Mechanical Engineering Department Third year class Gas Dynamics Tables Lecturer: Dr.Naseer Al-Janabi Ref. Fundamentals of Gas Dynamics, 2e - R. Zucker, O. Biblarz Appendixes A. B. C. D. E. F. G. H. I. J. K. L. Summary of the English Engineering (EE) System of Units Summary of the International System (SI) of Units Friction-Factor Chart Oblique-Shock Charts (γ = 1.4) (Two-Dimensional) Conical-Shock Charts (γ = 1.4) (Three-Dimensional) Generalized Compressibility Factor Chart Isentropic Flow Parameters (γ = 1.4) (including Prandtl Meyer Function) Normal-Shock Parameters (γ = 1.4) Fanno Flow Parameters (γ = 1.4) Rayleigh Flow Parameters (γ = 1.4) Properties of Air at Low Pressures Specific Heats of Air at Low Pressures 395
APPENDIX A Summary of the English Engineering (EE) System of Units 396
SUMMARY OF THE ENGLISH ENGINEERING (EE) SYSTEM OF UNITS Force Mass Length Time Temperature pound force pound mass foot second Rankine lbf lbm ft sec R NEVER say pound, as this is ambiguous! It is either a pound force (lbf) or a pound mass (lbm). A 1-pound force will give a 1-pound mass an acceleration of 32.174 feet/second2. F = 1(lbf) = ma gc 1 (lbm) 32.174 (ft/sec2 ) gc Thus gc = 32.174 lbm-ft/lbf-sec2 Temperature Gas constant Pressure Heat to work Power Standard gravity * M.M., molecular mass. T ( R) R 1 atm 1 Btu 1 hp g0 = = = = = = T ( F) + 459.67 1545/M.M.* ft-lbf/1bm- R 2116.2 lbf/ft2 778.2 ft-lbf 550 ft-lbf/sec 32.174 ft/sec2 397
398 APPENDIX A Useful Conversion Factors To convert from: To: Multiply by: meter meter newton kilogram K joule kwh joule watt m/s m/s km/h N/m2 N/m2 N/m2 kg/m3 N s/m2 m2/s J/kg K N m/kg K foot inch lbf lbm R Btu Btu ft-lbf horsepower ft/sec mph mph atmosphere lbf/in2 lbf/ft2 lbm/ft3 lbf-sec/ft2 ft2/sec Btu/lbm- R ft-lbf/lbm- R 3.281 3.937 10 2.248 10 1 2.205 1.800 9.479 10 4 3.413 103 7.375 10 1 1.341 10 3 3.281 2.237 6.215 10 1 9.872 10 6 1.450 10 4 2.089 10 2 6.242 10 2 2.089 10 2 1.076 10 2.388 10 4 1.858 10 1 (q) (q) (w) (V ) (V ) (V ) (p) (p) (p) (ρ) (µ) (ν) (cp ) (R) Source: The International System of Units, NASA SP-7012, 1973.
