Capillary Action Calculator
Calculate capillary rise or depression, surface tension, contact angle, tube inside diameter, or liquid density using Jurin’s law, with capillary-pressure checks and mixed-unit conversion.
Calculator is for informational purposes only. Terms and Conditions
This equilibrium cylindrical-capillary model gives positive height for capillary rise and negative height for capillary depression; it is most appropriate when the tube radius is small relative to the capillary length.
Choose the calculation setup
Select the unknown and a convenient display-unit set. Existing physical values are converted rather than reinterpreted.
Enter the known values
Use properties for the actual liquid, surface, and temperature. Contact angle is a liquid-solid-vapor property, not a liquid-only constant.
Fields marked required must be completed. Scientific notation such as 5e-4 is accepted. Height is signed: positive means rise and negative means depression.
Result
Primary answer first, followed by pressure, geometry, model checks, warnings, and calculation steps.
Result details
- Check—
Show calculation steps Review conversions, equation rearrangement, substitution, assumptions, and reverse checks
- Enter valid values to see the complete calculation.
Capillary rise and meniscus
The diagram changes between rise, neutral, and depression states and shows the signed height relative to the reservoir level.
Method, Sources, and Assumptions
Calculation basis, references, active assumptions, model limitations, and verification guidance.
The calculator balances the vertical surface-tension force against hydrostatic weight using Jurin’s law. Standard gravity is 9.80665 m/s² by convention; fluid properties and contact angle are user inputs because they depend on the actual system.
- The tube is treated as a uniform circular capillary and the meniscus is at static equilibrium.
- The model uses one equilibrium contact angle and constant liquid density and surface tension.
- Positive height denotes rise; negative height denotes depression. At 90° contact angle, the ideal capillary height is zero.
- Use measured or authoritative fluid and surface data for real engineering work; this calculator does not model dynamic wetting, contact-angle hysteresis, evaporation, porous-media tortuosity, or noncircular pores.
Calculator guide
What the Capillary Action Calculator Tells You
The Capillary Action Calculator above solves the equilibrium capillary relationship for capillary height, surface tension, contact angle, tube inside diameter, or liquid density. For the standard Earth-gravity case, enter the known surface tension, contact angle, inside diameter, and density; standard gravity is already loaded. A positive height means capillary rise, while a negative height means capillary depression.
The calculator uses Jurin’s law for a uniform circular tube. It is best suited to a static or equilibrium capillary problem, not to predicting how quickly a liquid wets a tube, paper, soil, or another porous material.
- Best for
- Equilibrium capillary rise or depression and reverse-solving the main Jurin variables.
- Primary relationship
- Jurin’s law, using tube inside diameter and a signed capillary height.
- Key limitation
- The cylindrical-tube equilibrium model does not reproduce real porous-media geometry or transient wicking.
How to Use the Calculator
Choose the quantity you want to solve for, then enter only the values that remain known. The calculator converts selected display units to a common internal unit system before applying the equation.
Choose the unknown
The Solve For control supports capillary rise or depression, surface tension, contact angle, tube inside diameter, and liquid density. The required fields and result units update with the selected mode.
Enter properties for the actual liquid and surface
Use liquid density and surface tension at conditions relevant to the problem. Enter the contact angle measured through the liquid at the tube wall; it represents the liquid-solid-vapor system, so it should not be treated as a universal property of the liquid alone.
Enter the tube inside diameter
The calculator uses inside diameter \(d\), not radius \(r\). If a reference gives radius, convert with \(d=2r\) before entering the geometry.
Read the sign as part of the answer
For height, a positive result is capillary rise and a negative result is capillary depression. The result details also report capillary pressure, the hydrostatic pressure check, capillary length, meniscus curvature radius, and the tube-radius-to-capillary-length ratio.
Capillary Action Formula and Jurin’s Law
Jurin’s law relates the equilibrium height of a liquid column to surface tension, wetting angle, liquid density, gravity, and tube size. The calculator uses the diameter form because inside diameter is usually the directly measured tube dimension.
