Thermodynamics

1 V 1 r 2 V 2 (17–50)Chapter 17 | 861as heat transfer through the duct wall at the same rate and disregarding anychanges in chemical composition. This simplified problem is still too complicatedfor an elementary treatment of the topic since the flow may involveQ .friction, variations in duct area, and multidimensional effects. In this section,we limit our consideration to one-dimensional flow in a duct of constantcross-sectional area with negligible frictional effects.Consider steady one-dimensional flow of an ideal gas with constant specificP 1 , T 1 , r 1V 1P 2 , T 2 , r 2V 2heats through a constant-area duct with heat transfer, but with negligibleControlvolumefriction. Such flows are referred to as Rayleigh flows after Lord Rayleigh(1842–1919). The conservation of mass, momentum, and energy equations FIGURE 17–51for the control volume shown in Fig. 17–51 can be written as follows:Control volume for flow in a constantareaMass equation Noting that the duct cross-sectional area A is constant, therelation ṁ 1 ṁ 2 or r 1 A 1 V 1 r 2 A 2 V 2 reduces toduct with heat transfer andnegligible friction.x-Momentum equation Noting that the frictional effects are negligibleand thus there are no shear forces, and assuming there are no externaland body forces, the momentum equation a F! aoutbm # V ! ainbm # V !in the flow (or x-) direction becomes a balance between static pressureforces and momentum transfer. Noting that the flows are high speedand turbulent, the momentum flux correction factor is approximately 1(b 1) and thus can be neglected. Then,orP 1 A 1 P 2 A 2 m # V 2 m # V 1 S P 1 P 2 1r 2 V 2 2V 2 1r 1 V 1 2V 1P 1 r 1 V 2 1 P 2 r 2 V 2 2(17–51)Energy equation The control volume involves no shear, shaft, or otherforms of work, and the potential energy change is negligible. If the rateof heat transfer is Q . and the heat transfer per unit mass of fluid is q Q . /ṁ, the steady-flow energy balance E . in E. out becomesQ # m # a h 1 V 2 1(17–52)2 b m# a h 2 V 2 22 b S q h 1 V 122 h 2 V 2 22For an ideal gas with constant specific heats, h c p T, and thusorq c p 1T 2 T 1 2 V 2 2 V 2 12q h 02 h 01 c p 1T 02 T 01 2(17–53)(17–54)Therefore, the stagnation enthalpy h 0 and stagnation temperature T 0change during Rayleigh flow (both increase when heat is transferred tothe fluid and thus q is positive, and both decrease when heat is transferredfrom the fluid and thus q is negative).Entropy change In the absence of any irreversibilities such as friction,the entropy of a system changes by heat transfer only: it increases withheat gain, and decreases with heat loss. Entropy is a property and thus