399 a 39.94 Ar CO2 CO He H2 CH4 N2 O2 H2O Argon Carbon dioxide Carbon monoxide Helium Hydrogen Methane Nitrogen Oxygen Water vapor 1.33 1.40 1.40 1.32 1.41 1.67 1.40 1.29 1.67 1.40 cp γ = cv 85.7 48.3 55.1 96.4 766 386 55.2 35.1 38.7 53.3 Gas Constant R ft-lbf/lbm- R Values for γ, R, cp, cv, and µ are for normal room temperature and pressure. 18.02 32.00 28.02 16.04 2.02 4.00 28.01 44.01 28.97 Symbol Air Gas Molecular Mass Properties of Gases English Engineering (EE) System a 0.445 0.218 0.248 0.532 3.42 1.25 0.248 0.203 0.124 0.240 0.335 0.156 0.177 0.403 2.43 0.750 0.177 0.157 0.074 0.171 Specific Heats Btu/lbm- R cp cv 343.9 227.1 278.6 1165.3 2.3 10 7 3.6 10 7 7 2.2 10 7 4.2 10 1.9 10 59.9 240 3.7 10 7 9.5 547.5 3.1 10 7 7 272 4.7 10 7 4.2 10 7 239 3.8 10 7 3204 736 492 673 188.1 33.2 507 1071 705 546 Critical Point Tc pc R psia Viscosity µ lbf-sec/ft2
APPENDIX B Summary of the International System (SI) of Units 400
SUMMARY OF THE INTERNATIONAL SYSTEM (SI) OF UNITS Force Mass Length Time Temperature newton kilogram meter second kelvin N kg m s K A 1-Newton force will give a 1-kilogram mass an acceleration of 1 meter/second2. F = 1(N) = ma gc 1 (kg) 1 (m/s2 ) gc Thus gc = 1 kg m/n s2 Temperature Gas constant Pressure Heat to work Power Standard gravity * M.M., molecular mass. T (K) R 1 atm 1 pascal (Pa) 1 bar (bar) 1 MPa 1 joule (J) 1 watt (W) g0 = = = = = = = = = T ( C) + 273.15 8314/M.M.* N m/kg K 1.013 105 N/m2 1 N/m2 1 105 N/m2 1 106 N/m2 1N m 1 J/s 9.81 m/s2 401
402 APPENDIX B Useful Conversion Factors To convert from: To: Multiply by: foot inch lbf lbm R Btu Btu ft-lbf horsepower ft/sec mph mph atmosphere lbf/in2 lbf/ft2 lbm/ft3 lbf-sec/ft2 ft2/sec Btu/lbm- R ft-lbf/lbm- R meter meter newton kilogram K joule kwh joule watt m/s m/s km/h N/m2 N/m2 N/m2 kg/m3 N s/m2 m2/s J/kg K N m/kg K 3.048 10 1 2.54 10 2 4.448 4.536 10 1 5.555 10 1 1.055 103 2.930 10 4 1.356 7.457 102 3.048 10 1 4.470 10 1 1.609 1.013 105 6.895 103 4.788 10 1.602 10 4.788 10 9.290 10 2 4.187 103 5.381 (q) (q) (w) (V ) (V ) (V ) (p) (p) (p) (ρ) (µ) (ν) (cp ) (R) Source: The International System of Units, NASA SP-7012, 1973.
403 a 39.94 Ar CO2 CO He H2 CH4 N2 O2 H2O Argon Carbon dioxide Carbon monoxide Helium Hydrogen Methane Nitrogen Oxygen Water vapor 1.33 1.40 1.40 1.32 1.41 1.67 1.40 1.29 1.67 1.40 cp γ = cv 461 260 296 519 4,120 2,080 297 189 208 287 Gas Constant R N m/kg K Values for γ, R, cp, cv, and µ are for normal room temperature and pressure. 18.02 32.00 28.02 16.04 2.02 4.00 28.01 44.01 28.97 Symbol Air Gas Molecular Mass Properties of Gases International System (SI) a 1,860 913 1,040 2,230 14,300 5,230 1,040 850 519 1,000 1,400 653 741 1,690 10,200 3,140 741 657 310 716 Specific Heats J/kg K cp cv 304.1 133.3 1.5 10 5 1.8 10 5 33.3 191.0 126.2 154.8 647.3 1.1 10 5 1.7 10 5 5 1.1 10 5 2.0 10 9.1 10 5 5.28 151.1 2.3 10 5 2.0 10 5 132.8 22.09 5.07 3.39 4.64 1.30 0.229 3.49 7.38 4.86 3.76 Critical Point Tc pc K MPa 1.8 10 5 Viscosity µ N s/m2
APPENDIX C Friction-Factor Chart 404
405 Figure AC.1 Moody diagram for determination of friction factor. (Adapted with permission from L. F. Moody, Friction factors for pipe flow, Transactions of ASME, Vol. 66, 1944.)
APPENDIX D Oblique-Shock Charts (γ = 1.4) (Two-Dimensional) 406
OBLIQUE-SHOCK CHARTS (γ = 1.4) 407 Figure AD.1 Shock-wave angle θ as a function of the initial Mach number M1 for different values of the flow deflection angle δ for γ = 1.4. (Adapted with permission from M. J. Zucrow and J. D. Hoffman, Gas Dynamics, Vol. I, copyright 1976, John Wiley & Sons, New York.)
