Which of the following statements regarding the efficiencies of air-breathing aircraft propulsion systems are **CORRECT**?

(A) The propulsive efficiency $\eta_p$ of a turbojet increases as the jet exhaust velocity $V_j$ approaches the flight speed $V_0$

(B) The thermal efficiency $\eta_{th}$ measures the conversion of fuel chemical energy into net mechanical kinetic energy of the propulsive jet

(C) High bypass ratio turbofan engines achieve higher propulsive efficiency than turbojets at subsonic flight speeds by accelerating a larger mass flow of air through a smaller velocity increment

(D) For a fixed flight speed, maximum engine thrust and maximum propulsive efficiency occur at the same exhaust velocity $V_j$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles For an air-breathing jet engine: 1. **Propulsive Efficiency:** $$\eta_p = \frac{\text{Thrust Power}}{\Delta \text{Kinetic Energy Rate}} = \frac{\dot{m} (V_j - V_0) V_0}{\frac{1}{2} \dot{m} (V_j^2 - V_0^2)} = \frac{2 V_0}{V_j + V_0} = \frac{2}{1 + \frac{V_j}{V_0}}$$ As $V_j \to V_0$, $\eta_p \to 100\%$. 2. **High Bypass Turbofans:** Thrust is $F = \dot{m}(V_j - V_0)$. To maintain high thrust while minimizing wasted kinetic energy in the exhaust $\frac{1}{2}\dot{m}(V_j - V_0)^2$, high bypass engines increase $\dot{m}$ and reduce $(V_j - V_0)$, resulting in dramatic increases in $\eta_p$ at subsonic flight Mach numbers. 3. **Thrust vs. Efficiency Conflict:** Thrust increases as $(V_j - V_0)$ increases, whereas propulsive efficiency *decreases* as $(V_j - V_0)$ increases. Hence, maximum thrust and maximum propulsive efficiency represent contradictory extremes, never occurring at the same $V_j$. ### 2. Statement Evaluation - **Statement A:** **CORRECT.** As $V_j / V_0 \to 1$, $\eta_p \to 1.0$. - **Statement B:** **CORRECT.** $\eta_{th} = \Delta KE / (\dot{m}_f Q_R)$. - **Statement C:** **CORRECT.** This is the core operational principle of modern high-bypass commercial turbofans. - **Statement D:** **INCORRECT.** Maximum thrust requires high $V_j$, which severely degrades propulsive efficiency. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Propulsive efficiency and thrust are inversely coupled with respect to exhaust jet velocity $V_j$. - **Standard Reference:** Mattingly, J.D., *Elements of Propulsion*, Chapter 1.

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