Capillary height using inside diameter
Plain language: signed capillary height equals four times surface tension times the cosine of contact angle, divided by density, gravity, and tube inside diameter.
The equivalent radius form is \(h=2\gamma\cos\theta/(\rho g r)\), because \(d=2r\). MIT OpenCourseWare derives the same equilibrium relationship from a force balance at the contact line and discusses capillary rise and fall in a tube.
Capillary pressure check
At equilibrium, the signed capillary pressure associated with the curved interface balances the signed hydrostatic pressure difference across the liquid column.
Capillary length
Capillary length is a characteristic scale comparing surface-tension and gravitational effects. Jurin’s narrow-capillary approximation is strongest when tube radius is much smaller than this length; the calculator reports the ratio so the user can judge how closely the geometry matches that assumption rather than treating one universal ratio as a hard cutoff.
- \(h\)
- Signed equilibrium capillary height relative to the outside liquid level; SI unit: m.
- \(\gamma\)
- Liquid-vapor surface tension; SI unit: N/m.
- \(\theta\)
- Contact angle measured through the liquid at the wall; entered in degrees or radians.
- \(\rho\)
- Liquid mass density; SI unit: kg/m³.
- \(g\)
- Gravitational acceleration; SI unit: m/s².
- \(d\)
- Tube inside diameter; SI unit: m.
| Solve for | Equation | Important domain check |
|---|---|---|
| Capillary height \(h\) | \(h=\dfrac{4\gamma\cos\theta}{\rho gd}\) | \(\gamma\), \(\rho\), \(g\), and \(d\) must be positive. |
| Surface tension \(\gamma\) | \(\gamma=\dfrac{h\rho gd}{4\cos\theta}\) | \(\cos\theta\neq0\), and the solved surface tension must be positive. |
| Contact angle \(\theta\) | \(\theta=\cos^{-1}\!\left(\dfrac{h\rho gd}{4\gamma}\right)\) | The inverse-cosine argument must lie from \(-1\) through \(+1\). |
| Inside diameter \(d\) | \(d=\dfrac{4\gamma\cos\theta}{\rho gh}\) | \(h\neq0\), and the solved diameter must be positive. |
| Liquid density \(\rho\) | \(\rho=\dfrac{4\gamma\cos\theta}{gdh}\) | \(h\neq0\), and the solved density must be positive. |
Worked Example: Capillary Rise in a 0.5 mm Tube
Consider a liquid with surface tension \(72\ \mathrm{mN/m}\), density \(1000\ \mathrm{kg/m^3}\), and a \(0^\circ\) contact angle in a circular tube with a \(0.5\ \mathrm{mm}\) inside diameter. These are example inputs for demonstrating the calculation, not universal property values for a named fluid.
Convert the units
Substitute the values
Result
\(h\approx58.74\ \mathrm{mm}\)
The positive sign indicates capillary rise: the equilibrium meniscus is predicted about 58.74 mm above the outside reservoir level under the stated idealized conditions.
How to Interpret Capillary Rise, Depression, and Pressure
The magnitude tells you how far the equilibrium meniscus is displaced from the outside reservoir level; the sign tells you the direction. A result can be mathematically valid while still being a poor physical model if the tube, interface, or fluid behavior violates the assumptions behind Jurin’s law.
| Contact angle | Sign of \(\cos\theta\) | Predicted behavior |
|---|---|---|
| \(0^\circ\le\theta<90^\circ\) | Positive | Capillary rise |
| \(\theta=90^\circ\) | Zero | No ideal rise or depression |
| \(90^\circ<\theta\le180^\circ\) | Negative | Capillary depression |
Rise, neutral, or depression
With all other inputs fixed, \(\theta<90^\circ\) gives rise, \(\theta=90^\circ\) gives zero ideal height, and \(\theta>90^\circ\) gives depression. A negative height is therefore a directional result, not automatically an error.
Diameter sensitivity
Holding \(\gamma\), \(\theta\), \(\rho\), and \(g\) constant, \(h\propto1/d\). Halving the tube diameter doubles the ideal height magnitude; doubling the diameter halves it.
Fast pressure check
For a valid equilibrium result, the calculator’s capillary pressure and \(\rho gh\) should agree in signed value apart from displayed rounding. A mismatch in a manual calculation usually points to a unit or diameter-versus-radius error.
| Quantity | Relationship | Practical interpretation |
|---|---|---|
| Surface tension \(\gamma\) | \(h\propto\gamma\) | Higher surface tension increases the height magnitude in the ideal model. |
| Inside diameter \(d\) | \(h\propto1/d\) | Smaller circular tubes produce larger equilibrium height magnitudes. |
| Density \(\rho\) | \(h\propto1/\rho\) | Higher density increases the gravitational load per unit height. |
| Gravity \(g\) | \(h\propto1/g\) | Lower gravity increases the equilibrium height magnitude, all else fixed. |
| Contact angle \(\theta\) | \(h\propto\cos\theta\) | The cosine controls both magnitude and whether the result is rise or depression. |
Common Capillary Action Calculation Mistakes
Most large errors come from geometry, units, or applying an equilibrium model to the wrong physical question. These checks catch the failures most likely to change the answer by an order of magnitude or reverse its sign.
Using radius where diameter is expected
The calculator uses inside diameter. Entering a radius directly as diameter makes the entered diameter half the intended value and doubles the calculated capillary-height magnitude.
Entering millimeters as meters
A tube diameter of \(0.5\ \mathrm{mm}\) is \(0.0005\ \mathrm{m}\), not \(0.5\ \mathrm{m}\). Use the unit selector rather than manually changing the unit label without changing the number.
Treating negative height as invalid
For contact angles greater than \(90^\circ\), \(\cos\theta\) is negative and Jurin’s law predicts depression. The calculator intentionally preserves that sign.
Using a liquid-only contact angle
Contact angle depends on the liquid, solid surface, and interface condition. Surface contamination, coatings, roughness, and wetting history can make a nominal literature value inappropriate for a specific experiment.
Mixing property conditions
Surface tension and density can vary with temperature and composition. Use properties that represent the same fluid state instead of combining unrelated reference conditions.
Using Jurin’s law for filling time
Jurin’s law predicts an equilibrium height. A time-dependent question requires a dynamic capillary-flow model that includes viscous effects and any other relevant forces.
Jurin’s Law vs. Young-Laplace vs. Lucas-Washburn
These relationships describe connected parts of capillary physics, but they answer different questions. Choosing the model based on the desired output is more useful than treating the equations as interchangeable.
Jurin’s law
Use it for the equilibrium rise or depression height in an idealized circular capillary. This is the governing model used by the calculator above.
Young-Laplace pressure
Use it to relate surface tension and interface curvature to pressure difference. In the ideal circular-tube equilibrium represented here, that pressure difference is consistent with \(\Delta P=\rho gh\).
Lucas-Washburn-type models
Use them for time-dependent capillary penetration or filling under their stated assumptions. Peer-reviewed microfluidic literature discusses Lucas-Washburn modeling as a dynamic capillary-flow framework rather than an equilibrium-height equation.
Porous-media models
Paper, soil, rock, textiles, and wick structures contain networks of nonuniform pores rather than one perfect tube. A single equivalent pore size can be useful for interpretation, but it does not fully represent tortuosity, pore-size distributions, saturation, or dynamic wetting.
Assumptions and Limits of Jurin’s Law
The equation is a compact equilibrium model. Its answer is most defensible when the real system resembles the circular, uniform, static capillary assumed in the derivation.
Static equilibrium
The calculation does not predict the time required to reach the reported height and does not include the transient viscous or inertial terms that control filling dynamics.
Uniform circular geometry
The diameter form assumes a cylindrical tube with a uniform inside diameter. Noncircular channels and changing cross-sections require a geometry-specific treatment.
Single contact angle
The model uses one equilibrium contact angle. Advancing and receding angles, contact-angle hysteresis, surface roughness, and contamination are not modeled.
Constant fluid properties
Density and surface tension are treated as constant inputs. Temperature gradients, concentration changes, reactions, evaporation, and multiphase effects can invalidate that simplification.
