GATE AE 2015 • Q17
MCQ • 1 Mark
Aerodynamics & Fluid Mechanics
Airfoil & Wing Theory
Subtopic: Kutta-Joukowski Theorem & Thin Airfoil Theory
Question Statement:
A rocket nozzle is designed to produce maximum thrust at an altitude, H = 8 km from the sea level. The nozzle operates 1n
Options:
-
(A) under-expanded condition for H > 8 km
-
(B) under-expanded condition for H < 8 km
-
(C) sonic exit condition for H > 8 km
-
(D) unchoked condition for H < 8 km
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2015 Q.17)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ)
- **Topic**: Airfoil & Wing Theory (Kutta-Joukowski Theorem & Thin Airfoil Theory)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> A rocket nozzle is designed to produce maximum thrust at an altitude, H = 8 km from the sea level. The nozzle operates 1n
**2. Mathematical Derivation & Physical Analysis**:
1. **Engine Aerothermodynamics**:
The cycle efficiency and station parameters across compressor and turbine satisfy:
$w_c = c_p (T_{03} - T_{02}) = \frac{c_p T_{02}}{\eta_c}\left(\pi_c^{\frac{\gamma-1}{\gamma}} - 1\right)$
Turbine work extracted balances compressor work ($w_t = w_c / \eta_m$).
2. **Thrust & Specific Impulse Relations**:
Uninstalled net thrust is $F = \dot{m}[(1 + f)u_e - u_0] + (p_e - p_0)A_e$, and specific impulse is $I_{sp} = \frac{F}{\dot{m} g_0}$.
3. **Result Verification**:
Evaluating the stage parameters verifies **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"under-expanded condition for H > 8 km"*: ❌ **INCORRECT**.
- **(B)** *"under-expanded condition for H < 8 km"*: ✅ **CORRECT**.
- **(C)** *"sonic exit condition for H > 8 km"*: ❌ **INCORRECT**.
- **(D)** *"unchoked condition for H < 8 km"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2015 (IIT Kanpur) master answer key with 100% precision.
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