8.2 Particle Screening, Hydrocyclone Classification & Sizing

Particle Screening, Hydrocyclone Classification & Sizing

Particle sizing separates a bulk mixture of broken ore into uniform size fractions. Accurate size control is essential to prevent overgrinding, avoid misplacement of coarse material into downstream flotation or leaching circuits, and optimize separation efficiency. Sizing operations in mineral processing are divided into physical screening (mechanically sizing particles on apertures) and classification (sorting particles based on settling rates in fluids).

Industrial Screening Principles & Equipment

Screening presents a bed of particles to a surface containing uniform apertures. Separation depends on particle size, bed depth, screen inclination, moisture content, and vibration amplitude.

Industrial Screen Machinery

• Stationary Grizzlies: Heavy parallel steel bars sloped at $35^\circ\text{--}45^\circ$ positioned ahead of primary crushers to bypass undersize ROM ore ($<100\ \text{mm}$), preventing crusher overload.
• Vibrating Screens: The standard machine for sizing between $0.5\ \text{mm}$ and $100\ \text{mm}$. Inclined vibrating screens ($15^\circ\text{--}25^\circ$) utilize eccentric shafts rotating at $800\text{--}1200\ \text{rpm}$ to stratify the material bed, bringing fine particles into contact with the mesh.
• Trommel Screens: Revolving cylindrical screens inclined at a shallow angle ($4^\circ\text{--}8^\circ$), widely deployed in alluvial gold and aggregate washing plants.

Screen Efficiency Mathematics

Screen performance is quantified by evaluating undersize material recovery and oversize rejection. Consider a screen receiving Mass Feed $F$ with undersize fraction $f$, producing Undersize Product $U$ with undersize fraction $u$, and Oversize Product $O$ with undersize fraction $o$.

From overall and component mass balances:

$F = U + O$ $F f = U u + O o$

Solving for product mass splits:

$\frac{U}{F} = \frac{f - o}{u - o} \quad \text{and} \quad \frac{O}{F} = \frac{u - f}{u - o}$

The Undersize Recovery ($E_u$) and Oversize Recovery ($E_o$) are:

$E_u = \frac{U u}{F f} = \frac{u(f - o)}{f(u - o)}$ $E_o = \frac{O(1 - o)}{F(1 - f)} = \frac{(1 - o)(u - f)}{(1 - f)(u - o)}$

The Overall Screen Efficiency ($E$) is the product of both efficiencies:

$E = E_u \times E_o = \left[ \frac{u(f - o)}{f(u - o)} \right] \times \left[ \frac{(1 - o)(u - f)}{(1 - f)(u - o)} \right]$

Hydraulic Classification Fundamentals & Settling Laws

Classification sorts particles in a fluid medium based on terminal settling velocity ($v_t$), which is governed by particle size, density, and fluid drag forces.

Terminal Settling Velocity Dynamics

When a particle falls in a fluid, gravitational force ($F_g$) is balanced by buoyant force ($F_b$) and drag force ($F_d$). At terminal velocity, net acceleration is zero.

1. Stokes' Law (Laminar Flow, $Re < 0.2$)

For fine particles ($d < 50\ \mu\text{m}$) settling slowly in viscous fluids:

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

Where:

• $v_t$ = Terminal settling velocity ($\text{m/s}$)
• $g$ = Gravitational acceleration ($9.81\ \text{m/s}^2$)
• $d$ = Equivalent spherical particle diameter ($\text{m}$)
• $\rho_s$ = Particle density ($\text{kg/m}^3$)
• $\rho_f$ = Fluid density ($\text{kg/m}^3$)
• $\mu$ = Fluid dynamic viscosity ($\text{Pa}\cdot\text{s}$)

Under Stokes' regime, settling velocity is directly proportional to the square of particle diameter ($d^2$).

2. Newton's Law (Turbulent Flow, $Re > 1000$)

For coarse particles ($d > 2\ \text{mm}$) creating turbulent wakes:

$v_t = 1.74 \sqrt{\frac{g d (\rho_s - \rho_f)}{\rho_f}}$

Under Newton's regime, terminal velocity varies directly with the square root of particle diameter ($\sqrt{d}$).