Small capillary relative to capillary length
MIT’s capillary-rise treatment identifies the narrow-tube limit as important to the Jurin approximation. The calculator reports tube radius divided by capillary length as a diagnostic, but this guide does not assign one unsourced universal ratio as a hard pass/fail limit.
Porous materials are not one tube
The U.S. Geological Survey describes capillary action in porous materials, but real soils and rocks contain connected pore networks. Do not interpret a single-tube result as an exact capillary-fringe prediction without a model appropriate to the porous medium.
Sources and Calculation Checks
The governing relationship, sign convention, physical interpretation, porous-media cautions, and gravity constant were checked against authoritative educational and government references. The worked example was also verified independently through the pressure balance \(\Delta P=\rho gh\).
- MIT OpenCourseWare — Interfacial Phenomena, Lecture 8: Capillary Rise — supports Jurin’s law, capillary rise and fall, the contact-line force balance, and the narrow-capillary framework.
- OpenStax College Physics 2e — Surface Tension and Capillary Action — supports the capillary-action, contact-angle, wetting, and rise-versus-fall interpretation.
- NIST Guide to the SI — Appendix B.9 — lists standard acceleration of free fall as \(9.80665\ \mathrm{m/s^2}\).
- U.S. Geological Survey — Capillary Action and Water — supports the description of capillary action in porous materials and the roles of adhesion, cohesion, and surface tension.
- Capillary flow of liquids in open microchannels — peer-reviewed open-access discussion of capillary-flow dynamics and Lucas-Washburn-based modeling.
The calculator uses a closed-form equilibrium equation rather than an empirical correlation or iterative solver. Reverse-solve modes are algebraic rearrangements of the same Jurin relationship, with domain checks applied where a positive physical property or a real inverse cosine is required.
Capillary Action Calculator FAQs
These answers address the most common interpretation and formula questions that remain after using the calculator.
What is the formula for capillary rise?
For a circular tube of radius \(r\), Jurin’s law is \(h=2\gamma\cos\theta/(\rho g r)\). Using inside diameter \(d=2r\), the same relationship is \(h=4\gamma\cos\theta/(\rho g d)\).
Why does a smaller tube produce greater capillary rise?
With surface tension, contact angle, density, and gravity held constant, Jurin’s law gives \(h\propto1/d\). A smaller diameter therefore increases the ideal equilibrium height magnitude.
Why is my capillary height negative?
A negative height means capillary depression. In the ideal equation, this occurs when the contact angle is greater than \(90^\circ\), making \(\cos\theta\) negative.
What happens at a 90° contact angle?
Because \(\cos90^\circ=0\), the ideal Jurin equation gives zero capillary rise or depression. Reverse-solving surface tension at exactly \(90^\circ\), or solving diameter or density from a zero-height condition, becomes singular in this formulation.
Why can a contact-angle calculation have no real solution?
The contact-angle rearrangement requires the inverse-cosine argument \(h\rho gd/(4\gamma)\) to lie between \(-1\) and \(+1\). A value outside that domain means the entered measurements are not mutually consistent with the ideal Jurin model.
What is capillary pressure?
Capillary pressure is the pressure difference associated with the curved liquid interface. For the ideal circular-tube model used here, the signed pressure is \(\Delta P=4\gamma\cos\theta/d\), and at equilibrium it matches the signed hydrostatic difference \(\rho gh\).
Does temperature affect capillary rise?
Temperature can change fluid properties such as surface tension and density and can also affect wetting behavior. Jurin’s equation does not contain temperature as a separate variable, so use property values and contact-angle data appropriate to the actual conditions.
Is capillary rise the same as wicking speed?
No. Jurin’s law gives an equilibrium height, while wicking or filling speed is a transient flow problem. Lucas-Washburn-type models include viscous resistance and are used for time-dependent capillary penetration under their applicable assumptions.
Can Jurin’s law be used for soil or paper?
It can provide an idealized equivalent-capillary interpretation, but soil, paper, rock, and textiles contain networks of pores with varying size, shape, connectivity, and wetting behavior. A single cylindrical-pore result should not be treated as an exact porous-media prediction.