An aircraft is cruising with a forward speed
aV and the jet exhaust speed relative
to the engine at the exit is
jV . If
2,ajVV = what is the propulsive efficiency?
Options:
(A) 0.50
(B) 1.00
(C) 0.33
(D) 0.67
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.22)**
**1. Intuitive Physical Setup & Given Parameters**:
An aircraft cruises at flight velocity $V_a$ with engine exhaust jet velocity $V_j$ relative to the engine. We are asked to express the propulsive efficiency $\eta_p$.
**2. Step-by-Step Mathematical Derivation**:
1. **Definition of Propulsive Efficiency**:
Propulsive efficiency $\eta_p$ is defined as the ratio of useful thrust power delivered to the aircraft to the total mechanical kinetic power added to the propellant stream:
$\eta_p = \frac{\text{Thrust Power}}{\Delta \text{Kinetic Energy per second}}$
2. **Formulation of Powers**:
- Thrust produced: $F = \dot{m}(V_j - V_a)$ (assuming ideal matched nozzle $p_e = p_a$).
- Useful thrust power: $P_{\text{thrust}} = F \cdot V_a = \dot{m}(V_j - V_a) V_a$.
- Rate of kinetic energy imparted to fluid: $\Delta KE = \frac{1}{2}\dot{m}(V_j^2 - V_a^2) = \frac{1}{2}\dot{m}(V_j - V_a)(V_j + V_a)$.
3. **Compute Efficiency Ratio**:
$\eta_p = \frac{\dot{m}(V_j - V_a) V_a}{\frac{1}{2}\dot{m}(V_j - V_a)(V_j + V_a)} = \frac{2 V_a}{V_j + V_a} = \frac{2}{1 + \frac{V_j}{V_a}}$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 2 / (1 + (V_a/V_j))**: ❌ **INCORRECT**. Inverts the velocity ratio.
- **(B) (V_j - V_a) / (V_j + V_a)**: ❌ **INCORRECT**. Missing factor of 2 and numerator contains relative velocity.
- **(C) 1 - (V_a/V_j)**: ❌ **INCORRECT**. Linear approximation.
- **(D) 2 / (1 + (V_j/V_a))**: ✅ **CORRECT**. Classical Froude propulsive efficiency formula: $\eta_p = \frac{2 V_a}{V_a + V_j}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Propulsive efficiency approaches 100% as $V_j \to V_a$, but thrust $F = \dot{m}(V_j - V_a)$ drops to 0. High bypass ratio turbofans achieve high $\eta_p$ by moving a huge mass flow $\dot{m}$ at a modest velocity increment $(V_j - V_a)$.
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