A large tank supplies air to a converging nozzle that discharges to atmospheric pressure. Assume the flow is reversible and adiabatic. For what range of tank pressures will the flow at the nozzle exit be sonic? If the tank pressure is 600 kPa (abs) and the temperature is 600 K, determine the mass flow rate through the nozzle, if the exit area is 0.00129 m².
To have sonic flow at an converging orifice, the head (tank) pressure must be (k + 1)/2]^(k/(k-1)) times atmospheric pressure. "k" is the specific heat ratio. For air with k = 1.4 , that ratio is 1.893
For more details, see
http://en.wikipedia.org/wiki/Choked_flow
To determine the range of tank pressures for which the flow at the nozzle exit will be sonic, we can use the critical flow equation.
The critical flow equation relates the cross-sectional area of the flow, the speed of sound, and the pressure in the flow. It can be written as:
A* / A = (σ / κ) ^ (1 / (κ - 1))
Where:
A* is the critical cross-sectional area
A is the actual cross-sectional area of the flow
σ is the specific heat ratio of the gas
κ is the specific heat ratio of the gas
In adiabatic and reversible flow, the specific heat ratio (κ) is equal to the ratio of specific heat capacities (cp / cv). For air, κ is approximately 1.4.
Assuming that the flow at the nozzle exit will be sonic, the critical cross-sectional area A* will be equal to the actual cross-sectional area A of the nozzle exit (0.00129 m²). Using this information, we can rearrange the equation to solve for the pressure in the flow:
P / P* = (A* / A) ^ (κ / (κ - 1))
Where:
P is the pressure in the flow
P* is the critical pressure
Since the flow is discharging to atmospheric pressure, the critical pressure P* is equal to atmospheric pressure, which can be approximated as 101.3 kPa.
Substituting the known values into the equation, we have:
P / 101.3 = (0.00129 / A) ^ (1.4 / (1.4 - 1))
Now, we can solve for the pressure P by rearranging the equation:
P = (101.3 kPa) * [(0.00129 / A) ^ (1.4 / 0.4)]
Substituting the value of A (0.00129 m²) into the equation, we can calculate the pressure P for which the flow at the nozzle exit will be sonic.
To determine the mass flow rate through the nozzle, we can use the mass flow rate equation:
ṁ = A * ρ * V
Where:
ṁ is the mass flow rate
A* is the cross-sectional area of the nozzle exit
ρ is the air density
V is the velocity of the flow
Since the flow is sonic at the nozzle exit, the velocity V will be equal to the speed of sound at the given temperature. The speed of sound can be calculated using the equation:
c = (κ * R * T) ^ 0.5
Where:
κ is the specific heat ratio (1.4 for air)
R is the gas constant for air (287 J/(kg*K))
T is the temperature (600 K)
Substituting the known values into the equation, we can calculate the speed of sound (c).
Once we have the speed of sound, we can calculate the air density (ρ) using the equation of state for ideal gases:
ρ = P / (R * T)
Substituting the known values into the equation, we can calculate the air density (ρ).
Finally, we can substitute the values of A*, ρ, and V into the mass flow rate equation to calculate the mass flow rate (ṁ).