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I've been trying to solve this (exam) question but keep getting wrong answers.

I need to calculate P1 (pressure before pump) and P2 (pressure after pump), the pump drives a fountain.

schematic view of fountain

Given information about the fountain:

The fountain is driven by 2 turbines, the two shafts at the bottom are at the same height (z1=z2) and have the same cross-section/diameter. The turbines each do half of the required work. Assume pressure/velocity are uniform in each cross-section.

Max_height_water = Ztop = 312 [meter]

A_turbine_tube = 225 [cm^2]

A_exit_tube = 60 [cm^2]

Z_turbine_tube = -3.5 [meter]

Z_exit_tube = 0 [meter]

P_exit = atmospheric pressure = 100 [kPa]

Rho_water = 998 [kg/m^3]

Velocities of water in the tubes:

Velocity_turbine_tube = 10.4 [m/s]

Velocity_exit_tube = 78.2 [m/s]

I have acces to my professor's solution to the problem (how he calculated P1 and P2). I understand how he calculated P2 (using bernoulli's equation).

When calculating P1 it looks likes once again he uses bernoulli's equation but this time the term 1/2*rho*v^2 is not present on the right hand side of the equation (which would make sense if the velocity were equal to zero at the exit point, the velocity is 78.2 m/s at this point however...).

enter image description here

Judging from the 'water surface' note the professor added to his solution it appears he is looking at point 1 (before turbine) and the exit point of the fountain (@ z=0).

I have tried finding how to apply bernoulli's eq. when there is a pump in between the two points of interest but found nothing which looked similar to my prof's solution.

Can anyone explain why the formula my prof uses to find P1 is correct (if it is indeed correct) and how to derive this formula from bernoulli's or another equation. Or if the formula is incorrect please explain how P1 can be determined.

Thank you very much :)

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closed as off-topic by John Rennie, Jon Custer, Yashas, honeste_vivere, Kyle Kanos Aug 13 '17 at 11:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – John Rennie, Jon Custer, Yashas, honeste_vivere, Kyle Kanos
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  • $\begingroup$ The pump vendor will have all of this information. You decide what upstream and downstream pressure you need, flow rate, etc., talk to a pump vendor, and he will sell you the pump that will do the job. $\endgroup$ – David White Aug 11 '17 at 16:32
  • $\begingroup$ Thats a good idea. As per you, the prof is the pump vendor?? ;-) $\endgroup$ – Wrichik Basu Aug 11 '17 at 18:12
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The water is being circulated by the pump, so water from the pool is flowing the the pump inlet. At the top of the pool, the pressure is 100 kPa, the velocity is zero, and the elevation is zero. At the pump inlet, the pressure is P1, the velocity is 10.4 m/s, and the elevation is -3.5 m. That is the motivation for your professor's equation.

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