The exit pressure, temperature, and density are approximately 1.21 atm, 289.5 K, and 0.415 kg/m³.
Let's find the exit pressure, temperature, and density for the given flow of air through a supersonic nozzle.
We are given the following information:
Reservoir pressure (P0) = 5 atm
Reservoir temperature (T0) = 500 K
Mach number at the nozzle exit (M) = 3
where:
p is the pressure at the nozzle exit
T is the temperature at the nozzle exit
ρ is the density at the nozzle exit
γ is the specific heat ratio of air (1.4 for air)
First, we can use the first isentropic relation to solve for the exit pressure:
p/p0 = (1 + (γ - 1)M^2)^(-γ/(γ - 1))
p/5 atm = (1 + (1.4 - 1)3^2)^(-1.4/(1.4 - 1))
p/5 atm ≈ 0.242 atm
p ≈ 1.21 atm
Next, we can use the second isentropic relation to solve for the exit temperature:
T/T0 = (1 + (γ - 1)M^2)^(-1)
T/500 K = (1 + (1.4 - 1)3^2)^(-1)
T/500 K ≈ 0.579
T ≈ 289.5 K
Finally, we can use the third isentropic relation to solve for the exit density:
ρ/ρ0 = (1 + (γ - 1)M^2)^(-1/(γ - 1))
ρ/ρ0 ≈ 0.622
ρ ≈ 0.415 kg/m³ .