Young-Laplace Equation Calculator

How to use this Young – Laplace equation calculator

What this calculator does

This calculator applies the Young – Laplace relation to a spherical meniscus in a cylindrical capillary tube. It estimates the pressure jump across the liquid – gas interface, the radius of curvature of the meniscus, the corresponding liquid-column height at hydrostatic equilibrium, and a consistent inside or outside pressure. It is useful for idealized laboratory tubes, pore-scale reasoning, wick design, soil and porous-material examples, and teaching surface-tension effects. It does not replace a full multiphase-flow model: real surfaces can be rough, contact angles can exhibit hysteresis, surface tension changes with temperature and contamination, and very small geometries may require additional physics.

When to use it

Use it to check a capillary-rise experiment, estimate the suction pressure created by a pore throat, compare wetting and non-wetting liquids, or test how tube size and contact angle influence a meniscus. The U.S. Geological Survey explanation of capillary action provides a practical overview of how adhesion, cohesion, and narrow spaces combine to move water.

How to calculate

1. The calculator opens with a complete water example and an immediately available XLSX workbook. Review the live pressure, curvature, and height results before changing anything.
2. Select a Fluid preset, or choose Custom and enter your own Surface tension (γ) and Density (ρ).
3. Enter the Inner radius of tube (a) in millimeters and the Contact angle (θ) in degrees. Results update live.
4. Optionally enter Pressure inside (pᵢ) or Pressure outside (pₒ). When both are present, the calculator reports the Young – Laplace-consistent values using the outside-minus-inside sign convention.
5. Keep or replace Gravitational acceleration (g). Read the calculation breakdown, then select Download Excel for a workbook built from the current canonical values.
6. Reset clears the demonstration data, results, and workbook state. Download Excel is then disabled until a complete valid input set is entered again.

Input guide

Fluid preset is a required selection. A preset supplies typical surface tension and density values; Custom lets you type both. Surface tension (γ) is a required positive decimal in newtons per meter, such as 0.07294 N/m for water near room temperature. Raising γ increases the pressure difference and equilibrium height in direct proportion. Do not enter millinewtons per meter without converting to N/m. Density (ρ) is a required positive decimal in kilograms per cubic meter, such as 998.2 kg/m³. Higher density lowers the equilibrium height but does not change the Young – Laplace pressure itself.

Inner radius of tube (a) is required and entered in millimeters, for example 1 mm. The code converts it to meters internally. Smaller radii produce larger pressure magnitudes and larger rise or depression. Enter radius, not diameter. Contact angle (θ) is required from 0° through 180°. Angles below 90° give a positive cosine and a wetting rise under this sign convention; angles above 90° give a negative pressure and depression. At 90°, pressure and height are zero, while the spherical meniscus radius is treated as unbounded.

Pressure inside (pᵢ) and Pressure outside (pₒ) are optional signed decimal pressures in pascals. They help anchor the pressure pair; the calculator enforces Δp = pₒ – pᵢ. Entering one lets the other be derived. Gravitational acceleration (g) is a required positive value in m/s²; 9.80665 is standard gravity. Lower gravity increases the equilibrium column height for the same capillary pressure.

Output guide

Pressure difference (Δp) is the primary result in pascals. It is an idealized identity from 2γcos(θ)/a, not a recommendation. Positive values indicate outside pressure above inside pressure and capillary rise in the chosen orientation; negative values indicate depression. Meniscus radius (R) is a/cos(θ), reported in millimeters with its sign retained to communicate curvature orientation. Equilibrium height (h) is Δp/(ρg), reported in millimeters. Its magnitude is driven by surface tension, radius, angle, density, and gravity. The calculated inside and outside pressures show a pair that exactly satisfies the displayed Δp.

The summary pills repeat the selected fluid, tube radius, contact angle, and whether the result is a rise, depression, or neutral state. The breakdown table lists the cosine term, meniscus radius, pressure difference, and equilibrium height. These rows use the same model values as the result cards and workbook.

Worked example

The startup example uses pure water with γ = 0.07294 N/m, ρ = 998.2 kg/m³, a tube radius of 1 mm, θ = 20°, and g = 9.80665 m/s². First, cos(20°) = 0.9396926. The pressure difference is 2 × 0.07294 × 0.9396926 ÷ 0.001 = 137.08 Pa. The meniscus radius is 1 mm ÷ 0.9396926 = 1.0642 mm. Dividing pressure by ρg gives 0.014006 m, or 14.006 mm of capillary rise. With outside pressure set to 101,325 Pa, the consistent inside pressure is 101,187.92 Pa.