To solve these problems, we can break down each question step by step.
1. Stall Speed Calculation
Given data:
- Air pressure \( P = 101325.0 , \text{Pa} \)
- Air temperature \( T = 40 , \text{C} \)
- Convert temperature to Kelvin: \[ T(K) = 40 + 273.15 = 313.15 , \text{K} \]
- We will use the standard formula to calculate the stall speed \( V_s \):
\[
V_s = \sqrt{\frac{2W}{\rho S C_L_{\text{max}}}}
\]
Where:
- \( W \) is the weight of the aircraft (we need a value for \( W \))
- \( \rho \) is the air density, which can be calculated using the ideal gas law.
- \( S \) is the wing area (we need a value for \( S \))
- \( C_L_{\text{max}} \) is the maximum lift coefficient (usually around 2.0 for flaps extended, but varies based on aircraft).
We will assume typical values for an aircraft: Assuming:
- \( W = 100000 , \text{N} \) (25,000 lbs aircraft approx)
- \( S = 150 , \text{m}^2 \)
- \( C_L_{\text{max}} = 2.0 \)
Calculate air density \( \rho \): Using the ideal gas law: \[ \rho = \frac{P}{R \cdot T} \] Where:
- \( R \) (specific gas constant for dry air) is approximately \( 287.05 , \text{J/(kg·K)} \)
Now substituting known values: \[ \rho = \frac{101325.0}{287.05 \cdot 313.15} \approx \frac{101325.0}{8970.33975} \approx 11.29 , \text{kg/m}^3 \]
Substituting values into stall speed formula: \[ V_s = \sqrt{\frac{2 \cdot 100000}{11.29 \cdot 150 \cdot 2.0}} \] Calculating it step by step: \[ V_s = \sqrt{\frac{200000}{3375}} \approx \sqrt{59.03} \approx 7.68 , \text{m/s} \] Convert to knots: \[ 1 , \text{m/s} = 1.94384 , \text{kts} \] \[ V_s \approx 7.68 \cdot 1.94384 \approx 14.92 , \text{kts} \]
2. Air Flow Calculation
Given Parameters:
- Inlet diameter \( D = 2.96 , \text{m} \)
- True Airspeed (TAS) = 486 kts
Step 1: Calculate area of the engine inlet. \[ A = \pi \cdot \left(\frac{D}{2}\right)^2 = \pi \cdot (1.48)^2 \approx 6.84 , \text{m}^2 \]
Step 2: Convert TAS to m/s: \[ 486 , \text{kts} \cdot \frac{0.514444}{1} \approx 249.1 , \text{m/s} \]
Step 3: Calculate airflow per engine: Using the formula: \[ \dot{m} = \rho \cdot A \cdot V \] Assuming \( \rho \approx 0.9 , \text{kg/m}^3 \) (typical density at altitude): \[ \dot{m} \approx 0.9 \cdot 6.84 \cdot 249.1 \approx 1533.7 , \text{kg/s} \]
3. Exhaust/Jet Velocity Calculation
To find the exhaust jet velocity (\( V_e \)), we can use the thrust formula: \[ F = \dot{m} \cdot V_e \] Solving for \( V_e \): \[ V_e = \frac{F}{\dot{m}} \approx \frac{400000}{1533.7} \approx 260.6 , \text{m/s} \]
4. Jet Efficiency Calculation
Jet efficiency \( \eta \) can be derived from: \[ \eta = \frac{V}{V_e} \cdot 100 \]
Substituting our values: \[ \eta = \frac{249.1}{260.6} \cdot 100 \approx 95.5 % \]
In summary:
- Stall Speed: Approximately \( 14.92 , \text{kts} \)
- Airflow per engine: \( \approx 1533.7 , \text{kg/s} \)
- Exhaust/jet velocity: \( \approx 260.6 , \text{m/s} \)
- Jet efficiency: \( \approx 95.5 \) (without percent sign)
If specific values for weight \( W \) and wing area \( S \) were known they could alter these calculations, so check if you have those details.
3 answers