CHAPTER 9 Compressible Flow

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CHPTER 9 Corssibl Flow Inrouion 9. v R. kv. R or R k k Rk k Char 9 / Corssibl Flow S of Soun 9.4 Subsiu Eq. 4.5.8 ino Eq. 4.5.7 an ngl onial nrgy hang: Q WS u~ u~. Enhaly is fin in Throynais as h u~ v u~ /. Thrfor, Q WS h h. ssu h flui is an ial gas wih onsan sifi ha so ha h T. Thn Q WS ( T T). Nx, l v R an k / v so ha / R k/( k ). Thn, wih h ial gas law RT, h firs law aks h for Q W S k k 9.6 Th s of soun is givn by /. For an isohral ross TR / K, whr K is a onsan. This an b iffrnia: K RT. Hn, h s of soun is RT.. 09 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 9.8 For war Bulk oulus = 0 0 6 Pa Sin 000 kg/, w s ha 6 0 0 000 45 /s 9.0 Sin = 450 /s for h sall wav, h i inrn is 0 450 0. 0069 sons 9. krt.4 87 6 5 /s. 56. 9. 9.4.4 87 6 56 /s. sin. M 000 000 sin 0.56. an 0.648. L 776 L 776 000. 776 s. L 9.6 Eq. 9..4: krt 0. 0.007.4 76 59 0. fs. Enrgy Eq: ( ) ( ) T ( T T). 0 T. T.4 76 59 f/s( 0. f/s) 60 f-lb/slug- R No: Us slug = lb-s /f ( = F/a). (Unis an b a ain!) 0.0 R or 0.0 F Isnroi Flow 9.8 a) s a 0 69.9 0 79.9 kpa abs. 69.9 kpa abs. s =0 s Fro s : s s s s / k /.4 79.9. 0.906 0.997 kg/. 69.9 69 900 79 900. 77. /s. 0.906 0.997 0 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 9.0 Is r 0.58 0? 0.58 00 05.7 kpa. a) b) r 0 0.58. hok flow. M. krt. 05.7 kpa..4 87T 000 98 000 T. T 48. K, 5.8 /s. 05.7 0.87 48..484 kg/..484 0.0 5.8 0.47 kg/s..4 0 000 0 r 0.58 0. M. 000 98=. 0.4 00.8 0 00 0.87 98.8..787 kg/. 57.9 /s..787 0.0 57.9 0.9 kg/s..4 9. a) r 0.58 0. M. 0.58 00 05.7 kpa. T 0.8 98 48. K. 05.7.48 kg/..4 87 48. 5.9 /s. 0.87 48..48 0.0 5.9 0.47 kg/s. b) r 0.58 0. 0 kpa, 0.65. M 0.8, T 0.884T0 0 0 0.87 6.4.79 kg/, 0.8.4 87 6.4 6.5 /s..79 0.0 6.5 0.4 kg/s. 9.4 0.58 400. kpa abs. T 0.8 0 5.5 K...4 87 5.5 8.5 /s. 0.05 8.5 7.9 kg/ s. 0.87 5.5 9.6 0.58 0 4.7 sia. 0 7.8 sia. T 0.8 500 46.6 R..4 76 46.6 000 fs. [4.7 44 / (76 46.6)] (.5/) 000 0.0 slug/s. 7.8. 0.58 9.4 sia, T 46.6 R, 000 fs. 0 0 [9.4 44 / (76 46.6)] (.5/) 000 0.0 slug/s. 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 9.8.667 077 T 5 59 00 59 T. T 5 K. 00 00.667/0.667 = 97.45 kpa abs. Nx, T 5 K, 97.45 kpa;.667 077 5 88.6 /s. 97.45.077 5 = 0.085 kg/. 0.085 0.0 88.6 = 0.075.667.667.667 59 00. 00 0 kpa. 0.667 00 /.077 00 0.667 4 0 9.54. or 6 0.667.6 0 6 40 0. Trial-an-rror: 9.8 /s. 0.0 kg/ an 99.4 kpa abs. 9.0 W n o rin h Mah nubr a h xi. Sin h M = a h hroa, hn hroa 9.7. Hn, h ara raio a h xi is 9.7.4. Using h air abls, w fin wo ossibl soluions, on for subsoni flow, an h ohr for sursoni flow in h ivrging sion of h nozzl. h xi: Subsoni Flow: M 0.5, T T0 0.954, an 0 0.840. Hn, M M krt 0.5.4 87 0.954 95 68 /s Sursoni Flow: M.76, T T0 0.675, an 0 0.850. Hn, M M krt.76.4 87 0.675 95 476 /s 9. / RT (45 4.7)44 / (76 50) 0.00964 slug/f. /.4 50.7 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 0.00964 (/).9 0.05 slug/s. 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 9.4.4 87 T krt. 000 9 000 T. T 44.0 K.. /s..4/0.4 44 6.5 500 6.5 kpa abs..76 kg/. 9 0.87 44.4 6 500 0.4.4.4.76 000 9..76 0.05. 0.075. 6 0.4 9 000=.04 0. Trial-an-rror:. /s, 659 /s. 5. 897, 0. 987 kg /. 494. kpa, 4. 9 kpa abs. 9.6 M. 0.58 0 6.4 sia, T 0.8 50 4. R. 0. 08 slug. f 5 0 0.08.4 76 4.. 0.9 f. 4 0.5. M.04, T 0.55 50 87 R,.04.4 76 87 = 684 fs. 0.9.708..708. 0.47 f. 4 4 9.8 Using orssibl flow abls for air, w rin h rssur raio an raur raio for M =.8 o b: T 0.0685, an 0.894 T 0 0. 0.0685 0 9 kpaabs an T 0.894 T 0 5 K M.8 krt.8.4 87 0 004 /s. 50 9.40 L M. Ngl visous ffs. M 0.40..4 87 0 0.05.5007.. 