Thermodynamics

Chapter 17 | 841Solution Nitrogen gas enters a converging nozzle. The properties at thenozzle exit are to be determined.Assumptions 1 Nitrogen is an ideal gas with k 1.4. 2 Flow through thenozzle is steady, one-dimensional, and isentropic.Analysis The schematic of the duct is shown in Fig. 17–25. For isentropicflow through a duct, the area ratio A/A* (the flow area over the area of thethroat where Ma 1) is also listed in Table A–32. At the initial Machnumber of Ma 1 0.3, we readA 1A* 2.0351 T 1T 0 0.9823P 1P 0 0.9395With a 20 percent reduction in flow area, A 2 0.8A 1 , andT 1 = 400 KP 1 = 100 kPaMa 1 = 0.3A 1N 2NozzleT 2P 2Ma 2A 2 = 0.8A 1FIGURE 17–25Schematic for Example 17–6 (not toscale).A 2A* A 2 A 1 10.82 12.03512 1.6281A 1 A*For this value of A 2 /A* from Table A–32, we readT 2T 0 0.9701P 2P 0 0.8993 Ma 2 0.391Here we chose the subsonic Mach number for the calculated A 2 /A* insteadof the supersonic one because the duct is converging in the flow directionand the initial flow is subsonic. Since the stagnation properties are constantfor isentropic flow, we can writeT 2T 1 T 2>T 0T 1 >T 0S T 2 T 1 a T 2>T 0T 1 >T 0b 1400 K2 a 0.97010.9823 b 395 KP 2 P 2>P 0S PP 1 P 1 >P 2 P 1 a P 2>P 0b 1100 kPa2a 0.8993 b 95.7 kPa0 P 1 >P 0 0.9395which are the temperature and pressure at the desired location.Discussion Note that the temperature and pressure drop as the fluid acceleratesin a converging nozzle.Converging–Diverging NozzlesWhen we think of nozzles, we ordinarily think of flow passages whosecross-sectional area decreases in the flow direction. However, the highestvelocity to which a fluid can be accelerated in a converging nozzle is limitedto the sonic velocity (Ma 1), which occurs at the exit plane (throat) of thenozzle. Accelerating a fluid to supersonic velocities (Ma 1) can be accomplishedonly by attaching a diverging flow section to the subsonic nozzle atthe throat. The resulting combined flow section is a converging–divergingnozzle, which is standard equipment in supersonic aircraft and rocket propulsion(Fig. 17–26).Forcing a fluid through a converging–diverging nozzle is no guaranteethat the fluid will be accelerated to a supersonic velocity. In fact, the fluidmay find itself decelerating in the diverging section instead of acceleratingif the back pressure is not in the right range. The state of the nozzle flow isdetermined by the overall pressure ratio P b /P 0 . Therefore, for given inletconditions, the flow through a converging–diverging nozzle is governed bythe back pressure P b , as will be explained.