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: State the power balance on the high-pressure spool.
The high pressure turbine (between stations 4 and 4.5) only exists to drive the compressor (between stations 2 and 3); no other device sits on that shaft. In an ideal engine with no mechanical loss, the power the turbine delivers must exactly equal the power the compressor absorbs:
\[ \dot{m}c_p(T_{t4}-T_{t4.5}) = \dot{m}c_p(T_{t3}-T_{t2}) \]
The mass flow rate and \(c_p\) cancel from both sides (ideal engine, same working fluid), leaving
\[ T_{t4}-T_{t4.5} = T_{t3}-T_{t2} \]
Step 2: Write the compressor and combustor-exit temperatures using the given ratios.
By definition \(\tau_r = T_{t0}/T_0\). Across an ideal inlet the total temperature does not change, so \(T_{t2}=T_{t0}=\tau_r T_0\).
By definition \(\tau_c = T_{t3}/T_{t2}\), so \(T_{t3} = \tau_c T_{t2} = \tau_c \tau_r T_0\).
By definition \(\tau_\lambda = T_{t4}/T_0\), so \(T_{t4} = \tau_\lambda T_0\).
Step 3: Substitute into the power balance from Step 1.
\[ T_{t4}-T_{t4.5} = T_{t3}-T_{t2} = \tau_c\tau_r T_0 - \tau_r T_0 = \tau_r T_0(\tau_c - 1) \]
So
\[ T_{t4.5} = T_{t4} - \tau_r T_0(\tau_c-1) = \tau_\lambda T_0 - \tau_r T_0(\tau_c-1) \]
Step 4: Divide by \(T_{t4}=\tau_\lambda T_0\) to get the required ratio.
\[ \frac{T_{t4.5}}{T_{t4}} = 1 - \frac{\tau_r T_0(\tau_c-1)}{\tau_\lambda T_0} = 1 - \frac{\tau_r}{\tau_\lambda}(\tau_c-1) \]
Step 5: Check the other options.
Option (D) has the right bracket \((\tau_c-1)\) but the wrong sign in front, which would mean the turbine adds temperature instead of extracting it, so it is wrong. Option (C) uses \((\tau_c+1)\) instead of \((\tau_c-1)\), which drops the correction for the compressor's own inlet temperature ratio, so it is wrong. Option (B) combines both mistakes together.
Final Answer:
\[ \boxed{\dfrac{T_{t4.5}}{T_{t4}} = 1 - \dfrac{\tau_r}{\tau_\lambda}(\tau_c-1)} \]
