Hydraulic and energy grade lines

Definitions

• Energy grade line (EGL): line showing the total head at each section: $H = \dfrac{p}{\rho g} + \dfrac{V^2}{2g} + z$.
• Hydraulic grade line (HGL): line showing the piezometric head: $\dfrac{p}{\rho g} + z$. It is the level to which water would rise in a piezometer tube.

$$\text{EGL} = \text{HGL} + \frac{V^2}{2g}$$

For a pipe of uniform diameter, the EGL and HGL are parallel and the vertical gap equals the velocity head.

Sketch (lower reservoir, pump, upper reservoir)

 head
  ^                                    upper reservoir
  |                                   ___ EGL = HGL
  |                           ,------'
  |                         ,'  <- Hp added by pump
  | ____ EGL = HGL         /
  |      `--._______      |   (EGL falls with friction,
  |  lower res.     `-.___|    HGL one V^2/2g below EGL)
  +-------------------[P]---------------------> x

Features

1. At a reservoir: velocity is nearly zero and pressure is atmospheric (gauge zero), so EGL and HGL both coincide with the free surface level.
2. Along the pipe: the EGL falls continuously because of friction losses, with slope $h_f/L = fV^2/(2gD)$. The HGL falls parallel to it for constant diameter.
3. At an entrance or valve (minor loss): the EGL drops suddenly by $K\,V^2/2g$.
4. At a pump: the pump adds head $H_p$, so both EGL and HGL rise suddenly by $H_p$ across it.
5. At a turbine: the turbine extracts head $H_t$, so both lines fall suddenly by $H_t$.
6. At a change of diameter: the gap between EGL and HGL changes, as $V^2/2g$ changes.
7. At a free discharge: the HGL passes through the centre of the outlet (pressure atmospheric) and the EGL lies a velocity head above it.
8. Pressure: the pressure head at any point is the vertical distance between the pipe centre line and the HGL. If the pipe is above the HGL, pressure is below atmospheric (negative); if it falls to vapour pressure, cavitation or siphon breaking can occur.