Thermodynamics

increase as the flow area of the duct decreases and must decrease as theflow area of the duct increases. Thus, at supersonic velocities, the pressuredecreases in diverging ducts (supersonic nozzles) and increases in convergingducts (supersonic diffusers).Another important relation for the isentropic flow of a fluid is obtained bysubstituting rV dP/dV from Eq. 17–14 into Eq. 17–16:Chapter 17 | 833dAA dV V 11 Ma2 2(17–17)This equation governs the shape of a nozzle or a diffuser in subsonic orsupersonic isentropic flow. Noting that A and V are positive quantities, weconclude the following:For subsonic flow 1Ma 6 12,For supersonic flow 1Ma 7 12,For sonic flow 1Ma 12,dAdV 6 0dAdV 7 0dAdV 0Thus the proper shape of a nozzle depends on the highest velocity desiredrelative to the sonic velocity. To accelerate a fluid, we must use a convergingnozzle at subsonic velocities and a diverging nozzle at supersonic velocities.The velocities encountered in most familiar applications are wellbelow the sonic velocity, and thus it is natural that we visualize a nozzle asa converging duct. However, the highest velocity we can achieve by a convergingnozzle is the sonic velocity, which occurs at the exit of the nozzle.If we extend the converging nozzle by further decreasing the flow area, inhopes of accelerating the fluid to supersonic velocities, as shown inFig. 17–16, we are up for disappointment. Now the sonic velocity will occurat the exit of the converging extension, instead of the exit of the originalnozzle, and the mass flow rate through the nozzle will decrease because ofthe reduced exit area.Based on Eq. 17–16, which is an expression of the conservation of massand energy principles, we must add a diverging section to a converging nozzleto accelerate a fluid to supersonic velocities. The result is a converging–diverging nozzle. The fluid first passes through a subsonic (converging) section,where the Mach number increases as the flow area of the nozzledecreases, and then reaches the value of unity at the nozzle throat. The fluidcontinues to accelerate as it passes through a supersonic (diverging) section.Noting that ṁ rAV for steady flow, we see that the large decrease in densitymakes acceleration in the diverging section possible. An example of thistype of flow is the flow of hot combustion gases through a nozzle in a gasturbine.The opposite process occurs in the engine inlet of a supersonic aircraft.The fluid is decelerated by passing it first through a supersonic diffuser,which has a flow area that decreases in the flow direction. Ideally, the flowreaches a Mach number of unity at the diffuser throat. The fluid is furtherP 0, T 0 Converging MaA = 1nozzle (sonic)P 0, T 0ConvergingnozzleAAMa A < 1BAttachmentMaB = 1(sonic)FIGURE 17–16We cannot obtain supersonic velocitiesby attaching a converging section to aconverging nozzle. Doing so will onlymove the sonic cross section fartherdownstream and decrease the massflow rate.