The Mach number immediately upstream from the shock (state3) is less thanM1of Fig. 13.17b; the corresponding temperature,T3, is higher thanT1. Since the shockstrength is reduced, the entropy rise across the shock is less, (s42s3),(s22s1). Thesubsonic diffusion following the shock results in a lower Mach number and highertemperature at the duct entrance. ThusM5,M2andT5.T2.When the heat addition rate is increased enough to drive the shock to the nozzlethroat, a further increase in heat addition will result in a decrease in mass flow rate.The Mach number at the channel inlet is reduced,M7,M5, and the channel flowshifts to another new Rayleigh line, as shown in Fig. 13.17d.Thus for specified mass flow rate, there is a maximum rate of heat addition forsupersonic flow throughout. For higher rates of heat addition, a shock occurs in thenozzle and flow is subsonic in the constant-area channel, but the exit flow remainssonic. If the shock position is specified, the heat addition along the Rayleigh line canbe calculated directly. If the heat addition is specified but the shock position or massflow rate are unknown, iteration is required to obtain a solution.Additional consideration of flow with shock waves is given in .13.5Flow in a Constant-Area Ductwith Friction (continued)Isothermal FlowGas flow in long constant-area ducts, such as natural gas pipelines, is essentially iso-thermal. Mach numbers in such flows are generally low, but significant pressurechanges can occur as a result of frictional effects acting over long duct lengths. Hence,such flows cannot be treated as incompressible. The assumption of isothermal flow ismuch more appropriate.For isothermal flow with friction (as opposed to the adiabatic flow with friction wepreviously discussed), the heat transferδQ/dmis not zero. On the other hand, we havethe simplification that the temperature is constant everywhere. As for adiabatic flow,we can start with our set of basic equations (Eqs. 13.1), describing one-dimensionalflow that is affected by area change, friction, heat transfer, and normal shocks,ρ1V1A15ρ2V2A25ρVA5³m5constantð13:1aÞRx1p1A12p2A25³mV22³mV1ð13:1bÞ13.5Flow in a Constant-Area Duct with Frictionw-29
δQdm1h11V2125h21V222ð13:1cÞ³mðs22s1Þ$ZCS1T_QA!dAð13:1dÞp5ρRTð13:1eÞΔh5h22h15cpΔT5cpðT22T1Þð13:1fÞΔs5s22s15cplnT2T12Rlnp2p1ð13:1gÞWe can simplify these equations by settingΔT50, soT15T2, andA15A25A. Inaddition we recall from Section 13.1 that the combination,h1V2/2 is thestagnationenthalpy,h0. Using these, our final set of equations (renumbered for convenience) isρ1V15ρ2V25ρV5G5³mA5constantð13:35aÞRx1p1A2p2A5³mV22³mV1ð13:35bÞq5δQdm5h022h015V222V212ð13:35cÞ³mðs22s1Þ$ZCS1T_QA!