To calculate the required lift coefficient for the Boeing 747 to fly at a given airspeed, we first need to determine the lift force required to maintain level flight.
Step 1: Calculate the required lift force
In steady horizontal flight, the lift force \( L \) must equal the weight \( W \) of the aircraft. The weight can be calculated using:
\[ W = m \cdot g \]
where:
- \( m = 250,000 \) kg (mass of the aircraft)
- \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity)
So,
\[ W = 250,000 , \text{kg} \times 9.81 , \text{m/s}^2 = 2,452,500 , \text{N} \]
Step 2: Calculate the required lift coefficient
The lift force equation for an aircraft can be expressed as:
\[ L = \frac{1}{2} \rho V^2 S C_L \]
where:
- \( \rho \) = air density at Flight Level 350
- \( V \) = velocity of the aircraft
- \( S \) = wing area
- \( C_L \) = lift coefficient
At Flight Level 350 (approximately 35,000 feet), the air density \( \rho \) is approximately \( 0. air density \approx 0.49 , \text{kg/m}^3\).
Assuming the airspeed \( V \) is provided (but missing here), let's say it was given as \( V , \text{m/s} \):
We can rearrange and solve for \( C_L \):
\[ C_L = \frac{L}{\frac{1}{2} \rho V^2 S} \]
Substituting the values:
\[ C_L = \frac{2,452,500}{\frac{1}{2} \times 0.49 \times V^2 \times 511} \]
Step 3: Calculating minimum required tail surface area
For longitudinal stability, we can consider the stability condition expressed as:
\[ \frac{\partial L}{\partial \alpha} \left(\frac{l_H}{S} \right) = \frac{\partial L_T}{\partial \alpha_T} \left(\frac{S_T}{S}\right) \]
where:
- \( \frac{\partial L}{\partial \alpha} \) = change in lift slope of the aircraft (given as \( \frac{dC_L}{d\alpha} = 0.098 , \text{per degree}\))
- \( \frac{\partial L_T}{\partial \alpha_T} \) = change in lift slope of the tail (given as \( \frac{dC_{L_T}}{d\alpha_T} = 0.047 , \text{per degree}\))
- \( l_H = 31.5 , \text{m} \) (distance from the center of the wing to the center of the tail)
- \( S_T \) = tail surface area
We also need to account for downwash. The effective angle of attack for the tail is reduced due to downwash:
\[ \alpha_T = \alpha - \epsilon \]
where \( \epsilon = \text{downwash} = \frac{1}{10} \alpha \).
A linear approximation yields:
\[ \epsilon = 0.1 \alpha = \text{proportional to angle of attack} , \alpha \]
Using the conditions of stability, we derive that:
\[ \frac{dC_L}{d\alpha} l_H = \frac{dC_{L_T}}{d\alpha_T} S_T S \]
This expression can be rearranged to calculate \( S_T \):
\[ S_T = \frac{dC_L}{d\alpha} l_H}{\frac{dC_{L}}{d\alpha_T}} \]
Substituting the known values:
\[ S_T = \frac{0.098 \times 31.5}{0.047} \]
Calculating the above expression gives the tail surface area required for stability.
This should provide approximate stability conditions for values influencing \( S_T \).
Please provide the specific airspeed value \( V \) for the full lift coefficient calculation and desired output for \( S_T \).
2 answers