Use of manometer and inverted manometer

Use of a manometer

A manometer measures pressure at a point (simple manometer) or the difference of pressure between two points (differential manometer) by balancing it against a column of liquid. Used for pipes, venturimeters, orifice meters and laboratory flow measurement; it is simple, cheap and accurate.

Inverted U-tube differential manometer

It is an inverted U-tube whose top part contains a light fluid (air or a light oil) lighter than the liquid in the pipes. It is used to measure small pressure differences, where a heavy liquid like mercury would give too small a reading.

        .---------------.
        | light liquid  |  (air or oil, rho_s)
  X ----+------.        |
        |      |  h     |
        |      '--------+---- right interface
   h_1  | liquid        |
        | rho_1         | h_2   liquid rho_2
        o A             |
                        o B

Let X–X be the left (higher) interface, A be $h_1$ below it, the right interface be $h$ lower than X–X, and B be $h_2$ below the right interface.

Working principle

Pressure at level X–X in the left limb:

$$p_A - \rho_1 g h_1$$

Pressure at level X–X in the right limb (go up $h_2$ of pipe liquid from B, then up $h$ of light liquid):

$$p_B - \rho_2 g h_2 - \rho_s g h$$

Equating:

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

If air is used at the top, $\rho_s g h \approx 0$.

Example: water in both pipes, oil $S = 0.8$ at top, $h_1 = 0.30$ m, $h_2 = 0.20$ m, $h = 0.12$ m: $p_A - p_B = 9.81(0.30 - 0.20 - 0.8\times0.12) = 0.039$ kN/m², a very small difference that an inverted manometer can read well.