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Metallurgical Accounting

Two Product Formula Calculator

The two-product formula turns three measured assays — the feed, concentrate, and tailings grades of one metal — into the four numbers that account for a single-concentrate circuit: the mass yield (mass pull) to concentrate, the metal recovery, the ratio of concentration, and the enrichment ratio. This calculator reconciles assays you have already measured. It accounts for measured streams, calculates accounting metrics, and reconciles assay relationships. It does NOT predict future plant performance, predict metallurgical recovery, predict flotation or leaching response, guarantee recovery, optimise a plant, or replace metallurgical testwork. The inputs must satisfy c > f > t (concentrate richer than feed, feed richer than tailings).

TypeInteractive engineering calculator

Calculator

% or g/t

assay of the feed stream

% or g/t

must be higher than the feed

% or g/t

must be lower than the feed

Result
Recovery81.6327 %
Concentrate yield (mass pull)8.16327 %
Ratio of concentration12.25 : 1
Enrichment ratio10 ×
  • !Accounting estimate only — this balances measured feed, concentrate, and tailings assays. It does NOT predict recovery, model flotation or leaching, optimise a plant, or guarantee performance.
  • !Use the same metal and the same assay units for all three streams, and confirm the samples are representative and reconciled. Results assume c > f > t.

Formulas

Mass yield (mass pull) to concentrate
Y = (f − t) / (c − t)
Metal recovery
R = Y × (c / f)
Ratio of concentration
K = 1 / Y = (c − t) / (f − t)
Enrichment ratio
ER = c / f

Diagram

Two-product formula — feed splits into concentrate and tailings by assay balancefeed fseparationconcentrate ctailings tY = (f − t)/(c − t) R = Y·c/fan assay accounting balance — not a recovery prediction

Worked example

A copper circuit assays feed 1.0% Cu, concentrate 10.0% Cu, and tailings 0.2% Cu. What are the recovery, yield, ratio of concentration, and enrichment ratio?

  1. 01Mass yield: Y = (1.0 − 0.2) / (10.0 − 0.2) = 0.8 / 9.8 = 0.08163 = 8.163%
  2. 02Recovery: R = 0.08163 × (10.0 / 1.0) = 0.8163 = 81.63%
  3. 03Ratio of concentration: K = 1 / 0.08163 = 12.25 (tonnes feed per tonne concentrate)
  4. 04Enrichment ratio: ER = 10.0 / 1.0 = 10.0
Result

Recovery 81.63%, yield 8.163%, ratio of concentration 12.25, enrichment ratio 10.0 — an accounting balance of measured assays, not a recovery prediction.

FAQ

Does the two-product formula predict recovery?
No. It accounts for recovery from assays you have already measured. It reconciles the feed, concentrate, and tailings grades into a recovery number for that measured period; it does not forecast what recovery a circuit will achieve, model flotation or leaching, or guarantee performance. Predicting recovery requires testwork and is out of scope.
What is the difference between yield (mass pull) and recovery?
Mass yield (mass pull) is the fraction of the feed mass that reports to the concentrate, Y = (f − t)/(c − t). Recovery is the fraction of the metal that reports to the concentrate, R = Y × c/f. A low-yield, high-grade concentrate can still carry most of the metal, which is why recovery and yield are different numbers.
Why must c > f > t?
The concentrate must be richer than the feed and the feed richer than the tailings for a single-concentrate upgrade to make physical sense. If the inputs violate c > f > t the balance gives meaningless (negative or >100%) results, so the calculator shows an input error instead.
What units should the assays be in?
Any consistent grade unit — percent for base metals or g/t for precious metals — as long as all three streams use the same metal and the same unit. The formula is a ratio of differences, so the units cancel.
Can I use this for metallurgical accounting and reconciliation?
Yes, for the basic two-product accounting relationship from a representative set of assays. It is not a full multi-stream reconciliation, a best-fit data-reconciliation engine, or a metallurgical-accounting dashboard — it reconciles one feed/concentrate/tailings assay set.

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