GATE AE 2025 • Q61 NAT • 2 Marks Propulsion

Propulsion

Thermodynamic Cycles of Gas Turbines

Subtopic: Turbojet, Turbofan & Turboprop Engine Performance

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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