Question

The mass flow rate in a supersonic wind tunnel is 2 kg/s when the stagnation pressure and stagnation temperature are 1 MPa and 800 K, respectively. If the stagnation pressure and stagnation temperature are changed to 3 MPa and 200 K, the mass flow rate in the tunnel changes to __________ kg/s (answer in integer).

Hint:

In a wind tunnel, the mass flow rate is proportional to the square root of the stagnation pressure and temperature. Use this relationship when conditions change to calculate the new mass flow rate.

Answer 1

Given Parameters

Initial conditions:
    Mass flow rate (\( \dot{m}_1 \)) = 2 kg/s
    Stagnation pressure (\( p_{0,1} \)) = 1 MPa = \( 1 \times 10^6 \) Pa
    Stagnation temperature (\( T_{0,1} \)) = 800 K

New conditions:
    Stagnation pressure (\( p_{0,2} \)) = 3 MPa = \( 3 \times 10^6 \) Pa
    Stagnation temperature (\( T_{0,2} \)) = 200 K

Key Concept
The mass flow rate in a choked nozzle (supersonic wind tunnel) is given by:
\[ \dot{m} = \frac{p_0 A^*}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}} \]
where:
    \( p_0 \) = stagnation pressure
    \( T_0 \) = stagnation temperature
    \( A^* \) = throat area (constant for the same tunnel)
    \( \gamma \), \( R \) = constants for the gas

Step 1: Mass Flow Rate Ratio
Since \( A^* \), \( \gamma \), and \( R \) are constant:
\[ \frac{\dot{m}_2}{\dot{m}_1} = \frac{p_{0,2}/\sqrt{T_{0,2}}}{p_{0,1}/\sqrt{T_{0,1}}} \]
Step 2: Calculate New Mass Flow Rate
Substitute the values:
\[ \frac{\dot{m}_2}{2} = \frac{3/\sqrt{200}}{1/\sqrt{800}} = 3 \times \sqrt{\frac{800}{200}} = 3 \times \sqrt{4} = 3 \times 2 = 6 \]
\[ \dot{m}_2 = 6 \times 2 = 12 \, \text{kg/s} \]
Final Answer
The new mass flow rate is \( \boxed{12} \, \text{kg/s} \).