To calculate the mass flow through the wind tunnel and the flow speed in the circular cross-section, we will follow these steps:
- Calculate the mass flow rate through the test section.
- Determine the flow speed in the circular cross-section.
Step 1: Calculate the Mass Flow Rate
The mass flow rate (\( \dot{m} \)) can be calculated using the equation: \[ \dot{m} = \rho \cdot A \cdot V \] where:
- \( \rho \) is the air density at standard sea level conditions (approximately \( 1.225 , \text{kg/m}^3 \)),
- \( A \) is the cross-sectional area of the test section,
- \( V \) is the flow velocity.
For a square test section with dimensions \( 2 \times 2 , \text{m} \): \[ A = 2 , \text{m} \times 2 , \text{m} = 4 , \text{m}^2 \]
Substituting the values into the equation: \[ \dot{m} = 1.225 , \text{kg/m}^3 \cdot 4 , \text{m}^2 \cdot 50 , \text{m/s} \] Calculating this gives: \[ \dot{m} = 1.225 \cdot 4 \cdot 50 = 245 , \text{kg/s} \]
Step 2: Determine the Flow Speed in the Circular Cross-Section
Next, we need to find the flow speed in the circular cross-section. The area of the circular cross-section is given by: \[ A_c = \pi r^2 \] Given that the radius (\( r \)) is 2 m: \[ A_c = \pi (2 , \text{m})^2 = 4\pi , \text{m}^2 \approx 12.566 , \text{m}^2 \]
Now, using the principle of conservation of mass, the mass flow rate must remain constant. Therefore, we can say: \[ \dot{m} = \rho \cdot A_c \cdot V_c \] where \( V_c \) is the flow speed in the circular cross-section.
Rearranging gives: \[ V_c = \frac{\dot{m}}{\rho \cdot A_c} \] Substituting for \( \dot{m} \), \( \rho \), and \( A_c \): \[ V_c = \frac{245 , \text{kg/s}}{1.225 , \text{kg/m}^3 \cdot 12.566 , \text{m}^2} \] Calculating this gives: \[ V_c \approx \frac{245}{15.372} \approx 15.95 , \text{m/s} \]
Summary of Results
- Mass Flow Rate through the Tunnel: \( \dot{m} \approx 245 , \text{kg/s} \)
- Flow Speed in the Circular Cross-Section: \( V_c \approx 15.95 , \text{m/s} \)
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