Physics Reference

The theoretical rotational speed at which centrifugal force on a particle at the shell equals the force of gravity. Beyond this speed, particles cease to fall — they centrifuge.

The simulator computes angular velocity as ω = −(%Nc/100) × Nc × 2π/60, where Nc = 42.3/√D and D is the effective mill diameter in metres. The negative sign sets the rotation direction.

Occurs below ~65% critical speed. Balls roll and slide down the surface of the charge in a continuous flow. Gentle grinding by attrition — suitable for coarse feed and rod mills.

The charge shoulder sits lower, and impact energy at the toe is minimal (green indicator).

Occurs at 65–82% critical speed. Balls are lifted by the rotating shell and lifters, then detach and fall in projectile trajectories — impacting the toe zone with significant energy.

This is the optimal grinding regime: maximum impact breakage combined with adequate cascading. Most SAG and ball mills operate in this range.

Above ~82% critical speed, centrifugal force exceeds gravity for most particles. Balls pin to the shell wall and rotate with it — no cascading, no cataracting, no effective grinding.

Avoid this regime. The simulator flags it in red.

Each lifter has a vertical leading face up to the Leading Edge Height (LEH), then an angled (raked) portion up to the full Lifter Height (hL). The rake angle (α) is measured from vertical.

The angular offset at the lifter top is (hL − LEH) × tan(α). A steeper face lifts balls more directly; a shallower face releases them earlier, affecting how high the charge shoulder reaches.

Lifter spacing S is the arc pitch between adjacent lifters: S = π·D_pitch / N, where D_pitch is the shell inner diameter minus 2×(rubber lining + lip). H_eff is the effective lifter height (reduced as the liner wears). S:H is a key lifter-design metric.

Per Malcolm Powell, the target S:H is aimed at mid-life of the liner — a new liner starts below the target and wears up through it. The working band is Target ± band width. The suggested row count places the new liner at the lower edge of the band so S:H lands on the target around 50% wear. Application presets (SAG ≈ 2.9 ± 1.0, Primary Ball ≈ 3.0 ± 1.5, Secondary Ball ≈ 2.5 ± 2.0) can be enabled in Default Settings, with a progressive-wear slider to confirm the ratio tracks through the target over the liner life.

The simulator detects impacts of the largest grinding-media balls against the liner and classifies them by impact speed. Above 3.5 m/s an impact is logged; Medium spans 3.5–4.5 m/s and High is ≥4.5 m/s.

Impacts are tallied into the two bands over a rolling window; when the alert throttle elapses, the dominant band (most impacts) is reported with the worst impact speed in that band. An expanding red flash marks the most severe recent impact point on the liner.

Fading yellow tracer lines follow a selectable fraction (25/50/75/100%) of the largest-diameter media balls, recording their shoulder→toe path so the cataracting trajectory is visible as a continuous streak rather than a single frame of positions.

Trails are stored per ball id and capped in length; newer segments render brighter and thicker, so the most recent flight reads as the brightest part of the trace.

The simulator models four independent friction coefficients: media-to-media (μbb), media-to-boundary (μb), ore-to-ore (μoo), and ore-to-boundary (μo). Each runs from 0.05 (slippery) to 0.7 (sticky).

Boundary friction scales how far the mill drags the charge up the shell before slippage — higher boundary friction lifts the shoulder higher. Particle-particle friction controls how the media and ore beds shear internally and how ore interlocks with media. All four take effect live without re-seeding.

At seed time, media and ore particles are over-seeded by a configurable factor (1–3×) and dropped into the mill under gravity with no rotation. They pack until motion drops below a threshold, producing a settled pile that fills to the target charge line.

Only once settling completes does rotation begin. This two-phase approach gives an accurate starting charge shape before cataracting motion develops.

The toggleable Total (green), Media (orange), and Ore (light blue) lines are the horizontal chords of the liner circle whose segment area equals that component's charge% × effective area.

The segment half-angle θ is solved from the target area, then the chord is drawn at height h2 = R − R(1 − cos(θ/2)) below centre. These show the target free-surface for each component.