Thermodynamics

840 | ThermodynamicsThe critical-pressure ratio is determined from Table 17–2 (or Eq. 17–22) tobe P*/P 0 0.5283.(a) The back pressure ratio for this case iswhich is greater than the critical-pressure ratio, 0.5283. Thus the exit planepressure (or throat pressure P t ) is equal to the back pressure in this case. Thatis, P t P b 0.7 MPa, and P t /P 0 0.670. Therefore, the flow is not choked.From Table A–32 at P t /P 0 0.670, we read Ma t 0.778 and T t /T 0 0.892.The mass flow rate through the nozzle can be calculated from Eq. 17–24.But it can also be determined in a step-by-step manner as follows:Thus,r t P tRT tV t Ma t c t Ma t 2kRT tm # r t A t V t 13.093 kg>m 3 2150 10 4 m 2 21437.9 m>s2 6.77 kg/s(b) The back pressure ratio for this case iswhich is less than the critical-pressure ratio, 0.5283. Therefore, sonic conditionsexist at the exit plane (throat) of the nozzle, and Ma 1. The flow ischoked in this case, and the mass flow rate through the nozzle can be calculatedfrom Eq. 17–25:m # 1k12>321k124k 2 A*P 0 aB RT 0 k 1 b2.4>0.8 150 10 4 m 2 1.42211045 kPa2 B 10.287 kJ>kg #aK21884 K2 1.4 1 b 7.10 kg/s 10.7782 B11.4210.287 kJ>kg # K21788.5 K2a1000 m 2 >s 21 kJ>kg b 437.9 m>sP b 0.7 MPaP 0 1.045 MPa 0.670T t 0.892T 0 0.892 1884 K2 788.5 K700 kPa10.287 kPa # m 3 >kg # K21788.5 K2 3.093 kg>m 3P b 0.4 MPaP 0 1.045 MPa 0.383since kPa # m 2 > 2kJ>kg 21000 kg>s.Discussion This is the maximum mass flow rate through the nozzle for thespecified inlet conditions and nozzle throat area.EXAMPLE 17–6Gas Flow through a Converging NozzleNitrogen enters a duct with varying flow area at T 1 400 K, P 1 100 kPa,and Ma 1 0.3. Assuming steady isentropic flow, determine T 2 , P 2 , and Ma 2at a location where the flow area has been reduced by 20 percent.