GATE AE 2017 • Q50 NAT • 2 Marks Propulsion

Propulsion

Compressors and Turbines

Subtopic: Centrifugal Compressors

Question Statement:

A centrifugal compressor requires 1800 kW of power to compress 10 kg/s of air entering at 300 K stagnation temperature. Taking specific heat $C_p = 1000$ J/(kg K), the stagnation temperature (in K) of air leaving the compressor is _________ K.

Official Answer Key:

480.0

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2017 Q.50)** **1. Intuitive Physical Setup & Given Parameters**: - Power input: $W_{comp} = 1800\text{ kW} = 1.8 \times 10^6\text{ W}$ - Mass flow rate: $\dot{m} = 10\text{ kg/s}$ - Inlet stagnation temperature: $T_{01} = 300\text{ K}$ - Specific heat: $C_p = 1000\text{ J/(kg K)}$ **2. Step-by-Step Thermodynamic Derivation**: 1. **Steady Flow Energy Equation across Compressor**: W_{comp} = \dot{m} C_p (T_{02} - T_{01}) 2. **Solving for Stagnation Temperature Rise ($\Delta T_0$)**: \Delta T_0 = T_{02} - T_{01} = \frac{W_{comp}}{\dot{m} C_p} = \frac{1800 \times 10^3\text{ W}}{10\text{ kg/s} \times 1000\text{ J/(kg K)}} = \frac{1800000}{10000} = 180\text{ K} 3. **Exit Stagnation Temperature ($T_{02}$)**: T_{02} = T_{01} + \Delta T_0 = 300\text{ K} + 180\text{ K} = 480\text{ K} The exit stagnation temperature is exactly **480 K**. **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Analytical Result**: **480.0 K** (Acceptable Range: [478, 482]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **480.0** - **Key Formula**: $T_{02} = T_{01} + \frac{\dot{W}}{\dot{m} C_p}$.
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