Thermodynamics

836 | ThermodynamicsEXAMPLE 17–4Critical Temperature and Pressure in Gas FlowP 0 = 1.4 MPaT 0 = 473 KP *T *CO 2FIGURE 17–19Schematic for Example 17–4.Calculate the critical pressure and temperature of carbon dioxide for the flowconditions described in Example 17–3 (Fig. 17–19).Solution For the flow discussed in Example 17–3, the critical pressure andtemperature are to be calculated.Assumptions 1 The flow is steady, adiabatic, and one-dimensional. 2 Carbondioxide is an ideal gas with constant specific heats.Properties The specific heat ratio of carbon dioxide at room temperature isk 1.289 (Table A–2a).Analysis The ratios of critical to stagnation temperature and pressure aredetermined to beT* 2T 0 k 1 21.289 1 0.8737k>1k121.289>11.28912P* 2 aP 0 k 1 b 2 a1.289 1 b 0.5477Noting that the stagnation temperature and pressure are, from Example17–3, T 0 473 K and P 0 1400 kPa, we see that the critical temperatureand pressure in this case areT* 0.8737T 0 10.87372 1473 K2 413 KP* 0.5477P 0 10.5477211400 kPa2 767 kPaDiscussion Note that these values agree with those listed in Table 17–1, asexpected. Also, property values other than these at the throat would indicatethat the flow is not critical, and the Mach number is not unity.ReservoirP r= P 0P eT r= T 0V r = 0P/P 01x12P b(Backpressure)P b = P 0P b> P*17–4 ■ ISENTROPIC FLOW THROUGH NOZZLESConverging or converging–diverging nozzles are found in many engineeringapplications including steam and gas turbines, aircraft and spacecraftpropulsion systems, and even industrial blasting nozzles and torch nozzles.In this section we consider the effects of back pressure (i.e., the pressureapplied at the nozzle discharge region) on the exit velocity, the mass flowrate, and the pressure distribution along the nozzle.P*P 00Lowest exitpressureFIGURE 17–20The effect of back pressure on thepressure distribution along aconverging nozzle.345P b =P*P b< P*P b =0xConverging NozzlesConsider the subsonic flow through a converging nozzle as shown inFig. 17–20. The nozzle inlet is attached to a reservoir at pressure P r andtemperature T r . The reservoir is sufficiently large so that the nozzle inletvelocity is negligible. Since the fluid velocity in the reservoir is zero andthe flow through the nozzle is approximated as isentropic, the stagnationpressure and stagnation temperature of the fluid at any cross sectionthrough the nozzle are equal to the reservoir pressure and temperature,respectively.