Work done by pump under water

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

• We have a container/volume of 1m3 100 m below mean sea level,


• A pump connected to it, pumping from inside the volume with outlet in the surrounding water at same height.


• And there is an pipeline to air, with a one-way valve allowing air to be sucked down to the volume and preventing anything from moving up.
  
  State 1 the volume is filled with water
  
  State 2 the volume is filled with air

What are the forces involved and how to calculate work done by a pump going from State 1 to state 2 - pumping water out of the closed volume and into the surrounding water at same height, hence creating a vacuum which sucks air down through the pipe.



Assumptions:



Ignore efficiency of the pump

Water as incompressible fluid



2. Relevant equations



Energy = Pressure*Volume

Pressure = Density * Gravity * Height

Force = Pressure * Area

Pressure = Force/Area



3. The attempt at a solution

My initial thought is that the work done that need to be done by the pump must equal the potential energy of State 2.



This energy is: E=PV, P=DGH



Pressure = ~1000 * 9,81 * 100m = ca. 10 bar = 10 000 N/m2



Energy= 1 m3 * 10 000 N/m2 = 10 000 Nm



Delta E = Heat transfer + Work done



Assuming no heat transfer and energy in state 1 is zero:



W= 1000 N/m







This however seems to me like a derived answer, I am looking for a different method, more direct calculation of actually moving the water.



Best regards Kihel