Thermodynamics

862 | Thermodynamicsa state function, and the entropy change of an ideal gas with constantspecific heats during a change of state from 1 to 2 is given bys 2 s 1 c p ln T 2 R ln P 2T 1 P 1The entropy of a fluid may increase or decrease during Rayleigh flow,depending on the direction of heat transfer.Equation of state Noting that P rRT, the properties P, r, and T of anideal gas at states 1 and 2 are related to each other by(17–55)T maxTMa < 1Cooling(Ma S 0)Heating(Ma S 1)Ma > 1Heating(Ma S 1)Cooling(Ma S )Ma b = 1/ kbaMa a = 1s maxsFIGURE 17–52T-s diagram for flow in a constant-areaduct with heat transfer and negligiblefriction (Rayleigh flow).aP 1 P 2(17–56)r 1 T 1 r 2 T 2Consider a gas with known properties R, k, and c p . For a specified inletstate 1, the inlet properties P 1 , T 1 , r 1 , V 1 , and s 1 are known. The five exitproperties P 2 , T 2 , r 2 , V 2 , and s 2 can be determined from the five equations17–50, 17–51, 17–53, 17–55, and 17–56 for any specified value of heattransfer q. When the velocity and temperature are known, the Mach numbercan be determined from Ma V>c V> 1kRT.Obviously there is an infinite number of possible downstream states 2corresponding to a given upstream state 1. A practical way of determiningthese downstream states is to assume various values of T 2 , and calculate allother properties as well as the heat transfer q for each assumed T 2 from theEqs. 17–50 through 17–56. Plotting the results on a T-s diagram gives acurve passing through the specified inlet state, as shown in Fig. 17–52. Theplot of Rayleigh flow on a T-s diagram is called the Rayleigh line, and severalimportant observations can be made from this plot and the results of thecalculations:1. All the states that satisfy the conservation of mass, momentum, andenergy equations as well as the property relations are on the Rayleighline. Therefore, for a given initial state, the fluid cannot exist at anydownstream state outside the Rayleigh line on a T-s diagram. In fact,the Rayleigh line is the locus of all physically attainable downstreamstates corresponding to an initial state.2. Entropy increases with heat gain, and thus we proceed to the right onthe Rayleigh line as heat is transferred to the fluid. The Mach number isMa 1 at point a, which is the point of maximum entropy (see Example17–13 for proof). The states on the upper arm of the Rayleigh lineabove point a are subsonic, and the states on the lower arm below pointa are supersonic. Therefore, a process proceeds to the right on the Rayleighline with heat addition and to the left with heat rejection regardlessof the initial value of the Mach number.3. Heating increases the Mach number for subsonic flow, but decreases itfor supersonic flow. The flow Mach number approaches Ma 1 in bothcases (from 0 in subsonic flow and from ∞ in supersonic flow) duringheating.4. It is clear from the energy balance q c p (T 02 T 01 ) that heatingincreases the stagnation temperature T 0 for both subsonic and supersonicflows, and cooling decreases it. (The maximum value of T 0 occursat Ma 1.) This is also the case for the thermodynamic temperature T