408 APPENDIX D Figure AD.2 Mach number downstream M2 for an oblique-shock wave as a function of the initial Mach number M1 for different values of the flow deflection angle δ for γ = 1.4. (Adapted with permission from M. J. Zucrow and J. D. Hoffman, Gas Dynamics, Vol. I, copyright 1976, John Wiley & Sons, New York.)
OBLIQUE-SHOCK CHARTS (γ = 1.4) 409 Figure AD.3 Static pressure ratio p2 /p1 across an oblique-shock wave as a function of the initial Mach number M1 for different values of the flow deflection angle δ for γ = 1.40. (Adapted with permission from M. J. Zucrow and J. D. Hoffman, Gas Dynamics, Vol. I, copyright 1976, John Wiley & Sons, New York.)
APPENDIX E Conical-Shock Charts (γ = 1.4) (Three-Dimensional) at 410
CONICAL-SHOCK CHARTS (γ = 1.4) 411 c c Figure AE.1 Shock wave angle θc for a conical-shock wave as a function of the initial Mach number M1 for different values of the cone angle δc for γ = 1.40. (Adapted with permission from M. J. Zucrow and J. D. Hoffman, Gas Dynamics, Vol. I, copyright 1976, John Wiley & Sons, New York.)
412 APPENDIX E c Figure AE.2 Surface Mach number Ms for a conical-shock wave as a function of the initial Mach number M1 for different values of the cone angle δc for γ = 1.40. (Adapted with permission from M. J. Zucrow and J. D. Hoffman, Gas Dynamics, Vol. I, copyright 1976, John Wiley & Sons, New York.)
CONICAL-SHOCK CHARTS (γ = 1.4) 413 c Figure AE.3 Surface static pressure ratio ps /p1 for a conical-shock wave as a function of the initial Mach number M1 for different values of the cone angle δc for γ = 1.40. (Adapted with permission from M. J. Zucrow and J. D. Hoffman, Gas Dynamics, Vol. I, copyright 1976, John Wiley & Sons, New York.)
APPENDIX F Generalized Compressibility Factor Chart 414
GENERALIZED COMPRESSIBILITY FACTOR CHART 415 Figure AF.1 Generalized compressibility factors (Zc = 0.27). (With permission from R. E. Sontag, C. Borgnakke, and C. J. Van Wylen, Fundamentals of Thermodynamics, 5th ed., copyright 1997, John Wiley & Sons, New York.)
APPENDIX G Isentropic Flow Parameters (γ = 1.4) (including Prandtl Meyer Function) 416
ISENTROPIC FLOW PARAMETERS (γ = 1.4) (INCLUDING PRANDTL MEYER FUNCTION) M p/pt T /Tt A/A pa/pt A ν 417 µ
418 M APPENDIX G p/pt T /Tt A/A pa/pt A ν µ
ISENTROPIC FLOW PARAMETERS (γ = 1.4) (INCLUDING PRANDTL MEYER FUNCTION) M p/pt T /Tt A/A pa/pt A ν 419 µ
420 M APPENDIX G p/pt T /Tt A/A pa/pt A ν µ
ISENTROPIC FLOW PARAMETERS (γ = 1.4) (INCLUDING PRANDTL MEYER FUNCTION) M p/pt T /Tt A/A pa/pt A ν 421 µ
422 M APPENDIX G p/pt T /Tt A/A pa/pt A ν µ
ISENTROPIC FLOW PARAMETERS (γ = 1.4) (INCLUDING PRANDTL MEYER FUNCTION) M p/pt T /Tt A/A pa/pt A ν 423 µ
424 M APPENDIX G p/pt T /Tt A/A pa/pt A ν µ
