To answer your questions, we start with the calculations step by step.
1. Calculate the Stall Speed of the Aircraft
The stall speed \( V_{\text{stall}} \) can be calculated using the following formula:
\[ V_{\text{stall}} = \sqrt{\frac{2 \cdot W}{\rho \cdot S \cdot C_{L_{\text{max}}}}} \]
where:
- \( W \) is the weight of the aircraft (in Newtons),
- \( \rho \) is the air density (in kg/m³),
- \( S \) is the wing area (in m²),
- \( C_{L_{\text{max}}} \) is the maximum lift coefficient when flaps are extended.
First, we calculate the weight \( W \):
\[ W = m \cdot g = 500,000 , \text{kg} \cdot 9.81 , \text{m/s}^2 = 4,905,000 , \text{N} \]
Next, we find the air density \( \rho \) at the given temperature and pressure.
Using the Ideal Gas Law:
\[ \rho = \frac{P}{R \cdot T} \]
Where:
- \( P = 101,325 , \text{Pa} \),
- \( R = 287.05 , \text{J/(kg·K)} \) (specific gas constant for air),
- \( T = 40 , \text{C} = 313.15 , \text{K} \).
Calculating density:
\[ \rho = \frac{101325.0}{287.05 \cdot 313.15} \approx 1.014 , \text{kg/m}^3 \]
Using \( C_{L_{\text{max}}} = 2.6 \) (with flaps extended):
Plugging in the values into the stall speed formula:
\[ V_{\text{stall}} = \sqrt{\frac{2 \cdot 4,905,000}{1.014 \cdot 845 \cdot 2.6}} \approx \sqrt{\frac{9,810,000}{2,2511.748}} \approx \sqrt{435} \approx 20.85 \text{ m/s} \]
Convert to knots (1 m/s ≈ 1.94384 kts):
\[ V_{\text{stall}} \approx 20.85 \times 1.94384 \approx 40.5 \text{ knots} \]
2. Calculate the Air Flow per Engine
To calculate the air flow per engine, we first need the total mass flow rate of air into the engines.
Assuming the aircraft is at flight level 280 (FL280), we can use the density from the previous calculation and True Airspeed (TAS).
\[ \dot{m} = \rho \cdot A \cdot V_{\text{TAS}} \]
Where:
- \( A \) is the total inlet area for all engines.
- The diameter of each engine is 2.96 m.
Inlet area of one engine:
\[ A_{\text{engine}} = \pi \left( \frac{D}{2} \right)^2 = \pi \left( \frac{2.96}{2} \right)^2 \approx 6.84 , \text{m}^2 \]
Total inlet area for 4 engines:
\[ A_{\text{total}} = 4 \cdot 6.84 = 27.36 , \text{m}^2 \]
Convert True Airspeed to m/s:
\[ V_{\text{TAS}} = 486 , \text{kts} \times 0.51444 \approx 250 , \text{m/s} \]
Now calculate the total mass flow rate:
\[ \dot{m} = 1.014 \cdot 27.36 \cdot 250 \approx 6,943.38 , \text{kg/s} \]
Now, mass flow rate per engine:
\[ \dot{m}_{\text{per engine}} = \frac{6,943.38}{4} \approx 1,735.85 , \text{kg/s} \]
3. Calculate the Exhaust/Jet Velocity of the Engines
Using the thrust equation for jet engines:
\[ F = \dot{m} \cdot (V_{jet} - V_{ambient}) \]
Where:
- \( F = 400,000 , \text{N} \)
- \( V_{ambient} \approx 250 , \text{m/s} \) (TAS).
Rearranging it to find \( V_{jet} \):
\[ V_{jet} = \frac{F}{\dot{m}} + V_{ambient} \]
Total mass flow \( \dot{m} \) is approximately 6,943.38 kg/s.
\[ V_{jet} = \frac{400,000}{6,943.38} + 250 \approx 57.6 + 250 \approx 307.6 \text{ m/s} \]
4. Calculate the Jet Efficiency
Jet efficiency \( \eta \) can be calculated using the formula:
\[ \eta = \frac{2 \cdot F}{\dot{m} \cdot (V_{jet} + V_{ambient})} \]
Substituting in our values:
\[ \eta = \frac{2 \cdot 400,000}{6,943.38 \cdot (307.6 + 250)} = \frac{800,000}{6,943.38 \cdot 557.6} \]
Calculating the denominator:
\[ 6,943.38 \cdot 557.6 \approx 3,865,120 \]
Thus:
\[ \eta \approx \frac{800,000}{3,865,120} \approx 0.207 \]
Converting this into percent:
\[ \eta \approx 20.7% \]
So the final answers are:
- Stall speed: 40.5 kts
- Air flow per engine: 1735.85 kg/s
- Exhaust velocity: 307.6 m/s
- Jet efficiency: 20.7 (without % sign).
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