A futuristic Aircraft manufacturer has come up with the idea to build a supersonic passenger jet, which is supposed to perform cruise flight at 24 kilometres altitude. For passenger comfort, the cabin pressure should remain at least 72 percent of the sea level value. To give the aircraft an aerodynamic shape its fuselage is rather long and slender, meaning the fuselage diameter is 'just' 4.5 metres.
The engineers have asked you, now experts in the field of pressure cabins and fatigue, to determine the required skin thickness to deal with the circular stress (which may not exceed 75MPa) in the fuselage skin. To be on the safe side, they ask you to take into account a safety factor of 1.5.
To determine the required skin thickness for the fuselage, we need to consider the circular stress on the fuselage skin. The formula for circular stress is:
σ = P * r / t
Where:
σ is the circular stress
P is the pressure difference
r is the radius of the fuselage
t is the skin thickness
To calculate the pressure difference, we need to consider the cabin pressure at altitude. The cabin pressure should be at least 72 percent of the sea level value. Assuming the sea level pressure is standard atmospheric pressure (101.325 kPa), the cabin pressure at 24 kilometers altitude would be:
P = 0.72 * 101.325 kPa = 72.99 kPa
Next, we need to calculate the radius of the fuselage. Since the diameter is given as 4.5 meters, the radius would be half of that:
r = 4.5 / 2 = 2.25 meters
Now, we can rearrange the formula to solve for the skin thickness:
t = P * r / (σ * safety factor)
Using the given data, with a circular stress limit of 75 MPa and a safety factor of 1.5, we can plug in the values to calculate the skin thickness:
t = (72.99 kPa * 2.25 m) / (75 MPa * 1.5)
t = 194.8 mm / 112.5
t ≈ 1.73 mm
Therefore, the required skin thickness for the fuselage is approximately 1.73 mm to withstand the circular stress limit of 75 MPa with a safety factor of 1.5.
To determine the required skin thickness to handle the circular stress in the fuselage skin, we can use the formula for stress in a thin-walled cylinder:
σ = (P * r) / (t * 2)
Where:
- σ is the stress
- P is the pressure difference between the inside and outside of the fuselage
- r is the radius of the fuselage
- t is the skin thickness
In this case, we need to determine the skin thickness (t). We know the stress value should not exceed 75MPa and the safety factor is 1.5.
First, we need to calculate the pressure difference (P). The cabin pressure should remain at least 72% of the sea level value. So, we need to find the pressure at 24 kilometers altitude.
Using the barometric formula, we know that the pressure decreases with altitude. At sea level (0 kilometers altitude), the average pressure is approximately 101.325 kilopascals (kPa).
To calculate the pressure at 24 kilometers altitude, we can use the formula:
P2 = P1 * exp(-h * (M * g) / (R * T))
Where:
- P2 is the pressure at 24 kilometers altitude
- P1 is the pressure at sea level (101.325 kPa)
- h is the altitude difference (24 kilometers)
- M is the molar mass of air (approximately 0.02897 kg/mol)
- g is the acceleration due to gravity (approximately 9.80665 m/s^2)
- R is the specific gas constant for air (approximately 8.314 J/(mol*K))
- T is the temperature at sea level (approximately 288.15 K)
Using these values in the formula, we can calculate P2.
P2 = 101.325 * exp(-24 * 0.02897 * 9.80665 / (8.314 * 288.15))
Now, we can calculate the pressure difference:
ΔP = P2 - P1
With this pressure difference, we can calculate the skin thickness:
t = (P * r) / (σ * 2)
Substituting the values, we can solve for t.
To determine the required skin thickness for the fuselage, we need to consider the circular stress that the fuselage skin will experience. Circular stress is the stress that acts perpendicular to the circumference of a cylinder, such as the fuselage.
We know that the circular stress should not exceed 75MPa, and we need to take into account a safety factor of 1.5. The safety factor is a multiplier used to ensure the design can handle unforeseen loads and variations.
To calculate the required skin thickness, we can use the formula for the hoop stress (also known as the circumferential or circular stress) in a cylindrical pressure vessel:
σ_h = P * r / t
Where:
σ_h is the hoop stress,
P is the internal pressure,
r is the radius of the cylinder, and
t is the thickness of the cylinder.
In this case, we need to solve for t.
First, let's calculate the internal pressure. We know that the cabin pressure should remain at least 72% of the sea level value. At 24 kilometers altitude, the atmospheric pressure decreases significantly. To determine the internal pressure, we need to find the equivalent sea level pressure at that altitude. Let's assume the sea level pressure is 101325 Pa (1 atmosphere).
The air density at 24 kilometers altitude can be estimated using the barometric formula:
ρ = ρ_0 * exp(-h/H)
Where:
ρ is the air density at altitude,
ρ_0 is the air density at sea level (1.225 kg/m^3),
h is the altitude (24000 m), and
H is the scale height (approximately 8000 m).
Substituting the values, we get:
ρ = 1.225 * exp(-24000/8000) ≈ 0.228 kg/m^3
The internal pressure can be calculated using the ideal gas law:
P = ρ * R * T
Where:
P is the pressure,
ρ is the air density,
R is the specific gas constant for air (approximately 287 J/kg·K), and
T is the temperature (let's assume 288.15 K, which is close to the standard temperature at sea level).
Substituting the values, we get:
P = 0.228 * 287 * 288.15 ≈ 19905 Pa
Now that we have the internal pressure, we can proceed with calculating the skin thickness.
Rearranging the formula for hoop stress (σ_h), we get:
t = P * r / (σ_h * safety_factor)
Substituting the known values:
t = 19905 * 4.5 / (75 * 1.5) ≈ 402.4 mm
Therefore, the required skin thickness for the fuselage to handle the circular stress, with a safety factor of 1.5, is approximately 402.4 mm.