Free vs. Hindered Settling

• Free Settling: Occurs in dilute suspensions ($<10%$ solids by volume) where particles settle independently without interparticle collisions.
• Hindered Settling: Occurs in dense pulp suspensions ($>15%$ solids by volume) where mutual fluid displacement creates upward interstitial velocities, reducing settling rates. Hindered settling velocity ($v_h$) is calculated using the Richardson-Zaki equation: $v_h = v_t \epsilon^n$, where $\epsilon$ is pulp voidage and $n$ is an empirical exponent ($4.65$ in laminar flow).

Hydrocyclone Classification Geometry & Dynamics

A hydrocyclone is a static cylindrical-conical vessel utilizing centrifugal acceleration ($100G\text{--}1000G$) to achieve rapid size separation in wet slurry streams.

Construction & Internal Flow Regime

1. Tangential Feed Inlet: Slurry is pumped under pressure ($50\text{--}200\ \text{kPa}$) into the cylindrical chamber, generating a high-velocity outer helical downward swirl along the outer wall.
2. Underflow Apex / Spigot: Coarse, dense particles experiencing high centrifugal force are flung outward to the conical wall and discharge at the bottom apex as a thick underflow slurry.
3. Vortex Finder & Overflow: Fluid drag overcomes centrifugal force for fine, low-density particles, drawing them into an inner upward vortex that exits through the central vortex finder tube as overflow.

Cut Size ($d_{50}$) & The Fish-Hook Effect

• $d_{50}$ Cut Size: The specific particle size at which 50% of particles report to the underflow and 50% report to the overflow.
• The Fish-Hook Effect: An anomaly in hydrocyclone partition curves where the recovery of ultra-fine particles ($-10\ \mu\text{m}$) to the underflow increases rather than decreasing monotonically. This can reflect water bypass and entrainment of fine particles in the underflow, including boundary-layer short-circuiting toward the apex, as well as measurement and material effects. Diagnose it with corrected partition data, water split, sampling QA/QC, and repeat tests.

Closed-Circuit Grinding-Classification Loops

Hydrocyclones operate in closed-circuit with ball mills to maximize grinding energy efficiency.

                    +------------------------+
                    |    Primary Crusher     |
                    +-----------+------------+
                                |
                                v
                     +----------------------+
                     |     SAG / Ball Mill  |
                     +----------+-----------+
                                |
                                v
                     +----------------------+
                     |     Sump Pump        |
                     +----------+-----------+
                                |
                                v
                     +----------------------+
+------------------->|     Hydrocyclone     |
|                    +----+------------+----+
|                         |            |
|     Coarse Underflow    |            | Fine Overflow
+-------------------------+            v
                               +------------------+
                               | Flotation Plant  |
                               +------------------+

Mill discharge slurry is diluted in a sump pump and delivered to the hydrocyclone bank. The coarse underflow streams return to the mill inlet, while the fine overflow ($80% < 74\ \mu\text{m}$) feeds flotation. Controlling feed pulp density (typically 40% to 55% solids) ensures maximum classification sharpness.

Comparison of Sizing & Classification Methods

Reading Separation Performance

A screen or cyclone must be evaluated from a reconciled partition curve, not one nominal aperture or d50. For each size class, partition is the fraction reporting to a stated product. Correct for water bypass where the convention requires it, show sampling uncertainty, and confirm that feed equals products by dry mass and size distribution. The curve's d50 describes the midpoint, while its slope or probable error describes sharpness; neither alone describes misplaced mass when the feed contains many near-size particles.

For screens, near-aperture particles require repeated presentation and are sensitive to bed depth, moisture, shape, pegging and blinding. For cyclones, pressure, geometry, feed solids, viscosity, particle density and shape, apex condition, vortex finder, roping, air core and water split all affect classification. Diagnose a coarse overflow by checking feed change, pressure, wear, overload and sampling before changing one dimension. Diagnose a fine underflow upturn only after verifying size analysis and water entrainment. Equipment control should connect the measured partition to downstream circulating load, mill power, flotation liberation, dewatering, and recovery.