CHAPTER 9 Compressible Flow

Transcription

1 CHPTER 9 Comrssibl Flow Char 9 / Comrssibl Flow Inroducion 9. c c cv + R. c kcv. c + R or c R k k Rk c k Sd of Sound 9.4 Subsiu Eq ino Eq and nglc onial nrgy chang: Q WS + + u~ u~. m ρ ρ Enhaly is dfind in Thrmodynamics as h u~ + v u~ + / ρ. Thrfor, Q WS + h h. m ssum h fluid is an idal gas wih consan scific ha so ha Δh cδt. Thn Q W S + c ( T T). m Nx, l c cv + R and k c/ cv so ha c / R k/( k ). Thn, wih h idal gas law ρrt, h firs law aks h form Q WS k +. m k ρ ρ Th sd of sound is givn by 9.6 c d/ dρ. For an isohrmal rocss TR / ρ K, whr K is a consan. This can b diffrniad: d Kdρ RTdρ. Hnc, h sd of sound is c RT. 9 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

2 Char 9 / Comrssibl Flow d For war Bulk modulus ρ d Pa ρ Sinc ρ 000 kg/m, w s ha d 0 0 c 45 m/s dρ 000 Sinc c 450 m/s for h small wav, h im incrmn is d Δ c sconds 6 9. c krt m/s. d c m. 9.4 c c m/s. sin α. M 000 sinα anα L 776 m L 776 Δ s. 000 m Δ Δ 0. Eq. 9..4: Δ 0. fs. ρc ρ krt Enrgy Eq: L 9.6 ( +Δ) ( Δ) + ct + c( T+ΔT). 0 Δ + + cδt. Δ c f/sc( 0. f/sc) Δ T 0.0 R or 0.0 F c 60 f-lb/slug- R No: Us slug lb-sc /f (m F/a). (Unis can b a ain!) 0 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

3 Char 9 / Comrssibl Flow Isnroic Flow 9.8 a) am kpa abs. s 69.9 kpa abs. From s : / k /.4 s s ρs ρ kg/m. ρ ρs m/s s 0 s Is r < ? kpa. 9.0 r 0 a) < chokd flow. M. krt kpa..4 87T T. T 48. K, 5.8 m/s ρ.484 kg/m. m.484 π kg/s b) ρ r > M < ρ ρ0.8. ρ.787 kg/m m/s m.787 π kg/s. a) r < M kpa. T K ρ.48 kg/m m/s m.48 π kg/s. 9. b) r > kpa, M 0.8, T 0.884T0 0 0 ρ.79 kg/m, m/s m.79 π kg/s. 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

4 Char 9 / Comrssibl Flow kpa abs. T K m/s. m π kg/s sia. 7.8 sia. T R fs. 9.6 m [ / ( )] π (.5 /) slug/sc sia, T 46.6 R, 000 fs. m [ / ( )] π (.5 /) slug/sc T T. T 5 K / kpa abs. 9.8 Nx, T 5 K, kpa; m/s ρ π ρ π kg/m ρ ρ kpa ρ 00 / or Trial-and-rror: 9.8 m/s. ρ 0.0 kg/m and 99.4 kpa abs. 9.0 W nd o drmin h Mach numbr a h xi. Sinc h M a h hroa, hn hroa 9.7 cm. Hnc, h ara raio a h xi is Using h air abls, w find wo ossibl soluions, on for subsonic flow, and h ohr for sursonic flow in h divrging scion of h nozzl. h xi: Subsonic Flow: M 0.5, T T , and Hnc, c krt ( ) M M m/s Sursonic Flow: M.76, T T , and Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

5 Char 9 / Comrssibl Flow Hnc, c krt ( ) M M m/s ρ / RT ( )44 / (76 50) slug/f. 9. / ρ slug/f fs m π (/) slug/sc T krt T. T 44.0 K.. m/s..4/ kpa abs. ρ.76 kg/m π ρ π 0.4 ρ.4.4 ρ Trial-and-rror:. m/s, 659 m/s , kg / m kpa, 4. 9 kpa abs. ρ M sia, T R. ρ. 08 slug. f π d m d 0.9 f M.04, T R, fs. 0.9 π d π d 0.47 f Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