ISENTROPIC FLOW PARAMETERS (γ = 1.4) (INCLUDING PRANDTL MEYER FUNCTION) M p/pt T /Tt A/A pa/pt A ν 425 µ
426 M APPENDIX G p/pt T /Tt A/A pa/pt A ν µ
ISENTROPIC FLOW PARAMETERS (γ = 1.4) (INCLUDING PRANDTL MEYER FUNCTION) M p/pt T /Tt A/A pa/pt A ν 427 µ
APPENDIX H Normal-Shock Parameters (γ = 1.4) 428
NORMAL-SHOCK PARAMETERS (γ = 1.4) M1 M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 429 pt2 /p1
430 M1 APPENDIX H M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 pt2 /p1
NORMAL-SHOCK PARAMETERS (γ = 1.4) M1 M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 431 pt2 /p1
432 M1 APPENDIX H M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 pt2 /p1
NORMAL-SHOCK PARAMETERS (γ = 1.4) M1 M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 433 pt2 /p1
434 M1 APPENDIX H M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 pt2 /p1
NORMAL-SHOCK PARAMETERS (γ = 1.4) M1 M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 435 pt2 /p1
436 M1 APPENDIX H M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 pt2 /p1
NORMAL-SHOCK PARAMETERS (γ = 1.4) M1 M2 p2 /p1 T2 /T1 7V /a1 pt2 /pt1 437 pt2 /p1
APPENDIX I Fanno Flow Parameters (γ = 1.4) 438
FANNO FLOW PARAMETERS (γ = 1.4) M T /T p/p pt /pt V /V f Lmax /D 439 Smax /R
440 M APPENDIX I T /T p/p pt /pt V /V f Lmax /D Smax /R
FANNO FLOW PARAMETERS (γ = 1.4) M T /T p/p pt /pt V /V f Lmax /D 441 Smax /R
442 M APPENDIX I T /T p/p pt /pt V /V f Lmax /D Smax /R
FANNO FLOW PARAMETERS (γ = 1.4) M T /T p/p pt /pt V /V f Lmax /D 443 Smax /R
444 M APPENDIX I T /T p/p pt /pt V /V f Lmax /D Smax /R
FANNO FLOW PARAMETERS (γ = 1.4) M T /T p/p pt /pt V /V f Lmax /D 445 Smax /R
446 M APPENDIX I T /T p/p pt /pt V /V f Lmax /D Smax /R
FANNO FLOW PARAMETERS (γ = 1.4) M T /T p/p pt /pt V /V f Lmax /D 447 Smax /R
448 M APPENDIX I T /T p/p pt /pt V /V f Lmax /D Smax /R
FANNO FLOW PARAMETERS (γ = 1.4) M T /T p/p pt /pt V /V f Lmax /D 449 Smax /R
APPENDIX J Rayleigh Flow Parameters (γ = 1.4) 450
RAYLEIGH FLOW PARAMETERS (γ = 1.4) M Tt /Tt T /T p/p pt /pt V /V 451 Smax /R
452 APPENDIX J M Tt /Tt T /T p/p pt /pt V /V Smax /R
RAYLEIGH FLOW PARAMETERS (γ = 1.4) M Tt /Tt T /T p/p pt /pt V /V 453 Smax /R
454 APPENDIX J M Tt /Tt T /T p/p pt /pt V /V Smax /R
RAYLEIGH FLOW PARAMETERS (γ = 1.4) M Tt /Tt T /T p/p pt /pt V /V 455 Smax /R
456 APPENDIX J M Tt /Tt T /T p/p pt /pt V /V Smax /R
RAYLEIGH FLOW PARAMETERS (γ = 1.4) M Tt /Tt T /T p/p pt /pt V /V 457 Smax /R
458 APPENDIX J M Tt /Tt T /T p/p pt /pt V /V Smax /R
RAYLEIGH FLOW PARAMETERS (γ = 1.4) M Tt /Tt T /T p/p pt /pt V /V 459 Smax /R
460 APPENDIX J M Tt /Tt T /T p/p pt /pt V /V Smax /R
RAYLEIGH FLOW PARAMETERS (γ = 1.4) M Tt /Tt T /T p/p pt /pt V /V 461 Smax /R