Introduction & Context

The critical (cut) particle diameter is the size at which a dispersed solid phase is just captured by a gravity-driven separator. In process engineering, it marks the boundary between particles that leave with the overflow and those that settle to the underflow. Typical applications include:

Primary clarifiers in water treatment
Yeast settling tanks in breweries
API recovery basins in pharmaceutical plants
Desanders protecting downstream equipment

Knowing this diameter allows engineers to predict removal efficiency, size new units, or diagnose poor performance in existing settlers.

Methodology & Formulas

The calculation couples the terminal-settling equation with an explicit drag correlation. The objective is to find the diameter \(d_\text{crit}\) for which the terminal velocity equals the allowable up-flow velocity imposed by the throughput.

Allowable superficial velocity
\[ v_\text{allow}= \dfrac{Q}{A} \]
Particle Reynolds number
\[ Re_\text{p}= \dfrac{\rho_\text{f}\,v_\text{allow}\,d}{\mu} \]
Drag coefficient (Haider & Levenspiel)
\[ C_\text{D}= \dfrac{24}{Re_\text{p}}\!\left(1+0.173\,Re_\text{p}^{\,0.657}\right) + \dfrac{0.413}{1+16\,300\,Re_\text{p}^{\,-1.09}} \]
Force balance (gravity vs drag)
\[ v_\text{allow}= \sqrt{\dfrac{4\,g\,(\rho_\text{p}-\rho_\text{f})\,d}{3\,\rho_\text{f}\,C_\text{D}}} \]

The above single-variable equation is driven to zero by a Newton-like fixed-point iteration; the Stokes analytical estimate supplies the initial guess.

Flow regime limits based on particle Reynolds number
Regime	Range of \(Re_\text{p}\)	Drag behaviour
Stokes (laminar)	\(Re_\text{p}\le 1\)	\(C_\text{D}\approx 24/Re_\text{p}\)
Intermediate	\(1\lt Re_\text{p}\le 1000\)	Correlation required
Newton (turbulent)	\(Re_\text{p}\gt 1000\)	\(C_\text{D}\approx 0.44\)

The returned root is the critical particle diameter \(d_\text{crit}\) in metres.

Critical particle size (CPS) is the diameter above which particles will segregate, settle, or otherwise misbehave in your process, and below which they will remain suspended, mixed, or pneumatically conveyed. Knowing CPS lets you:

Design cyclones, hoppers, and pneumatic lines that avoid saltation or unwanted classification.
Set milling or granulation targets so every downstream step sees a consistent bulk.
Predict whether fines will flood a filter, blind a screen, or create dust-explosion hazards.

In short, CPS is the boundary between “behaves like the bulk” and “does its own thing,” so sizing equipment without it is guesswork.

Use the Stokes–Muschelknautz modified settling equation rearranged for critical diameter:

\(d_c = \sqrt{\frac{18\, \mu\, v_t}{g\, (\rho_p - \rho_g)}}\)
\(\mu\) = gas viscosity, \(v_t\) = terminal velocity you can tolerate, \(\rho_p\) = particle density, \(\rho_g\) = gas density.
If \(v_t\) equals your superficial velocity, anything bigger than \(d_c\) will settle; anything smaller stays entrained.

For rough estimates in air at ambient conditions, the rule-of-thumb is \(d_c\) (μm) ≈ 1000 / (\(\rho_p\) – 1) for 1 m s^-1 gas velocity.

Run a two-step “bucket & balance” test:

Take a 1 kg representative sample from the process stream, split it into 250 g sub-samples.
Pass each sub-sample through a stack of sieves spaced at ½ φ intervals around your calculated CPS.
Weigh the retained fractions; plot mass vs. diameter. The size at which the cumulative curve drops sharply is your real CPS.
Compare to your prediction; adjust gas velocity, cyclone vortex finder, or mill screen accordingly.

A handheld laser diffraction unit or even a good microscope with image analysis can confirm the sieve data within 5%.

No; surface forces dominate when particles are below ≈ 100 μm and moisture exceeds 3% (w/w). Switch to the Rumpf–Molerus cohesion-modified equation:

\(d_{c,\text{cohesive}} = d_{c,\text{Stokes}} \times \exp\left(\frac{0.3\, H_a}{\rho_p\, g\, d_p^2}\right)\)
\(H_a\) = Hamaker constant for the liquid bridge, typically \(10^{-19}\)–\(10^{-20}\) J for water.
If the exponent > 1.5, expect lumps and defluidization; reduce moisture or add flow aid before relying on the dry CPS.

Worked Example: Critical Particle Size for Grit Removal in a Horizontal Flow Settler

A small wastewater treatment plant uses a horizontal-flow grit chamber to remove dense mineral particles before primary clarification. The chamber must capture all particles larger than the critical size that would otherwise escape with the effluent. Determine this critical diameter for the given flow and fluid conditions.

Gravitational acceleration, \(g = 9.806\ \text{m s}^{-2}\)
Flow rate, \(Q = 0.0069\ \text{m}^3\ \text{s}^{-1}\)
Horizontal cross-sectional area, \(A = 12\ \text{m}^2\)
Fluid density, \(\rho_f = 1000\ \text{kg m}^{-3}\)
Fluid dynamic viscosity, \(\mu = 0.0012\ \text{Pa s}\)
Particle density, \(\rho_p = 1050\ \text{kg m}^{-3}\)

Compute the overflow rate \(v_0\) (settling velocity that must be achieved for 100% capture): \[ v_0 = \frac{Q}{A} = \frac{0.0069}{12} = 0.001\ \text{m s}^{-1} \]
Assume Stokes’ law applies (check Reynolds number later). Equate the Stokes settling velocity to the required overflow rate and solve for the unknown diameter \(d\): \[ v_0 = \frac{(\rho_p - \rho_f)\, g\, d^2}{18\,\mu} \quad\Rightarrow\quad d^2 = \frac{18\,\mu\,v_0}{(\rho_p - \rho_f)\,g} \]
Insert the known values: \[ d^2 = \frac{18\,(0.0012)\,(0.001)}{(1050 - 1000)\,(9.806)} = \frac{2.16\times10^{-5}}{490.3} = 4.405\times10^{-8}\ \text{m}^2 \]
Take the square root to obtain the critical diameter: \[ d = \sqrt{4.405\times10^{-8}} = 0.000210\ \text{m}\ (0.210\ \text{mm}) \]
Verify the Reynolds number to confirm Stokes’ regime: \[ Re = \frac{\rho_f\,v_0\,d}{\mu} = \frac{1000\,(0.001)\,(0.000210)}{0.0012} = 0.175\ (<1,\ \text{hence Stokes valid}) \]

Final Answer: The critical particle size that will be completely removed in the grit chamber is 0.210 mm.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle