Q. 4.9

A single-spool afterburning turbojet engine is powering a fighter airplane flying at Mach number $M_{\text{a}}$ = 2.0 at an altitude of 16,200 m where the temperature is 216.6°K and the pressure is 10.01 kPa. The inlet is of the axisymmetric type and is fitted with a spike having a deflection angle 12°. The air mass flow rate is 15 kg/s. The compressor has a pressure ratio of 5 and isentropic efficiency of 0.85. The pressure loss in the combustion chamber is 6% and the heating value of fuel is 45,000 kJ/kg. The burner efficiency is $\eta_{\text{b}}$ = 0.96. The TIT is 1200 K and its isentropic efficiency is 0.9. The maximum temperature in the afterburner is 2000 K. The pressure drop in the afterburner is 3% and the afterburner efficiency is 0.9. The nozzle efficiency is $\eta_{\text{n}}$ = 0.96. Calculate

1. The stagnation pressure ratio of the diffuser ($r_{\text{d}}$) and its isentropic efficiency
2. The thrust force

Take $C_{\text{pt}}=1.157, C_{\text{pc}}=1.005 \text{ kJ/kg.K}$.

Verified Solution

1. Intake: The intake here is identical to that in Example 3, Chapter 3. Two shock waves are developed in the intake; the first is an oblique one, while the second is normal. The important results from this example are
The total conditions for air upstream of the engine are $T_{0\text{a}}=390 \text{ K and }P_{0\text{a}}=7.824 P_{\text{a}}$ .
The flight speed is $V=M \sqrt{\gamma RT_{\text{a}}}=590 \text{ m/s}$.
The stagnation pressure ratio in the intake is $r_{\text{d}}=\frac{P_{02}}{P_{0\text{a}}} =0.884$.
From Equation 4.61 , the diffuser efficiency is

$\eta_{\text{d}}=\frac{(1+((\gamma-1)/2)M^2)(r_{\text{d}})^{(\gamma-1)/\gamma}-1}{((\gamma-1)/2)M^2} \quad \quad \quad (4.61) \\ \eta_{\text{d}}=\frac{(1+0.2 \times 4)(0.884)^{0.286}-1}{0.2 \times 4} =0.922$

The stagnation temperature of air leaving the diffuser is $T_{02}=T_{0\text{a}}=390 \text{ K}$.
The stagnation pressure of air leaving the intake is

$P_{02}=0.884 \ P_{0\text{a}}=0.884 \times 7.824 \ P_{\text{a}}=0.884 \times 7.824 \times 10.01=69.23 \text{ kPa}$

2. Compressor:

$P_{03}=\pi_{\text{c}}P_{02}=5 \times 69.23 =346.15 \text{ kPa} \\ T_{03}=T_{02}\left[1+\frac{\pi_{\text{C}}^{(\gamma_{\text{C}}-1)/\gamma_{\text{C}}}-1}{\eta_{\text{C}}} \right] =658.2 \text{ K}$

3. Combustion chamber:

$\begin{matrix} P_{04}&=& (1- \Delta P_{\text{cc}}),P_{03}&=&0.94 P_{03}=325.38 \text{ kPa} \\ T_{04}&=& 1200 \text{ K, then} \\ f&=&\frac{Cp_{\text{t}}T_{04}-Cp_{\text{c}}T_{03}}{\eta_{\text{b}}Q_{\text{R}}-Cp_{\text{t}}T_{04}}&=&0.0174 \end{matrix}$

4. Turbine: Energy balance for compressor and turbine

$Cp_{\text{t}}(1+f)(T_{04}-T_{05})=Cp_{\text{c}}(T_{03}-T_{02}) \\ T_{05}=T_{04}-\frac{Cp_{\text{c}}(T_{03}-T_{02})}{Cp_{\text{t}}(1+f)} =1200-\frac{1.005 \times (658.2-390)}{1.157 \times 1.0174} =970.9 \text{ K} \\ P_{05}=P_{04}\left[1-\frac{T_{04}-T_{05}}{\eta_{\text{t}}T_{04}} \right] ^{(\gamma_{\text{t}}/\gamma_{\text{t}}-1)}=325.38\left[1-\frac{1200-970.9}{0.9 \times 1200} \right] ^4=125.37 \text{ kPa}$

5. Tail pipe: A pressure drop of 3% is encountered in the afterburner; thus

$P_{06 \text{A}}=0.97 \ P_{05}=121.6 \text{ kPa}$

The maximum temperature is now $P_{06 \text{A}}$ = 2000 K

$f_{\text{ab}}=\frac{Cp_{\text{t}}(T_{06 \text{A}}-T_{05})}{\eta_{\text{ab}}Q_{\text{R}}-Cp_{\text{t}}T_{06\text{A}}} =\frac{1.157 \times (2000-970.9)}{0.9 \times 45,000-1.157 \times 2000} =0.0312$

Check nozzle choking with

$\frac{P_{06\text{A}}}{P_{\text{c}}} =\frac{1}{(1-(1/\eta_{\text{n}})(\gamma_{\text{n}}-1)/(\gamma_{\text{n}}+1))^{\gamma_{\text{n}}/(\gamma_{\text{n}-1})}} =1.907 \\ \therefore P_{\text{c}}=63.77 \text{ kPa} \\ \because P_{\text{c }}\gt P_{\text{a}}$

Nozzle is choked also.

$\begin{matrix} P_{7\text{A}}&=&P_{\text{c}}&=&63.77 \text{ kPa} \\ T_{7\text{a}}&=&\frac{T_{06\text{A}}}{(\gamma_{\text{n}}+1)/2} &=&1716.9 \text{ K} \\ V_{7\text{A}}&=&a_{7\text{A}}&=&\sqrt{\gamma_{\text{n}}RT_{7\text{A}}}&=&809.5 \text{ m/s} \end{matrix}$

The nozzle area is calculated from the relation

$\begin{matrix} A_{7\text{A}} &=&\frac{\dot{m}_{\text{a}}(1+f+f_{\text{ab}})}{\rho_{7\text{A}}V_{7\text{A}}} =\frac{RT_{7\text{A}}\dot{m}_{\text{a}}(1+f+f_{\text{ab}})}{P_{7\text{A}}V_{7\text{A}}} \\ &=& \frac{287 \times (1716.9)\times (15)\times (1+0.017+0.0312)}{63.77 \times 10^3 \times (809.5)} \\ &=& 0.15 \text{ m}^2 \end{matrix}$

The thrust force for operative afterburner is

$\begin{matrix} T_{\text{ab}} &=& \dot{m}_{\text{a}}[(1+f+f_{\text{ab}})V_{7\text{A}}-V]+A_{7\text{A}}(P_{7\text{A}}-P_{\text{a}}) \\ &=& 15[1.0486 \times 809.5-590]+0.15 \times 10^3(63.77-10.01)=11,946 \text{ N} \\ &=& 11.946 \text{ kN} \end{matrix}$