Air enters a converging nozzle at 3.5 m/s at 1.2 kg/m3. The inlet area is 5 cm2. If the outlet area is 1.7 cm2 and velocity of 7.5 m/s, determine the volume flow rate and air density at the nozzle
outlet.
To determine the volume flow rate and air density at the nozzle outlet, we can use the principle of mass conservation and the relationship between velocity and density.
1. Calculate the mass flow rate through the nozzle:
The mass flow rate (ṁ) remains constant along a streamline in a steady flow. It can be calculated as the product of density (ρ) and velocity (V) multiplied by the cross-sectional area (A) of flow.
ṁ = ρAV
Given:
Inlet velocity (V1) = 3.5 m/s
Inlet density (ρ1) = 1.2 kg/m^3
Inlet area (A1) = 5 cm^2 = 0.0005 m^2
Using the above formula, we can calculate the mass flow rate at the inlet:
ṁ1 = ρ1 * A1 * V1
2. Determine the density at the nozzle outlet:
Since we are assuming ideal gas behavior, we can use the ideal gas law to relate density, pressure, and temperature:
ρ = p / (R * T)
Where:
p is the pressure in Pascal (Pa)
R is the specific gas constant for air (287 J/kg*K)
T is the temperature in Kelvin (K)
We don't have the pressure or temperature at the outlet, but since the flow is adiabatic (no heat transfer), we can use the isentropic flow relations to calculate the outlet density in terms of the inlet density and velocity.
The isentropic relation for density and velocity is given by:
(ρ2 / ρ1) = (A1 / A2) * (V1 / V2)^2
Given:
Outlet area (A2) = 1.7 cm^2 = 0.00017 m^2
Outlet velocity (V2) = 7.5 m/s
Using the isentropic relation, we can calculate the outlet density (ρ2) in terms of the inlet density (ρ1):
ρ2 = ρ1 * (A1 / A2) * (V1 / V2)^2
3. Calculate the volume flow rate at the outlet:
The volume flow rate (Q) is the product of density (ρ) and velocity (V) multiplied by the cross-sectional area (A) of flow.
Q = ρAV
Now that we have the outlet density (ρ2), we can calculate the volume flow rate at the outlet:
Q2 = ρ2 * A2 * V2
By calculating ṁ1 and Q2, you can obtain the required volume flow rate and air density at the nozzle outlet.