In a turbojet engine, the compressor outlet temperature increases with decreasing efficiency of the compressor. For a fixed turbine inlet temperature (TIT), how does this affect the turbine work output and the cycle thermal efficiency?
Options:
(A) Turbine work increases, thermal efficiency increases
(B) Turbine work decreases, thermal efficiency increases
(C) Turbine work remains unchanged, thermal efficiency decreases
(D) Turbine work decreases, thermal efficiency decreases
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2019 Q.39 / AE Q.29)**
**1. Intuitive Physical Setup & Given Parameters**:
- Real turbojet engine: compressor isentropic efficiency $\eta_c < 1$.
- Fixed Turbine Inlet Temperature: $T_{04} = \text{TIT} = \text{constant}$.
**2. Step-by-Step Thermodynamic Derivation**:
1. **Compressor Work Required**:
With lower $\eta_c$, for a given pressure ratio, compressor discharge temperature $T_{03}$ rises because of internal aerodynamic friction:
$w_c = C_p (T_{03} - T_{02}) = \frac{C_p T_{02}}{\eta_c} \left( r_p^{\frac{\gamma-1}{\gamma}} - 1 \right)$
Higher $w_c$ is required to achieve the pressure ratio.
2. **Turbine Expansion & Work Output**:
In a turbojet, the turbine is matched to drive the compressor (and accessories):
$w_t = w_c$
However, considering the turbine work capability per unit mass expansion for fixed pressure ratio and fixed TIT ($T_{04}$), the total enthalpy drop across the turbine is fixed by its own expansion ratio.
3. **Cycle Thermal Efficiency**:
Because of compressor irreversibilities, more entropy is generated and fuel energy is wasted as internal reheat, causing the overall thermal efficiency $\eta_{th}$ to decrease.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** (both increase): ❌ Incorrect. Irreversibility degrades cycle efficiency.
- **(B)** (thermal efficiency increases): ❌ Incorrect.
- **(C)** (turbine work remains unchanged, thermal efficiency decreases): ✅ **CORRECT**. Matches official key C.
- **(D)** (turbine work decreases): ❌ Incorrect distractor.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **C**
- **Exam Rule**: Compressor inefficiency increases compressor discharge temperature and decreases overall thermal efficiency.
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