Stokes’ Law Calculator
Calculate the terminal velocity of spherical particles settling in a fluid using Stokes’ Law
Comprehensive Guide to Stokes’ Law: Theory, Applications, and Example Calculations
Stokes’ Law describes the frictional force (drag force) acting on spherical objects moving through a viscous fluid. This fundamental principle in fluid dynamics has extensive applications in environmental engineering, sedimentology, and various industrial processes. Understanding Stokes’ Law enables precise calculations of particle settling velocities, which are crucial for designing sedimentation tanks, analyzing air pollution, and developing pharmaceutical formulations.
The Mathematical Foundation of Stokes’ Law
The drag force (Fd) on a spherical particle moving through a viscous fluid is given by:
Fd = 6πμrv
Where:
- Fd = Drag force (N)
- μ = Dynamic viscosity of the fluid (Pa·s or kg/(m·s))
- r = Radius of the spherical particle (m)
- v = Velocity of the particle (m/s)
At terminal velocity, the drag force equals the gravitational force minus the buoyant force:
6πμrv = (4/3)πr³(ρp – ρf)g
Solving for terminal velocity (v):
v = [2(ρp – ρf)gr²] / (9μ)
Key Assumptions and Limitations
Stokes’ Law applies under specific conditions:
- Laminar flow regime: Reynolds number (Re) must be less than 1 (Re < 1)
- Spherical particles: The equation assumes perfect spheres
- Infinite fluid medium: No wall effects from container boundaries
- Uniform particle density: Homogeneous internal composition
- Steady-state velocity: Terminal velocity has been reached
For non-spherical particles, shape factors must be incorporated. The Reynolds number is calculated as:
Re = (ρfvd)/μ
Practical Applications of Stokes’ Law
| Application Field | Specific Use Case | Typical Particle Size Range |
|---|---|---|
| Environmental Engineering | Design of sedimentation tanks in water treatment | 1-100 micrometers |
| Atmospheric Science | Modeling aerosol particle deposition in lungs | 0.1-10 micrometers |
| Pharmaceutical Industry | Formulation of suspensions and emulsions | 0.1-50 micrometers |
| Mining Engineering | Gravity separation of minerals | 10-500 micrometers |
| Food Processing | Separation of cream from milk | 1-20 micrometers |
Step-by-Step Example Calculation
Let’s calculate the terminal velocity of a quartz particle (density = 2650 kg/m³) with diameter 50 micrometers settling in water at 20°C (density = 998 kg/m³, viscosity = 0.001 Pa·s):
- Convert diameter to radius: r = 50 × 10⁻⁶ m / 2 = 25 × 10⁻⁶ m
- Calculate density difference: (2650 – 998) = 1652 kg/m³
- Apply Stokes’ Law equation:
v = [2 × 1652 × 9.81 × (25 × 10⁻⁶)²] / (9 × 0.001)
- Compute numerator:
2 × 1652 × 9.81 × 625 × 10⁻¹² = 2.022 × 10⁻⁵
- Final calculation:
v = (2.022 × 10⁻⁵) / 0.009 = 2.247 × 10⁻³ m/s = 2.25 mm/s
Verification of Reynolds Number
To ensure Stokes’ Law applicability, we must verify Re < 1:
Re = (998 × 2.247 × 10⁻³ × 50 × 10⁻⁶) / 0.001 = 0.112
Since 0.112 < 1, Stokes' Law is valid for this calculation.
Comparison of Settling Velocities in Different Fluids
| Fluid | Density (kg/m³) | Viscosity (Pa·s) | Terminal Velocity (mm/s) | Reynolds Number |
|---|---|---|---|---|
| Water (20°C) | 998 | 0.001002 | 2.25 | 0.112 |
| Air (20°C) | 1.204 | 1.81 × 10⁻⁵ | 123.5 | 0.415 |
| Glycerin | 1260 | 1.49 | 0.0015 | 7.4 × 10⁻⁵ |
| SAE 30 Oil | 890 | 0.29 | 0.042 | 0.0012 |
Advanced Considerations
For more accurate real-world applications, several factors must be considered:
- Particle shape factors: Non-spherical particles require correction factors (typically 0.7-1.3)
- Hindered settling: In concentrated suspensions, particles interfere with each other
- Temperature effects: Fluid viscosity changes significantly with temperature
- Wall effects: In confined spaces, settling velocity decreases near walls
- Electrostatic forces: Can affect very small particles (colloidal range)
For particles where Re > 1, the drag coefficient becomes more complex and requires empirical correlations such as:
CD = 24/Re (1 + 0.15Re0.687) for 1 < Re < 1000
Experimental Validation Methods
Laboratory techniques to measure settling velocities include:
- Settling column method: Direct observation of particles in a graduated cylinder
- Laser diffraction: Non-intrusive measurement of particle size and velocity
- Particle image velocimetry (PIV): High-speed camera tracking of particles
- Centrifuge methods: Accelerated settling for small particles
- Electrozone sensing: Individual particle counting and sizing
These methods typically show good agreement with Stokes’ Law predictions for spherical particles in the appropriate Re range, with experimental errors generally under 5% for carefully controlled conditions.
Environmental Applications
Stokes’ Law plays a crucial role in environmental engineering:
- Water treatment: Design of clarifiers and sedimentation basins where typical overflow rates are 1-2 m/h based on Stokes’ Law calculations
- Air pollution control: Efficiency of electrostatic precipitators and fabric filters depends on particle settling velocities
- Soil erosion studies: Modeling of silt transport in rivers and streams
- Oceanography: Understanding marine snow settling in ocean waters
- Climate science: Aerosol particle deposition affects cloud formation and albedo
For example, in wastewater treatment plants, the surface loading rate (m³/m²·h) is calculated based on the minimum settling velocity of particles that need to be removed, which is directly derived from Stokes’ Law calculations.
Industrial Process Optimization
Manufacturing processes leverage Stokes’ Law for:
- Pharmaceutical suspensions: Ensuring uniform distribution of active ingredients
- Paint formulations: Preventing pigment settling during storage
- Mineral processing: Gravity separation of ores based on density differences
- Food industry: Cream separation and clarification of juices
- Nanotechnology: Controlled deposition of nanoparticles
In the pharmaceutical industry, the Hatch-Choate equation extends Stokes’ Law to polydisperse systems:
d(ln n)/dD = -kDm
Where n is the number of particles, D is diameter, and k and m are empirical constants.
Common Calculation Errors and Pitfalls
Avoid these mistakes when applying Stokes’ Law:
- Unit inconsistencies: Always use SI units (kg, m, s, Pa)
- Incorrect viscosity values: Viscosity changes dramatically with temperature
- Assuming sphericity: Most real particles have irregular shapes
- Ignoring concentration effects: Hindered settling occurs above 5% volume fraction
- Neglecting wall effects: Significant in containers with diameter < 100× particle diameter
- Misapplying Reynolds number: Always verify Re < 1 for Stokes' Law validity
For example, using the viscosity of water at 0°C (0.00179 Pa·s) instead of 20°C (0.00100 Pa·s) would result in a 44% underestimation of settling velocity.
Authoritative Resources
For further study, consult these authoritative sources:
- U.S. EPA Air Quality Modeling – Government guidelines on particle deposition modeling
- Engineering Toolbox Stokes’ Law – Practical calculations and examples
- MIT Stokes’ Law Lecture Notes – Comprehensive theoretical treatment from MIT