Question
An aircraft is cruising at a speed of M = 0.77. The freestream temperature is , and . The pressure is measured on a point along the wing, and it is found to be .
Calculate the for this point.
Assume isentropic flow.
unanswered
Calculate the stagnation temperature in Kelvin under these conditions.
Assume isentropic flow.
Calculate the for this point.
Assume isentropic flow.
unanswered
Calculate the stagnation temperature in Kelvin under these conditions.
Assume isentropic flow.
Answers
GPT-4o mini
To calculate the stagnation temperature for an aircraft cruising at a certain Mach number and given conditions, we can use the following formula:
\[
T_0 = T + \frac{V^2}{2c_p}
\]
Where:
- \( T_0 \) is the stagnation temperature.
- \( T \) is the freestream temperature.
- \( V \) is the freestream velocity.
- \( c_p \) is the specific heat at constant pressure.
However, since we want to apply the conditions of isentropic flow, we can derive \( T_0 \) from Mach number \( M \) directly using a more straightforward relation:
\[
T_0 = T \left(1 + \frac{\gamma - 1}{2} M^2 \right)
\]
Where:
- \( \gamma \) is the ratio of specific heats (approximately \( 1.4 \) for air).
- \( M \) is the Mach number.
### Step 1: Calculate Freestream Temperature
Assuming you have a value for the freestream temperature \( T \). If not provided, let’s give an example temperature \( T = 288.15 \, K \) (which is roughly 15°C).
### Step 2: Calculate Stagnation Temperature
Plugging the given values into the formula, let's compute the stagnation temperature.
Using \( M = 0.77 \) and \( \gamma = 1.4 \):
\[
T_0 = T \left(1 + \frac{1.4 - 1}{2} (0.77)^2 \right)
\]
First, we calculate \( \frac{1.4 - 1}{2} \):
\[
\frac{1.4 - 1}{2} = 0.2
\]
Now compute \( (0.77)^2 \):
\[
(0.77)^2 = 0.5929
\]
Substituting this back in:
\[
T_0 = 288.15 \left(1 + 0.2 \times 0.5929 \right)
\]
\[
T_0 = 288.15 \left(1 + 0.11858 \right)
\]
\[
T_0 = 288.15 \left(1.11858 \right)
\]
\[
T_0 \approx 322.39 \, K
\]
### Final Answer
The stagnation temperature \( T_0 \) at this point is approximately **322.39 K**.
If you have any specific values for freestream temperature and pressure, feel free to share, and I'll adjust the calculation accordingly.
\[
T_0 = T + \frac{V^2}{2c_p}
\]
Where:
- \( T_0 \) is the stagnation temperature.
- \( T \) is the freestream temperature.
- \( V \) is the freestream velocity.
- \( c_p \) is the specific heat at constant pressure.
However, since we want to apply the conditions of isentropic flow, we can derive \( T_0 \) from Mach number \( M \) directly using a more straightforward relation:
\[
T_0 = T \left(1 + \frac{\gamma - 1}{2} M^2 \right)
\]
Where:
- \( \gamma \) is the ratio of specific heats (approximately \( 1.4 \) for air).
- \( M \) is the Mach number.
### Step 1: Calculate Freestream Temperature
Assuming you have a value for the freestream temperature \( T \). If not provided, let’s give an example temperature \( T = 288.15 \, K \) (which is roughly 15°C).
### Step 2: Calculate Stagnation Temperature
Plugging the given values into the formula, let's compute the stagnation temperature.
Using \( M = 0.77 \) and \( \gamma = 1.4 \):
\[
T_0 = T \left(1 + \frac{1.4 - 1}{2} (0.77)^2 \right)
\]
First, we calculate \( \frac{1.4 - 1}{2} \):
\[
\frac{1.4 - 1}{2} = 0.2
\]
Now compute \( (0.77)^2 \):
\[
(0.77)^2 = 0.5929
\]
Substituting this back in:
\[
T_0 = 288.15 \left(1 + 0.2 \times 0.5929 \right)
\]
\[
T_0 = 288.15 \left(1 + 0.11858 \right)
\]
\[
T_0 = 288.15 \left(1.11858 \right)
\]
\[
T_0 \approx 322.39 \, K
\]
### Final Answer
The stagnation temperature \( T_0 \) at this point is approximately **322.39 K**.
If you have any specific values for freestream temperature and pressure, feel free to share, and I'll adjust the calculation accordingly.