Mineral Processing, Concentrator Mass Balance & Froth Flotation
Two-Product Froth Flotation Mass Pull & Metal Recovery Formula
Calculate Concentrate Mass Pull C/F (%), Metallurgical Metal Recovery R (%), Ratio of Concentration K, and Hourly Concentrate Filter-Cake Tonnage (t/hr) from Feed, Concentrate & Tailings Assays.
Three XRF chemical assays (f, c, t) lock metallurgical recovery R without weighing concentrate slurry!
In copper, nickel, lithium-spodumene, and zinc froth flotation concentrators, online X-ray fluorescence (XRF) analyzers continuously sample Feed Ore Grade f (%), Froth Concentrate Grade c (%), and Barren Tailings Grade t (%). Solving simultaneous ore-tonnage conservation (F = C + T) and contained-metal conservation (F * f = C * c + T * t) eliminates uncertain slurry mass-flow meters completely: Concentrate Mass Pull (%) is (C / F) * 100 = ((f - t) / (c - t)) * 100, Ratio of Concentration (tons of ore milled per ton of concentrate shipped) is K = (c - t) / (f - t), and true Metallurgical Metal Recovery (%) is R = ((c * (f - t)) / (f * (c - t))) * 100! Subtracting (f - t) / f as a naive shortcut overestimates recovery because it ignores the mass fraction pulled into concentrate!
Verify on a 500.0 t/hr Copper Sulfide Concentrator—Mill_Feed_Rate F = 500.0 t/hr (B2), Feed_Assay f = 0.80% Cu (B3), Concentrate_Assay c = 26.00% Cu (B4), and Tailings_Assay t = 0.08% Cu (B5): Concentrate Mass Pull hits ((0.80 - 0.08) / (26.00 - 0.08)) * 100 = 2.78% (E2), Metallurgical Copper Recovery reaches 90.28% (E3), Concentration Ratio K equals 36.00 tons ore/ton concentrate (E4), and Concentrate Shipped equals 500 * 0.027778 = 13.89 dry t/hr (E5)!
| Flotation Circuit KPI (500 t/hr Ore, f = 0.80% Cu, c = 26.0%, t = 0.08%) | Formula | Verified Output |
|---|---|---|
| Concentrate Mass Pull C/F (%) | ((f - t) / (c - t)) * 100 | 2.78% (2.33% @ t = 0.20% Cu) |
| Metallurgical Cu Recovery (%) | ((c * (f - t)) / (f * (c - t))) * 100 | 90.28% (75.58% @ t = 0.20% Cu!) |
| Ratio of Concentration K (t/t) | (c - t) / (f - t) | 36.00 (43.00 @ t = 0.20% Cu) |
| Concentrate Tonnage C (t/hr) | Feed_t_hr * ((f - t) / (c - t)) | 13.89 t/hr (11.63 @ t = 0.20% Cu) |
Exact Three-Assay Metallurgical Recovery Formula
Validate Feed_Rate > 0 and strict grade ordering 0 <= Tailings_t < Feed_f < Concentrate_c <= 100!
=LET(fd_t, B2, fg_p, B3, cg_p, B4, tg_p, B5, IF(OR(fd_t <= 0, tg_p < 0, fg_p <= tg_p, cg_p <= fg_p, cg_p > 100), "Invalid Assay Ordering", ROUND(((cg_p * (fg_p - tg_p)) / (fg_p * (cg_p - tg_p))) * 100, 2)))Coarse Grind Unliberated Sulfide Tailings Loss Check (Set Tailings Grade t = 0.20% Cu)
Simulate an over-ground/under-ground collector upset where Tailings_Grade B5 rises from 0.08% to 0.20% Cu: Metallurgical Recovery E3 plunges 14.7 percentage points from 90.28% down to 75.58%, losing 2.26 t/hr of concentrate in E5 (dropping to 11.63 t/hr)!
=IF(ISTEXT(E3), E3, ROUND(B2 * ((B3 - B5) / (B4 - B5)), 2))Try the example
Use the original values or change them to test your understanding.