Question Statement:
How does the propulsive efficiency of a turbofan engine, operating at a given cruise altitude and Mach number, vary with bypass ratio (BPR)?
Options:
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(A) Decreases monotonically with increasing BPR
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(B) Increases monotonically with increasing BPR
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(C) Remains constant with BPR
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(D) First decreases and then increases with BPR
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.17 / AE Q.7)**
**1. Intuitive Physical Setup & Given Parameters**:
- Turbofan engine operating at constant cruise altitude and flight speed $V_0$.
- Bypass ratio $\text{BPR} = \frac{\dot{m}_{fan}}{\dot{m}_{core}}$.
**2. Step-by-Step Propulsion Derivation**:
1. **Froude Propulsive Efficiency Formulation**:
For any air-breathing jet engine with effective average exhaust jet velocity $V_j$:
$\eta_p = \frac{2 V_0}{V_0 + V_j} = \frac{2}{1 + \frac{V_j}{V_0}}$
2. **Effect of Increasing Bypass Ratio (BPR)**:
- For a fixed thrust requirement $F = \dot{m}_{total}(V_j - V_0)$, increasing BPR allows a larger total mass flow rate $\dot{m}_{total}$ of air to be accelerated by a smaller velocity increment $\Delta V = (V_j - V_0)$.
- Consequently, the average jet exhaust velocity $V_j$ decreases closer to the flight velocity $V_0$.
- As $V_j / V_0 \to 1$, the denominator $(1 + V_j/V_0)$ decreases towards 2, so the propulsive efficiency $\eta_p$ increases towards 100%.
- Therefore, propulsive efficiency increases monotonically with increasing BPR.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** (Decreases): ❌ Incorrect.
- **(B)** (Increases monotonically with increasing BPR): ✅ **CORRECT**. Directly validated by Froude propulsive efficiency $\eta_p = \frac{2}{1 + V_j/V_0}$.
- **(C)** (Remains constant): ❌ Incorrect.
- **(D)** (Non-monotonic): ❌ Incorrect.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: Higher Bypass Ratio (BPR) $\implies$ Lower exhaust velocity $V_j$ $\implies$ Higher propulsive efficiency $\eta_p$.
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