Question 12.14: Rayleigh Flow in a Tubular Combustor A combustion chamber co...
Rayleigh Flow in a Tubular Combustor
A combustion chamber consists of tubular combustors of 15-cm diameter. Compressed air enters the tubes at 550 K, 480 kPa, and 80 m/s (Fig. 12–54). Fuel with a heating value of 42,000 kJ/kg is injected into the air and is burned with an air–fuel mass ratio of 40. Approximating combustion as a heat transfer process to air, determine the temperature, pressure, velocity, and Mach number at the exit of the combustion chamber.
Fuel is burned in a tubular combustion chamber with compressed air. The exit temperature, pressure, velocity, and Mach number are to be determined.
Assumptions 1 The assumptions associated with Rayleigh flow (i.e., steady onedimensional flow of an ideal gas with constant properties through a constant crosssectional area duct with negligible frictional effects) are valid. 2 Combustion is complete, and it is treated as a heat addition process, with no change in the chemical composition of the flow. 3 The increase in mass flow rate due to fuel injection is disregarded.
Properties We take the properties of air to be k = 1.4, c_p = 1.005 kJ/kg·K, and R = 0.287 kJ/kg·K.
Analysis The inlet density and mass flow rate of air are
\rho_1=\frac{P_1}{R T_1}=\frac{480 kPa }{(0.287 kJ / kg \cdot K )(550 K )}=3.041 kg / m ^3
\dot{m}_{\text {air }}=\rho_1 A_1 V_1=\left(3.041 kg / m ^3\right)\left[\pi(0.15 m )^2 / 4\right](80 m / s )=4.299 kg / s
The mass flow rate of fuel and the rate of heat transfer are
\dot{m}_{\text {fuel }}=\frac{\dot{m}_{ air }}{A F}=\frac{4.299 kg / s }{40}=0.1075 kg / s
\dot{Q}=\dot{m}_{\text {fuel }} HV =(0.1075 kg / s )(42,000 kJ / kg )=4514 kW
q=\frac{\dot{Q}}{\dot{m}_{\text {air }}}=\frac{4514 kJ / s }{4.299 kg / s }=1050 kJ / kg
The stagnation temperature and Mach number at the inlet are
T_{01}=T_1+\frac{V_1^2}{2 c_p}=550 K +\frac{(80 m / s )^2}{2(1.005 kJ / kg \cdot K )}\left(\frac{1 kJ / kg }{1000 m ^2 / s ^2}\right)=553.2 K
c_1=\sqrt{k R T_1}=\sqrt{(1.4)(0.287 kJ / kg \cdot K )(550 K )\left(\frac{1000 m ^2 / s ^2}{1 kJ / kg }\right)}=470.1 m / sMa _1=\frac{V_1}{c_1}=\frac{80 m / s }{470.1 m / s }=0.1702
The exit stagnation temperature is, from the energy equation q = c_p (T_{02} − T_{01}),
T_{02}=T_{01}+\frac{q}{c_p}=553.2 K +\frac{1050 kJ / kg }{1.005 kJ / kg \cdot K }=1598 K
The maximum value of stagnation temperature T_0^* occurs at Ma = 1, and its value can be determined from Table A–15 or from Eq. 12–65. At Ma_1 = 0.1702 we read T_0 /T_0^* = 0.1291. Therefore,
\frac{T_0}{T_0{ }^*}=\frac{(k+1) Ma ^2\left[2+(k-1) Ma ^2\right]}{\left(1+k Ma ^2\right)^2} (12.65)
T_0^*=\frac{T_{01}}{0.1291}=\frac{553.2 K }{0.1291}=4284 K
The stagnation temperature ratio at the exit state and the Mach number corresponding to it are, from Table A–15,
\frac{T_{02}}{T_0^*}=\frac{1598 K }{4284 K }=0.3730 \rightarrow Ma _2=0.3142 \cong 0 . 3 1 4
The Rayleigh flow functions corresponding to the inlet and exit Mach numbers are (Table A–15):
Ma _1=0.1702: \quad \frac{T_1}{T^*}=0.1541 \quad \frac{P_1}{P^*}=2.3065 \quad \frac{V_1}{V^*}=0.0668
Ma _2=0.3142: \quad \frac{T_2}{T^*}=0.4389 \quad \frac{P_2}{P^*}=2.1086 \quad \frac{V_2}{V^*}=0.2082
Then the exit temperature, pressure, and velocity are determined to be
\frac{T_2}{T_1}=\frac{T_2 / T^*}{T_1 / T^*}=\frac{0.4389}{0.1541}=2.848 \rightarrow T_2=2.848 T_1=2.848(550 K )=1570 K
\frac{P_2}{P_1}=\frac{P_2 / P^*}{P_1 / P^*}=\frac{2.1086}{2.3065}=0.9142 \rightarrow P_2=0.9142 P_1=0.9142(480 kPa )=439 kPa
\frac{V_2}{V_1}=\frac{V_2 / V^*}{V_1 / V^*}=\frac{0.2082}{0.0668}=3.117 \rightarrow V_2=3.117 V_1=3.117(80 m / s )=249 m / s
Discussion Note that the temperature and velocity increase and pressure decreases during this subsonic Rayleigh flow with heating, as expected. This problem can also be solved using appropriate relations instead of tabulated values, which can likewise be coded for convenient computer solutions.