Terminal Velocity Calculation for Settling Particles

Reference ID: MET-B074 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

In many process‑engineering operations—such as sedimentation tanks, mineral flotation, and pharmaceutical suspension handling—engineers need to predict how quickly solid particles settle through a fluid. The terminal settling velocity determines the residence time required for particles to be removed or to reach a desired concentration. This reference sheet outlines the theoretical basis for calculating the low‑Reynolds‑number (Stokes) terminal velocity of a spherical particle, the associated Reynolds number, the transient time needed to approach a specified fraction of that velocity, and a quick check on the validity of the Stokes approximation.

Methodology & Formulas

The calculation proceeds through a series of physics‑based steps, each expressed in closed‑form algebraic equations. All symbols are defined in the table that follows.

Symbol	Description	Units	Typical Range (process context)
\(g\)	Gravitational acceleration	m·s\(^{-2}\)	9.8 (standard)
\(d\)	Particle diameter (converted to metres)	m	10\(^{-6}\) – 10\(^{-3}\)
\(\rho_s\)	Solid particle density	kg·m\(^{-3}\)	1000 – 5000
\(\rho_f\)	Fluid density (e.g., water)	kg·m\(^{-3}\)	~1000
\(\mu\)	Dynamic viscosity of the fluid (Pa·s)	Pa·s	10\(^{-3}\) – 10\(^{-1}\)
\(\alpha\)	Desired fraction of the terminal velocity (dimensionless, 0 < α < 1)	–	0.90 – 0.99
\(v_t\)	Steady‑state (terminal) settling velocity	m·s\(^{-1}\)	–
\(Re\)	Particle Reynolds number based on \(v_t\)	–	Typically < 1 for Stokes regime
\(b\)	Exponential‑approach constant governing transient acceleration	s\(^{-1}\)	–
\(t_{\alpha}\)	Time required for the particle speed to reach \(\alpha\,v_t\)	s	–
\(F_{\text{Stokes}}\)	Viscous drag force predicted by Stokes law	N	–
\(F_{\text{inertial}}\)	Quadratic (inertial) drag contribution	N	–
\(\varepsilon\)	Inertial‑drag error expressed as a percentage of the Stokes drag	%	–

Step 1 – Convert practical units to SI. The particle diameter supplied in micrometres (\(d_{\mu m}\)) and the fluid viscosity supplied in centipoise (\(\mu_{cP}\)) are converted to metres and pascal‑seconds, respectively:

\[ d = d_{\mu m}\times10^{-6},\qquad \mu = \mu_{cP}\times10^{-3} \]

Step 2 – Compute the low‑Reynolds‑number terminal velocity (Stokes law). Balancing the net gravitational force with the Stokes viscous drag yields:

\[ v_t = \frac{d^{2}\,g\,(\rho_s-\rho_f)}{18\,\mu} \]

Step 3 – Evaluate the Reynolds number. Using the terminal velocity as the characteristic velocity:

\[ Re = \frac{\rho_f\,v_t\,d}{\mu} \]

Step 4 – Determine the exponential‑approach constant. The linearized equation of motion for a sphere in a viscous fluid gives a first‑order decay constant:

\[ b = \frac{18\,\mu}{\rho_s\,d^{2}} \]

Step 5 – Calculate the time to reach a specified fraction \(\alpha\) of the terminal velocity. Solving the first‑order differential equation for velocity with the initial condition \(v(0)=0\) provides:

\[ t_{\alpha} = -\frac{1}{b}\,\ln\!\bigl(1-\alpha\bigr) \]

Step 6 – Verify the Stokes‑law assumption. The inertial (quadratic) drag term can be compared with the Stokes term:

\[ F_{\text{Stokes}} = 3\pi\,\mu\,d\,v_t \] \[ F_{\text{inertial}} = \frac{9\pi}{16}\,\rho_f\,v_t^{2}\,d^{2} \] \[ \varepsilon = \frac{F_{\text{inertial}}}{F_{\text{Stokes}}}\times100\% \] If \(\varepsilon\) is well below a few percent, the Stokes approximation is justified; larger values indicate that higher‑order drag correlations may be required.

These equations together provide a complete, analytically tractable framework for estimating particle settling behavior in the low‑Re regime, assessing the time scale for velocity development, and confirming the validity of the underlying assumptions.

The terminal velocity (V_t) for a small, spherical particle in the laminar‑flow regime can be obtained from Stokes' law:

V_t = ( (ρ_p - ρ_f) * g * d^2 ) / (18 * μ )

Where:

ρ_p = particle density (kg m⁻³)
ρ_f = fluid density (kg m⁻³)
g = gravitational acceleration (9.81 m s⁻²)
d = particle diameter (m)
μ = dynamic viscosity of the fluid (Pa·s)

Steps to apply the equation:

Measure or obtain the particle size distribution and select a representative diameter (often the equivalent spherical diameter).
Determine the fluid temperature; look up ρ_f and μ at that temperature.
Calculate the density difference (ρ_p – ρ_f).
Insert the values into the formula and solve for V_t.
Validate that the Reynolds number Re = (ρ_f * V_t * d)/μ is < 0.1; otherwise Stokes' law is not valid.

