Stokes' Law Sedimentation Rate Calculator
Standard sand/soil is approx 2650 kg/m³.
Pure water is approx 1000 kg/m³.
Water at 20°C is approx 0.001 Pa·s (or 1 cP).
1 µm = 0.000001 meters. Enter radius, not diameter.
How to Calculate Sedimentation Rate
Understanding how particles settle in a fluid is critical for various fields including environmental engineering, geology, wastewater treatment, and industrial chemical processing. The sedimentation rate refers to the terminal velocity at which a particle falls through a stationary fluid under the influence of gravity.
What is Stokes' Law?
The primary method for calculating sedimentation rate for small spherical particles in a laminar flow regime is Stokes' Law. Derived by George Gabriel Stokes in 1851, this formula balances the gravitational force pulling the particle down against the drag force and buoyancy of the fluid pushing it up.
v = [2/9] × [(ρp – ρf) / μ] × g × R²
Where the variables represent:
- v: Terminal sedimentation velocity (m/s)
- ρp (Rho_p): Density of the particle (kg/m³)
- ρf (Rho_f): Density of the fluid (kg/m³)
- μ (Mu): Dynamic viscosity of the fluid (Pa·s)
- g: Acceleration due to gravity (9.81 m/s²)
- R: Radius of the particle (meters)
Step-by-Step Calculation Guide
To calculate the sedimentation rate manually, follow these steps:
- Determine Densities: Identify the density of your particle (e.g., quartz sand is ~2650 kg/m³) and the fluid (water is ~1000 kg/m³).
- Check Viscosity: Determine the fluid's viscosity at the specific operating temperature. Viscosity drops significantly as temperature rises.
- Measure Radius: Ensure you have the radius of the particle (half the diameter). Convert this value to meters for the formula.
- Apply the Formula: Square the radius, multiply by gravity and the density difference, then divide by 9 times the viscosity. Multiply the result by 2.
Factors Affecting Sedimentation Rate
Several variables can significantly alter how fast a particle settles:
- Particle Size: Because the radius is squared in Stokes' Law, a small increase in size leads to a massive increase in settling speed.
- Temperature: Warmer fluids are less viscous, allowing particles to settle faster. This is why sedimentation tanks in wastewater treatment may perform differently in summer versus winter.
- Particle Shape: Stokes' Law assumes perfect spheres. Irregular, flat, or needle-like particles settle slower than the formula predicts due to increased drag.
- Concentration: If the fluid is crowded with particles, they hinder each other, leading to "hindered settling" which requires more complex modifications to the standard formula.
Applications of Sedimentation Calculation
Calculating sedimentation rates is not just a theoretical exercise; it has practical real-world uses:
- Wastewater Treatment: Designing settling tanks and clarifiers requires knowing how long it takes for solids to drop to the bottom so clear water can be removed from the top.
- Geology & Soil Science: The hydrometer test uses sedimentation rates to determine the texture of soil (percentage of sand, silt, and clay).
- Beverage Industry: Clarifying wine or beer involves the sedimentation of yeast and other particulates.
- Medical Labs: The Erythrocyte Sedimentation Rate (ESR) is a common blood test that measures how quickly red blood cells settle, indicating inflammation in the body.
Understanding the Results
If your calculation results in a negative number, it means the particle density is lower than the fluid density. In this scenario, the particle will float (rise) rather than settle. This is common in oil-water separation processes.
Error
Please fill in all fields with valid numbers.';
return;
}
// Convert strings to floats
var p_density = parseFloat(rho_p);
var f_density = parseFloat(rho_f);
var mu = parseFloat(viscosity);
var r_microns = parseFloat(radius_microns);
if (isNaN(p_density) || isNaN(f_density) || isNaN(mu) || isNaN(r_microns)) {
resultBox.style.display = 'block';
resultBox.style.borderLeft = '5px solid #dc3545';
resultBox.innerHTML = 'Error
Please enter numeric values only.';
return;
}
if (mu <= 0) {
resultBox.style.display = 'block';
resultBox.style.borderLeft = '5px solid #dc3545';
resultBox.innerHTML = 'Error
Viscosity must be greater than 0.';
return;
}
// Physics Constants
var g = 9.81; // m/s^2
// Unit conversions: Microns to Meters
var r_meters = r_microns * 1e-6;
// Stokes Law Calculation: v = (2/9) * ( (rho_p – rho_f) / mu ) * g * r^2
// Formula breakdown:
// 1. Density difference
var delta_rho = p_density – f_density;
// 2. Main term
var velocity_mps = (2 / 9) * (delta_rho / mu) * g * Math.pow(r_meters, 2);
// Unit Conversions for display
// Meters per second is usually very small for sedimentation, so we convert to practical units
var velocity_mm_s = velocity_mps * 1000;
var velocity_cm_min = velocity_mps * 100 * 60;
var velocity_m_h = velocity_mps * 3600;
// Determine if settling or floating
var state = velocity_mps >= 0 ? "Settling Downwards" : "Floating Upwards";
var displayColor = velocity_mps >= 0 ? "#28a745" : "#17a2b8";
// Handle negative values for floating (display absolute speed)
var abs_velocity_mps = Math.abs(velocity_mps);
var abs_velocity_mm_s = Math.abs(velocity_mm_s);
var abs_velocity_cm_min = Math.abs(velocity_cm_min);
// Scientific notation threshold check
var display_mps = abs_velocity_mps < 0.0001 ? abs_velocity_mps.toExponential(4) : abs_velocity_mps.toFixed(6);
// Render Result
resultBox.style.display = 'block';
resultBox.style.borderLeft = '5px solid ' + displayColor;
var htmlOutput = '' + state + '
';
htmlOutput += '' + display_mps + ' m/s
';
htmlOutput += 'Terminal Velocity (Meters/Second)
';
htmlOutput += ''; htmlOutput += '
' + abs_velocity_mm_s.toFixed(4) + ' mm/s
';
htmlOutput += 'Millimeters per Second
';
htmlOutput += '' + abs_velocity_cm_min.toFixed(4) + ' cm/min
';
htmlOutput += 'Centimeters per Minute
';
resultBox.innerHTML = htmlOutput;
}