Subtopic: Higher Order Linear ODEs with Constant Coefficients
Question Statement:
In a closed-circuit supersonic wind tunnel, the c onvergent-divergent (C-D) nozzle and test section
are followed by a C-D diffuser to swallow the starting shock. Here, we should have the
Options:
(A) diffuser throat larger than the nozzle throat and the shock located just at the diffuser throat.
(B) diffuser throat larger than the nozzle throat and the shock located downstream of the diffuser
throat.
(C) diffuser throat of the same size as the nozzle throat and the shock located just at the diffuser
throat.
(D) diffuser throat of the same size as the nozzle throat and the shock located downstream of the
diffuser throat.
2012 AEROSPACE ENGINEERING - AE
AE 6/12
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.38)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (2 Mark, MCQ)
- **Topic**: Differential Equations (Higher Order Linear ODEs with Constant Coefficients)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> In a closed-circuit supersonic wind tunnel, the c onvergent-divergent (C-D) nozzle and test section are followed by a C-D diffuser to swallow the starting shock. Here, we should have the
**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)** *"diffuser throat larger than the nozzle throat and the shock located just at the diffuser throat."*: ❌ **INCORRECT**.
- **(B)** *"diffuser throat larger than the nozzle throat and the shock located downstream of the diffuser
throat."*: ✅ **CORRECT**.
- **(C)** *"diffuser throat of the same size as the nozzle throat and the shock located just at the diffuser
throat."*: ❌ **INCORRECT**.
- **(D)** *"diffuser throat of the same size as the nozzle throat and the shock located downstream of the
diffuser throat.
2012 AEROSPACE ENGINEERING - AE
AE 6/12"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.
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