Thermodynamics

846 | ThermodynamicsMa 1 > 1FlowV 1 V 2P P1 2h1 h 2r1 r2ss1 2ControlvolumeShock waveMa 2 < 1FIGURE 17–29Control volume for flow across anormal shock wave.(1886–1975) and develop relationships for the flow properties before andafter the shock. We do this by applying the conservation of mass, momentum,and energy relations as well as some property relations to a stationarycontrol volume that contains the shock, as shown in Fig. 17–29. The normalshock waves are extremely thin, so the entrance and exit flow areas for thecontrol volume are approximately equal (Fig 17–30).We assume steady flow with no heat and work interactions and nopotential energy changes. Denoting the properties upstream of the shockby the subscript 1 and those downstream of the shock by 2, we have thefollowing:Conservation of mass: r 1 AV 1 r 2 AV 2(17–29)orr 1 V 1 r 2 V 2Conservation of energy: h 1 V 2 12 h 2 V 2 22(17–30)orh 01 h 02(17–31)Conservation of momentum:Rearranging Eq. 17–14 and integrating yieldA 1P 1 P 2 2 m # 1V 2 V 1 2(17–32)Increase of entropy: s 2 s 1 0(17–33)FIGURE 17–30Schlieren image of a normal shock ina Laval nozzle. The Mach number inthe nozzle just upstream (to the left) ofthe shock wave is about 1.3. Boundarylayers distort the shape of the normalshock near the walls and lead to flowseparation beneath the shock.Photo by G. S. Settles, Penn State University. Usedby permission.We can combine the conservation of mass and energy relations into a singleequation and plot it on an h-s diagram, using property relations. Theresultant curve is called the Fanno line, and it is the locus of states thathave the same value of stagnation enthalpy and mass flux (mass flow perunit flow area). Likewise, combining the conservation of mass and momentumequations into a single equation and plotting it on the h-s diagram yielda curve called the Rayleigh line. Both these lines are shown on the h-s diagramin Fig. 17–31. As proved later in Example 17–8, the points of maximumentropy on these lines (points a and b) correspond to Ma 1. Thestate on the upper part of each curve is subsonic and on the lower partsupersonic.The Fanno and Rayleigh lines intersect at two points (points 1 and 2),which represent the two states at which all three conservation equations aresatisfied. One of these (state 1) corresponds to the state before the shock,and the other (state 2) corresponds to the state after the shock. Note thatthe flow is supersonic before the shock and subsonic afterward. Thereforethe flow must change from supersonic to subsonic if a shock is to occur.The larger the Mach number before the shock, the stronger the shock willbe. In the limiting case of Ma 1, the shock wave simply becomes a soundwave. Notice from Fig. 17–31 that s 2 s 1 . This is expected since the flowthrough the shock is adiabatic but irreversible.