Sedimentation Rate Calculator

A comprehensive tool to calculate and understand particle settling velocity in a fluid.

Sedimentation Rate Calculation

Variable Definitions and Units

Variable	Meaning	Unit (Metric)	Unit (Imperial)
Particle Diameter (dp)	Diameter of the settling particle	mm	in
Particle Density (ρp)	Density of the particle material	kg/m³	lb/ft³
Fluid Density (ρf)	Density of the fluid medium	kg/m³	lb/ft³
Dynamic Viscosity (μ)	Measure of the fluid's resistance to flow	Pa·s	lb/(ft·s)
Gravitational Acceleration (g)	Acceleration due to gravity	m/s²	ft/s²
Terminal Velocity (Vs)	Constant speed reached during freefall	m/s	ft/s
Drag Coefficient (Cd)	Dimensionless force coefficient	Unitless	Unitless
Reynolds Number (Re)	Dimensionless parameter for flow regime	Unitless	Unitless
Buoyancy Force (Fb)	Upward force exerted by the fluid	N	lbf

What is Sedimentation Rate?

Sedimentation rate, often referred to as settling velocity or terminal velocity, is the constant speed that a particle reaches when falling through a fluid under the influence of gravity. When a particle is introduced into a fluid, it initially accelerates due to gravity. However, as its speed increases, the drag force exerted by the fluid opposing the motion also increases. Eventually, the drag force plus the buoyant force equals the gravitational force acting on the particle, and the net force becomes zero. At this point, the particle stops accelerating and falls at a constant velocity, known as its terminal velocity. This concept is fundamental in fields like environmental engineering (water treatment, soil erosion), chemical engineering (separation processes), and geology.

Who should use it: Environmental engineers, process engineers, geologists, hydrologists, and anyone dealing with particle transport and separation in liquid or gas media. It's crucial for designing settling tanks, predicting pollutant transport, and understanding natural processes like riverbed deposition.

Common misunderstandings: A frequent misunderstanding is that particles continuously accelerate. In reality, terminal velocity is reached relatively quickly. Another confusion arises with units, as densities and viscosities can be expressed in various systems, impacting the final velocity calculation. Some may also incorrectly assume Stokes' Law applies universally, neglecting the importance of the Reynolds number and drag correction factors for larger or faster-moving particles.

Sedimentation Rate Formula and Explanation

The calculation of sedimentation rate (terminal velocity, $V_s$) is an iterative process because the drag coefficient ($C_d$) depends on the Reynolds number ($Re$), which in turn depends on the velocity itself. A common approach involves using empirical correlations for $C_d$. For spherical particles, the fundamental principle is the balance of forces:

Gravitational Force ($F_g$) + Buoyant Force ($F_b$) = Drag Force ($F_d$)

Where:

• $F_g = V_p \times \rho_p \times g$ (Volume of particle * Density of particle * Gravitational acceleration)
• $F_b = V_p \times \rho_f \times g$ (Volume of particle * Density of fluid * Gravitational acceleration)
• $F_d = \frac{1}{2} \times \rho_f \times V_s^2 \times C_d \times A_p$ (0.5 * Density of fluid * Velocity squared * Drag Coefficient * Projected Area of particle)

For a sphere, $V_p = \frac{\pi}{6} d_p^3$ and $A_p = \frac{\pi}{4} d_p^2$. Substituting these and rearranging for the drag force component:

$V_s = \sqrt{\frac{4gd_p(\rho_p – \rho_f)}{3\rho_f C_d}}$

The challenge lies in $C_d$. For low Reynolds numbers ($Re < 1$), Stokes' Law applies, where $C_d = \frac{24}{Re}$. For higher $Re$, corrections are needed. A widely used correction is the Schiller-Naumann equation (for $Re < 1000$):

$C_d = \frac{24}{Re} (1 + 0.15 \times Re^{0.687})$

The Reynolds number is defined as:

$Re = \frac{\rho_f \times V_s \times d_p}{\mu}$

Our calculator uses an iterative method or an approximation to solve for $V_s$ given the inputs and the relationship between $C_d$ and $Re$. The "Primary Result" often displays the calculated terminal velocity ($V_s$), while intermediate values show $C_d$, $Re$, and the buoyancy force ($F_b$).

Variables Table:

Variable	Meaning	Unit (Metric)	Unit (Imperial)
$d_p$	Particle Diameter	mm	in
$\rho_p$	Particle Density	kg/m³	lb/ft³
$\rho_f$	Fluid Density	kg/m³	lb/ft³
$\mu$	Dynamic Viscosity of Fluid	Pa·s	lb/(ft·s)
$g$	Gravitational Acceleration	m/s²	ft/s²
$V_s$	Terminal Velocity	m/s	ft/s
$C_d$	Drag Coefficient	Unitless	Unitless
$Re$	Reynolds Number	Unitless	Unitless
$F_b$	Buoyancy Force	N	lbf

Practical Examples

Example 1: Sand Settling in Water

• Particle Diameter ($d_p$): 0.5 mm
• Particle Density ($\rho_p$): 2650 kg/m³
• Fluid Density ($\rho_f$): 1000 kg/m³ (Water)
• Dynamic Viscosity ($\mu$): 0.001 Pa·s (Water at ~20°C)
• Gravitational Acceleration ($g$): 9.81 m/s²

Calculation Result: The calculator might yield a terminal velocity of approximately 0.025 m/s. The Reynolds number would likely be low, indicating Stokes' Law is a reasonable approximation.

