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# Appendix 2: Potential Energy of a Fluid

Consider a column of a fluid of height h and area A, as shown in Figure 11.

To raise this column of water to a total height of h, means doing work against gravity. The force required to do this is given by Equation 3, except we replace the “a” term with the acceleration due to gravity “g”: $F = M.g$

The mass of the column is given by Equation (5): $M = \rho.A.h$

So we can rewrite the equation for the force required to raise this column of a fluid against gravity as: $F=\rho.A.h.g$

Now from Equation (6), the amount of energy required to carry out this action is: $E=F.S$

In this case not all the fluid is being raised to a height h, instead the average distance moved by the fluid is h/2:

So we can write: $E_p=\rho.A.h.g.S$

where: S = h/2

This is the potential energy (Ep )of the column.

In the Bernoulli equation, the energies are represented as pressures. From Equation (4) we get: $P = F/A$

And from Equation (6): $E = F.S$

Combining these two equations we get: $P = E/(A.S)$

So if we divide the potential energy Equation (25) by (A.S) we get it in terms of pressure : $P_p=\rho.g.h$

(26)

This equation represents the potential energy of a fluid in terms of a pressure