Subtopic: Dutch Roll, Spiral & Roll Subsidence Modes
Question Statement:
A perfect gas flows through a frictionless constant-area duct with heat addition.
The inlet conditions are as follows: pressure 100 kPa, density 1 kg/m3, and
velocity 100 m/s. At a particular downstream location, the gas velocity is 200 m/s.
The static pressure at the downstream location is ______ kPa (answer in integer).
Official Answer Key:
90.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.50)**
**1. Intuitive Physical Setup & Given Parameters**:
Rayleigh flow (frictionless constant-area duct with heat addition):
- Inlet conditions (Station 1):
- Pressure: $p_1 = 100\text{ kPa}$
- Density: $\rho_1 = 1\text{ kg/m}^3$
- Velocity: $V_1 = 100\text{ m/s}$
- Downstream location (Station 2):
- Velocity: $V_2 = 200\text{ m/s}$
We need to determine the downstream static pressure $p_2$ in $\text{kPa}$ (integer).
**2. Step-by-Step 1D Gas Dynamic Derivation**:
1. **Conservation of Mass in Constant-Area Duct**:
$\dot{m} = \rho_1 A V_1 = \rho_2 A V_2 \implies \rho_1 V_1 = \rho_2 V_2$
Mass flux:
$G = \rho_1 V_1 = 1\text{ kg/m}^3 \times 100\text{ m/s} = 100\text{ kg/m}^2\cdot\text{s}$
2. **Conservation of Linear Momentum (Rayleigh Line)**:
In a frictionless constant-area duct with heat transfer, the momentum equation states that impulse function is strictly conserved:
$p_1 + \rho_1 V_1^2 = p_2 + \rho_2 V_2^2$
3. **Substitute Mass Flux $G = \rho V$**:
$\rho_1 V_1^2 = G V_1 = 100 \times 100 = 10,000\text{ Pa} = 10\text{ kPa}$
$\rho_2 V_2^2 = G V_2 = 100 \times 200 = 20,000\text{ Pa} = 20\text{ kPa}$
4. **Solve for $p_2$**:
$p_1 + \rho_1 V_1^2 = p_2 + \rho_2 V_2^2$
$100\text{ kPa} + 10\text{ kPa} = p_2 + 20\text{ kPa}$
$110 = p_2 + 20 \implies p_2 = 110 - 20 = 90\text{ kPa}$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $p_2 = 90\text{ kPa}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{90}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **90.0** (Acceptable Range: **90 to 90**)
- **Exam Takeaway**: Along any Rayleigh line, the total momentum flux $p + \rho V^2$ is conserved. Heat addition accelerates subsonic flow and reduces static pressure: $p_2 = p_1 - \rho_1 V_1(V_2 - V_1)$.
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