Particle Size Calculation Formula Calculator
Calculate particle diameter based on its settling velocity using the Stokes’ Law formula.
Calculated Particle Diameter
Particle Size vs. Settling Velocity
What is the Formula Used for Calculating Particle Size?
The calculation of particle size from fluid dynamics is most commonly performed using **Stokes’ Law**. This law provides a formula that relates the size of a small, spherical particle to its terminal settling velocity in a viscous fluid. The principle is fundamental in fields like geology, environmental science, pharmacology, and materials science. It assumes that the flow around the falling particle is laminar (smooth and non-turbulent), which is valid for small particles at low velocities. The formula for what is the formula used for calculating particle size is a powerful tool for indirect measurement where direct observation is impractical. This calculator specifically applies Stokes’ Law to determine particle diameter.
The Particle Size Calculation Formula (Stokes’ Law) and Explanation
When a particle is released in a fluid, it accelerates due to gravity until the downward gravitational force is balanced by the upward buoyant force and the viscous drag force. At this point, it reaches a constant “terminal velocity”. By rearranging Stokes’ Law, we can solve for the particle’s diameter.
This equation forms the core of our particle size calculation. A deep understanding of each variable is crucial for accurate results.
Variables in the Formula
| Variable | Meaning | Common Unit (SI) | Typical Range |
|---|---|---|---|
| d | Particle Diameter | meters (m) | 1 µm – 100 µm (for Stokes’ Law) |
| η (eta) | Dynamic Viscosity of Fluid | Pascal-seconds (Pa·s) | 0.001 Pa·s (Water) – 1 Pa·s (Glycerol) |
| v | Settling Velocity of Particle | meters/second (m/s) | 10⁻⁶ m/s – 10⁻² m/s |
| ρp (rho_p) | Density of Particle | kilograms/cubic meter (kg/m³) | 1100 kg/m³ (Latex) – 7850 kg/m³ (Steel) |
| ρf (rho_f) | Density of Fluid | kilograms/cubic meter (kg/m³) | 1.2 kg/m³ (Air) – 1000 kg/m³ (Water) |
| g | Acceleration due to Gravity | meters/second² (m/s²) | 9.81 m/s² (on Earth) |
For more detailed information on viscosity, our Stokes’ Law Calculator provides further context.
Practical Examples of Particle Size Calculation
Example 1: Silt Particle in Water
Imagine a tiny silt particle settling in a calm lake at 20°C. We want to find its size.
- Inputs:
- Settling Velocity (v): 0.05 mm/s (or 0.00005 m/s)
- Fluid (Water) Viscosity (η): 0.001 Pa·s
- Particle (Silt) Density (ρp): 2650 kg/m³
- Fluid (Water) Density (ρf): 998 kg/m³
- Calculation:
- d = √[ (18 × 0.001 × 0.00005) / ( (2650 – 998) × 9.81 ) ]
- d ≈ √[ 0.0000009 / 16206.12 ] ≈ 7.45 x 10⁻⁶ meters
- Result: The particle diameter is approximately **7.45 micrometers (µm)**.
Example 2: Volcanic Ash in Air
Let’s calculate the size of a fine volcanic ash particle falling through air.
- Inputs:
- Settling Velocity (v): 1 cm/s (or 0.01 m/s)
- Fluid (Air) Viscosity (η): 1.81 x 10⁻⁵ Pa·s
- Particle (Ash) Density (ρp): 2400 kg/m³
- Fluid (Air) Density (ρf): 1.225 kg/m³ (negligible compared to ash)
- Calculation:
- d = √[ (18 × 1.81e-5 × 0.01) / ( (2400 – 1.225) × 9.81 ) ]
- d ≈ √[ 3.258e-6 / 23532 ] ≈ 1.17 x 10⁻⁵ meters
- Result: The particle diameter is approximately **11.7 micrometers (µm)**. This relates to concepts discussed in our guide on Sedimentation Analysis.
How to Use This Particle Size Calculation Formula Calculator
- Enter Settling Velocity: Input the measured terminal velocity of the particle. Select the appropriate units (m/s or cm/s).
- Enter Fluid Viscosity: Provide the dynamic viscosity of the fluid the particle is in. You can choose between Pascal-seconds (Pa·s) or centiPoise (cP).
