In an ideal turbofan engine shown in the figure below, the compressor is driven by the high pressure turbine, and the fan is driven by the low pressure turbine. The stations 0, 2, 3, 4, 4.5, and 5 refer to free-stream, compressor inlet, compressor outlet, combustor exit, high pressure turbine exit, and low pressure turbine exit, respectively, and the subscript 't' refers to the total condition. Also, \(\tau_r = T_{t0}/T_0\), \(\tau_c = T_{t3}/T_{t2}\) and \(\tau_\lambda = T_{t4}/T_0\). The total temperature ratio of the high pressure turbine \((T_{t4.5}/T_{t4})\) is given by ______.
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The HP turbine only drives the compressor, so equate the compressor's temperature rise to the HP turbine's temperature drop.
Step 1: Non-dimensionalise every station temperature by the free-stream static temperature $T_0$.
Define $\theta_i = T_{ti}/T_0$ for each station. From the given definitions:
$\theta_2 = T_{t2}/T_0 = \tau_r$ (inlet is ideal, so $T_{t2}=T_{t0}$)
$\theta_3 = T_{t3}/T_0 = (T_{t3}/T_{t2})(T_{t2}/T_0) = \tau_c\tau_r$
$\theta_4 = T_{t4}/T_0 = \tau_\lambda$
Step 2: Write the shaft power balance directly in these non-dimensional temperatures.
The compressor and HP turbine sit on the same shaft, so the temperature rise across the compressor equals the temperature drop across the turbine (same $\dot m c_p$ throughout an ideal single-stream core):
\[
\theta_3 - \theta_2 = \theta_4 - \theta_{4.5}
\]
Step 3: Solve for the unknown station value $\theta_{4.5}$.
\[
\theta_{4.5} = \theta_4 - (\theta_3-\theta_2) = \tau_\lambda - (\tau_c\tau_r - \tau_r) = \tau_\lambda - \tau_r(\tau_c-1)
\]
Step 4: Convert back to the requested ratio $T_{t4.5}/T_{t4}$.
Since $T_{t4.5}/T_{t4} = \theta_{4.5}/\theta_4$,
\[
\frac{T_{t4.5}}{T_{t4}} = \frac{\tau_\lambda - \tau_r(\tau_c-1)}{\tau_\lambda} = 1 - \frac{\tau_r}{\tau_\lambda}(\tau_c-1)
\]
This is the same result reached by working entirely with the dimensionless station temperatures instead of carrying $T_0$ through every line, confirming option (A).
\[
\boxed{1-\dfrac{\tau_r}{\tau_\lambda}(\tau_c-1)}
\]
