The mach number immediately upstream from the shock

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The Mach number immediately upstream from the shock (state 3 ) is less than M 1 of Fig. 13.17 b ; the corresponding temperature, T 3 , is higher than T 1 . Since the shock strength is reduced, the entropy rise across the shock is less, ( s 4 2 s 3 ) , ( s 2 2 s 1 ). The subsonic diffusion following the shock results in a lower Mach number and higher temperature at the duct entrance. Thus M 5 , M 2 and T 5 . T 2 . When the heat addition rate is increased enough to drive the shock to the nozzle throat, a further increase in heat addition will result in a decrease in mass flow rate. The Mach number at the channel inlet is reduced, M 7 , M 5 , and the channel flow shifts to another new Rayleigh line, as shown in Fig. 13.17 d . Thus for specified mass flow rate, there is a maximum rate of heat addition for supersonic flow throughout. For higher rates of heat addition, a shock occurs in the nozzle and flow is subsonic in the constant-area channel, but the exit flow remains sonic. If the shock position is specified, the heat addition along the Rayleigh line can be calculated directly. If the heat addition is specified but the shock position or mass flow rate are unknown, iteration is required to obtain a solution. Additional consideration of flow with shock waves is given in [10]. 13.5 Flow in a Constant-Area Duct with Friction (continued) Isothermal Flow Gas 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 pressure changes 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 is much more appropriate. For isothermal flow with friction (as opposed to the adiabatic flow with friction we previously discussed), the heat transfer δ Q/dm is not zero. On the other hand, we have the 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-dimensional flow that is affected by area change, friction, heat transfer, and normal shocks, ρ 1 V 1 A 1 5 ρ 2 V 2 A 2 5 ρ VA 5 ³ m 5 constant ð 13 : 1a Þ R x 1 p 1 A 1 2 p 2 A 2 5 ³ mV 2 2 ³ mV 1 ð 13 : 1b Þ 13.5 Flow in a Constant-Area Duct with Friction w-29
δ Q dm 1 h 1 1 V 2 1 2 5 h 2 1 V 2 2 2 ð 13 : 1c Þ ³ m ð s 2 2 s 1 Þ $ Z CS 1 T _ Q A ! dA ð 13 : 1d Þ p 5 ρ RT ð 13 : 1e Þ Δ h 5 h 2 2 h 1 5 c p Δ T 5 c p ð T 2 2 T 1 Þ ð 13 : 1f Þ Δ s 5 s 2 2 s 1 5 c p ln T 2 T 1 2 R ln p 2 p 1 ð 13 : 1g Þ We can simplify these equations by setting Δ T 5 0, so T 1 5 T 2 , and A 1 5 A 2 5 A . In addition we recall from Section 13.1 that the combination, h 1 V 2 /2 is the stagnation enthalpy , h 0 . Using these, our final set of equations (renumbered for convenience) is ρ 1 V 1 5 ρ 2 V 2 5 ρ V 5 G 5 ³ m A 5 constant ð 13 : 35a Þ R x 1 p 1 A 2 p 2 A 5 ³ mV 2 2 ³ mV 1 ð 13 : 35b Þ q 5 δ Q dm 5 h 0 2 2 h 0 1 5 V 2 2 2 V 2 1 2 ð 13 : 35c Þ ³ m ð s 2 2 s 1 Þ $ Z CS 1 T _ Q A !

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