Spiral Concentrator Bands (2D)

What a spiral concentrator does

A spiral concentrator is a helical trough, about 1 m across, down which a dilute mineral slurry runs as a thin film. Heavy grains (for example cassiterite or ilmenite) settle into a slow bed layer near the bottom and drift toward the inner edge; light gangue (quartz) stays suspended in the fast upper layer and is carried outward. Splitters at the end cut the stream into concentrate, middlings and tailings. This page is an educational physics model, not a process design.

Equations in the model

• Film flow: laminar-type profile with an effective (eddy) viscosity: u(z) = (g sin a / nu_e)(h z - z^2/2), flow Q = g sin a h^3 W / (3 nu_e), so the film depth is h = (3 nu_e Q / (g sin a W))^(1/3). Friction velocity u* = sqrt(g h sin a).
• Settling: terminal speed from the drag balance with CD = 24/Re (1 + 0.15 Re^0.687) (Schiller-Naumann), which reduces to Stokes' law v = (rho_p - rho_f) g d^2 / (18 mu) for Re < 1. Hindered settling in a concentrated pulp: v_h = v (1 - phi)^4.65 (Richardson-Zaki).
• Vertical mixing: each grain does a random walk with turbulent diffusivity D = f_mix x 0.068 u* h and reflecting walls at the bed and the surface, giving an equilibrium profile c(z) ~ exp(-v z / D). The Rouse number v / (0.41 u*) tells how bed-hugging a grain is.
• Secondary flow: the radial momentum balance nu_e w'' = -(u^2/R - c) with no slip at the bed, zero stress at the surface and zero net radial flux gives water moving outward near the surface and inward along the bed.
• Grains: along-trough speed is the local fluid speed reduced by bed friction near the floor; radial speed is the secondary flow plus a small centrifugal slip tau (1 - rho_f/rho_p) u^2 / R plus turbulent diffusion.

Honest simplifications

• Real spiral flow is turbulent, three-dimensional and carries a wash-water layer; the eddy viscosity, the mixing factor and the bed-friction factor are tuning parameters, not measured constants. The secondary-flow factor 1 means the laminar-profile theory above.
• The film depth is uniform across the width, grains do not interact except through the hindered-settling factor, and the Monte-Carlo ensemble uses about 1200 weighted grains per run, so recovery and grade have a statistical error of a few percent (the seed button shows it).
• Grains are drawn much larger than real size so that they are visible. Plant results depend on feed preparation, slimes, and many spirals in series.

Checks run in the page

(1) Terminal velocity from the drag balance against Stokes' law for fine quartz. (2) The integral of u(z) against the feed flow, and zero net secondary flux. (3) A closed-box Monte-Carlo run against the analytic exponential settling profile, plus product mass balances.

• Load Cassiterite from quartz, wait for the statistics, then press Best splitters and read grade and recovery. Move splitter 1 outward and watch grade fall while recovery rises (graph 3).
• Choose Narrow density gap: separation almost vanishes because the settling velocities become similar (graph 4).
• Choose Too fast and steep: a thicker, faster film lowers the Rouse numbers and heavy grains are carried to the tail.
• Raise the solids to 22 % and see the hindered-settling penalty. Set the secondary flow factor to 0 and see what the radial flow adds.