Understanding Bernoulli's Equation in a Vent with a Fan

[Red_CCF]

Hi

I made up this problem in an attempt to understand the Bernoulli equation better.

Homework Statement

If there is a constant cross section vent that is open to atmosphere on both sides and have a fan in the middle of the vent, and define state 1 to be at the inlet plane of the vent, state 2 to be at the inlet of the fan, state 3 to be at the exit of the fan, and state 4 to be right at the exit of the vent, what are the relative values of pressure and velocity at state 2-4 relative to state 1? Neglect friction/viscous and compressibility effects.

Homework Equations

Modified Bernoulli's equation:

va^2/2+pa/ρ + wfan = vb^2/2+pb/ρ where state b is after the fan state a is before entry into the fan

The Attempt at a Solution

I would think that p1 = p4 = patm (since these two states are right at the inlet and exit) and that velocity at any point in the system would be the same if density is the same at the inlet and exit (incompressible flow) by conservation of mass. In fact I would think that p1 = p2 and p3 = p4 which implies atmospheric pressure throughout given the assumption of no friction.

However, I also think that by applying the equation, p2/ρ + wfan = p3/ρ which implies that pressure of state 3 is higher than that of state 2. However, p4 should equal p3 given that all variables between these two states are the same and since p4 must equal patm, the two statements above seem to contradict which is the source of my confusion.

Thanks very much

 

[Aero51]

The fan is doing work on the system, therefore the velocities at the inlet and the exit cannot be the same. You have set up your bounds correctly, that is a good start. The next step will be to include the addition of work done by the fan into your system. The bernoulli equation will no longer be balanced. In
other words:
(P2-P1)+p/2(V2^2+V1^2)+work=0. I'm on my phone now so I can't help you more. Good luck.

 

[Red_CCF]

The fan is doing work on the system, therefore the velocities at the inlet and the exit cannot be the same. You have set up your bounds correctly, that is a good start. The next step will be to include the addition of work done by the fan into your system. The bernoulli equation will no longer be balanced. In
other words:
(P2-P1)+p/2(V2^2+V1^2)+work=0. I'm on my phone now so I can't help you more. Good luck.

Hi

Thanks for responding.

If the velocities before and after the fan are not the same, then would that mean that the flow has to be compressible or else I do not see how the mass flow rates at inlet and exit can balance? Also, at the exit the density of the fluid must be equal to that of the atmosphere, so is there a gradient in density along the vent as well if what I said above is true?

Thanks

 

[Aero51]

I should be more specific in my last post. If you wish to account for the velocities across the fan, you should employ actuator disk theory as a first approximation.

 

[Aero51]

Also, your model does not take into account the change in the velocity profile. The effective cross sectional area behind the fan is reduced thus conserving mass flow.

 

[Red_CCF]

Also, your model does not take into account the change in the velocity profile. The effective cross sectional area behind the fan is reduced thus conserving mass flow.

Hi

But if the vent is long enough, would the flow "converge" such that the effective cross sectional area becomes equal to that of the inlet and velocity decrease? If I look at this from just an inlet to exit perspective and take the vent as a control volume, I would see the same pressure at both ends but a higher velocity at the exist across the same cross sectional area as the inlet?

Thanks

 

[Aero51]

You are correct in saying that if the vent was "long enough" that the effective cross sectional area would equal to the vent area. However, modeling this flow with bernoulli's equation will be insufficient. At this point one must also take into account effects of turbulence and friction. If you chose to model this flow with potential flow, the cross sectional area of the velocity profile behind the fan will not change and mass flow will be conserved. This is why potential flow and Bernoulli's equation are only used as preliminary models.