GATE AE 2018 • Q29 MCQ • 1 Mark Propulsion

Propulsion

Compressors and Turbines

Subtopic: Axial Flow Compressor Stage Dynamics & Stall

Question Statement:

The degree of reaction ($R$) of an axial flow compressor stage is defined as the ratio of:

Options:

  • (A) Total enthalpy rise in rotor to total enthalpy rise in stage
  • (B) Static pressure rise across stage to total pressure rise across stage
  • (C) Static enthalpy rise in stator to static enthalpy rise in stage
  • (D) Static enthalpy rise in rotor to static enthalpy rise in stage

Official Answer Key:

D

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2018 Q.29)** **1. Intuitive Physical Setup & Given Parameters**: - An axial compressor stage consists of a rotor followed by a stator. - The degree of reaction $R$ quantifies how much of the stage static enthalpy (or pressure) rise occurs in the moving rotor versus the stationary stator. **2. Step-by-Step Mathematical Derivation**: 1. **Definition of Degree of Reaction ($R$)**: R = \frac{\Delta h_{rotor}}{\Delta h_{stage}} = \frac{h_2 - h_1}{h_3 - h_1} where $h_1$ is rotor inlet static enthalpy, $h_2$ is rotor exit static enthalpy, and $h_3$ is stator exit static enthalpy. 2. For an incompressible or calorically perfect gas with constant axial velocity: R = \frac{T_2 - T_1}{T_3 - T_1} \approx \frac{\Delta P_{rotor}}{\Delta P_{stage}} **3. Comprehensive Options & Distractor Analysis**: - **(A)** (Total enthalpy in rotor / total in stage): ❌ Incorrect. In the stator, no work is done, so total enthalpy rise in stage equals total enthalpy rise in rotor (ratio would always be 1). - **(B)** (Static to total pressure rise): ❌ Incorrect. - **(C)** (Static in stator / static in stage): ❌ Incorrect. That would be $(1 - R)$. - **(D)** (Static enthalpy rise in rotor to stage): ✅ **CORRECT**. Matches the standard thermodynamic definition $R = \Delta h_{rotor} / \Delta h_{stage}$. **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **D** - **Key Formula**: $R = \frac{h_2 - h_1}{h_3 - h_1} = \frac{\text{Static enthalpy rise in rotor}}{\text{Static enthalpy rise in stage}}$.
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