Manometry & Hydrostatic Pressure

Use static fluid columns to turn height differences into pressure differences.

fluid-staticspressuremeasurement

⏱️ About 18 min

A few centimeters of liquid in a glass tube can measure thousands of pascals—no electronics required.

💡

The big idea: In a fluid at rest, pressure increases with depth. Manometers exploit this to measure pressure differences with geometry and density.

🎯 By the end, you'll be able to

Apply the hydrostatic pressure relation to vertical depth changes
Set up manometer balance equations using a consistent sign convention
Compute pressure differences from U-tube readings and fluid densities
Explain when an inclined manometer improves resolution

📎 Helpful to know first

Density & Viscosity of Newtonian Fluids

Hydrostatic pressure in a resting fluid

For a fluid at rest with approximately constant density, pressure increases linearly with depth. Moving downward a vertical distance h increases pressure by ρgh.

This result is the backbone of manometry: differences in column heights correspond to differences in pressure.

\[ P = P_0 + \rho g h \]

Hydrostatic pressure increase with depth h below a reference level.

🔑 Sign convention that prevents mistakes

Pick a path through the manometer and keep this rule: going downward increases pressure by +ρgΔh; going upward decreases pressure by −ρgΔh. Switch ρ when you move into a different fluid.

Common manometer types

U-tube manometer: compares an unknown pressure to a reference (often atmosphere).

Differential manometer: connects to two points and measures their pressure difference.

Inclined manometer: uses a shallow angle so a small pressure change produces a larger readable length change.

U-tube manometer: the height difference h in a dense manometer fluid corresponds to a pressure difference.

Relating height difference to pressure difference

For a simple U-tube with the same fluid in both legs above the manometer fluid (or negligible gas density), the pressure difference is often approximated by ΔP = ρ_m g h, where ρ_m is the manometer fluid density.

If multiple fluids are involved, write the pressure balance step-by-step through each segment.

\[ \Delta P = \rho_m g h \]

Common special case (dense manometer fluid, negligible gas density).

📝 Worked example: A U-tube manometer containing mercury (ρ = 13534 kg/m³ at 20°C) is connected between a pressurized gas line and atmosphere. The mercury level difference is h = 120 mm (gas side lower mercury level). Assume gas density is negligible. What is the gauge pressure in the gas line?

Convert height: h = 120 mm = 0.120 m.
For this configuration, gauge pressure equals ΔP = ρHg g h.
ΔP = (13534 kg/m³)(9.81 m/s²)(0.120 m) = 15932 Pa.
Convert to kPa: 15932 Pa ÷ 1000 = 15.93 kPa.

✓ 15.93 kPa (gauge)

✏️ Practice: A vertical water manometer (ρ = 998 kg/m³) shows a height difference of 0.35 m between the two legs. Assuming negligible gas density, what pressure difference does this correspond to?

kPa

Solution

Use ΔP = ρ g h.
ΔP = (998 kg/m³)(9.81 m/s²)(0.35 m) = 3427 Pa.
Convert to kPa: 3427 Pa ÷ 1000 = 3.43 kPa.

Check your understanding

1. In a static fluid with constant density, pressure changes with depth according to:

Moving downward increases pressure by ρgh, so P = P0 + ρgh.

2. Why are dense fluids like mercury used in manometers?

Higher density means the same pressure corresponds to a smaller height.

3. An inclined manometer is most useful because it:

A shallow angle converts a small vertical rise into a larger length along the tube.

✅ Key takeaways

In static fluids with constant density, pressure increases linearly with depth
Manometers convert a measurable height difference into a pressure difference
A consistent sign convention (down: +ρgΔh, up: −ρgΔh) prevents errors
Dense manometer fluids allow large pressures to be measured with small height changes

➡️ Next you’ll connect these hydrostatic ideas to practical devices like Bourdon gauges and pressure transducers.

Want to test yourself on this? Try the Chemical Aptitude test →