Centrifugal Concentrator - 2D Bowl Cross-Section and Recovery Curves

Controls

Live readouts

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Numerical check

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Performance graph

How to use

Half of a spinning concentrator bowl, seen in cross-section from the axis (left dashed line) to the wall (right). The bowl throws heavy particles outward against the wall with a field of tens of times gravity, while water fed through holes in the wall pushes particles back inward. Heavy gold and denser minerals (yellow) settle outward faster than light quartz (grey) and are caught at the wall; the rest is washed out at the inner edge into the tailings.

On the right, the partition curves show the probability that a particle of each size is retained. The bars give the feed-weighted recovery of the heavy mineral, the share of quartz that is also retained (mass pull) and the resulting enrichment.

Try: raise the bowl speed to catch finer gold; raise the water to wash more quartz out (and lose some fine gold); switch the mineral to pyrite and watch recovery fall as the density contrast shrinks; slide the feed d50 down to see why very fine gold is hard to recover.

The science

Centrifugal field. A particle at radius r in a bowl spinning at angular speed omega feelsa = omega^2 r, G = omega^2 r / g900 rpm at a 66 mm radius gives about 60 G.

Settling in the field. Balance of centrifugal force and fluid drag, with the Schiller-Naumann drag law over the whole particle Reynolds number range:(pi/6) d^3 (rho_p - rho_f) a = (1/2) rho_f C_D (pi/4) d^2 v^2, C_D = (24/Re)(1 + 0.15 Re^0.687)At small Re this reduces to Stokes law v = d^2 (rho_p - rho_f) a / (18 mu); two minerals settle equally fast when d_1/d_2 = sqrt((rho_2-rho_f)/(rho_1-rho_f)), the equal-settling ratio (3.33 for quartz against gold).

Counter-flow and hindered settling. Water enters through the wall and flows inward with the superficial velocity u = Q/(2 pi r h) (continuity). Solids in the pulp slow settling by the Richardson-Zaki factor (1 - phi)^4.65.

Capture probability. Each particle drifts radially with velocity mu(r) = v_settle(r) - u(r) plus turbulent diffusion D. The probability to reach the wall before the inner edge is the exact one-dimensional first-passage result P(r0) = integral from r_in to r0 of exp(-integral mu/D) dr divided by the same integral over the whole zone. The curves use that solution, the animated particles use a stochastic simulation of the same equation, and the page checks the two methods against each other.

Bed capacity and displacement. Light gangue that reaches the wall can only fill a bed of finite capacity C. When the bed is full, a denser arriving particle displaces the lightest one in the bed, which is flushed away; a lighter arrival is washed out. So the concentrate is heavy mineral plus at most C kg of gangue: the mass balance is m_conc = m_heavy + min(C - m_heavy, m_quartz arriving), and the grade enrichment is (m_heavy / m_conc) divided by the feed grade.

Simplified: a one-dimensional radial model with a lognormal feed size distribution, liberated spherical particles, no riffle geometry and a fixed bed capacity. Water speed and diffusion are uniform in height. It illustrates the physics of gravity concentration, not the throughput of a specific machine.

Keyboard, touch and accessibility

Space pauses or resumes the simulation.
R resets it to its initial state.
C re-runs the numerical check.
All sliders are native range inputs and work with arrow keys, Home and End; on touch devices drag on the canvas where the sim supports pointer input.
Rendering is capped at a device pixel ratio of 2 so phones stay smooth; the frame-time graph above shows the cost of the current settings (green line is 16.7 ms, i.e. 60 frames per second).

What is checked and why you can trust it

Every time the page loads, and whenever you press Re-run check, the simulation evaluates its own mathematics against an independent reference, such as a closed-form formula, a finite-difference derivative or an exhaustive count. Each line above reports PASS or FAIL together with the measured numbers, so you can see by how much it agrees. Results are also published on window.__msChecks for automated testing. The check never uses the same code path as the animation it validates, otherwise it would only confirm itself.

How to read a failure: a FAIL line means the measured discrepancy exceeded the stated tolerance for the current slider values. Try Reset, then move the sliders back toward their defaults; if a check still fails at the defaults it is a bug worth reporting. Tolerances are chosen from the numerical method itself, never tuned to make a test pass: round-off level (1e-9) for exact algebra, about 1e-3 for finite differences and Monte Carlo estimates with a few hundred thousand samples, and a documented margin for stochastic learning runs.

Reproducibility: the particle and pen simulations use the browser random generator for initial positions, so exact pixels differ between visits, but the checks depend only on mathematics and not on those positions. Fixed wave tables (curl noise) and fixed grammars (L-system) are deterministic.

This is an educational model. It shows the mechanism clearly, but it is deliberately simplified; the science panel lists what was left out.