If you need to know how fast a particle sinks in a fluid, Stokes’ law offers a straightforward way to estimate it—but only if you stay within its limitations. Undershoot your calculation, and your tank plugs up or underperforms; overshoot, and you waste capital on oversizing. This Settling Velocity Stokes Calculator lets you quickly work out how long a given particle actually takes to settle, provided you know the main properties—particle size, density, viscosity, and so on. This isn’t just for water plants; it matters any time you need to separate solids from fluids by gravity, from mineral processing right through to air quality assessments. Below you'll find the equations, a real design walkthrough, guidance on when the assumptions break down, and honest answers on topics like particle shape and hindered settling.
What is settling velocity?
Settling velocity is the steady speed a particle reaches when dropped in a fluid. At this point, the pull of gravity, the push of buoyancy, and the resistance from the fluid’s drag have balanced out. The particle doesn’t speed up or slow down—it just keeps falling at the same rate.
Simple Explanation
If you’ve dropped a marble into honey and then into water, you’ve seen settling velocity at work. Bigger and denser particles settle faster, and thick fluids like honey slow everything down. Stokes’ law puts numbers to this effect, but only when the particle is small and the flow around it is smooth (laminar). Once the particle or flow gets larger or faster, these simple calculations start breaking down, and you have to use different approaches.
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Diagram
Settling Velocity Stokes Calculator
How to Use This Calculator
This calculator is intended for education, concept evaluation, and preliminary design. Results are based on the equations and assumptions described on this page, but cannot account for every real-world load case, tolerance, material property, environmental condition, installation detail, safety factor, code, or regulatory requirement. Verify all inputs, assumptions, units, and results independently before selecting components or using the result in a real application. Safety-critical, structural, medical, lifting, transportation, or regulated applications must be reviewed by a qualified engineer.
- Pick the calculation mode for what you need: settling velocity, particle diameter, density, viscosity, Reynolds number, or time.
- Fill in what you know—particle diameter (m), densities (kg/m³), viscosity (Pa·s), settling distance (m) if needed.
- Check that your particle is heavier than your fluid—otherwise, it won’t settle and the calculator will tell you there’s a problem.
- Click Calculate and get the result.
Settling Velocity Stokes Interactive Calculator
Watch particles fall through fluid at their terminal settling velocity. Adjust particle size, density, and fluid properties to see how each parameter affects the balance between gravity, buoyancy, and drag forces.
SETTLING VELOCITY
9.0 mm/s
REYNOLDS NUMBER
0.90
SETTLING TIME
111 s
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Equations
To find how fast a particle settles, use the Stokes formula below. Plug in the particle size, the density difference (particle minus fluid), and the viscosity. Double-check units—errors here are easy and will throw off results by orders of magnitude.
Stokes Settling Velocity
vs = g · d² · (ρp - ρf)⁄18μ
Where:
- vs = settling (terminal) velocity (m/s)
- g = gravitational acceleration (9.81 m/s²)
- d = particle diameter (m)
- ρp = particle density (kg/m³)
- ρf = fluid density (kg/m³)
- μ = dynamic viscosity of fluid (Pa·s)
Particle Diameter from Settling Velocity
d = √ 18μ · vs⁄g · (ρp - ρf)
This inverted form allows determination of particle size based on observed settling behavior in classification or sizing operations.
Reynolds Number for Settling Particles
Re = ρf · vs · d⁄μ
Where:
- Re = Reynolds number (dimensionless)
Validity Criteria:
- Re < 0.1: Stokes law highly accurate (creeping flow)
- 0.1 < Re < 1: Stokes law approximate (transitional)
- Re > 1: Stokes law invalid, use empirical drag correlations
Settling Time
t = h⁄vs
Where:
- t = time required for particle to settle distance h (s)
- h = vertical settling distance (m)
This assumes the particle reaches terminal velocity immediately and experiences no turbulence or horizontal flow interference.
Simple Example
Take a quartz sand particle (diameter = 0.0001 m, density = 2650 kg/m³) settling in water (density = 1000 kg/m³, viscosity = 0.001 Pa·s):
vs = (9.81 × 0.0001² × (2650 − 1000)) / (18 × 0.001) = 9.016×10⁻³ m/s ≈ 9.0 mm/s
Reynolds number Re = (1000 × 9.016×10⁻³ × 0.0001) / 0.001 = 0.90 — that puts you squarely in the transition regime, where Stokes' law starts to drift from reality but can still give you a ballpark estimate.
