Problem Statement
An aeroplane is flying at an altitude of 5000 m. Calculate the pressure around the aeroplane, given the lapse-rate in the atmosphere is 0.0065 K/m. Neglect variation of g with altitude. Take pressure and temperature at ground level as 10.143 N/cm² and 15°C and density of air as 1.285 kg/m³.
Given Data
- Altitude, \(Z = 5000 \, \text{m}\)
- Ground Level Pressure, \(p_0 = 10.143 \, \text{N/cm}^2\)
- Ground Level Temperature, \(t_0 = 15^\circ\text{C}\)
- Temperature Lapse Rate, \(L = 0.0065 \, \text{K/m}\)
- Ground Level Density, \(\rho_0 = 1.285 \, \text{kg/m}^3\) (Note: unit corrected from original text)
Solution
1. Convert Initial Conditions to SI Units
2. Calculate Temperature at 5000 m (\(T\))
The temperature at altitude Z is found using the lapse rate.
3. Calculate the Gas Constant (\(R\))
The specific gas constant for air can be determined from the ground-level conditions.
4. Calculate Pressure at 5000 m (\(p\))
Using the pressure-temperature relationship for a polytropic atmosphere with a constant lapse rate:
First, calculate the exponent:
Now, calculate the pressure:
Converting to N/cm²:
The pressure around the aeroplane at an altitude of 5000 m is approximately \( 52,509 \, \text{N/m}^2 \) or \( 5.25 \, \text{N/cm}^2 \).
Explanation of the Atmospheric Model
To determine atmospheric properties at altitude, we use a model that accounts for the fact that air gets colder as you go up. The temperature lapse rate (\(L\)) defines this rate of cooling. Assuming this rate is constant allows us to model the atmosphere as a polytropic process.
The calculation first determines the temperature at 5000 m. Then, using the ground-level data, it finds the specific gas constant (\(R\)) for air. Finally, it uses the barometric formula, which relates the pressures at two different altitudes to the temperatures at those altitudes, to find the final pressure.
Physical Meaning
The result shows that at an altitude of 5000 m (about 16,400 feet), the atmospheric pressure drops to approximately 52,509 N/m². This is only about 52% of the sea-level pressure of 101,430 N/m².
This dramatic pressure drop is why modern aircraft have pressurized cabins. The air at this altitude is too thin to breathe comfortably and sustain consciousness. The pressure difference between the inside of the cabin (kept near sea-level pressure) and the outside air creates significant structural loads on the aircraft’s fuselage. This calculation is a fundamental part of the aerospace engineering required to design safe and functional aircraft.
