Sedimentation rate refers to the terminal velocity at which a particle settles through a fluid under the influence of gravity. This process is fundamental in various fields, including geology (sediment transport), engineering (wastewater treatment), and medicine (erythrocyte sedimentation rate).
For spherical particles settling in a viscous fluid, this rate is best calculated using Stokes' Law, derived by George Gabriel Stokes in 1851.
The Stokes' Law Formula
The calculator above uses the following physics formula to determine the settling velocity ($v$):
v = [2 · (ρp – ρf) · g · r²] / (9 · μ)
Where:
v = Terminal settling velocity (m/s)
ρp (Rho p) = Density of the particle (kg/m³)
ρf (Rho f) = Density of the fluid (kg/m³)
g = Gravitational acceleration (9.81 m/s²)
r = Radius of the particle (m)
μ (Mu) = Dynamic viscosity of the fluid (Pa·s)
How to Use This Calculator
To perform a calculation, you need to know the physical properties of both the particle and the liquid it is suspended in.
Enter Particle Density: For soil or sand, this is typically around 2650 kg/m³. For oil droplets, it might be around 900 kg/m³.
Enter Fluid Density: Pure water is 1000 kg/m³. Saltwater is roughly 1025 kg/m³.
Enter Viscosity: This measures the fluid's resistance to flow. Water at 20°C has a viscosity of approximately 0.001 Pa·s.
Enter Radius: Input the size of the particle in micrometers (μm). For example, fine sand might have a radius of 50-100 μm.
Factors Affecting Sedimentation Speed
Several variables dramatically change how fast particles settle:
Particle Size: Because the radius is squared ($r^2$) in the formula, doubling the particle size increases the sedimentation rate by a factor of four. This is the most sensitive variable.
Density Difference: The speed is proportional to the difference between the particle and fluid density $(\rho_p – \rho_f)$. If the particle is lighter than the fluid, the result will be negative, indicating the particle will rise (flotation) rather than settle.
Viscosity: Higher viscosity fluids (like honey or oil) drastically slow down sedimentation compared to water. Temperature changes can alter viscosity significantly, thereby changing the settling rate.
Limitations
Stokes' law assumes the particle is a smooth, rigid sphere and that the flow of fluid around the particle is laminar (smooth). If the particle is very large or falling very fast, turbulence occurs, and this formula becomes less accurate. The calculator provides the Reynolds Number (Re); generally, if Re is less than 0.1, Stokes' Law is highly accurate.
function calculateSedimentation() {
// 1. Get Input Values
var particleDensity = parseFloat(document.getElementById('particleDensity').value);
var fluidDensity = parseFloat(document.getElementById('fluidDensity').value);
var viscosity = parseFloat(document.getElementById('viscosity').value);
var radiusMicrometers = parseFloat(document.getElementById('particleRadius').value);
var gravity = parseFloat(document.getElementById('gravity').value);
// 2. Validate Inputs
if (isNaN(particleDensity) || isNaN(fluidDensity) || isNaN(viscosity) || isNaN(radiusMicrometers) || isNaN(gravity)) {
alert("Please enter valid numerical values for all fields.");
return;
}
if (viscosity <= 0) {
alert("Viscosity must be greater than 0.");
return;
}
// 3. Convert Units for Calculation (SI Units)
// Convert radius from micrometers to meters
var radiusMeters = radiusMicrometers * 1e-6;
// 4. Apply Stokes' Law Formula
// v = (2 * (rho_p – rho_f) * g * r^2) / (9 * mu)
var densityDifference = particleDensity – fluidDensity;
var numerator = 2 * densityDifference * gravity * Math.pow(radiusMeters, 2);
var denominator = 9 * viscosity;
var velocityMetersPerSecond = numerator / denominator;
// 5. Convert Results to other units
var velocityCentimetersPerSecond = velocityMetersPerSecond * 100;
var velocityMillimetersPerHour = velocityMetersPerSecond * 1000 * 3600;
// Calculate Reynolds Number: Re = (2 * r * v * rho_f) / mu
// Using absolute velocity because Re is about flow regime
var diameterMeters = 2 * radiusMeters;
var reynoldsNumber = (diameterMeters * Math.abs(velocityMetersPerSecond) * fluidDensity) / viscosity;
// 6. Display Results
// Format numbers for scientific notation if very small, or fixed decimals otherwise
function formatNumber(num) {
if (Math.abs(num) < 0.0001 && num !== 0) {
return num.toExponential(4);
} else {
return num.toFixed(6); // High precision for small settling rates
}
}
function formatLargeNumber(num) {
if (Math.abs(num) < 0.01 && num !== 0) {
return num.toExponential(4);
} else {
return num.toFixed(2);
}
}
document.getElementById('res-ms').innerHTML = formatNumber(velocityMetersPerSecond);
document.getElementById('res-cms').innerHTML = formatNumber(velocityCentimetersPerSecond);
document.getElementById('res-mmh').innerHTML = formatLargeNumber(velocityMillimetersPerHour);
document.getElementById('res-re').innerHTML = reynoldsNumber.toExponential(3);
// Show result section
document.getElementById('results').style.display = 'block';
}