To solve the provided problem, we'll go through the calculations step-by-step.
Step 1: Calculate the Stagnation Temperature
The stagnation temperature (\( T_0 \)) for a gas can be calculated using the formula:
\[ T_0 = T + \frac{V^2}{2c_p} \]
However, since we have supersonic conditions, we can use the relationship for isentropic flow:
\[ T_0 = T \left(1 + \frac{\gamma - 1}{2} M^2\right) \]
Where:
- \( M \) is the Mach number
- \( T \) is the freestream temperature
- \( \gamma \) is the ratio of specific heats (for air, \( \gamma \approx 1.4 \))
Given:
- Freestream temperature (\( T \)) = 222.77 K
- Mach number (\( M \)) = 0.77
Plugging in these values into the formula:
\[ T_0 = 222.77 \left(1 + \frac{1.4 - 1}{2} (0.77^2)\right) \]
Calculating \((0.77^2)\):
\[ 0.77^2 = 0.5929 \]
Then substituting this into the equation:
\[ T_0 = 222.77 \left(1 + \frac{0.4}{2} \times 0.5929\right) \] \[ T_0 = 222.77 \left(1 + 0.2 \times 0.5929\right) \] \[ T_0 = 222.77 \left(1 + 0.11858\right) \] \[ T_0 = 222.77 \times 1.11858 \approx 249.29 , \text{K} \]
Step 2: Calculate \( C_p \)
To find the specific heat capacity at constant pressure (\( C_p \)), we can use the relation between pressure, density, and stagnation temperature.
First, we find the correct equation for \( C_p \).
For air, \( C_p \) can be expressed as:
\[ C_p = \frac{R}{\gamma - 1} \]
Where:
- \( R \) is the specific gas constant for air, about \( R \approx 287 , \text{J/(kg·K)} \)
- \( \gamma \approx 1.4 \)
Calculating \( C_p \):
\[ C_p = \frac{287}{1.4 - 1} = \frac{287}{0.4} \approx 717.5 , \text{J/(kg·K)} \]
Summary of Results
- Stagnation Temperature: \( T_0 \approx 249.29 , \text{K} \)
- Specific Heat Capacity at Constant Pressure: \( C_p \approx 717.5 , \text{J/(kg·K)} \)
These calculations assume isentropic flow conditions.
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