0.086 or 8.6..5007.5007 4 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 9.4 Isnroi flow. Sin k =.4 for nirogn, h isnroi flow abl ay b us. M, 4.5. 00 i.4 97 7 8 /s. i 0.907 kg/. 0.97 7 i M > M = M < ~ = 0 i i i 0 0.0098 0.0098. 0.00. 0.907 8 4.5 M, T 0.57 T0, 0.07 0. 7 00 T0 T 044 K or 77 C. 0 670 kpa abs. 0.57 0.07 9.44 ssu 0 0 kpa. Thn 0.498 kg/. 0.89 7 80 000 9.8 F. 0.498 0.5. 60 /s. 6 9.46 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 0 0 F B.94.4 87 09.95 68 /s. F B 00 0.87 09.95 0.05 68 56 000 0. 4 000 N. Noral Shok 9.48 a) 0.9850 000. 80 000 0.985 000( 000) 000.4 80 87 8 0. 0.9850 kg/. 0.4 0.87 8 000.4 ( 985 065 000) 84 00 = 0 0.4 985 784 784 00 0. 6 /s..774 kg/. 4 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow Subsiu in an fin 808 kpa abs. M 000 808.966. T 746 K or 47 C..4 87 8 0.87.774 6 M 0. 477.. 4 87 746 9.50 T km k ( k ) M ( k )M. T k k ( k )M M (4kM k ) k k M k. (This is Eq. 9.4.). Subsiu ino abov: k ( k ) ( k ) ( k ) ( ) ( ) ( ) 4k k k k k ( k ) / k ( k ) / For a srong shok in whih. k,. k ( k ) ( k ) k ( k ) ( k )( k ). 9.5 If M 0.5, hn M.645..645.4 87 9 908 /s. 600 8.00 00 600 kpa abs. 8. kg/. 0.87 (.85 9) 9.54 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 isnroi flow fro 0: For M = 0.458, = 0.866 0 an T 0.960 T0. 0 9 / 0.866 9 kpa abs. T 0 69.5 / 0.96 7 K or 448 C. 5 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 0 9.56 4. M 0.47. 0.985 0 0 0.5 kpa abs. 0.985 M. 0.58 0.5 54.5 kpa. T 0.8 98 48. K. rains a, 54.5 0.87 48. 0.7599 kg/..4 87 48. 5.9 /s. 0.7599 0.05 5.9 0.47 kg/s. If hroa ara is ru, M 0.7599 kg/ an 0.7599 0.0 5.9 0.0 kg/s. 4.7 sia. 4. M.94, an / 9.98. 4.7.48 sia. M.94, / 0 0.098. 0 9.98 9.58.48 0.098 M, 0.58 49.7 6. sia. T 0.8 50 4. R.. 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. 49.7 sia. aor Flow 9.60 0./. 655 0.546 0 0.546 00 655 kpa. T 67 585 K. 00 655.4 kg/.. 46 585 59 /s. (M.) 0.46 585. 4. 4 59. 0. 060 or 6. 4 T 0./. 0 0 67 80. K 0.575 kg/. 00 0.46 80. 87 80. 87 67. (Enrgy fro 0.) ( 87 J/kg K) 050 /s. 4 0.575( /4) 050. 0.09 or 9.. 6 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow 9.6 0./. 8.9 M, 0.546 50 8.9 sia. T 60 009 R. 50 Obliqu Shok Wav 8.9 44 76 009 0.004 slug/f.. 760 009 90 fs. 0.5 0.004 90. 0.99 f. or.9". 4 800 9.64 M. 9.. 4 87 0 Fro Fig. 9.5, 46, 79. a) 46. Mn.9sin 46.65. Mn 0.654 M sin(46 0 ). M. 49.. 0 40 0. 4 kpa abs. T. 4 0 4 K. = 0 o ).4 87 4.49 60 /s. a ah shok = 5 o 9.66 Mn.5sin 5.0. Mn 0.576. T.696 0 54 K. M 0.576/ sin(5 0 ).6. 0. 47. Mn.6sin 47.65. Mn 0.654 M sin(47 0 ). M.44. T.4 54 7 K. M krt.44.4 87 7 780 /s. 9.68 M, 0. 8. Mn sin 8.4. Mn 0.76..5 40 86. kpa abs. 0.76 M.8. 6.44 86. 555 kpa abs. sin(8 0 ) ( ) noral 0. 40 4 kpa abs. 7 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

Char 9 / Corssibl Flow Exansion Wavs 9.70 6. 4. For M 4, 65. 8. (S Fig. 9.8.) T 65. 8 6. 4 9. 4. T T T T T 0 0 7 0.8 7 K. 0.5556 4.4 87 7 867 /s. T 56 C. 9.7 a) 9.. 9. 5 44.. Mu.7. u (0/0.0585) 0.0465 = 4.4 kpa abs. For 5 an M.5, 7. Mn.5sin 7.. Mn 0.889.. 0 6.4 kpa abs. M M 0.889/ sin(7 5 ).7. 9.74 If 5 wih M 4, hn Fig. 9.5 = 8. n M 4sin8.4. M 0.88. n.67 0.5 kpa. shok M M l M u shok 0.88 M.64. sin(8 5 ) 0.0077 M 4, 65.8. 75.8, M 4.88. 0 0.006586 0 u u 0 = 6.6 kpa. C L Lif.5 os 5 0 / 6.6 ( /) os0.4 4 0 0.0854. C D Drag.5sin 5 6.6 ( /) sin0.4 4 0 0.00. 8 0 Cngag Larning. ll Righs Rsrv. May no b sann, oi or ulia, or os o a ublily assibl wbsi, in whol or in ar.

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