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Question 61
Question Statement:
An ideal turbofan with a bypass ratio of 5 has core mass flow rate, ṁ a,c= 100 kg/s.
The core and the fan exhausts are separate and optimally expanded. The core
exhaust speed is 600 m/s and the fan exhaust speed is 120 m/s. If the fuel mass
flow rate is negligible in comparison to ṁ a,c, the static specific thrust (T/ṁ a,c)
developed by the engine is ____ Ns/kg (rounded off to the nearest integer).
Official Answer Key:
1200.0
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.61)**
**1. Intuitive Physical Setup & Given Parameters**:
Ideal unmixed separate-exhaust turbofan engine:
- Bypass ratio: $BPR = \alpha = \frac{\dot{m}_{\text{fan}}}{\dot{m}_{\text{core}}} = 5$
- Core mass flow rate: $\dot{m}_{\text{core}} = 100\text{ kg/s}$
- Core exhaust speed: $V_{jc} = 600\text{ m/s}$
- Fan exhaust speed: $V_{jf} = 120\text{ m/s}$
- Static condition: flight speed $V_0 = 0$
- Optimally expanded exhausts ($p_e = p_a$)
- Negligible fuel mass flow rate ($\dot{m}_f \ll \dot{m}_{\text{core}}$)
We need to find the **static specific thrust** based on core air mass flow rate: $T_{\text{sp}} = \frac{T}{\dot{m}_{\text{core}}}$ in $\text{N}\cdot\text{s/kg}$ (nearest integer).
**2. Step-by-Step Propulsion Derivation**:
1. **Mass Flow Rates**:
$\dot{m}_{\text{core}} = 100\text{ kg/s}$
$\dot{m}_{\text{fan}} = \alpha \dot{m}_{\text{core}} = 5 \times 100 = 500\text{ kg/s}$
2. **Total Thrust Produced at Static Condition ($V_0 = 0$)**:
Since nozzles are optimally expanded and $V_0 = 0$:
$T = \dot{m}_{\text{core}} V_{jc} + \dot{m}_{\text{fan}} V_{jf}$
$T = (100 \times 600) + (500 \times 120) = 60,000 + 60,000 = 120,000\text{ N} = 120\text{ kN}$
3. **Specific Thrust Based on Core Flow**:
$T_{\text{sp}} = \frac{T}{\dot{m}_{\text{core}}} = \frac{120,000\text{ N}}{100\text{ kg/s}} = 1200\text{ N}\cdot\text{s/kg}$
**3. Comprehensive Options & Distractor Analysis**:
- **Evaluation Criteria**: Numerical Answer Type (NAT).
- **Exact Analytical Result**: $T_{\text{sp}} = 1200\text{ N}\cdot\text{s/kg}$.
- **Rounding Requirement**: Nearest integer $\implies \mathbf{1200}$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **1200.0** (Acceptable Range: **1200 to 1200**)
- **Exam Takeaway**: Core-normalized specific thrust: $\frac{T}{\dot{m}_c} = V_{jc} + BPR \times V_{jf} = 600 + 5(120) = 1200\text{ N}\cdot\text{s/kg}$.
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