Question

Mass 250.000 kg
Wing Area 511 m2
Wingspan 60.0 m
Number of Engines 4
Distance from aerodynamic center of wing-body to the tails aerodynamic center (lH) 31.5 m
dCL/dalpha of aircraft 0.098 per degree
dCL/dalpha of tail 0.047 per degree
Downwash at tail 1.00 degree per 10.0 degrees alpha


A Boeing 747 Airliner is flying in steady, horizontal flight at Flight Level 350.The aircraft is flying at .

What is the required lift coefficient to fly at the given airspeed


The design requirement is that the aircraft should be stable when the c.g. is less than 1.8 meters behind the aerodynamic center of central wing.

What is the minimum required tail surface area (in m^2 ) to achieve this?
Tip: first derivate the expression for longitudinal stability

Answer 1

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 \( S_T \).