Manometers

A manometer is a device that uses a liquid column to measure pressure differences. The simplest form is the U-tube manometer, where a dense indicating fluid (often mercury, $\rho = 13\,546\,\text{kg/m}^3$) fills the bottom of a U-shaped tube.

Simple U-Tube Manometer

For a U-tube connecting a pressure source on one side to atmosphere on the other, the gauge pressure at the source equals:

$\Delta P = \rho_m \, g \, \Delta h$

where $\rho_m$ is the manometer fluid density and $\Delta h$ is the height difference between the two fluid columns.

Differential Manometer

When connecting two pipes at different elevations, the pressure difference accounting for both the manometer fluid and the pipe fluid is:

$P_1 - P_2 = \rho_m \, g \, h_m - \rho_f \, g \, (z_2 - z_1)$

where:

• $\rho_m$ — manometer fluid density (kg/m³)
• $h_m$ — manometer reading (height difference, m)
• $\rho_f$ — pipe fluid density (kg/m³)
• $z_1, z_2$ — elevations of the two pipe centrelines (m)

When the pipes are at the same elevation ($z_1 = z_2$), this reduces to the simple formula $\Delta P = \rho_m g h_m$.

Example: Mercury Manometer

A mercury manometer ($\rho_m = 13\,546\,\text{kg/m}^3$) reads $\Delta h = 0.1\,\text{m}$:

$\Delta P = 13\,546 \times 9.81 \times 0.1 = 13\,288.6\,\text{Pa}$

Your Task

Implement:

• manometer_pressure(rho_m, delta_h) — pressure difference for a simple U-tube (Pa)
• gauge_pressure_manometer(rho_m, h_m, rho_fluid, z1, z2) — differential manometer reading (Pa)