CHAPTER 9 Compressible Flow

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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. 4.5.8 ino Eq. 4.5.7 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.

Char 9 / Comrssibl Flow 9.8 9.0 d For war Bulk modulus ρ d 0 0 6 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 0 450 0. 0069 sconds 6 9. c krt.4 87 6 5 m/s. d c 56. 9 m. 9.4 c c.4 87 6 56 m/s. sin α. M 000 sinα 0.56. anα 0.648. L 776 m L 776 Δ 000. 776 s. 000 m Δ Δ 0. Eq. 9..4: Δ 0. fs. ρc ρ krt 0.007.4 76 59 Enrgy Eq: L 9.6 ( +Δ) ( Δ) + ct + c( T+ΔT). 0 Δ + + cδt. Δ c.4 76 59 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.

Char 9 / Comrssibl Flow Isnroic Flow 9.8 a) am + 0 69.9 + 0 79.9 kpa abs. s 69.9 kpa abs. From s : / k /.4 s s 79.9 +. ρs ρ 0.906 0.997 kg/m. ρ ρs 69.9 + 69 900 79 900. 77. m/s. 0.906 0.997 s 0 s Is r < 0.58 0? 0.58 00 05.7 kpa. 9.0 r 0 a) < 0.58. chokd flow. M. krt. 05.7 kpa..4 87T 000 98 + 000 T. T 48. K, 5.8 m/s. 05.7 ρ.484 kg/m. m.484 π 0.0 5.8 0.47 kg/s. 0.87 48. b).4.4 0 000 0 ρ r > 0.58 0. M <. 000 98 +. 0.4 ρ 00.8 00 ρ0.8. ρ.787 kg/m. 57.9 m/s. 0.87 98 m.787 π 0.0 57.9 0.9 kg/s. a) r < 0.58 0. M. 0.58 00 05.7 kpa. T 0.8 98 48. K. 05.7 ρ.48 kg/m..4 87 48. 5.9 m/s. 0.87 48. m.48 π 0.0 5.9 0.47 kg/s. 9. b) r > 0.58 0. 0 kpa, 0.65. M 0.8, T 0.884T0 0 0 ρ.79 kg/m, 0.8.4 87 6.4 6.5 m/s. 0.87 6.4 m.79 π 0.0 6.5 0.4 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.

Char 9 / Comrssibl Flow 9.4 0.58 400. kpa abs. T 0.8 0 5.5 K...4 87 5.5 8.5 m/s. m π 0.05 8.5 7.9 kg/s. 0.87 5.5 0.58 4.7 sia. 7.8 sia. T 0.8 500 46.6 R. 0 0.4 76 46.6 000 fs. 9.6 m [4.7 44 / (76 46.6)] π (.5 /) 000 0.0 slug/sc. 0 0 7.8. 0.58 9.4 sia, T 46.6 R, 000 fs. m [9.4 44 / (76 46.6)] π (.5 /) 000 0.0 slug/sc..667 077 T 5 59 00 + 59 T. T 5 K. 00 00.667/0.667 97.45 kpa abs. 9.8 Nx, T 5 K, 97.45 kpa;.667 077 5 88.6 m/s. 97.45 ρ π ρ π.077 5 0.085 kg/m. 0.085 0.0 88.6 0.075.667 ρ.667 59 00 +. 00 0 ρ kpa. 0.667 ρ 00 /.077 00 0.667 + 4 0 9.54..667 6 0.667 or.6 0 + 6 40 0. 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 9.7.4. 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 T0 0.954, and 0 0.840. Hnc, c krt ( ) M M 0.5.4 87 0.954 95 68 m/s Sursonic Flow: M.76, T T0 0.675, and 0 0.850. 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.

Char 9 / Comrssibl Flow Hnc, c krt ( ) M M.76.4 87 0.675 95 476 m/s ρ / RT (45 + 4.7)44 / (76 50) 0.00964 slug/f. 9. /.4 50.7 ρ 0.00964 0.00857 slug/f. 59.7 0.00964 4 0.00857. 4.495. 4.495.4 59.7 44.4 50.7 44 + +..9 fs. 0.4 0.00964 0.4 0.00857 m 0.00964 π (/).9 0.05 slug/sc. 9.4 9.6.4 87 T krt. 000 9 + 000 T. T 44.0 K.. m/s..4/0.4 44 6.5 500 6.5 kpa abs. ρ.76 kg/m. 9 0.87 44.4 π ρ 6 500 π 0.4 ρ.4.4 ρ.76 000 9 +..76 0.05. 0.075. 6 0.4 9 000 +.04 0. Trial-and-rror:. m/s, 659 m/s. 5. 897, 0. 987 kg / m. 494. kpa, 4. 9 kpa abs. ρ M. 0.58 0 6.4 sia, T 0.8 50 4. R. ρ. 08 slug. f π d m 0.08.4 76 4.. d 0.9 f. 4 5 0.5. M.04, T 0.55 50 87 R,.04.4 76 87 0 0 684 fs. 0.9 π d π.708..708. d 0.47 f. 4 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.

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 0.0685, and 0.894. 0.0685 0 9 kpa abs T 0 0 and T 0.894 T 5 K 0 Mc.8 krt.8.4 87 0 004 m/s. 9.40 9.4 50 L M. Nglc viscous ffcs. M 0.40..4 87 0 π 0.05 π d.5007.. 0.086 m or 8.6 cm. d.5007.5007 4 Isnroic flow. Sinc k.4 for nirogn, h isnroic flow abl may b usd. M, i 4.5. M > M 00 i.4 97 7 8 m/s. ρ i 0.907 kg/m. 0.97 7 m 0 0.0098 i 0.0098 m. 0.00 m. ρ 0.907 8 4.5 i i M, T 0.57 T, 0.07. 0 0 M < ~ 0 7 00 T0 T 044 K or 77 C. 0 670 kpa abs. 0.57 0.07 9.44 9.46 ssum 0 0 kpa. Thn ρ 0.498 kg/m. 0.89 7 80 000 9.8 F m ρ. 0.498π 0.5. 60 m/s. 6 M. 4; M.94, 0.0980 0. T 0.665 T 0.665 00 0.0 K, 0 00 0.098. 56 kpa abs. 0 0.94.4 87 09.95 68 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.

