Problem Statement
Find the Mach number when an aeroplane is flying at 1000 km/hour through still air having pressure of 7 N/cm² and temperature of -5°C. Take R = 287.14 J/kg K. Calculate the pressure and temperature of air at stagnation point. Take k = 1.4.
Given Data & Constants
- Speed of aeroplane, \(V = 1000 \, \text{km/hr}\)
- Static pressure, \(P_1 = 7.0 \, \text{N/cm}^2\)
- Static temperature, \(T_1 = -5^\circ\text{C}\)
- Adiabatic index, \(k = 1.4\)
- Gas constant, \(R = 287.14 \, \text{J/kg K}\)
Solution
1. Convert Units and Calculate Speed of Sound
First, we convert the speed and temperature to SI units.
$$ V = 1000 \, \frac{\text{km}}{\text{hr}} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{hr}}{3600 \, \text{s}} \approx 277.78 \, \text{m/s} $$
$$ T_1 = -5^\circ\text{C} + 273.15 = 268.15 \, \text{K} $$
Now, calculate the local speed of sound (\(c\)).
$$ c = \sqrt{kRT_1} = \sqrt{1.4 \times 287.14 \times 268.15} \approx 328.28 \, \text{m/s} $$
2. Calculate the Mach Number (M)
$$ M = \frac{V}{c} = \frac{277.78}{328.28} \approx 0.846 $$
3. Calculate Stagnation Temperature (\(T_0\))
The stagnation temperature is the temperature the air reaches when it is brought to rest adiabatically.
$$ T_0 = T_1 \left(1 + \frac{k-1}{2}M^2\right) $$
$$ T_0 = 268.15 \left(1 + \frac{1.4-1}{2}(0.846)^2\right) = 268.15 (1 + 0.2 \times 0.7157) $$
$$ T_0 = 268.15 \times 1.14314 \approx 306.5 \, \text{K} \quad (33.35^\circ\text{C}) $$
4. Calculate Stagnation Pressure (\(P_0\))
First, convert the static pressure to Pascals: \(P_1 = 7.0 \, \text{N/cm}^2 = 70000 \, \text{N/m}^2\).
$$ P_0 = P_1 \left(1 + \frac{k-1}{2}M^2\right)^{\frac{k}{k-1}} $$
$$ P_0 = 70000 \times (1.14314)^{\frac{1.4}{0.4}} = 70000 \times (1.14314)^{3.5} $$
$$ P_0 \approx 70000 \times 1.605 \approx 112350 \, \text{N/m}^2 \quad (11.235 \, \text{N/cm}^2) $$
Final Results:
Mach Number: \( \approx 0.846 \)
Stagnation Pressure: \( \approx 11.24 \, \text{N/cm}^2 \)
Stagnation Temperature: \( \approx 306.5 \, \text{K} \) or \(33.35^\circ\text{C}\)
