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Homework Help: Work done by pump under water

  1. Sep 18, 2014 #1
    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.


    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
  2. jcsd
  3. Sep 18, 2014 #2


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    Science Advisor

    I think 1 bar = 100 000 Pa = 100 000 N/m2.

    Well, it amounts to the same thing as what you've done, but you could think of it this way:

    Your container is basically a cylinder with height 1 m and cross-sectional area 1 m2.
    Your pump needs to push the water out against the ambient pressure, which would be just like pushing a piston into the cylinder to push out the water.

    The force is therefore F = P * area of piston, and the total work would be F*(height) = P(area)(height).
  4. Sep 18, 2014 #3
    To understand what is happening mechanistically, note that, as soon as you pump the slightest amount of water out of the volume, the pressure within the volume will drop to 1 atm. This is because the volume is connected by a column of air (with negligible static head) directly to the air at the surface. So the water inside the volume is being pumped from a pressure of 1 atm (1 bar) to a pressure of 11 bars (static head + surface pressure) at depth.

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