The pressure coefficient \(C_p\) is defined as:
\[ C_p = \frac{p - p_{\infty}}{q_{\infty}} \]
where:
- \(p\) is the local static pressure,
- \(p_{\infty}\) is the freestream static pressure, and
- \(q_{\infty}\) is the dynamic pressure.
To calculate \(C_p\), we first need to determine the freestream static pressure \(p_{\infty}\) and the dynamic pressure \(q_{\infty}\).
Step 1: Calculate the freestream static pressure (\(p_{\infty}\))
First, we need to calculate the dynamic pressure \(q_{\infty}\):
\[ q_{\infty} = \frac{1}{2} \rho V^2 \]
Since \(V\) is given in terms of Mach number (\(M\)), we can find the freestream velocity using:
\[ V = M \cdot a \]
where \(a\) is the speed of sound, which can be calculated using the relation:
\[ a = \sqrt{\gamma R T} \]
Assuming air as an ideal gas, we take \(\gamma \approx 1.4\) (ratio of specific heats for air) and \(R \approx 287 , \text{J/(kg·K)}\).
- Calculate the speed of sound \(a\):
\[ a = \sqrt{1.4 \cdot 287 \cdot 222.77} \approx \sqrt{ 1.4 \cdot 287 \cdot 222.77 } \approx \sqrt{ 113433.9114 } \approx 336.57 , \text{m/s} \]
- Calculate velocity \(V\):
\[ V = M \cdot a = 0.77 \cdot 336.57 \approx 259.57 , \text{m/s} \]
- Calculate the dynamic pressure \(q_{\infty}\):
\[ q_{\infty} = \frac{1}{2} \rho V^2 = \frac{1}{2} (0.4097) (259.57^2) \]
First, calculate \(V^2\):
\[ V^2 \approx 259.57^2 \approx 67454.41 , \text{m}^2/\text{s}^2 \]
Now compute \(q_{\infty}\):
\[ q_{\infty} \approx \frac{1}{2} (0.4097) (67454.41) \approx \frac{1}{2} (27685.499) \approx 13842.75 , \text{Pa} \]
Step 2: Calculate \(p_{\infty}\) using the barometric formula
To calculate \(p_{\infty}\), we can use the hydrostatic equation or the ideal gas law:
\[ p_{\infty} = \rho R T \]
\[ p_{\infty} = (0.4097 , \text{kg/m}^3) \times (287 , \text{J/(kg·K)}) \times (222.77 , \text{K}) \]
\[ p_{\infty} \approx 0.4097 \times 287 \times 222.77 \approx 24567.046 , \text{Pa} \approx 24567 , \text{Pa} \]
Step 3: Now calculate \(C_p\)
Given:
- Local static pressure \(p = 9120 , \text{Pa}\),
- Freestream static pressure \(p_{\infty} \approx 24567 , \text{Pa}\),
- Dynamic pressure \(q_{\infty} \approx 13842.75 , \text{Pa}\).
\[ C_p = \frac{p - p_{\infty}}{q_{\infty}} = \frac{9120 - 24567}{13842.75} \]
Calculating the numerator:
\[ 9120 - 24567 \approx -15447 \]
Now calculate \(C_p\):
\[ C_p = \frac{-15447}{13842.75} \approx -1.115 \]
Final Result:
The pressure coefficient \(C_p\) at this point is approximately:
\[ \boxed{-1.115} \]