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Metallurgical Accounting

Three Product Formula Calculator

The three-product formula reconciles a circuit that makes two concentrates plus a tailings — for example a Pb-Zn circuit producing a lead concentrate and a zinc concentrate. Using two metals (A and B) assayed across all four streams, it solves the two-metal mass balance on a feed basis (F = 1) for the mass fraction of feed reporting to each concentrate and the tailings, then the recovery of each metal to each concentrate. It accounts for measured streams, calculates accounting metrics, and reconciles assay relationships. It does NOT predict future plant performance, predict recovery, predict flotation or leaching response, guarantee recovery, optimise a plant, or replace metallurgical testwork. The two concentrates must differ in composition (a non-degenerate balance).

TypeInteractive engineering calculator

Calculator

Feed
% or g/t
% or g/t
Concentrate 1
% or g/t
% or g/t
Concentrate 2
% or g/t
% or g/t
Tailings
% or g/t
% or g/t
Result
Concentrate 1 yield5.28401 %
Concentrate 2 yield9.36459 %
Tailings yield85.3514 %
Metal A recovery → Con 188.0668 %
Metal A recovery → Con 26.24306 %
Metal B recovery → Con 17.04535 %
Metal B recovery → Con 285.842 %
  • !Accounting estimate only — this is a two-metal, four-stream mass balance of measured assays. It does NOT predict recovery, model flotation/separation, optimise a circuit, or guarantee performance.
  • !The two concentrates must differ in composition (a non-degenerate balance). Use the same metals and units across all four streams, and confirm samples are representative and reconciled.

Formulas

Concentrate 1 mass fraction of feed
C₁/F = [(f−t)(c₂′−t′) − (f′−t′)(c₂−t)] / [(c₁−t)(c₂′−t′) − (c₁′−t′)(c₂−t)]
Concentrate 2 mass fraction of feed
C₂/F = [(f−t)(c₁′−t′) − (f′−t′)(c₁−t)] / [(c₂−t)(c₁′−t′) − (c₂′−t′)(c₁−t)]
Tailings mass fraction of feed
T/F = 1 − C₁/F − C₂/F
Recovery of a metal to a concentrate
R = (Cᵢ/F) × (concentrate grade / feed grade)

Diagram

Three-product formula — two-metal balance into two concentrates and tailingsfeed f, f′circuitcon 1 (c1, c1′)con 2 (c2, c2′)tails (t, t′)solve two-metal balance for C1/F, C2/Fan accounting balance — not a separation model

Worked example

A Pb-Zn circuit (metal A = Pb, metal B = Zn): feed Pb 3.0% / Zn 6.0%; lead concentrate Pb 50% / Zn 8%; zinc concentrate Pb 2% / Zn 55%; tailings Pb 0.2% / Zn 0.5%. Find each concentrate yield and the metal recoveries.

  1. 01C₁/F = [(2.8)(54.5) − (5.5)(1.8)] / [(49.8)(54.5) − (7.5)(1.8)] = 142.7 / 2700.6 = 0.05284 = 5.28%
  2. 02C₂/F = [(2.8)(7.5) − (5.5)(49.8)] / [(1.8)(7.5) − (54.5)(49.8)] = −252.9 / −2700.6 = 0.09365 = 9.36%
  3. 03T/F = 1 − 0.05284 − 0.09365 = 0.85351 = 85.35%
  4. 04Pb recovery → Con 1: 0.05284 × (50 / 3.0) = 88.07%
  5. 05Zn recovery → Con 2: 0.09365 × (55 / 6.0) = 85.84%
  6. 06Mass-closure check (per 100 t feed): Pb 5.284·50 + 9.365·2 + 85.351·0.2 = 300 ✓; Zn 5.284·8 + 9.365·55 + 85.351·0.5 = 600 ✓
Result

Con 1 yield 5.28%, Con 2 yield 9.36%, tailings 85.35%; Pb→Con 1 88.07%, Zn→Con 2 85.84% — a mass-balance accounting of measured assays, not a separation prediction.

FAQ

When do I use the three-product formula instead of the two-product formula?
Use the three-product formula when the circuit makes two concentrates plus a tailings (e.g. a Pb concentrate and a Zn concentrate). The two-product formula handles a single concentrate plus a tailings. The three-product case needs two metals assayed across all four streams to close the balance.
Why does it need two metals?
With three products (two concentrates and tailings) on a feed basis there are two unknown mass fractions to solve. One metal gives one balance equation; a second metal gives the second independent equation needed to solve for both concentrate yields. The two metals must distinguish the concentrates.
Does this predict how the circuit will separate the metals?
No. It accounts for the split implied by the measured assays — it reconciles grades you have measured into mass fractions and recoveries. It does not predict flotation or separation behaviour, optimise the circuit, or guarantee performance; those require testwork and modelling that are out of scope.
What is a degenerate balance?
If the two concentrates have nearly the same composition the determinant of the balance is near zero and the solution is unstable or undefined. The calculator detects this and shows an input error rather than returning a misleading result. Check that the two concentrate assays genuinely differ.
Is this a metallurgical reconciliation or n-stream balance tool?
No. It is the classic three-product (two-concentrate) closed-form balance only. Full multi-stream reconciliation, n-stream balances, and metallurgical-accounting dashboards are deliberately out of scope.

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