967 θ 6 Fr, θ 65 8 θ 65 8 6 9 T T T T 9 8 R r 5 F T T 5556 76 8 fs 968 a) θ 9 θ 9 + 5 7 585 65 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 875 875 u 56 sin( 5 5 ) Fr 7, θ 6 Fr θ 6 + 5, l 58 c) Frc n la ( 6 ) F F C cs 5 996 L 9 5 F d) C sin 5 87 D 5 irfil surfac F Drag Lif 969 β 9 n sin 9 85 6 Pa n 786 786 5 θ 5 6 θ 59 6 55 sin( 9 5 ) shc 6 Pa 88 C D 6 sin 5 6 5 87 5 7
97 If θ 5 wih, hn Fig 95 β 8 n sin8 n 88 67 5 Pa l 88 l 6 sin( 8 5 ) shc l u shc 65 8 75 8 u 88 u 77, θ 6586 66 Pa Lif 5cs 5 / 6 6 / cs CL 85 C Drag 5 sin 5 6 6 / sin D 8