To perform the calculations needed, we will follow these steps:
Step 1: Calculate the air flow per engine at FL280
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Convert True Airspeed (TAS) from knots to m/s: \[ TAS = 486 \text{ kts} \times 0.51444 \text{ (m/s per knots)} = 249.78 \text{ m/s} \]
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Calculate air density at FL280 (28,000 ft). The approximate air density at this altitude can be calculated using the barometric formula or standard atmosphere assumptions. At FL280, the density is approximately \( \rho \approx 0.905 \) kg/m³.
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Calculate the total mass flow rate of air entering the engines. The total thrust is 400 kN, and the thrust can be expressed in terms of mass flow and exhaust velocity: \[ Thrust = \dot{m} \cdot (V_{e} - V_{0}) \] where:
- \( \dot{m} \) is the mass flow rate (kg/s).
- \( V_{e} \) is the exhaust velocity (m/s).
- \( V_{0} \) is the flight velocity (249.78 m/s).
Since we don't know \( V_{e} \) yet, we'll need to compute it in the next step but we can temporarily assume ideal conditions for calculating mass flow rate.
- Calculate the mass flow rate of air entering one engine. With \( V_{0} \) known and assuming equal distribution of thrust among four engines: \[ \text{Total Thrust} = 400,000 \text{ N} \] Each engine produces: \[ Thrust_{per , engine} = \frac{400,000 \text{ N}}{4} = 100,000 \text{ N} \]
Now, we use the thrust equation with an assumed exhaust velocity which will be derived as we progress.
- Inlet Area: Calculate the area of the inlet of one engine using the diameter: \[ A = \pi \left( \frac{D}{2} \right)^2 = \pi \left( \frac{2.96}{2} \right)^2 \approx 6.835 \text{ m}^2 \]
Assuming equal distribution, and far away from the intake velocity, we can compute, for simplifying conditions: \[ \text{Total mass flow rate} = air , density \times velocity \times area \] Per engine: \[ \dot{m}{engine} = \rho \times V{0} \times A_{inlet} \] \[ \dot{m}{engine} = 0.905 , \text{kg/m}^3 \times 249.78 , \text{m/s} \times 6.835 , \text{m}^2 \approx 14.379 \text{ kg/s per engine.} \] Overall, we would distribute the total mass flow: \[ \dot{m}{total} = 4 \cdot \dot{m}_{engine} \Rightarrow 4 \cdot 14.379 \approx 57.516 \text{ kg/s} \]
Step 2: Calculate the exhaust velocity of the engines
Now using the thrust equation to resolve \( V_{e} \): \[ Thrust = \dot{m} \cdot (V_{e} - V_{0}) \] Rearranging gives: \[ V_{e} = \frac{Thrust}{\dot{m}} + V_{0} \] Inserting the numbers, \[ V_{e} \approx \frac{100,000}{14.379} + 249.78 \approx 2482.3 \text{ m/s} \]
Step 3: Calculate the jet efficiency (η)
- Using the thrust equation: Jet efficiency can be approximated as: \[ \eta = \frac{V_{0}}{V_{e}} \times 100% \] Thus, \[ \eta \approx \frac{249.78}{2482.3} \times 100 \approx 10.06 , \text{percent (without % sign)} \]
Summary
- Air flow per engine: 14.379 kg/s
- Exhaust/jet velocity of the engines: 2482.3 m/s
- Jet efficiency of the engines: 10.06
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