Measuring pressure difference in a pipe

The pressure difference between two points in a pipe (or between two pipes) is measured with a differential manometer: a U-tube containing a heavy manometric liquid (usually mercury), with its two limbs connected to the two points.

U-tube differential manometer (points at different levels)

     A o                     
       |  pipe liquid S1      o B
       |                      |  pipe liquid S2
       | h_A                  | h_B
       |       h              |
   X --+-----.  .-------------+
       |  Hg |  | Hg surface
        \____|__|____/
          mercury (S_g)

Let A be $h_A$ above the datum X–X (the lower mercury surface), B be $h_B$ above the higher mercury surface, and $h$ the difference between the mercury levels.

Pressure at X–X in the left limb $= p_A + \rho_1 g h_A$. Pressure at X–X in the right limb $= p_B + \rho_2 g h_B + \rho_g g h$.

$$p_A - p_B = \rho_g g h + \rho_2 g h_B - \rho_1 g h_A$$

Both points at the same level, same liquid

If A and B are in the same horizontal pipe carrying a liquid of density $\rho$:

$$p_A - p_B = g h(\rho_g - \rho)$$

For water and mercury: $p_A - p_B = 12.6\,\rho_w g h$.

Inverted U-tube manometer

For small pressure differences, a light liquid (air or oil) is used at the top of an inverted U-tube:

$$p_A - p_B = \rho_1 g h_1 - \rho_2 g h_2 - \rho_s g h$$

where $h_1$ and $h_2$ are the depths of A and B below their interfaces with the light liquid, $h$ is the level difference between the two interfaces, and $\rho_s$ is the density of the light liquid.

Steps of measurement

1. Connect each limb to its pipe tapping, filling the leads with the pipe liquid (no air bubbles).
2. Let the manometer settle and read the level difference $h$.
3. Choose a datum at the lower interface and equate the pressures in both limbs.

Other devices: a pair of Bourdon gauges, or electronic differential pressure transducers, for large differences.