How this calculation works
The vertical surface-tension component around the tube balances the liquid-column weight. Contact angle below 90° gives a rise; above 90° gives a depression relative to the outside surface.
h = 2γcosθ/(ρgr)A worked example
With surface tension 72 mN/m, contact angle 0°, density 1,000 kg/m³, radius 0.5 mm and g = 10 m/s², the ideal height is 28.8 mm.
Good to know
Ideal clean narrow circular tube in equilibrium with a large reservoir. Radius is not diameter. Fluid properties and wetting angle are supplied; contamination, dynamic wetting, evaporation and pore networks are excluded.
Reference: OpenStax: surface tension and capillary action (opens in a new tab) ↗
The contact angle determines the sign
Surface tension acts around the contact line. Its vertical component depends on the cosine of the contact angle measured through the liquid. Below 90° the model gives a rise; above 90° it gives a depression.
At exactly 90°, the vertical component is zero and the modeled height difference is zero. This special case is displayed as zero rather than a tiny artifact from floating-point trigonometry.
The hydrostatic pressure calculator evaluates the liquid-column pressure that balances surface tension in the capillary model.
A smaller radius supports a taller column
The ideal height is inversely proportional to tube radius. Halving radius doubles height when the other inputs are fixed. The default 72 mN/m, 0° contact angle, 1,000 kg/m³ density, 0.5 mm radius and g = 10 give 28.8 mm.
The hydrostatic pressure calculator relates column height to density and gravity. Capillary rise supplies a surface-tension pressure difference that balances that hydrostatic column under the equilibrium assumptions.
Radius and diameter are easy to confuse
The formula uses radius. Entering a measured diameter in that field doubles the denominator and halves the predicted height. Surface tension is entered in mN/m and converted to N/m separately.
A clean narrow tube and a large surrounding reservoir are assumed. Real contact angles can vary with contamination and wetting history; porous materials contain networks rather than one ideal cylindrical tube. This static result does not give a wicking time.
A few questions, answered
Can the result be negative?
Yes. A contact angle above 90° gives a modeled depression below the outside liquid surface.
Does zero surface tension give any rise?
No. The modeled capillary pressure and height are zero.
Can I use it as a soil-water model?
A soil pore network has additional geometry and behavior; this page represents one ideal circular capillary tube.