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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