Thermodynamics

872 | Thermodynamics(b) The velocity of sound and the Mach numbers at the throat and the exitof the nozzle are determined by replacing differential quantities withdifferences,The velocity of sound at the throat is determined by evaluating the specificvolume at s t 7.1292 kJ/kg · K and at pressures of 1.115 and 1.065 MPa(P t 25 kPa):The Mach number at the throat is determined from Eq. 17–12 to beThus, the flow at the throat is sonic, as expected. The slight deviation ofthe Mach number from unity is due to replacing the derivatives bydifferences.The velocity of sound and the Mach number at the nozzle exit are determinedby evaluating the specific volume at s 2 7.2019 kJ/kg · K and atpressures of 325 and 275 kPa (P 2 25 kPa):andc Bc Bc a 0P0r b 1>211115 10652 kPa11>0.23776 1>0.246332 kg>m a 1000 m2 >s 2b 584.6 m>s3 1 kPa # m 3 >kgMa V c1325 2752 kPa11>0.63596 1>0.722452 kg>m a 1000 m2 >s 2b 515.4 m>s3 1 kPa # m 3 >kgMa V c585.8 m>s584.6 m>s 1.002929.8 m>s515.4 m>s 1.804Thus the flow of steam at the nozzle exit is supersonic.s1>2¢P c¢ 11>v2 dsSUMMARYIn this chapter the effects of compressibility on gas flow areexamined. When dealing with compressible flow, it is convenientto combine the enthalpy and the kinetic energy of thefluid into a single term called stagnation (or total) enthalpyh 0 , defined ash 0 h V 22The properties of a fluid at the stagnation state are calledstagnation properties and are indicated by the subscript zero.The stagnation temperature of an ideal gas with constant specificheats isT 0 T V 22c pwhich represents the temperature an ideal gas would attain ifit is brought to rest adiabatically. The stagnation properties ofan ideal gas are related to the static properties of the fluid byP 0P a T 0T b k>1k12and r 0r a T 0T b 1>1k12The speed at which an infinitesimally small pressure wavetravels through a medium is the speed of sound. For an idealgas it is expressed asc a 0PB 0r b 2kRTsThe Mach number is the ratio of the actual velocity of thefluid to the speed of sound at the same state:Ma V cThe flow is called sonic when Ma 1, subsonic when Ma 1,supersonic when Ma 1, hypersonic when Ma 1, andtransonic when Ma 1.