HGL and TEL definition and sketch

Definitions

• Hydraulic Gradient Line (HGL) is the line joining the levels to which water would rise in piezometer tubes along the pipe. Its height above the datum is the piezometric head $\dfrac{p}{\gamma} + z$, where $z$ is the elevation head (height of the pipe axis above the datum) and $\dfrac{p}{\gamma}$ is the pressure head.
• Total Energy Line (TEL) (also called EGL) is the line showing the total head $\dfrac{p}{\gamma} + z + \dfrac{V^2}{2g}$. It lies above the HGL by the velocity (kinetic energy) head $\dfrac{V^2}{2g}$.

Features for a real fluid

• The TEL always falls in the direction of flow because of head losses. Its slope equals the energy gradient $S_f = h_f/L$.
• In a uniform pipe $V$ is constant, so the HGL is parallel to the TEL, a fixed distance $V^2/2g$ below it.
• Minor losses (entrance, expansion, valve, bend) show as sudden vertical drops of the TEL.
• Where the HGL is below the pipe axis the pressure is below atmospheric (negative); where it is above, the pressure is positive.
• A pump gives a sudden rise of both lines; a turbine gives a drop.
• At a free outlet to the atmosphere the HGL meets the pipe axis (gauge pressure zero). At a reservoir both lines coincide with the free surface ($V = 0$).

Uniform pipe (constant diameter)

 TEL  (reservoir surface)
  \__ entrance loss 0.5V²/2g
      `-.__
           `-.__ TEL (slope = hf/L)
  ^             `-.__
  | V²/2g (same all along)  `-.__
  v                              `-.__
 HGL ......`-.__                      (parallel to TEL)
                `-.__
                     `-.__
 pipe axis ===========================> outlet (HGL meets axis)
   ^ z (elevation head)
 datum ------------------------------------------

The TEL and HGL are two parallel sloping lines; the gap between them is constant at $V^2/2g$. The height of the HGL above the pipe axis is the pressure head $p/\gamma$, and the height of the axis above the datum is $z$.

Non-uniform pipe (diameter changes from $D_1$ to $D_2 > D_1$)

 TEL  ---.__
            `-.__          steep slope (small pipe)
                 `-.__
   ^                  `-.__|
   | V1²/2g                |  drop (V1-V2)²/2g
   v                       `-.__
 HGL  ---.__                    `-.__ TEL, flatter slope
            `-.__                    ^
                 `-.__               | V2²/2g (smaller gap)
                      `-.__ |   ____ v
                           `+--'     `-.__ HGL
 axis ==========[ D1 ]=====|==[ D2 > D1 ]=========

• In the smaller pipe the velocity head is large, so the HGL is far below the TEL and the slope $h_f/L$ is steeper.
• At a sudden expansion the TEL drops by $(V_1 - V_2)^2/2g$. As $V$ falls, the velocity head $V^2/2g$ falls too, so the HGL can rise (pressure recovery) even though energy is lost.
• In the larger pipe the TEL is flatter and the gap between the two lines is smaller.
• In a gradual contraction or expansion the lines change slope smoothly instead of jumping.