Thermodynamics

stagnation pressure, stagnation density, etc.). The stagnation state and thestagnation properties are indicated by the subscript 0.The stagnation state is called the isentropic stagnation state when thestagnation process is reversible as well as adiabatic (i.e., isentropic). Theentropy of a fluid remains constant during an isentropic stagnation process.The actual (irreversible) and isentropic stagnation processes are shown onthe h-s diagram in Fig. 17–4. Notice that the stagnation enthalpy of the fluid(and the stagnation temperature if the fluid is an ideal gas) is the same forboth cases. However, the actual stagnation pressure is lower than the isentropicstagnation pressure since entropy increases during the actual stagnationprocess as a result of fluid friction. The stagnation processes are oftenapproximated to be isentropic, and the isentropic stagnation properties aresimply referred to as stagnation properties.When the fluid is approximated as an ideal gas with constant specificheats, its enthalpy can be replaced by c p T and Eq. 17–1 can be expressed asorc p T 0 c p T V 2T 0 T V 2(17–4)Here T 0 is called the stagnation (or total) temperature, and it representsthe temperature an ideal gas attains when it is brought to rest adiabatically.The term V 2 /2c p corresponds to the temperature rise during such a processand is called the dynamic temperature. For example, the dynamic temperatureof air flowing at 100 m/s is (100 m/s) 2 /(2 1.005 kJ/kg · K) 5.0 K.Therefore, when air at 300 K and 100 m/s is brought to rest adiabatically (atthe tip of a temperature probe, for example), its temperature rises to thestagnation value of 305 K (Fig. 17–5). Note that for low-speed flows, thestagnation and static (or ordinary) temperatures are practically the same.But for high-speed flows, the temperature measured by a stationary probeplaced in the fluid (the stagnation temperature) may be significantly higherthan the static temperature of the fluid.The pressure a fluid attains when brought to rest isentropically is calledthe stagnation pressure P 0 . For ideal gases with constant specific heats, P 0is related to the static pressure of the fluid byP 0P a T 0T b k>1k12(17–5)By noting that r 1/v and using the isentropic relation Pv k P 0 v 0k , theratio of the stagnation density to static density can be expressed asr 0r a T 0T b 1>1k12(17–6)When stagnation enthalpies are used, there is no need to refer explicitly tokinetic energy. Then the energy balance E # in E # out for a single-stream,steady-flow device can be expressed as2c pq in w in 1h 01 gz 1 2 q out w out 1h 02 gz 2 22(17–7)Chapter 17 | 825FIGURE 17–3Kinetic energy is converted toenthalpy during a stagnation process.© Reprinted with special permission of KingFeatures Syndicate.hh0hIsentropicstagnationstateV 22P 0P 0,actActualstagnationstatePActual stateFIGURE 17–4The actual state, actual stagnationstate, and isentropic stagnation stateof a fluid on an h-s diagram.s