Thermodynamics

Another parameter sometimes used in the analysis of one-dimensionalisentropic flow of ideal gases is Ma*, which is the ratio of the local velocityto the speed of sound at the throat:Chapter 17 | 839Ma* V c*(17–27)It can also be expressed asMa* V cc c* Macc* Ma2kRT2kRT* Ma TB T*where Ma is the local Mach number, T is the local temperature, and T* isthe critical temperature. Solving for T from Eq. 17–18 and for T* fromEq. 17–21 and substituting, we getMa* Ma Bk 12 1k 12Ma 2(17–28)Values of Ma* are also listed in Table A–32 versus the Mach number fork 1.4 (Fig. 17–23). Note that the parameter Ma* differs from the Machnumber Ma in that Ma* is the local velocity nondimensionalized withrespect to the sonic velocity at the throat, whereas Ma is the local velocitynondimensionalized with respect to the local sonic velocity. (Recall that thesonic velocity in a nozzle varies with temperature and thus with location.)Ma.0.901.001.10.Ma * A P r TA * P0r0 T 0... . .0.9146 1.0089 0.59131.0000 1.0000 0.52831.0812.1.0079.0.4684. . .FIGURE 17–23Various property ratios for isentropicflow through nozzles and diffusers arelisted in Table A–32 for k 1.4 forconvenience.EXAMPLE 17–5Effect of Back Pressure on Mass Flow RateAir at 1 MPa and 600°C enters a converging nozzle, shown in Fig. 17–24,with a velocity of 150 m/s. Determine the mass flow rate through the nozzlefor a nozzle throat area of 50 cm 2 when the back pressure is (a) 0.7 MPaand (b) 0.4 MPa.Solution Air enters a converging nozzle. The mass flow rate of air throughthe nozzle is to be determined for different back pressures.Assumptions 1 Air is an ideal gas with constant specific heats at roomtemperature. 2 Flow through the nozzle is steady, one-dimensional, andisentropic.Properties The constant-pressure specific heat and the specific heat ratio ofair are c p 1.005 kJ/kg K and k 1.4, respectively (Table A–2a).Analysis We use the subscripts i and t to represent the properties at thenozzle inlet and the throat, respectively. The stagnation temperature andpressure at the nozzle inlet are determined from Eqs. 17–4 and 17–5:T 0i T i V 2 i1150 m>s2 2 873 K 2c p 2 11.005 kJ>kg #a 1 kJ>kgK2 1000 m 2 >s b 884 K2P 0i P i a T k>1k120ib 11 MPa2 a 884 K 1.4>11.412T i 873 K b 1.045 MPaThese stagnation temperature and pressure values remain constant throughoutthe nozzle since the flow is assumed to be isentropic. That is,T 0 T 0i 884 KandP 0 P 0i 1.045 MPaAIRP i = 1 MPaT i = 600°CV i = 150 m/sConvergingnozzleFIGURE 17–24Schematic for Example 17–5.P bA t = 50 cm 2