Subtopic: Higher Order Linear ODEs with Constant Coefficients
Question Statement:
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, 𝜏r = Tt0 T0⁄ ,
𝜏c = Tt3 Tt2⁄ and 𝜏𝜆 = Tt4 T0.⁄ The total temperature ratio of the high pressure
turbine (Tt4.5 Tt4⁄ ) is given by ______.
Options:
(A) 1 - 𝜏r
𝜏𝜆
(𝜏c - 1)
(B) 1 + 𝜏r
𝜏𝜆
(𝜏c + 1)
(C) 1 - 𝜏r
𝜏𝜆
(𝜏c + 1)
(D) 1 + 𝜏r
𝜏𝜆
(𝜏c - 1)
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.38)**
**1. Intuitive Physical Setup & Given Parameters**:
For an ideal turbofan engine:
- High Pressure Turbine (HPT) drives the compressor (stations 2 to 3).
- Low Pressure Turbine (LPT) drives the fan.
- Station notations: $0$ (freestream), $2$ (compressor inlet), $3$ (compressor exit), $4$ (combustor exit / HPT inlet), $4.5$ (HPT exit / LPT inlet).
- Temperature ratios:
$\tau_r = \frac{T_{t0}}{T_0}, \quad \tau_c = \frac{T_{t3}}{T_{t2}}, \quad \tau_\lambda = \frac{T_{t4}}{T_0}$
We need to find the total temperature ratio across the HPT: $\tau_{\text{HPT}} = \frac{T_{t4.5}}{T_{t4}}$.
**2. Step-by-Step Thermodynamic Derivation**:
1. **Shaft Power Balance for the High-Pressure Spool**:
In an ideal cycle, the power required by the compressor must be provided entirely by the high-pressure turbine:
$\dot{W}_{\text{HPT}} = \dot{W}_{\text{comp}}$
$\dot{m}_c c_p (T_{t4} - T_{t4.5}) = \dot{m}_c c_p (T_{t3} - T_{t2})$
$T_{t4} - T_{t4.5} = T_{t3} - T_{t2}$
2. **Divide by $T_{t4}$**:
$1 - \frac{T_{t4.5}}{T_{t4}} = \frac{T_{t3} - T_{t2}}{T_{t4}} = \frac{T_{t2}}{T_{t4}} \left(\frac{T_{t3}}{T_{t2}} - 1\right) = \frac{T_{t2}}{T_{t4}} (\tau_c - 1)$
3. **Relate $T_{t2}$ and $T_{t4}$ to Given Ratios**:
- Assuming an ideal inlet diffuser, $T_{t2} = T_{t0} = T_0 \tau_r$.
- By definition of $\tau_\lambda$: $T_{t4} = T_0 \tau_\lambda$.
- Therefore:
$\frac{T_{t2}}{T_{t4}} = \frac{T_0 \tau_r}{T_0 \tau_\lambda} = \frac{\tau_r}{\tau_\lambda}$
4. **Substitute into HPT Temperature Ratio**:
$1 - \frac{T_{t4.5}}{T_{t4}} = \frac{\tau_r}{\tau_\lambda} (\tau_c - 1)$
$\frac{T_{t4.5}}{T_{t4}} = 1 - \frac{\tau_r}{\tau_\lambda} (\tau_c - 1)$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 1 - (𝜏r/𝜏𝜆)(𝜏c - 1)**: ✅ **CORRECT**. Directly derived from the high-pressure spool shaft power balance.
- **(B) 1 + (𝜏r/𝜏𝜆)(𝜏c + 1)**: ❌ **INCORRECT**. Sign errors; turbine expansion must decrease total temperature ($T_{t4.5} < T_{t4}$).
- **(C) 1 - (𝜏r/𝜏𝜆)(𝜏c + 1)**: ❌ **INCORRECT**. Incorrect $(\tau_c + 1)$ instead of $(\tau_c - 1)$.
- **(D) 1 + (𝜏r/𝜏𝜆)(𝜏c - 1)**: ❌ **INCORRECT**. Predicts turbine exit temperature higher than inlet.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Spool power balance equates $\Delta T_t$ across turbine and driven component: $\Delta T_{t,\\text{turb}} = \frac{\dot{m}_c}{\dot{m}_t} \Delta T_{t,\\text{comp}}$. Normalizing by $T_0$ yields $\tau_{\text{turb}} = 1 - \frac{\tau_r}{\tau_\lambda}(\tau_c - 1)$.
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