Pressure energy in a liquid at rest

We have in a roundabout way come to recognise pressure energy but we have done it for a liquid that is flowing steadily. Now we must decide whether the idea can be extended to liquid that is at rest. Figure 3-6 shows a tank filled with liquid that is at rest. An element of liquid having volume  and having the shape of a thin disc forms part of the free surface. The depth of liquid in the tank is  and so the potential energy of the element relative to the bottom of the tank is  where r is the density of the liquid.

The element could be moved very slowly to a new position at a depth of  and, as the movement involves no net force outside the liquid, no work would be done during the movement. This means that the element has lost potential energy equal to  without doing any work nor apparently gaining any other form of mechanical energy in exchange. As this statement is contrary to the law of the conservation of energy we must again face the need to look for, and try to recognise, a form of mechanical energy other than potential energy and kinetic energy. As the only observable change in the water in the element is the rise in its pressure, we must expect this new form of energy to be associated with pressure.

The absolute pressure at the surface is the atmospheric pressure  and so, for the element when it is on the surface, the product of pressure and volume is . At depth  the absolute pressure is  and the product of pressure and volume is . Clearly, for the element, the product of pressure and volume has increased by  which is numerically equal to the loss of potential energy. It is also equal to the mass of the element times the same expression as we used to define pressure energy/unit mass. So it seems that the concept of pressure energy, which has been devised to permit us to quantify the energy entering or leaving a closed system, can also be used to quantify the energy in store in a liquid at rest. However it is still only true because the liquid is continuous and able to flow in the gravitational field.

When the element is at depth  its potential energy relative to the datum is . Our interest then lies in the fact that the sum of this potential energy and the pressure energy possessed by the element is:-

and that this is the same wherever the element may be in the tank. We can deduce that in a liquid at rest the sum of the potential energy/unit weight and the pressure energy/unit weight at every point is the same and equal to H. These two forms of energy are interchangeable.