Thermodynamics

Also,P 0 P 0 P P* a 1 k 1 k>1k12Ma 2 b aP* 0 P P* P* 0 2which simplifies to1 k1 kMa ba1 k 12 2k>1k12b(17–66)Chapter 17 | 867P 0P* 0 k 11 kMa2 1k 12Ma2c2k 1(17–67)The five relations in Eqs. 17–63, 17–65, and 17–67 enable us to calculatethe dimensionless pressure, temperature, density, velocity, stagnation temperature,and stagnation pressure for Rayleigh flow of an ideal gas with aspecified k for any given Mach number. Representative results are given intabular form in Table A–34 for k 1.4.Choked Rayleigh FlowIt is clear from the earlier discussions that subsonic Rayleigh flow in a ductmay accelerate to sonic velocity (Ma 1) with heating. What happens ifwe continue to heat the fluid? Does the fluid continue to accelerate to supersonicvelocities? An examination of the Rayleigh line indicates that the fluidat the critical state of Ma 1 cannot be accelerated to supersonic velocitiesby heating. Therefore, the flow is choked. This is analogous to not beingable to accelerate a fluid to supersonic velocities in a converging nozzle bysimply extending the converging flow section. If we keep heating the fluid,we will simply move the critical state further downstream and reduce theflow rate since fluid density at the critical state will now be lower. Therefore,for a given inlet state, the corresponding critical state fixes the maximumpossible heat transfer for steady flow (Fig. 17–57). That is,q max h* 0 h 01 c p 1T* 0 T 01 2k>1k12d(17–68)Further heat transfer causes choking and thus the inlet state to change (e.g.,inlet velocity will decrease), and the flow no longer follows the same Rayleighline. Cooling the subsonic Rayleigh flow reduces the velocity, and the Machnumber approaches zero as the temperature approaches absolute zero. Notethat the stagnation temperature T 0 is maximum at the critical state of Ma 1.In supersonic Rayleigh flow, heating decreases the flow velocity. Furtherheating simply increases the temperature and moves the critical state furtherdownstream, resulting in a reduction in the mass flow rate of the fluid.It may seem like supersonic Rayleigh flow can be cooled indefinitely, but itturns out that there is a limit. Taking the limit of Eq. 17–65 as the Machnumber approaches infinity givesq maxT 1 TRayleigh 2 T *flowT 01T 02 T*01ChokedflowFIGURE 17–57For a given inlet state, the maximumpossible heat transfer occurs whensonic conditions are reached at the exitstate.Lim MaS T 0T* 0 1 1 k 2(17–69)which yields T 0 /T* 0 0.49 for k 1.4. Therefore, if the critical stagnationtemperature is 1000 K, air cannot be cooled below 490 K in Rayleigh flow.Physically this means that the flow velocity reaches infinity by the time thetemperature reaches 490 K—a physical impossibility. When supersonic flowcannot be sustained, the flow undergoes a normal shock wave and becomessubsonic.