Total energy line and hydraulic gradient line

Total energy line (TEL or EGL)

The total energy line is the line joining the total heads $\left(\frac{p}{\rho g} + \frac{V^2}{2g} + z\right)$ at points along the pipe, plotted above the datum.

• It always falls in the direction of flow (due to friction and minor losses), except where a pump adds energy.
• It has sudden drops at entry, sudden expansion, contraction, valves and bends.

Hydraulic gradient line (HGL)

The hydraulic gradient line joins the piezometric heads $\left(\frac{p}{\rho g} + z\right)$, i.e. the levels to which water would rise in piezometers fitted along the pipe.

• It lies below the TEL by the velocity head $V^2/2g$.
• It is parallel to the TEL in a uniform pipe; it can rise where the pipe enlarges (velocity falls).
• Where the pipe rises above the HGL, the pressure is negative (siphon or cavitation risk).

Sketch: two reservoirs joined by a pipe with a sudden enlargement

 ===== reservoir 1
  \__  entry loss 0.5 V1^2/2g
     \___  TEL (steep in small pipe)
  HGL  \_ \___
   (V1^2/2g) \_|  drop (V1-V2)^2/2g
        \___   |____  TEL (flatter)
            \__ ^    \______
               \| HGL rises   \___  exit loss V2^2/2g
                |\____ HGL ______\_|
                |                  ===== reservoir 2
  small pipe d1 |  large pipe d2
 ===============+==================

• At the entrance, the TEL drops by the entry loss $0.5V_1^2/2g$ and the HGL drops further by $V_1^2/2g$.
• Along each pipe both lines slope down by $h_f = fLV^2/2gD$; the slope is steeper in the small pipe.
• At the sudden enlargement, the TEL drops by $(V_1 - V_2)^2/2g$ but the HGL rises, since velocity head falls.
• At the exit, the whole velocity head $V_2^2/2g$ is lost; the HGL meets the lower reservoir surface.