Bernoulli's equation in real practice

Statement

For steady, ideal, incompressible flow along a streamline, the total energy per unit weight (sum of pressure, velocity and datum heads) is constant:

$$\frac{p}{\rho g} + \frac{V^2}{2g} + z = \text{constant}$$

Assumptions

1. Ideal fluid (no viscosity, so no friction loss).
2. Steady flow.
3. Incompressible flow.
4. Irrotational flow along a streamline, with uniform velocity over the section.

Modification for real flow

Real fluids are viscous, and velocity is not uniform across a section. So:

1. A head loss term $h_L$ (friction loss $h_f = \frac{fLV^2}{2gD}$ plus minor losses at bends, valves, entry and exit) is added on the downstream side.
2. A kinetic energy correction factor $\alpha$ is applied to the velocity head ($\alpha \approx 2$ for laminar, 1.03–1.1 for turbulent pipe flow).
3. Energy added by a pump ($H_p$) or extracted by a turbine ($H_t$) is included.

$$\frac{p_1}{\rho g} + \alpha_1\frac{V_1^2}{2g} + z_1 + H_p = \frac{p_2}{\rho g} + \alpha_2\frac{V_2^2}{2g} + z_2 + H_t + h_L$$

Example: in a hydropower penstock, $H_{net} = H_{gross} - h_L$.