Fluids Problem - Atmospheric Tank Pressure Vent

[dropdeadmarc]

Homework Statement

Theres a atmospheric tank with a goose neck vent on the top and also an inlet pipe [a diameter 2 inch schedule 40] on the top. There is a compressed air line [from a 50 psi compressor] going into the inlet. The pressure inside the tank should not be above 2 psi. Size the diameter of the goose neck vent.

Inlet:
P = 50 psi
D_inner = 2.07 inches
ρ = 4.32 lb/ft^3

Goose Neck Vent:
P = 2 psi
D_inner = ?

Homework Equations

Bernoulli's
0.5ρV² + P = 0.5ρV² + P
Q = V*A [volumetric flow rate]
V*A = V*A [in = out]

The Attempt at a Solution

I assumed I could use bernoullis to find the V out from the compressed air line. [Pinside = 50, Poutside = 0 [atmsophere], Vin = 0, Vout=?]
I got a Velocity of 68.31300511 ft/s

From that and the cross sectional area of the 2.07in pipe I got a Q of 95.79056195 CFM.

Next I moved onto the tank. I substituted
V_inA_in/A_out = V_out into Bernoulli's and Solved for A_out.
A_out = sqrt( Q² / (V_in² - 2*ΔP/ρ)

Problem is I ended up getting the same cross sectional area as the inlet. I'm not sure where I went wrong. Perhaps incompressible flow? I appreciate any help.

 

[SteamKing]

What's wrong with having an inlet and a vent with the same area?

 

[dropdeadmarc]

Nothing. But that doesn't satisfy the problem statement or make sense with my numbers.

if Qin = Qout
VA in = VA out
and therefore Vin = Vout

if this is true, according to Bernoulli's there would be no pressure change. My P_in and P_out are different. they need to be according to the problem statement. The ΔP should equal the 2 psi, if I'm not mistaken?

 

[SteamKing]

If you are trying to maintain less than 2 psi pressure differential above atmospheric, then it seems your chosen method of solution is somewhat simplified.

I think you are ignoring the change in pressure inside the tank as air is added from the 50 psi source. It seems that there should be an additional relation such that:

net mass of air added to the tank = mass of air flowing in - mass of air flowing out of the vent

The net mass of air added must be less than the amount of air which would produce a rise of 2 psi inside the tank above atmospheric. The vent would be sized to allow enough air to flow out so there would not be a buildup of additional air in the tank.

When the air starts to flow from the 50 psi source, compressibility must be checked before assuming that Bernoulli applies.

 

[rezeew]

I would first like to commend you on your thorough and detailed approach to solving this problem. It is clear that you have a strong understanding of Bernoulli's equation and its applications.

However, I would like to point out that in this scenario, the flow is not incompressible. The compressed air from the inlet will expand as it enters the tank, resulting in a change in density and velocity. Therefore, Bernoulli's equation may not accurately represent the flow in this system.

In order to accurately determine the required diameter of the goose neck vent, I would suggest using the ideal gas law to calculate the density of the air inside the tank at 2 psi. This can then be used to calculate the required mass flow rate of air through the vent. From there, you can determine the required cross-sectional area of the vent using the equation Q = VA.

Additionally, it is important to consider the effect of friction and other losses in the system, which may require a larger vent diameter to compensate for. I would suggest using a flow rate slightly higher than the calculated value to ensure that the pressure inside the tank does not exceed 2 psi.

Overall, I believe your approach is on the right track, but taking into account the compressibility of the air and potential losses in the system will lead to a more accurate and reliable solution.