Char 9 / Comrssibl Flow 00 FB π 0.05 68 + 56 000π 0. 4 000 N. 0.87 09.95 Normal Shock a) ρ 0.9850 000. 80 000 0.985 000( 000) 9.48 000.4 80 + 0.4 0.87 8 000.4 87 8 0. ρ 0.9850 kg/m. ρ + ( 985 + 065 000) 84 00 0 0.4 985 784 + 784 00 0. 6 m/s. ρ.774 kg/m. Subsiu in and find 808 kpa abs. ρ ρ 000 808 M.966. T 746 K or 47 C..4 87 8 0.87.774 6 M. 4 87 746 0477.. 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.. 9.5 If M 0.5, hn M.645..645.4 87 9 908 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.

Char 9 / Comrssibl Flow 600 8.00 00 600 kpa abs. ρ 8. kg/m. 0.87 (.85 9) 9.54 9.56 0.65 0 6.4 kpa. T. K. M 000 /.4 87..4. M 0.4578..85 6.4 9 kpa. T.0. 69.5 K. For isnroic flow from 0: For M 0.458, 0.866 0 and T 0.960 T. 9 / 0.866 9 kpa abs. T 69.5 / 0.96 7 K or 448 C. 0 0 0 0 4. M 0.47. 0.985 0.5 kpa abs. 0 0 0.985 M. 0.58 0.5 54.5 kpa. T 0.8 98 48. K. 54.5 ρ 0.7599 kg/m..4 87 48. 5.9 m/s. 0.87 48. m 0.7599 π 0.05 5.9 0.47 kg/s. If hroa ara is rducd, M rmains a, ρ 0.7599 kg/m and m 0.7599 π 0.0 5.9 0.0 kg/s. 4.7 sia. 4. M.94, and / 9.98. 4.7 9.98.48 sia. M.94, / 0.098. 0 0 M, 0.58 49.7 6. sia. T 0.8 50 4. R..48 49.7 sia. 0.098 9.58. 4 76 4. 00 fs. M.94,.48 sia. T 0.665 50 90.6 R.. 94. 4 76 90. 6 989 fs. M 0.4788, 4.7 sia. T T.609 90.6 497. R. 0.4788.4 76 497. 5 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.

Char 9 / Comrssibl Flow aor Flow 9.60 0./. 655 0.546 0 0.546 00 655 kpa. T 67 585 K. 00 655 ρ.4 kg/m.. 46 585 59 m/s. (M.) 0.46 585 π d m ρ. 4 4. 59. d 0. 060 m or 6 cm. 4 T 0./. 0 0 67 80. K ρ 0.575 kg/m. 00 0.46 80. + 87 80. 87 67. (Enrgy from 0.) ( c 87 J/kg K) π 050 m/s. 4 0.575( d /4) 050. d 0.09 m or 9. cm. 9.6 0./. 8.9 M, 0.546 50 8.9 sia. T 60 009 R. 50 8.9 44 ρ 0.004 slug/f.. 760 009 90 fs. 76 009 πd 0.5 0.004 90. d 0.99 f. or.9". 4 Obliqu Shock Wav 800 M 9... 4 87 0 From Fig. 9.5, β 46, 79. 0 o 9.64 a) n β 46. M.9sin 46.65. M 0.654 M sin(46 0 ). n M 49... 0 40 0. 4 kpa abs. T. 4 0 4 K..4 87 4.49 60 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.

Char 9 / Comrssibl Flow c) a dachd shock 5o n n T M.5sin 5.0. M 0.576..696 0 54 K. 9.66 θ θ β M 0.576/ sin(5 0 ).6. 0. 47. n n M.6sin 47.65. M 0.654 M sin(47 0 ). M.44. T.4 54 7 K. M krt.44.4 87 7 780 m/s. 9.68 θ β n n M, 0. 8. M sin 8.4. M 0.76..5 40 86. kpa abs. 0.76 M.8. 6.44 86. 555 kpa abs. sin(8 0 ) ( ) normal 0. 40 4 kpa abs. Exansion Wavs 9.70 θ 6. 4. For M 4, θ 65. 8. (S Fig. 9.8.) θ 65. 8 6. 4 9. 4. T T0 T T 7 0.8 7 K. T T 0.5556 0 4.4 87 7 867 m/s. T 56 C. a) θ θ u u 9.. 9.+ 5 44.. M.7. (0/0.0585) 0.0465 9.7 4.4 kpa abs. For θ β n n 5 and M.5, 7. M.5sin 7.. M 0.889.. 0 6.4 kpa abs. M M 0.889/ sin(7 5 ).7. 8 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.

Char 9 / Comrssibl Flow If θ 5 wih M 4, hn Fig. 9.5 β 8. M 4sin8.4. M 0.88. n n.67 0.5 kpa. 0.88 M.64. sin(8 5 ) shock M M l M u shock C C L D 0.0077 0 M 4, θ 65.8. 75.8, Mu 4.88. u 0 0 0.006586 6.6 kpa. Lif.5cos5 0 / 6.6 ( /) cos0 0.0854. ρ.4 4 0 Drag.5sin 5 6.6 ( /) sin0 0.00. ρ.4 4 0 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.

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