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

1. The problem statement, all variables and given/known data

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.

2. Relevant 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

3. 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

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# Homework Help: Bernoulli's Equation Question

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