Thermodynamics

870 | ThermodynamicshWilson line (x = 0.96)12P 1P 2SaturationlineFIGURE 17–59The h-s diagram for the isentropicexpansion of steam in a nozzle.sordinarily one would expect the steam to start condensing when it strikesthe saturation line. However, this is not always the case. Owing to the highspeeds, the residence time of the steam in the nozzle is small, and there maynot be sufficient time for the necessary heat transfer and the formation ofliquid droplets. Consequently, the condensation of the steam may bedelayed for a little while. This phenomenon is known as supersaturation,and the steam that exists in the wet region without containing any liquid iscalled supersaturated steam. Supersaturation states are nonequilibrium (ormetastable) states.During the expansion process, the steam reaches a temperature lower thanthat normally required for the condensation process to begin. Once the temperaturedrops a sufficient amount below the saturation temperature correspondingto the local pressure, groups of steam moisture droplets ofsufficient size are formed, and condensation occurs rapidly. The locus ofpoints where condensation takes place regardless of the initial temperatureand pressure at the nozzle entrance is called the Wilson line. The Wilsonline lies between the 4 and 5 percent moisture curves in the saturationregion on the h-s diagram for steam, and it is often approximated by the4 percent moisture line. Therefore, steam flowing through a high-velocitynozzle is assumed to begin condensation when the 4 percent moisture line iscrossed.The critical-pressure ratio P*/P 0 for steam depends on the nozzle inlet stateas well as on whether the steam is superheated or saturated at the nozzle inlet.However, the ideal-gas relation for the critical-pressure ratio, Eq. 17–22, givesreasonably good results over a wide range of inlet states. As indicated inTable 17–2, the specific heat ratio of superheated steam is approximated ask 1.3. Then the critical-pressure ratio becomesk>1k12P* 2 aP 0 k 1 b 0.546When steam enters the nozzle as a saturated vapor instead of superheatedvapor (a common occurrence in the lower stages of a steam turbine), thecritical-pressure ratio is taken to be 0.576, which corresponds to a specificheat ratio of k 1.14.EXAMPLE 17–16Steam Flow through aConverging–Diverging NozzleSteam enters a converging–diverging nozzle at 2 MPa and 400°C with a negligiblevelocity and a mass flow rate of 2.5 kg/s, and it exits at a pressure of300 kPa. The flow is isentropic between the nozzle entrance and throat, andthe overall nozzle efficiency is 93 percent. Determine (a) the throat and exitareas and (b) the Mach number at the throat and the nozzle exit.Solution Steam enters a converging–diverging nozzle with a low velocity.The throat and exit areas and the Mach number are to be determined.Assumptions 1 Flow through the nozzle is one-dimensional. 2 The flow isisentropic between the inlet and the throat, and is adiabatic and irreversiblebetween the throat and the exit. 3 The inlet velocity is negligible.