Example 2: Clay Particle Settling in Water

• Particle Diameter ($d_p$): 0.01 mm
• Particle Density ($\rho_p$): 2400 kg/m³
• Fluid Density ($\rho_f$): 1000 kg/m³ (Water)
• Dynamic Viscosity ($\mu$): 0.001 Pa·s (Water at ~20°C)
• Gravitational Acceleration ($g$): 9.81 m/s²

Calculation Result: For this smaller particle, the terminal velocity would be much lower, perhaps around 0.0002 m/s (0.2 mm/s). The Reynolds number will be very small, confirming the applicability of Stokes' Law.

How to Use This Sedimentation Rate Calculator

1. Select Unit System: Choose either 'Metric (SI)' or 'Imperial (US)' from the dropdown to match your input data. The calculator will automatically adjust units for display.
2. Input Particle Diameter ($d_p$): Enter the diameter of the particle you are analyzing. The unit displayed (e.g., mm or inches) will reflect your selected unit system.
3. Input Particle Density ($\rho_p$): Enter the density of the material the particle is made of.
4. Input Fluid Density ($\rho_f$): Enter the density of the fluid the particle is settling in.
5. Input Fluid Viscosity ($\mu$): Enter the dynamic viscosity of the fluid. Ensure it's the dynamic viscosity (Pa·s or lb/(ft·s)), not kinematic.
6. Input Gravitational Acceleration ($g$): Enter the value for gravitational acceleration relevant to your location (typically 9.81 m/s² on Earth).
7. Click Calculate: The calculator will compute the terminal velocity ($V_s$), Reynolds number ($Re$), drag coefficient ($C_d$), and buoyancy force ($F_b$).
8. Interpret Results: The primary result is the terminal velocity. The Reynolds number helps determine the flow regime (laminar vs. turbulent). The assumptions listed are important for the accuracy of the calculation.
9. Copy Results: Use the 'Copy Results' button to get a text summary of your inputs, outputs, and assumptions for documentation.

Key Factors That Affect Sedimentation Rate

1. Particle Size (Diameter): Larger particles generally settle faster due to a higher gravitational force relative to drag. This is a primary factor in $V_s$.
2. Particle Density: Denser particles experience a greater gravitational force, leading to a higher settling velocity, assuming fluid density remains constant.
3. Fluid Density: A higher fluid density increases the buoyant force, opposing gravity, thus reducing the net downward force and decreasing settling velocity.
4. Fluid Viscosity: Higher viscosity means greater resistance to flow (drag), which significantly slows down the particle's settling velocity. This is particularly important at low Reynolds numbers (Stokes' regime).
5. Particle Shape: While this calculator assumes spherical particles, non-spherical shapes experience different drag forces and turbulence, affecting their settling rate. Irregular shapes often settle slower.
6. Flow Conditions (Reynolds Number): The relationship between inertial forces (related to velocity) and viscous forces (Reynolds number) dictates the drag coefficient. At low $Re$, drag is proportional to velocity (Stokes' Law); at higher $Re$, it becomes proportional to velocity squared, changing the dynamics.
7. Gravitational Force: While constant on Earth's surface for practical purposes, changes in gravity (e.g., on different celestial bodies) would directly impact the driving force for sedimentation.

FAQ

• Q: What is the difference between dynamic and kinematic viscosity?
  A: Dynamic viscosity ($\mu$) is the fluid's resistance to shear flow (units like Pa·s or cP). Kinematic viscosity ($\nu$) is dynamic viscosity divided by density ($\nu = \mu / \rho_f$), representing momentum diffusivity (units like m²/s or Stokes). This calculator requires dynamic viscosity.
• Q: Does the calculator handle non-spherical particles?
  A: No, this calculator is based on formulas derived for spherical particles. Non-spherical shapes have different drag characteristics.
• Q: My Reynolds number is very high (e.g., > 1000). Is the result accurate?
  A: The Schiller-Naumann correction used here is generally valid for $Re < 1000$. For very high $Re$, different drag models or empirical data are needed, and the terminal velocity might approach a constant value determined by a near-constant $C_d$.
• Q: How do I convert between unit systems?
  A: Use the 'Unit System' dropdown. For manual conversion: 1 inch = 0.0254 m, 1 lb/ft³ ≈ 16.0185 kg/m³, 1 lb/(ft·s) ≈ 1.48816 Pa·s, 1 ft/s ≈ 0.3048 m/s.
• Q: What does a negative result for terminal velocity mean?
  A: A negative result is not physically possible in this context. It likely indicates an error in input, specifically if particle density is less than fluid density (meaning the particle floats, not sinks).
• Q: How accurate is the Schiller-Naumann correction?
  A: It provides a good approximation for spherical particles in the transitional flow regime ($1 < Re < 1000$). Accuracy decreases outside this range.
• Q: Can I use this for gas sedimentation?
  A: Yes, the principles are the same, but you would input the density and viscosity of the gas. Ensure units are consistent.
• Q: What if the particle floats instead of sinks?
  A: If particle density ($\rho_p$) is less than fluid density ($\rho_f$), the net force is upward, and the particle will float or rise. The formula might produce non-physical results (like negative velocity), indicating buoyancy dominates.

Related Tools and Resources

• Fluid Dynamics Calculator – Explore related fluid properties.
• Particle Size Distribution Analyzer – Understand how particle size impacts various properties.
• Water Treatment Design Calculator – See sedimentation applications in action.
• Sediment Transport Model – Advanced modeling for environmental applications.
• Rheology Calculator – For non-Newtonian fluids where viscosity is not constant.
• Density Conversion Tool – Quickly convert between density units.