To determine the required reservoir pressure, temperature, flow speeds, and expansion ratio for a wind tunnel with a test section at sea-level conditions and a Mach number of 4 (M=4), we can use the relationships and equations from compressible flow.
Given:
- Mach Number (M): 4
- Sea-level conditions (typical values):
- Pressure \( P_0 \): 101.325 kPa (or 1.01325 bar)
- Temperature \( T_0 \): 288.15 K
- Specific heat ratio \( \gamma \): 1.4 (for air)
1. Required Reservoir Pressure
The relationship between reservoir and test section pressures for supersonic flow can be derived from the isentropic flow relations. The following can be used:
\[ P_t = P_0 \left(1 + \frac{\gamma - 1}{2}M^2\right)^{\frac{\gamma}{\gamma - 1}} \]
Where:
- \( P_t \) is the total (reservoir) pressure
- \( M \) is the Mach number (M=4)
- Pressures must be in the same units.
Substituting the values:
\[ P_t = 101.325 \left(1 + \frac{1.4 - 1}{2} \cdot 4^2\right)^{\frac{1.4}{1.4 - 1}} \]
Calculating inside the parentheses:
\[ = 101.325 \left(1 + 0.2 \cdot 16\right)^{3.5} = 101.325 \left(1 + 3.2\right)^{3.5} = 101.325 \left(4.2\right)^{3.5} \]
Now calculate \(4.2^{3.5}\): \[ 4.2^{3.5} \approx 152.114 \] Calculating \( P_t \): \[ P_t \approx 101.325 \cdot 152.114 \approx 15473.31 , \text{Pa} \approx 15.47 , \text{bar} \]
2. Required Reservoir Temperature
Using the relationship between reservoir temperature and Mach number: \[ T_t = T_0 \left(1 + \frac{\gamma - 1}{2} M^2\right) \]
Substituting values: \[ T_t = 288.15 \left(1 + \frac{1.4 - 1}{2} \cdot 4^2\right) = 288.15 \left(1 + 0.2 \cdot 16\right) = 288.15 \left(1 + 3.2\right) = 288.15 \cdot 4.2 \] Calculating: \[ T_t \approx 288.15 \cdot 4.2 = 1216.23 , \text{K} \]
3. Flow Speed in the Test Section
The flow speed in the test section can be calculated using: \[ V = M \cdot a \]
Where \( a \) is the speed of sound at sea-level conditions:
\[ a = \sqrt{\gamma R T} \quad \text{(where R is the specific gas constant)} \] For air, \( R \approx 287 , \text{J/(kg·K)} \):
Calculating \( a \): \[ a = \sqrt{1.4 \cdot 287 \cdot 288.15} \approx \sqrt{116388.6} \approx 341 , \text{m/s} \]
Now calculate \( V \): \[ V = 4 \cdot 341 \approx 1364 , \text{m/s} \]
4. Flow Speed in the Throat
The flow speed in the throat (where Mach number is 1) can be calculated as: \[ V_t = a_t = 341 , \text{m/s} \quad \text{(at M = 1)} \]
5. Expansion Ratio
The expansion ratio \( \epsilon \) can be determined as follows: \[ \epsilon = \frac{A_t}{A_0} = \frac{1}{M} \cdot \left(\frac{P_0}{P_t}\right)^{\frac{(1 - 1/\gamma)}{2}} \]
For M=4, we first calculate the area ratio from the isentropic relations, which typically follow the trend of high Mach flow ratios.
But since this is a theoretical evaluation, the specific method to determine this is not straightforward without simulation or empirical data but can be approximated as \( A_t \) approaching the limit.
Summary
- Reservoir Pressure (P_t): ≈ 15.47 bar
- Reservoir Temperature (T_t): ≈ 1216.23 K
- Flow Speed in the Test Section: ≈ 1364 m/s
- Flow Speed in the Throat: ≈ 341 m/s
- Expansion Ratio: Theoretical calculations (would require more details for exact numbers).
Please verify all calculations and assumptions based on your specific context and settings.