Hydrometallurgy

Two Product vs Three Product Formula

When to use the two-product formula (one concentrate, one metal, three assays) versus the three-product formula (two concentrates, two metals, four streams) — the data each needs, their assumptions, strengths, weaknesses, and a worked Cu vs Pb-Zn contrast. A decision guide for accounting balances, not a recovery-prediction model.

TypeEngineering guide — concept explainer

Definition

The two-product and three-product formulas are both metallurgical accounting balances, but they fit different circuit shapes. The two-product formula applies when a feed splits into one concentrate and one tailings: from a single metal assayed in those three streams (c > f > t) it gives mass yield, recovery, ratio of concentration, and enrichment ratio. The three-product formula applies when a feed splits into two concentrates and a tailings — for example a lead concentrate and a zinc concentrate from the same feed. Because three products on a feed basis leave two unknown mass fractions to solve, the three-product formula needs two metals assayed across all four streams: one metal gives one balance equation, the second metal gives the second independent equation. The choice is therefore set by the number of saleable products the circuit makes and the number of metals you can assay, not by preference.

Why it matters

Choosing the wrong formula gives wrong accounting. If a circuit really makes two concentrates and you force a two-product balance on just one of them, you ignore the metal splitting to the other concentrate and the recovery numbers are wrong. Conversely, the three-product formula needs two metals that genuinely distinguish the two concentrates — if the concentrates have nearly the same composition the balance becomes degenerate (its determinant approaches zero) and the solution is unstable. Knowing which formula the circuit calls for, and what data each demands, is the difference between a balance that closes and one that quietly misallocates metal. What does not change between them is the boundary: both are accounting tools that reconcile measured assays. Neither predicts recovery, models flotation or separation, optimises a circuit, or guarantees performance — they account for what the sampled streams show.

Formula

Two-product — mass yield

Y = (f − t) / (c − t)

Two-product — recovery

R = Y × (c / f)

Three-product — concentrate 1 yield

C₁/F = [(f−t)(c₂′−t′) − (f′−t′)(c₂−t)] / [(c₁−t)(c₂′−t′) − (c₁′−t′)(c₂−t)]

Three-product — concentrate 2 yield

C₂/F = [(f−t)(c₁′−t′) − (f′−t′)(c₁−t)] / [(c₂−t)(c₁′−t′) − (c₂′−t′)(c₁−t)]

Three-product — recovery to a concentrate

R = (Cᵢ/F) × (concentrate grade / feed grade)

Units involved

•Two-product: f, c, t — one metal across three streams, same units
•Three-product: f/f′, c₁/c₁′, c₂/c₂′, t/t′ — two metals across four streams
•Y, C₁/F, C₂/F, T/F — mass fractions of feed (×100 for %)
•R — recovery, a fraction (×100 for %)
•Assays in consistent grade units (% or g/t)

Concept diagram

Worked example

Contrast a single-concentrate Cu circuit (two-product) with a Pb-Zn two-concentrate circuit (three-product).

01Two-product Cu: f = 1.0%, c = 10.0%, t = 0.2% → Y = 0.8/9.8 = 8.163%, R = 8.163% × 10 = 81.63%
02One metal and three assays were enough to close the single-concentrate balance
03Three-product Pb-Zn: feed Pb 3.0 / Zn 6.0; Pb-con 50/8; Zn-con 2/55; tails 0.2/0.5
04Con 1 (Pb) yield = 142.7/2700.6 = 5.28%; Con 2 (Zn) yield = 9.36%
05Pb→Con 1 = 5.28% × (50/3.0) = 88.07%; Zn→Con 2 = 9.36% × (55/6.0) = 85.84%
06Two metals across four streams were required because there are two concentrates to allocate

Result

The single-concentrate circuit closes with the two-product formula; the two-concentrate circuit needs the three-product formula and a second metal — both reconcile measured assays, neither predicts performance.

Common mistakes

•Using the two-product formula on a circuit that actually makes two concentrates.
•Trying the three-product formula with only one metal, leaving the balance unsolvable.
•Choosing two metals that do not distinguish the concentrates, giving a degenerate balance.
•Mixing assay units or metals between streams.
•Reading either result as a prediction or guarantee rather than an accounting of measured assays.

When to use the calculator

Use the two-product formula calculator for a single-concentrate circuit, and the ratio of concentration calculator when you only need the feed-per-concentrate ratio. Use the three-product formula calculator when the circuit makes two concentrates and you have two metals assayed across all four streams. For the concept background — recovery, yield, ratio of concentration, and enrichment ratio — see the metallurgical recovery formulas explained guide. For the recovery a circuit will actually achieve, use testwork and qualified review.

Two Product Formula Calculator →Three Product Formula Calculator →Ratio of Concentration Calculator →

FAQ

How do I decide between the two-product and three-product formula?

Count the products. One concentrate plus a tailings → two-product formula (one metal, three assays). Two concentrates plus a tailings → three-product formula (two metals across four streams). The circuit shape and the metals you can assay decide it, not preference.

Why does the three-product formula need two metals?

With three products on a feed basis there are two unknown mass fractions to solve. Each metal provides one independent mass-balance equation, so two metals are needed to solve for both concentrate yields. The two metals must distinguish the concentrates or the balance is degenerate.

Can the two-product formula handle a two-concentrate circuit if I combine the concentrates?

Only if you treat the two concentrates as one combined product, which discards the split between them. To account for each concentrate separately — and the recovery of each metal to each — you need the three-product formula.

Do either of these formulas predict plant performance?

No. Both are accounting balances that reconcile measured assays for a sampled period. Neither predicts recovery, models flotation or separation, optimises the circuit, or guarantees performance. Those require metallurgical testwork and modelling, which are out of scope here.