Find Mass Flow Rate and the Exit Flow Area

[Northbysouth]

Homework Statement

Air enters aone-inlet, one-exit contreol volume at 6 bar, 500K and 30m/s through a flow area of 28cm³. At the exit the pressure is 3 bar, the temperature is 456.5 K and the velocity is 300 m/s. The air behaves as an ideal gas. For steady state operation determine

a) the mass flow rate, in kg/s
b) the exit flow area, in cm³

Homework Equations

m' = dm/dt = ρvA

The Attempt at a Solution

I have the area of the cross section, A, and the velocity at both ends, but I'm not sure about the ρ.

Can I use:

ρ = p/RT to find the air density?

 

[Northbysouth]

I've managed to find the mass flow rate, using the density equation given above, I found the air density and then plugged this number into m' =ρvA

m' = (4.1803 kg/m³)(30 m/s)(28cm²)

m' = 0.351 kg/s

Does the mass flow rate stay constant? Can I rework these equations to find the Exit flow area?

 

[haruspex]

Does the mass flow rate stay constant?

Mass is a a conserved quantity. In steady state, the mass within the volume is neither increasing nor decreasing, so the mass flow rate in must equal the mass flow rate out.

 

[SteamKing]

Areas are usually measured in square cm, rather than cubic cm.

 

[Nickie]

Yes, you can use the ideal gas law to find the air density. The ideal gas law is expressed as:

pV = nRT

Where:
p = pressure
V = volume
n = number of moles
R = gas constant
T = temperature

To find the air density, you can rearrange the ideal gas law to solve for density, ρ:

ρ = (pM)/(RT)

Where:
p = pressure
M = molar mass of the gas
R = gas constant
T = temperature

In this problem, the air is behaving as an ideal gas, so you can use the ideal gas law to find the air density at both the inlet and exit conditions. Once you have the air density, you can use the mass flow rate equation, ρvA, to calculate the mass flow rate. Then, you can use the ideal gas law again to find the exit flow area, since you have all the other variables in the equation.

Hope this helps!