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.
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...).
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 :)