Potential Energy of A Fluid

Question

For Bernoulli's theorem, the equation is $$P + \rho gh + \frac{\rho v^2}2 = \text{const}.$$ I know that these terms are pressure energy per volume, potential energy per volume and kinetic energy per volume.

I have a doubt: I think potential energy per unit volume should be $\rho gh/2$ ($\rho$ is density).

My derivation: Take a cuboid container of base area $A$ and fill it up to height $h$ with liquid of density $\rho$. The mass of liquid is $\rho Ah$ and its center of mass is at height $h/2$. So the potential energy is $$E = \rho A h\frac{gh} 2.$$ So, according to me, potential energy per unit volume is $$\frac{E}{Ah} = \frac{\rho gh} 2.$$

Answer 1

The terms are not the averaged energy per volume as you derive for your container, but the energy per volume for an infinitesimally small parcel of liquid at some point in the liquid (and the equation is valid along a stream line of the liquid).