Theory & Engineering Applications
Fundamental Principles of Stokes Settling
Stokes’ law boils down to balancing three forces on a sinking sphere: gravity pulls it down, buoyancy pushes it up, and viscous drag holds it back. When these add to zero, you’ve reached terminal velocity. The main equation assumes your particle is a perfect sphere, there’s plenty of fluid around (no container effects), you’re at low Reynolds number (so the flow is “smooth” all around the particle), and the particles don’t crowd or tangle with each other. Once you move away from these assumptions, results can diverge quickly from real-world measurements.
Critical Non-Obvious Limitation: Wall Effects
One tripwire in using Stokes’ law is walls or confining spaces. The standard setup assumes an infinite tank, but most columns and basins aren’t big compared to particle size. If your particle is much more than 1/10 the width of your container, drag from the wall slows it down. For example, with d/D = 0.1 (particle diameter/container diameter), the real settling speed is about 79% what Stokes’ law predicts. Ignore this and your pilot columns or lab tests will overstate performance.
Temperature Dependence and Viscosity Variations
Viscosity changes fast with temperature—especially in water. At 10°C, water’s viscosity is about 1.3 mPa·s. By 40°C, it’s dropped to 0.65 mPa·s. Practically, a particle will settle roughly twice as fast in warm water as cold water, assuming everything else is the same. This isn’t an edge case—seasonal changes and process heat can radically shift settling rates between summer and winter. For accurate numbers, don’t rely on “room temperature” values: look up viscosity for your operating conditions or use a correlation like Andrade’s equation. A small mistake on temperature can easily swing settling rates by 25% or more in water-based systems.
Particle Shape Factors and Non-Spherical Corrections
Most real-life particles are not perfect spheres. Clays, fragments, and even manufactured granules are nearly always a bit odd in shape, which increases drag. Sphericity (ψ) gives you a way to estimate how non-spherical your particles are. ψ ranges from 1 (perfect sphere) to about 0.6 for flaky or angular particles. For quick calculations, you can adjust the Stokes velocity by multiplying by ψn, with n usually around 1.5–2. That’s a heavy penalty: even moderate departures from spherical shape can nearly halve settling velocity. For fibers, disks, or irregular shapes, expect further deviation or do a bench-top settling test.
Transition to Higher Reynolds Numbers
Stokes’ law only covers creeping flow. As your Reynolds number moves above 0.1 (bigger particles, lower viscosity, or faster velocities), the simple drag balance fails. Past Re ≈ 1, empirical corrections like the Schiller-Naumann equation are more accurate. In these cases, drag goes roughly as the square of velocity, not linearly. For large or fast-settling particles—or low viscosity fluids—use iterative solutions and drag coefficient charts instead of the Stokes formula.
Applications in Water and Wastewater Treatment
Primary and secondary settling tanks in wastewater plants rely heavily on these principles. The size of the tank—and whether it can remove target particles—depends directly on calculated settling velocities. You size the area so even the slowest particle you want to catch can reach the bottom before the water flows out. But if you design using summer temperatures, don’t count on the same efficiency in winter. Also, incoming water can contain large, fluffy, or jammed particles, and these don't always follow the theory. Always leave safety margin and check with pilots or field trials before finalizing major designs.
Mineral Processing and Classification
In mining, you use settling velocity to split particles based on size or density. A fluidized classifier, for example, sets a water upflow so some particles settle and others are washed over. That upflow gets set with settling velocity calculations: anything slower goes overhead, anything faster settles out. If your minerals have very different densities (say, quartz vs. magnetite), even smaller heavy particles can settle as fast as larger light ones—so double-check your calculations when processing mixed materials.
Atmospheric Particulate Matter and Air Quality
Fine dusts in air settle extremely slowly. Using Stokes’ law gives you an idea of just how persistent small particles are: a 10 μm particle might take 9 hours to settle 100 meters, assuming zero wind. If you’re working with air filters or pollution models, that’s why “PM2.5” is such a headache—it just doesn't want to fall out by gravity.
Worked Example: Sedimentation Tank Design
Say you need to design a clarifier to handle sand in river water (15,000 m³/day, 78 μm particle median, 2650 kg/m³ density, water at 15°C). First, calculate settling velocity: plug into Stokes’ law and you get about 17.3 m/h. Check the Reynolds number—here, it’s 0.329 (transitional range). Surface area needs to be at least flow/velocity, which works out to ~36 m². Add a generous safety factor, especially since you’re in the transition regime, and round up to 54 m². If you try to combine both a realistic detention time and the surface area, you’ll probably have to upsize the tank or deepen it, otherwise you’ll have insufficient residence time for proper settling. Rework the math until both requirements are met—don’t trust the textbook example; check your target against your real site numbers and process constraints.