- Enter Particle Density: Input the density of the material the particle is made of. Common units like kg/m³ and g/cm³ are available.
- Enter Fluid Density: Input the density of the fluid. The density difference between the particle and the fluid is a key driver of sedimentation.
- Interpret the Results: The calculator instantly provides the particle’s diameter in micrometers (µm). It also shows intermediate calculations like the density difference to help you understand the formula. The chart provides a visual representation of your result. A deeper dive into these concepts can be found in our article on Particle Settling Velocity.
Key Factors That Affect Particle Size Calculation
- Fluid Viscosity: Higher viscosity creates more drag, meaning for the same velocity, the particle must be larger. Temperature significantly affects viscosity. You can learn more with a Dynamic Viscosity Explained tool.
- Density Difference (ρp – ρf): A larger difference between particle and fluid density increases the net gravitational force, leading to a higher settling velocity for a given particle size. If particle density is less than fluid density, the particle will float, and this formula does not apply.
- Particle Shape: Stokes’ Law assumes perfectly spherical particles. Irregularly shaped particles (e.g., flat or needle-like) experience more drag and will settle slower than a sphere of equivalent mass, leading to an underestimation of their “equivalent spherical diameter.”
- Turbulence: The formula is only valid for laminar flow (Reynolds Number < 1). Larger particles or higher velocities can create turbulent flow, where drag increases significantly and Stokes’ Law becomes inaccurate.
- Particle Interactions: In concentrated suspensions, particles can interfere with each other’s settling, a phenomenon known as hindered settling. This calculator assumes a dilute suspension where particles do not interact.
- Brownian Motion: For very small particles (typically < 1 µm), random collisions with fluid molecules (Brownian motion) can interfere with or even prevent gravitational settling, making this calculation method less reliable. Our guide on Brownian Motion Impact discusses this.
Frequently Asked Questions (FAQ)
- 1. What is the primary assumption of this particle size calculation formula?
- The primary assumption is that the particle is spherical and settling in a state of laminar flow, where viscous forces dominate over inertial forces. This is quantified by a low Reynolds number (Re < 1).
- 2. What happens if the particle density is lower than the fluid density?
- If the particle density is lower than the fluid density, the particle will be buoyant and float upwards instead of settling. The calculator will show an error, as Stokes’ Law for settling is not applicable in this scenario.
- 3. How accurate is the particle size calculation?
- The accuracy depends on how well the real-world conditions match the assumptions of Stokes’ Law. It is highly accurate for small, spherical particles in dilute suspensions. For non-spherical particles, it provides an “equivalent spherical diameter.”
- 4. Can I use this calculator for any fluid?
- Yes, as long as you know the fluid’s dynamic viscosity and density. It works for liquids (like water, oil) and gases (like air).
- 5. Why does the calculator output size in micrometers (µm)?
- The typical particle sizes for which Stokes’ Law is valid fall into the micrometer range (millionths of a meter). It is the most common and convenient unit for this type of analysis.
- 6. Does temperature affect the calculation?
- Yes, indirectly but significantly. Fluid viscosity and density are both temperature-dependent. You must use the viscosity and density values that correspond to the temperature of your fluid for an accurate particle size calculation.
- 7. What is the difference between dynamic and kinematic viscosity?
- Dynamic viscosity (η or μ) is the measure of a fluid’s internal resistance to flow. Kinematic viscosity is the dynamic viscosity divided by the fluid density. Stokes’ Law specifically uses dynamic viscosity.
- 8. What are the limitations of using Stokes’ Law?
- The main limitations are its assumptions of spherical particles, laminar flow, and dilute suspensions. It does not work well for large particles, high velocities, concentrated slurries, or irregularly shaped particles. For more, read about Sieve Analysis vs Sedimentation.
Related Tools and Internal Resources
Explore these related resources for more in-depth analysis and conversion tools:
- Stokes’ Law Calculator: A tool focused on calculating settling velocity when particle size is known.
- Reynolds Number Calculator: Determine if your fluid flow is laminar or turbulent, a key assumption for this calculation.
- Guide to Sedimentation Testing: An in-depth look at the practical methods of measuring particle settling.
- Particle Characterization Methods: An overview of different techniques used to measure particle size.