Understanding Sedimentation

When tiny particles are suspended in a fluid, they eventually settle due to gravity if they are denser than the surrounding medium. The settling velocity is a key factor in numerous scientific and engineering disciplines, from water treatment and environmental science to pharmaceuticals and geology. For very small, spherical particles in a viscous fluid, the settling behavior is described by Stokes’ law, which assumes laminar flow around the particle. This calculator focuses on that regime, letting you estimate how fast particles sink under idealized conditions.

Stokes’ Law Formula

In the laminar regime, the drag force on a sphere is proportional to its velocity, radius, and the fluid’s viscosity. Balancing gravitational and drag forces leads to $v=\frac{2(\mathrm{\rho \_p}-\mathrm{\rho \_f})g{r}^2}{9\eta }$, where $\mathrm{\rho \_p}$ and $\mathrm{\rho \_f}$ are the densities of the particle and fluid, respectively, $g$ is gravitational acceleration, $r$ is particle radius, and $\eta$ is dynamic viscosity. This relation shows that larger, denser particles settle faster, while higher viscosity slows them down. It is valid only when the particle Reynolds number is very small (typically below 0.1).

Importance in Industry

Engineers rely on Stokes’ law to design sedimentation tanks, clarifiers, and centrifuges. In wastewater treatment, predicting how fast solids settle helps determine tank dimensions and retention times. In the pharmaceutical industry, controlling particle size ensures consistent drug delivery, so understanding sedimentation rates is crucial during formulation. Geologists study sedimentation to reconstruct past climates and depositional environments. Even food scientists examine how particles like cocoa powder settle in chocolate milk. Each application hinges on predicting how quickly suspended materials separate from liquids.

Using the Calculator

Enter the particle radius, densities, viscosity, and gravitational acceleration. The defaults correspond to a micron-sized mineral grain settling in water. After clicking Compute, the calculator applies Stokes’ law to deliver the terminal settling velocity in meters per second. You can alter the inputs to explore how density contrasts or fluid viscosity impact sedimentation. For instance, increasing the fluid’s viscosity by an order of magnitude will slow settling by the same factor, illustrating why honey pours much more slowly than water.

Limitations and Reynolds Number

Stokes’ law assumes the flow around the particle remains laminar. If the particle is too large or the fluid velocity too high, turbulence arises, and the simple formula no longer holds. The dimensionless Reynolds number $\mathrm{Re}=\frac{2\mathrm{\rho \_f}vr}{\eta }$ helps determine validity. When $\mathrm{Re}$ is less than about 0.1, Stokes’ law applies. Our calculator provides the settling velocity first; you can then compute $\mathrm{Re}$ to verify if the assumption is reasonable for your scenario.

Environmental and Biological Systems

Natural waters are rarely perfectly still, so real sedimentation often deviates from the ideal. Nevertheless, Stokes’ law gives a foundational estimate that guides research on lake sediments, river deposits, and even airborne particulates settling from the atmosphere. In biology, cells and organelles settle according to similar principles, making Stokes’ law relevant for centrifugation protocols used to separate cellular components. Understanding these basics can help you interpret a wide array of phenomena, from cloud formation to the design of industrial separators.

Non‑Spherical Particles

Most real particles are not perfect spheres. Flakes, rods, and irregular grains experience more drag than spheres of the same volume, reducing their settling velocities. Engineers often introduce a shape factor or equivalent spherical diameter to adapt Stokes’ law to such particles. While this calculator assumes sphericity, you can approximate non-spherical behavior by increasing the effective radius or viscosity to reflect the additional drag.

Hindered Settling and Concentration Effects

When particle concentrations are high, settling slows because rising fluid must move past neighboring particles. This hindered settling phenomenon is common in thickeners and sludge beds. Empirical corrections, such as the Richardson–Zaki equation, modify the settling velocity based on solids volume fraction. For dilute suspensions the correction is minor, but at high concentrations the terminal velocity can drop dramatically.

Laboratory Tips

In bench experiments, ensuring the fluid is quiescent before timing settlement is critical. Vibrations or convection currents can invalidate Stokes’ law assumptions. Using tall, narrow cylinders minimizes wall effects, and recording temperature allows accurate viscosity selection. For very small particles, dynamic light scattering or centrifugation methods may offer more precise measurements than simple column tests.

Example Velocities

Particle Radius (μm)	Fluid	Velocity (mm/s)
1	Water	0.0002
5	Water	0.005
10	Oil (η=0.05)	0.0008

Conclusion

The Stokes Settling Velocity Calculator offers a straightforward way to predict how fast particles fall through viscous fluids under laminar conditions. By adjusting the variables, you can quickly see how size, density, viscosity, and gravity influence sedimentation. Use this tool as a starting point for more detailed fluid dynamics analysis or to gain intuition for laboratory experiments and industrial processes.