GATE AE 2021 • Q17 MCQ • 1 Mark Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Compressible Flow & Shocks

Subtopic: 1D Isentropic Flow & Area-Mach Relations

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:

  • (A) Decreases monotonically with increasing BPR
  • (B) Increases monotonically with increasing BPR
  • (C) Remains constant with BPR
  • (D) First decreases and then increases with BPR

Official Answer Key:

B

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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