An aircraft is cruising at a speed of M = 0.77. The freestream temperature is 222.77K , and Density is 0.4097 kg/m^3 . The pressure is measured on a point along the wing, and it is found to be 9120 pa

Calculate the Cp for this point.
Assume isentropic flow.

Calculate the stagnation temperature in Kelvin under these conditions.
Assume isentropic flow.

1 answer

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

  1. Stagnation Temperature: \( T_0 \approx 249.29 , \text{K} \)
  2. Specific Heat Capacity at Constant Pressure: \( C_p \approx 717.5 , \text{J/(kg·K)} \)

These calculations assume isentropic flow conditions.