Thermodynamics

Chapter 17 | 851The fluid properties after the shock (denoted by subscript 2) are relatedto those before the shock through the functions listed in Table A–33. ForMa 1 2.0, we readMa 2 0.5774 P 02P 01 0.7209 P 2P 1 4.5000 T 2T 1 1.6875 r 2r 1 2.6667Then the stagnation pressure P 02 , static pressure P 2 , static temperature T 2 ,and static density r 2 after the shock areP 02 0.7209P 01 10.7209211.0 MPa2 0.721 MPaP 2 4.5000P 1 14.50002 10.1278 MPa2 0.575 MPaT 2 1.6875T 1 11.687521444.5 K2 750 Kr 2 2.6667r 1 12.6667211.002 kg>m 3 2 2.67 kg/m 3(b) The entropy change across the shock iss 2 s 1 c p ln T 2T 1 R ln P 2P 1 11.005 kJ>kg # K2ln 11.68752 10.287 kJ>kg # K2ln 14.50002 0.0942 kJ/kg # KThus, the entropy of the air increases as it experiences a normal shock,which is highly irreversible.(c) The air velocity after the shock can be determined from V 2 Ma 2 c 2 ,where c 2 is the speed of sound at the exit conditions after the shock:V 2 Ma 2 c 2 Ma 2 2kRT 2 10.57742 B11.42 10.287 kJ>kg # K21750 K2 a1000 m 2 >s 21 kJ>kg b 317 m/s(d) The mass flow rate through a converging–diverging nozzle with sonicconditions at the throat is not affected by the presence of shock waves inthe nozzle. Therefore, the mass flow rate in this case is the same as thatdetermined in Example 17–7:m # 2.86 kg/sDiscussion This result can easily be verified by using property values at thenozzle exit after the shock at all Mach numbers significantly greater than unity.Example 17–9 illustrates that the stagnation pressure and velocitydecrease while the static pressure, temperature, density, and entropyincrease across the shock. The rise in the temperature of the fluid downstreamof a shock wave is of major concern to the aerospace engineerbecause it creates heat transfer problems on the leading edges of wings andnose cones of space reentry vehicles and the recently proposed hypersonicspace planes. Overheating, in fact, led to the tragic loss of the space shuttleColumbia in February of 2003 as it was reentering earth’s atmosphere.