Stokes' law assumes laminar flow (Re < 0.1) around a smooth, spherical particle. It becomes inaccurate when:

Particle diameter exceeds roughly 0.1 mm, raising the Reynolds number above 0.1.
The particle shape deviates significantly from a sphere (high sphericity factor).
The fluid exhibits non‑Newtonian behavior or large viscosity variations.
Inter‑particle interactions or high solids concentration cause hindered settling.
Surface roughness or porosity alters drag characteristics.

In these cases, use empirical correlations (e.g., Richardson‑Zaki, Wen‑Yu) or CFD‑based drag models.

Replace the simple diameter in Stokes' law with an equivalent spherical diameter (d_eq) that preserves the particle’s volume, and introduce a shape factor (C_s) to modify the drag term:

d_eq = (6 * V_p / π)^(1/3), where V_p is the particle volume measured by displacement or image analysis.
C_s = (1 / Φ), where Φ (sphericity) ranges from 0 (highly irregular) to 1 (perfect sphere). Typical values: 0.6–0.9 for crushed ore, 0.9–1.0 for granules.
Modified terminal velocity: V_t = ( (ρ_p - ρ_f) * g * d_eq^2 ) / (18 * μ * C_s )

For highly irregular particles, consider using the drag coefficient correlations of Haider‑Levenspiel or the empirical Richardson‑Zaki exponent to capture hindered settling effects.

Temperature influences both fluid density (ρ_f) and viscosity (μ), which appear directly in the Stokes' equation. To incorporate temperature effects:

Obtain temperature‑dependent property data from standard tables or equations of state (e.g., μ(T) = A · 10^(B/(T‑C)) for water).
Re‑calculate ρ_f and μ at the operating temperature before inserting them into the terminal‑velocity formula.
If the temperature gradient is significant across the settling column, perform a segment‑wise analysis: divide the column into zones, compute V_t for each zone, and use the smallest V_t as the design‑basis velocity.
Check the Reynolds number in each zone; a rise in temperature usually lowers μ, increasing Re and potentially moving the flow out of the Stokes regime.

By updating the fluid properties for the actual temperature, the predicted settling rate remains accurate across process conditions.

Worked Example: Terminal Velocity of a Fine Particle in Water

A process engineer at a water‑treatment plant needs to evaluate how quickly a 30‑µm solid particle will settle in the clarifier. The particle density, fluid properties, and operating conditions are known. The engineer must calculate the terminal settling velocity and verify that Stokes’ law is applicable.

Knowns

Gravitational acceleration, \(g = 9.81\) m s\(^{-2}\)
Particle diameter, \(d = 30.0\) µm = \(3.00 \times 10^{-5}\) m
Particle density, \(\rho_s = 1550\) kg m\(^{-3}\)
Fluid density, \(\rho_f = 998\) kg m\(^{-3}\)
Dynamic viscosity of fluid, \(\mu = 1.002 \times 10^{-3}\) Pa s
Shape correction factor, \(\alpha = 0.99\) (dimensionless)

Step‑by‑Step Calculation

Compute the density difference: \(\Delta\rho = \rho_s - \rho_f = 1550 - 998 = 552\) kg m\(^{-3}\).
Assume an initial terminal velocity using Stokes’ law: \[v_{t,0} = \frac{(\rho_s - \rho_f) g d^{2}}{18 \mu}\]. Substituting the numbers, \[v_{t,0} = \frac{552 \times 9.81 \times (3.00\times10^{-5})^{2}}{18 \times 1.002\times10^{-3}} = 2.70 \times 10^{-4}\ \text{m s}^{-1}.\]
Determine the Reynolds number with the provisional velocity: \[\text{Re} = \frac{\rho_f v_{t,0} d}{\mu} = \frac{998 \times 2.70\times10^{-4} \times 3.00\times10^{-5}}{1.002\times10^{-3}} = 0.008.\] Since \(\text{Re}<0.1\), Stokes’ regime is valid.
Compute the Stokes drag force: \[F_{\text{Stokes}} = 3 \pi \mu d v_{t,0} = 3 \pi (1.002\times10^{-3})(3.00\times10^{-5})(2.70\times10^{-4}) = 7.66 \times 10^{-11}\ \text{N}.\]
Estimate the inertial drag force (using \(\tfrac{1}{2} C_D \rho_f A v_{t,0}^{2}\) with \(C_D \approx 0.47\) for a sphere): \[F_{\text{inertial}} = \tfrac{1}{2}(0.47)\rho_f\left(\frac{\pi d^{2}}{4}\right)v_{t,0}^{2}=1.16 \times 10^{-13}\ \text{N}.\]
Calculate the percent error if inertial drag were ignored: \[\text{error} = \frac{F_{\text{inertial}}}{F_{\text{Stokes}}}\times100 = 0.151\%.\]
Apply the shape correction factor: \[v_t = \alpha\,v_{t,0} = 0.99 \times 2.70\times10^{-4} = 2.70\times10^{-4}\ \text{m s}^{-1}.\] (The correction changes the value only in the third significant figure.)

Final Answer

The terminal settling velocity of the 30‑µm particle in water at 20 °C is 2.70 × 10⁻⁴ m s⁻¹ (≈ 0.270 mm s⁻¹). The Reynolds number is 0.008, confirming that Stokes’ regime applies, and the inertial drag contribution is negligible (<0.2 %).

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