Thermodynamics

The conservation of energy principle (Eq. 17–31) requires that the stagnationenthalpy remain constant across the shock; h 01 h 02 . For ideal gasesh h(T ), and thusThat is, the stagnation temperature of an ideal gas also remains constantacross the shock. Note, however, that the stagnation pressure decreasesacross the shock because of the irreversibilities, while the thermodynamictemperature rises drastically because of the conversion of kinetic energyinto enthalpy due to a large drop in fluid velocity (see Fig. 17–32).We now develop relations between various properties before and after theshock for an ideal gas with constant specific heats. A relation for the ratio ofthe thermodynamic temperatures T 2 /T 1 is obtained by applying Eq. 17–18twice:T 01 1 a k 1T 1 2SHOCK WAVEDividing the first equation by the second one and noting that T 01 T 02 ,wehaveFrom the ideal-gas equation of state,hP 01h 01 = h 02h 01 h 02h2V12 2h110s1T 01 T 02 (17–34)Ma decreasesChapter 17 | 8472 V 22Subsonic flow(Ma < 1)2b Ma = 1aMa = 1Supersonic flow(Ma > 1)FIGURE 17–31The h-s diagram for flow across as2snormal shock.NormalshockP increasesP 0 decreasesT increasesT 0 remains constantr increasess increasesFanno lineRayleigh lineb Ma 2 1and T 02 1 a k 1 b Ma 2 2T 2 2T 2 1 Ma2 1 1k 12>2T 1 1 Ma 2 2 1k 12>2r 1 P 1RT 1andr 2 P 2RT 2(17–35)Substituting these into the conservation of mass relation r 1 V 1 r 2 V 2 andnoting that Ma V/c and c 1kRT, we haveP 02FIGURE 17–32Variation of flow properties across anormal shock.T 2 P 2V 2 P 2Ma 2 c 2 P 2Ma 2 2T 2 a P 22b a Ma 22bT 1 P 1 V 1 P 1 Ma 1 c 1 P 1 Ma 1 2T 1P 1 Ma 1(17–36)