The figure below shows a compressor stage with station numbers 1, 2, and 3 as
indicated. If p0i, T0i, and Ci refer to the average values of total pressure, total
temperature, and absolute flow speed, respectively, at the ith station, select the
CORRECT option considering losses.
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.42)**
**1. Intuitive Physical Setup & Given Parameters**:
The problem depicts a single stage of an axial compressor:
- Station 1: Rotor inlet
- Station 2: Rotor exit / Stator inlet
- Station 3: Stator exit
We are given total pressure $p_{0i}$, total temperature $T_{0i}$, and absolute flow speed $C_i$ at each station, and asked to identify the correct inequalities considering real losses.
**2. Step-by-Step Turbomachinery Aerodynamic Derivation**:
1. **Total Pressure ($p_{0i}$) Behavior**:
- In the **rotor** ($1 \to 2$), mechanical work is input into the air stream via rotating blades. The total pressure rises significantly:
$p_{02} > p_{01}$
- In the **stator** ($2 \to 3$), the blades are stationary (no work transfer, $\dot{W} = 0$). Fluid flows through diffusing passages where boundary layer friction and turbulence cause unavoidable stagnation pressure loss:
$p_{03} < p_{02} \implies p_{02} > p_{03}$
2. **Total Temperature ($T_{0i}$) Behavior**:
- In the **rotor** ($1 \to 2$), work addition increases the stagnation enthalpy and total temperature:
$T_{02} > T_{01}$
- In the **stator** ($2 \to 3$), since no shaft work is added or extracted and heat transfer through the casing is negligible (adiabatic stationary vane):
$h_{03} = h_{02} \implies T_{02} \approx T_{03}$
3. **Absolute Velocity ($C_i$) Behavior**:
- The rotor accelerates absolute flow in the tangential direction while diffusing relative velocity, resulting in a net increase in absolute velocity:
$C_2 > C_1 \implies C_1 < C_2$
- The stator converts dynamic pressure (kinetic energy) into static pressure by decelerating the absolute flow:
$C_3 < C_2 \implies C_2 > C_3$
4. Combining all three criteria:
$p_{01} < p_{02}, \quad p_{02} > p_{03}; \quad T_{01} < T_{02}, \quad T_{02} \approx T_{03}; \quad C_1 < C_2, \quad C_2 > C_3$
**3. Comprehensive Options & Distractor Analysis**:
- **(A)**: ✅ **CORRECT**. Accurately satisfies rotor work input ($p_{01}<p_{02}, T_{01}<T_{02}$), stator friction loss ($p_{03}<p_{02}$), stator adiabatic conservation ($T_{02}\approx T_{03}$), and stator diffusion ($C_3<C_2$).
- **(B)**: ❌ **INCORRECT**. Claims $p_{02} = p_{03}$, which ignores stator frictional losses.
- **(C)**: ❌ **INCORRECT**. Claims $T_{02} < T_{03}$, implying work addition in the stationary stator.
- **(D)**: ❌ **INCORRECT**. Claims $p_{02} < p_{03}$, which violates second law of thermodynamics in a stationary duct.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Rotors add shaft work ($p_0 \uparrow, T_0 \uparrow$); stators only diffuse flow without work ($T_0 = \text{const}$, $p_0 \downarrow$ due to entropy generation).
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