Answer:
The isentropic stagnation pressure is approximately 201.15 kilopascals.
Step-by-step explanation:
To determine the isentropic stagnation pressure, you can use the isentropic relation for stagnation properties in a compressible flow. The isentropic stagnation pressure (P0) can be calculated using the following formula:
P0 = P * [1 + ((γ - 1) / 2) * M^2]^(γ / (γ - 1))
Where:
P0 is the isentropic stagnation pressure.
P is the static pressure (150 kPa).
γ (gamma) is the specific heat ratio or the ratio of specific heats for air, which is approximately 1.4.
M is the Mach number, which can be calculated using the formula:
M = V / a
Where:
V is the velocity of air (247 m/s).
a is the speed of sound in air, which depends on temperature and can be calculated using the formula:
a = sqrt(γ * R * T)
Where:
γ (gamma) is the specific heat ratio (1.4).
R is the specific gas constant for air (approximately 287 J/(kg·K)).
T is the temperature in Kelvin (25°C = 298.15 K).
Now, let's calculate the values step by step:
Calculate the speed of sound (a):
a = sqrt(1.4 * 287 J/(kg·K) * 298.15 K)
a ≈ 343.35 m/s
Calculate the Mach number (M):
M = 247 m/s / 343.35 m/s
M ≈ 0.719
Now, substitute the values into the formula for isentropic stagnation pressure (P0):
P0 = 150 kPa * [1 + ((1.4 - 1) / 2) * (0.719^2)]^(1.4 / (1.4 - 1))
P0 ≈ 150 kPa * [1 + (0.2 * 0.518161)]^2.4
P0 ≈ 150 kPa * [1 + 0.1036322]^2.4
P0 ≈ 150 kPa * (1.1036322)^2.4
P0 ≈ 150 kPa * 1.3409832
P0 ≈ 201.15 kPa
So, the isentropic stagnation pressure is approximately 201.15 kilopascals.