Thermodynamics

848 | ThermodynamicsCombining Eqs. 17–35 and 17–36 gives the pressure ratio across the shock:P 2 Ma 1 21 Ma 2 1 1k 12>2P 1 Ma 2 21 Ma 2 2 1k 12>2(17–37)Equation 17–37 is a combination of the conservation of mass and energyequations; thus, it is also the equation of the Fanno line for an ideal gas withconstant specific heats. A similar relation for the Rayleigh line can beobtained by combining the conservation of mass and momentum equations.From Eq. 17–32,However,P 1 P 2 m# A 1V 2 V 1 2 r 2 V 2 2 r 1 V 2 1rV 2 a P RT b1Mac22 a P RT b1Ma2kRT22 PkMa 2Thus,orP 1 11 kMa 2 12 P 2 11 kMa 2 22P 2P 1 1 kMa2 11 kMa 2 2Combining Eqs. 17–37 and 17–38 yieldsMa 2 2 Ma2 1 2>1k 122Ma 2 1k>1k 12 1(17–38)(17–39)This represents the intersections of the Fanno and Rayleigh lines and relatesthe Mach number upstream of the shock to that downstream of the shock.The occurrence of shock waves is not limited to supersonic nozzles only.This phenomenon is also observed at the engine inlet of a supersonic aircraft,where the air passes through a shock and decelerates to subsonicvelocities before entering the diffuser of the engine. Explosions also producepowerful expanding spherical normal shocks, which can be verydestructive (Fig. 17–33).Various flow property ratios across the shock are listed in Table A–33 foran ideal gas with k 1.4. Inspection of this table reveals that Ma 2 (theMach number after the shock) is always less than 1 and that the larger thesupersonic Mach number before the shock, the smaller the subsonic Machnumber after the shock. Also, we see that the static pressure, temperature,and density all increase after the shock while the stagnation pressuredecreases.The entropy change across the shock is obtained by applying the entropychangeequation for an ideal gas across the shock:s 2 s 1 c p ln T 2 R ln P 2(17–40)T 1 P 1which can be expressed in terms of k, R, and Ma 1 by using the relationsdeveloped earlier in this section. A plot of nondimensional entropy change