Subtopic: Turbine Power Output & Steady Flow Energy Equation
Question Statement:
Air, at 450 K stagnation temperature and at a rate of 50 kg/s, enters an axial turbine stage and leaves at 350 K stagnation temperature. Taking specific heat at constant pressure $C_p = 1005$ J/(kg K), the power output (in MW) from the turbine is _________ (round off to 2 decimal places).
Official Answer Key:
5.03
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.48)**
**1. Intuitive Physical Setup & Given Parameters**:
- Mass flow rate of air: $\dot{m} = 50\text{ kg/s}$
- Inlet total temperature: $T_{01} = 450\text{ K}$
- Exit total temperature: $T_{02} = 350\text{ K}$
- Specific heat of air: $C_p = 1005\text{ J/(kg K)}$
**2. Step-by-Step Mathematical Derivation**:
1. **Steady Flow Energy Equation (SFEE) across Adiabatic Turbine**:
$P_{output} = \dot{m} \Delta h_0 = \dot{m} C_p (T_{01} - T_{02})$
2. **Total Temperature Drop**:
$\Delta T_0 = 450 - 350 = 100\text{ K}$
3. **Evaluating Power Output**:
$P_{output} = 50\text{ kg/s} \times 1005\text{ J/(kg K)} \times 100\text{ K} = 5025000\text{ W} = 5.025\text{ MW}$
Rounding to two decimal places:
$P_{output} \approx 5.03\text{ MW}$
Official acceptable range: **[5.00, 5.05] MW**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Official Range**: **5.00 to 5.05 MW**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **5.03**
- **Key Formula**: $P = \dot{m} C_p \Delta T_0$.
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