6 Char 9 / Comrssibl Flow 9.8 Using comrssibl flow abls for air, w drmin h rssur raio and mraur raio for M.8 o b: T , and kpa abs T 0 0 and T T 5 K 0 Mc.8 krt m/s L M. Nglc viscous ffcs. M π 0.05 π d m or 8.6 cm. d Isnroic flow. Sinc k.4 for nirogn, h isnroic flow abl may b usd. M, i 4.5. M > M 00 i m/s. ρ i kg/m m i m m. ρ i i M, T 0.57 T, M < ~ T0 T 044 K or 77 C kpa abs ssum 0 0 kpa. Thn ρ kg/m F m ρ π m/s. 6 M. 4; M.94, T T K, kpa abs m/s. 0 0 F 4 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

7 Char 9 / Comrssibl Flow 00 FB π π N Normal Shock a) ρ ( 000) ρ kg/m. ρ + ( ) m/s. ρ.774 kg/m. Subsiu in and find 808 kpa abs. ρ ρ M.966. T 746 K or 47 C M T km k+ ( k+ ) M ( k+ )M. T k+ k M ( )M [4kM k ] + k + + k + + k. (This is Eq. 9.4.). Subsiu ino abov: k k M 9.50 ρ ρ ( k+ ) ( k+ ) + ( k ) ( k+ ) ( k+ ) + k k+ ( k ) ( k+ ) + ( k ) ( k+ ) + ( k )( k+ ) 4 k + ( k + ) / k + + ( k ) / For a srong schock in which. k >>, ρ + ρ k If M 0.5, hn M m/s. 5 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

8 Char 9 / Comrssibl Flow kpa abs. ρ 8. kg/m (.85 9) kpa. T. K. M 000 / M kpa. T K. For isnroic flow from 0: For M 0.458, and T T. 9 / kpa abs. T 69.5 / K or 448 C M kpa abs M kpa. T K ρ kg/m m/s m π kg/s. If hroa ara is rducd, M rmains a, ρ kg/m and m π kg/s. 4.7 sia. 4. M.94, and / sia. M.94, / M, sia. T R sia fs. M.94,.48 sia. T R fs. M , 4.7 sia. T T R fs. 6 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

9 Char 9 / Comrssibl Flow aor Flow / kpa. T K ρ.4 kg/m m/s. (M.) π d m ρ d m or 6 cm. 4 T 0./ K ρ kg/m (Enrgy from 0.) ( c 87 J/kg K) π 050 m/s ( d /4) 050. d 0.09 m or 9. cm /. 8.9 M, sia. T R ρ slug/f fs πd d 0.99 f. or.9". 4 Obliqu Shock Wav 800 M From Fig. 9.5, β 46, o 9.64 a) n β 46. M.9sin M M sin(46 0 ). n M kpa abs. T K m/s. 7 0 Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

10 Char 9 / Comrssibl Flow c) a dachd shock 5o n n T M.5sin 5.0. M K θ θ β M 0.576/ sin(5 0 ) n n M.6sin M M sin(47 0 ). M.44. T K. M krt m/s θ β n n M, M sin 8.4. M kpa abs M kpa abs. sin(8 0 ) ( ) normal kpa abs. Exansion Wavs 9.70 θ For M 4, θ (S Fig. 9.8.) θ T T0 T T K. T T m/s. T 56 C. a) θ θ u u M.7. (0/0.0585) kpa abs. For θ β n n 5 and M.5, 7. M.5sin 7.. M kpa abs. M M 0.889/ sin(7 5 ) Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.

11 Char 9 / Comrssibl Flow If θ 5 wih M 4, hn Fig. 9.5 β 8. M 4sin8.4. M n n kpa M.64. sin(8 5 ) shock M M l M u shock C C L D M 4, θ , Mu u kpa. Lif.5cos5 0 / 6.6 ( /) cos ρ Drag.5sin ( /) sin ρ Cngag Larning. ll Righs Rsrvd. May no b scannd, coid or dulicad, or osd o a ublicly accssibl wbsi, in whol or in ar.