Find the Mach number when an aeroplane is flying at 900 km/hour through still air having a pressure of 8.0 N/cm² and temperature -15°C. Take k = 1.4 and R = 287 J/kg K. Calculate the pressure, temperature and density of air at the stagnation point on the nose of the plane.

Aeroplane Stagnation Properties Calculation

Problem Statement

Given Data & Constants

Speed of aeroplane, $V = 900 \, \text{km/hr}$
Static pressure, $P_1 = 8.0 \, \text{N/cm}^2$
Static temperature, $T_1 = -15^\circ\text{C}$
Adiabatic index, $k = 1.4$
Gas constant, $R = 287 \, \text{J/kg K}$

Solution

1. Convert Units and Calculate Speed of Sound

First, we convert the speed and temperature to SI units.

$$ V = 900 \, \frac{\text{km}}{\text{hr}} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{hr}}{3600 \, \text{s}} = 250 \, \text{m/s} $$ $$ T_1 = -15^\circ\text{C} + 273.15 = 258.15 \, \text{K} $$

Now, calculate the local speed of sound ($c$).

$$ c = \sqrt{kRT_1} = \sqrt{1.4 \times 287 \times 258.15} \approx 322.06 \, \text{m/s} $$

2. Calculate the Mach Number (M)

$$ M = \frac{V}{c} = \frac{250}{322.06} \approx 0.776 $$

3. Calculate Stagnation Temperature ($T_0$)

The stagnation temperature is the temperature the air reaches when it is brought to rest adiabatically.

$$ T_0 = T_1 \left(1 + \frac{k-1}{2}M^2\right) $$ $$ T_0 = 258.15 \left(1 + \frac{1.4-1}{2}(0.776)^2\right) = 258.15 (1 + 0.2 \times 0.602) $$ $$ T_0 = 258.15 \times 1.1204 \approx 289.23 \, \text{K} \quad (16.08^\circ\text{C}) $$

4. Calculate Stagnation Pressure ($P_0$)

First, convert the static pressure to Pascals: $P_1 = 8.0 \, \text{N/cm}^2 = 80000 \, \text{N/m}^2$.

$$ P_0 = P_1 \left(\frac{T_0}{T_1}\right)^{\frac{k}{k-1}} = P_1 (1.1204)^{\frac{1.4}{0.4}} $$ $$ P_0 = 80000 \times (1.1204)^{3.5} \approx 80000 \times 1.486 $$ $$ P_0 \approx 118880 \, \text{N/m}^2 \quad (11.89 \, \text{N/cm}^2) $$

5. Calculate Stagnation Density ($\rho_0$)

Using the ideal gas law with the stagnation properties.

$$ \rho_0 = \frac{P_0}{R T_0} = \frac{118880}{287 \times 289.23} \approx 1.432 \, \text{kg/m}^3 $$

Final Results:

Mach Number: $ \approx 0.776 $

Stagnation Pressure: $ \approx 11.89 \, \text{N/cm}^2 $

Stagnation Temperature: $ \approx 289.23 \, \text{K} $ or $16.08^\circ\text{C}$

Stagnation Density: $ \approx 1.432 \, \text{kg/m}^3 $

Find the Mach number when an aeroplane is flying at 900 km/hour through still air having a pressure of 8.0 N/cm² and temperature -15°C. Take k = 1.4 and R = 287 J/kg K. Calculate the pressure, temperature and density of air at the stagnation point on the nose of the plane.

Problem Statement

Given Data & Constants

Solution

1. Convert Units and Calculate Speed of Sound

2. Calculate the Mach Number (M)

3. Calculate Stagnation Temperature (\(T_0\))

4. Calculate Stagnation Pressure (\(P_0\))

5. Calculate Stagnation Density (\(\rho_0\))

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Find the Mach number when an aeroplane is flying at 900 km/hour through still air having a pressure of 8.0 N/cm² and temperature -15°C. Take k = 1.4 and R = 287 J/kg K. Calculate the pressure, temperature and density of air at the stagnation point on the nose of the plane.

Problem Statement

Given Data & Constants

Solution

1. Convert Units and Calculate Speed of Sound

2. Calculate the Mach Number (M)

3. Calculate Stagnation Temperature (\(T_0\))

4. Calculate Stagnation Pressure (\(P_0\))

5. Calculate Stagnation Density (\(\rho_0\))

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