Hindered Settling and Particle Concentration Effects
Stokes’ law is for isolated particles. If you have a concentrated suspension (more than about 0.1% solids by volume), particles interfere with each other's fall. This slows things down a lot, especially as you approach the 1–10% solids region common in thickeners and some sludge systems. The Richardson-Zaki formula lets you apply a correction, but at higher concentrations, you’re always better off running a settling test on the real mixture rather than trusting calculated values. In these cases, the slow-down can reach 50% or more.
For more calculators or fluid/particle property tables, see the full engineering calculators library.
Practical Applications
Scenario: Designing a Water Treatment Clarifier
Jennifer, a civil engineer at a municipal water utility, is designing a new primary clarifier to handle increased flow from a growing suburb. The raw water contains suspended clay and silt particles averaging 45 micrometers in diameter with a density of 2680 kg/m³. She needs to determine if her proposed rectangular basin measuring 12 meters by 8 meters with 3.2 meters depth can handle 8,500 m³ per day while achieving 80% removal efficiency at the worst-case summer temperature of 25°C (where water viscosity drops to 0.000891 Pa·s). Using the Stokes settling velocity calculator, she finds the particles settle at 0.00254 m/s or 9.14 m/h. The basin surface area of 96 m² yields an overflow rate of 8,500/(24×96) = 3.69 m/h, well below the particle settling velocity of 9.14 m/h, confirming the design will effectively capture the target particle size with substantial safety margin. The 2.3-hour detention time (volume 307 m³ divided by flow 354 m³/h) provides adequate settling duration, validating the basin dimensions before construction begins.
Scenario: Mineral Processing Classification
Carlos, a metallurgical engineer at a copper concentrator plant, needs to optimize the performance of a hydrocyclone classifier separating fine ore particles. The process requires removing minus-20-micrometer particles that interfere with downstream flotation. His laboratory measured settling velocities of various size fractions in process water at 18°C, but he needs to predict behavior at the plant's operating temperature of 32°C where viscosity changes significantly. Using the calculator's settling velocity mode with particle diameter 20 μm, particle density 4200 kg/m³ (copper-bearing mineral), fluid density 1035 kg/m³ (process water with dissolved salts), and viscosity 0.000765 Pa·s at 32°C, he calculates a settling velocity of 0.000282 m/s. The Reynolds number of 0.0076 confirms Stokes regime validity. Comparing this to the 0.000198 m/s settling velocity at 18°C (viscosity 0.00105 Pa·s), Carlos finds the 14°C temperature increase accelerates settling by 42%, explaining recent improvement in classifier performance during the summer months and helping him adjust upflow rates seasonally to maintain consistent separation.
Scenario: Environmental Compliance for Airborne Particles
Dr. Lisa Chen, an environmental consultant investigating dust dispersion from a cement plant, must determine how far 15-micrometer limestone dust particles can travel from the emission stack before settling. The particles have density 2710 kg/m³ and are released at 65 meters height. Using ambient conditions of 20°C (air density 1.204 kg/m³, viscosity 1.825×10⁻⁵ Pa·s), she calculates the settling velocity as 0.00118 m/s or 4.25 m/h using the Stokes calculator. The Reynolds number of 0.00116 confirms creeping flow conditions. In still air, the settling time calculator shows these particles would take 15.3 hours to settle from 65 meters height, but with typical wind speeds of 3-5 m/s, horizontal transport of 165 to 275 kilometers is theoretically possible before deposition. This analysis supports her recommendation for enhanced dust suppression systems and helps establish appropriate monitoring station locations downwind. The quantitative settling data strengthens the environmental impact assessment and demonstrates compliance planning is based on rigorous engineering calculations rather than arbitrary buffer zones.
Frequently Asked Questions
When is Stokes' law invalid and what should I use instead? +
How does water temperature affect settling velocity in practical applications? +
What particle size range is appropriate for Stokes settling calculations? +
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What is hindered settling and when does it become important? +
How can I use Stokes' law to estimate particle size from settling tests? +
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About the Author
Robbie Dickson — Chief Engineer & Founder, FIRGELLI Automations
Robbie Dickson brings over two decades of engineering expertise to FIRGELLI Automations. With a distinguished career at Rolls-Royce, BMW, and Ford, he has deep expertise in mechanical systems, actuator technology, and precision engineering.
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