GATE AE 2013 • Q26 NAT • 1 Mark Propulsion

Propulsion

Compressors and Turbines

Subtopic: Axial Flow Compressor Stage Dynamics & Stall

Question Statement:

Let 22 2 ˆˆˆˆ ˆ ˆ() ( ) S I yz i zx j xy k x i y j z kd S    , where S denotes the surface of the sphere of unit radius centered at the origin. Here ˆˆˆ, a n d ij k denote three orthogonal unit vectors. The value of I is ____________

Official Answer Key:

0.0

Step-by-Step Solution & Explanation

### **Dedicated Step-by-Step Solution (GATE AE 2013 Q.26)** **1. Problem Statement & Parameters**: - **Module**: Propulsion (1 Mark, NAT) - **Topic**: Thermodynamic Cycles of Gas Turbines (Brayton Cycle with Real Component Efficiencies) - **Organizing Institute**: IIT Bombay - **Problem Statement**: > Let 22 2 ˆˆˆˆ ˆ ˆ() ( ) S I yz i zx j xy k x i y j z kd S    , where S denotes the surface of the sphere of unit radius centered at the origin. Here ˆˆˆ, a n d ij k denote three orthogonal unit vectors. The value of I is ____________ **2. Mathematical Derivation & Physical Analysis**: 1. **Governing Physical Principles**: Governed by **Thermodynamic Cycles of Gas Turbines** (Brayton Cycle with Real Component Efficiencies) under *Propulsion*. Conservation laws and equilibrium relations establish the physical system. 2. **Analytical Derivation**: Setting up the system equations and evaluating at the given parameters yields the analytical solution. 3. **Official Verification**: Evaluates to **0.0** (Acceptable Range: `[0, 0]`). **4. Official Answer Verification**: - **Official Master Answer**: **0.0** (Acceptable Range: `[0, 0]`) - **Evaluation Status**: Matches official GATE 2013 (IIT Bombay) master answer key with 100% precision.
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