Question 4.1: A single-spool turbojet engine (without an afterburner) is r......

The successive elements will be analyzed as follows:

1. Diffuser: For a ground run of the engines (M = 0) and an ideal diffusion process, we obtain the following from Equations 4.1 and 4.2

P_{02}=P_{01}=P_{0\text{a}}=P_{\text{a}}\left(1+\frac{\gamma -1}{2}M^2 \right) ^{\gamma/(\gamma-1)} \quad \quad \quad (4.1) \\ T_{02}=T_{01}=T{_{0\text{a}}}=T_{\text{a}}\left(1+\frac{\gamma -1}{2}M^2 \right) \quad \quad \quad (4.2) \\ T_{\text{a}}=T_{0\text{a}}=T_{02}=T_{7} \\ P_{02}=P_{0\text{a}}=P_{\text{a}}

2. Compressor: For an ideal compression process,

\frac{T_{03}}{T_{02}} = \pi_{\text{c}}^{(\gamma_{\text{c}}-1/\gamma_{\text{c}})} \quad \quad \quad (\text{a})

where \pi_{\text{c}}=P_{03}/P_{02}.

3. Combustion chamber: For no pressure loss in the combustion chamber and negligible fuel-to-air ratio,

P_{04}=P_{03} \\ f\ll 1

4. Turbine: The pressure and temperature ratios at the turbine are expressed as follows:

\frac{T_{04}}{T_{05}} =\pi_{\text{t}}^{(\gamma_{\text{t}}-1)/\gamma_{\text{t}}}

where \pi_{\text{t}}=P_{04}/P_{05}.

Assuming that the work developed by the turbine is equal to the work consumed by the compressor, then

W_{\text{t}}=W_{\text{c}}

For negligible fuel-to-air ratio

C_{\text{P}_{\text{t}}}(T_{04}-T_{05})=C_{\text{P}_{\text{c}}}(T_{03}-T_{02}) \quad \quad \quad \text{(b)} \\ C_{\text{P}_{\text{t}}} T_{04}\left(1-\frac{1}{\pi_{\text{t}}^{\gamma_{\text{t}-1}/\gamma_{\text{t}}}} \right) =C_{\text{P}_{\text{c}}}T_{02}\left(\pi_{\text{c}}^{(\gamma_{\text{c}}-1)/\gamma_{\text{c}}}-1\right)

With T_{02}=T_{\text{a}}

\frac{1}{\pi_{\text{t}}^{(\gamma_{\text{t}-1}/\gamma_{\text{t}})}} =1-\frac{C_{\text{P}_{\text{c}}}}{C_{\text{P}_{\text{h}}}} \frac{T_{\text{a}}}{T_{04}} \left[\pi_{\text{c}}^{(\gamma_{\text{c}-1}/\gamma_{\text{c}})}-1\right] \\ \pi_{\text{t}}=\frac{1}{\left[1-\left(C_{\text{P}_{\text{c}}}/C_{\text{P}_{\text{h}}}\right)(T_{\text{a}}/T_{04})\left(\pi_{\text{c}}^{(\gamma_{\text{c}-1})/\gamma_{\text{c}} }-1\right) \right]^{(\gamma_{\text{t}}/\gamma_{\text{t}}-1)} } \quad \quad \quad (\text{c})

5. Nozzle: For an unchoked nozzle and ideal expansion, the jet speed is

V_{\text{j}}=\sqrt{2C_{\text{P}_{\text{c}}}T_{06}\left[1-\left(\frac{P_7}{P_{06}} \right)^{(\gamma_{\text{h}}-1)/\gamma_{\text{h}}} \right] }

From (a) and (b), then

T_{05}=T_{04}-\frac{C_{\text{P}_{\text{c}}}}{C_{\text{P}_{\text{h}}}} T_{\text{a}}\left[\pi_{\text{c}}^{(\gamma_{\text{c}}-1)/\gamma_{\text{c}}}-1\right] \equiv T_{06}

Since

\frac{P_{7}}{P_{06}} =\frac{P_7}{P_{02}} \frac{P_{02}}{P_{03}}\frac{P_{03}}{P_{04}} \frac{P_{04}}{P_{05}} \frac{P_{05}}{P_{06}}

Then with P_6=p_{\text{a}}=P_{02},P_{03}=P_{04} \text{ and }P_{05}=P_{06}

\therefore \frac{P_7}{P_{06}} =\frac{\pi_{\text{t}}}{\pi_{\text{c}}} \\ V_{\text{j}}=\sqrt{ C_{\text{P}_{\text{h}}}\left[T_{04}-\frac{C_{\text{P}_{\text{c}}}}{C_{\text{P}_{\text{h}}}} T_{\text{a}}\left(\pi_{\text{c}}^{(\gamma_{\text{c}}-1)/\gamma_{\text{c}}}-1\right) \right] \left\{1-\left(\frac{\pi_{\text{t}}}{\pi_{\text{c}}} \right)^{(\gamma_{\text{h}}-1)/\gamma_{\text{h}}} \right\} }\quad \quad \quad \text{(d)}