What is pressure energy? Derive an expression for it.

Pressure energy is the energy possessed by liquid by virtue of pressure.

To derive the expression for pressure energy, take a wide tank connected to a narrow pipe fitted with frictionless piston at the base as shown in figure.
                          

Let A and a be the area of cross-sections of tank and pipe respectively.
By holding the piston, fill the tank to a height h above the pipe with a liquid of density ρ.
The pressure of liquid on the piston is,

                        P = ρgh

Force on the piston in outward direction is,
                       F= ρgha

Force to be exerted on the piston to keep it in equilibrium is,

                       F = ρgha

As A >>a, therefore if the piston is pushed inward by a small distance x, the height of liquid in the tank remains constant and hence the force on the piston is constant.
The work done to push the piston inward by distance x is, W = Fx = ρghax
This work done against pressure is stored in the form of pressure energy.
This energy is stored in the volume of liquid pushed inside i.e. ax.
Therefore, pressure energy stored per unit volume of liquid is,