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Two-product mass balance: calculating mass yield and recovery in mineral processing

Understand and verify mass-yield and component-recovery calculations from feed, product and reject assays.

6-minute read Published on 13 October 2026 Written by Minerallum
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For a simple separation process with one feed and two outputs, product mass yield and the distribution of a tracked component can be estimated from three assays: feed, product and reject.

The calculation does not depend on a Minerallum product database or supplier-specific property. It follows directly from conservation of mass.

The Minerallum calculator applies these equations to user-entered values and exposes the calculation so the result can be checked independently.

Direct answer

Let:

  • F = dry-solids feed mass rate;
  • P = dry-solids product mass rate;
  • T = dry-solids reject/tailings mass rate;
  • f = tracked-component assay in the feed;
  • p = tracked-component assay in the product;
  • t = tracked-component assay in the reject.

For a steady-state two-output system:

F=P+TF=P+T

and for the tracked component:

Ff=Pp+TtFf=Pp+Tt

Product mass yield therefore is:

Y=PF=f-tp-tY=\frac{P}{F}=\frac{f-t}{p-t}

The component distribution, or recovery, to product is:

RP=PpFfR_P=\frac{Pp}{Ff}

therefore:

RP=pff-tp-tR_P=\frac{p}{f}\frac{f-t}{p-t}

These are the classical two-product mass-balance relationships used in mineral processing. A 2024 Minerals Engineering paper explicitly presents them as equations derived from conservation of mass for a steady-state separator.

Why use assays instead of measuring every stream directly?

In a real plant, accurate measurement of dry-solids feed, product and reject rates can be difficult. Processes fluctuate, some flows are discontinuous, and sampling itself introduces uncertainty.

The two-product calculation infers the fraction of solids reporting to product from the assays of the three streams.

That does not make the assays perfect. The quality of the calculated result depends directly on the quality and representativeness of the inputs.

Step-by-step derivation

Start with total mass:

F=P+TF=P+T

Therefore:

T=F-PT=F-P

The tracked-component balance is:

Ff=Pp+TtFf=Pp+Tt

Substitute T:

Ff=Pp+(F-P)tFf=Pp+(F-P)t

Expand:

Ff=Pp+Ft-PtFf=Pp+Ft-Pt

Group terms:

F(f-t)=P(p-t)F(f-t)=P(p-t)

Divide by F(p-t):

PF=f-tp-t\frac{P}{F}=\frac{f-t}{p-t}

P/F is the fraction of the feed dry-solids mass reporting to product.

Mass yield and component recovery are different

This distinction matters.

Mass yield is the fraction of total feed mass reporting to product:

Y=PFY=\frac{P}{F}

Component recovery is the fraction of the tracked component in the feed that reports to product:

RP=PpFfR_P=\frac{Pp}{Ff}

A product may represent a small fraction of total feed mass while recovering a large fraction of a strongly enriched component.

Other useful outputs

Reject mass fraction

YT=1-YY_T=1-Y

Component distribution to reject

When f > 0 and the component is conserved:

RT=1-RPR_T=1-R_P

Feed-to-product mass ratio

K=FP=1YK=\frac{F}{P}=\frac{1}{Y}

This indicates how many units of feed mass correspond to one unit of product mass.

Product/feed assay ratio

E=pfE=\frac{p}{f}

A value above 1 means the tracked component is enriched in the product. A value below 1 means it is depleted.

Must the tracked component be a metal?

No.

Mathematically, the essential requirement is that the tracked quantity is conserved across the selected balance boundary.

Depending on the process and assumptions, it may be:

  • a chemical constituent;
  • a mineral species;
  • a particle-size fraction, provided breakage, agglomeration or reaction does not significantly create or destroy that tracked fraction.

A 2025 Minerals Engineering paper, for example, applies the two-product yield relationship to particle-size fractions in a classifier under explicit steady-state and no-relevant-transformation assumptions.

Conditions required before using the result

1. One feed and two outputs

The model assumes one incoming stream and two outgoing streams.

If there is a third product, middlings, dust loss, an unmeasured dissolved stream, or another output carrying a significant amount of the tracked component, the simple model is incomplete.

2. Representative data from the same operating condition

The three assays should represent the same time period or a sufficiently steady operating condition.

Combining a feed assay taken before a process change with a product assay taken after that change can produce a number with no valid physical interpretation.

3. Conservation of the tracked component

The calculation assumes the tracked quantity is not created or destroyed within the selected boundary.

Reaction, dissolution, precipitation, volatilisation or an unmeasured transformation may invalidate the balance.

4. Same assay unit and analytical basis

The three assays must be directly comparable:

  • all wt%;
  • all g/t;
  • all ppm;
  • or another common consistent unit.

Do not mix, for example, a wet-basis assay with two dry-basis assays.

A useful consistency check

For a physically consistent balance, the feed assay must lie between the two output assays:

min(p,t)≤f≤max(p,t)\min(p,t)\le f\le\max(p,t)

This follows because:

f=Yp+(1-Y)tf=Yp+(1-Y)t

If the feed assay lies outside the interval defined by the two outputs, the three values cannot exactly describe a simple conserved-component one-feed/two-output balance.

The Minerallum calculator should therefore flag the data instead of artificially forcing a 0–100% result.

What if product and reject assays are identical?

If:

p=tp=t

the denominator in the yield equation is zero:

Y=f-tp-tY=\frac{f-t}{p-t}

The tracked component then provides no information from which the mass split can be inferred.

The calculator should report that the yield is not identifiable from those values.

Why can assay error matter so much?

The equations are exact for exact data satisfying the model assumptions. Industrial measurements always carry uncertainty.

Iveson and Galvin showed in 2024 that the two-product formula can amplify assay uncertainty non-linearly, particularly when the selected component does not strongly distinguish the relevant streams.

Wills and Finch likewise explain that simple product formulas propagate measurement error; they do not statistically correct it.

More advanced data reconciliation methods are available for redundant and noisy datasets. That is a different task from the transparent deterministic calculation provided here.

What the Minerallum calculator does

It:

  1. receives the three user-entered assays;
  2. checks basic mathematical consistency;
  3. calculates product mass yield;
  4. calculates component distribution;
  5. calculates product and reject flow rates when a feed rate is supplied;
  6. shows every equation with substituted values;
  7. lets the user independently verify the result.

It uses no Minerallum product property and no supplier-specific dataset.

What it does not do

It does not:

  • certify the assays;
  • correct sampling error;
  • predict future plant performance;
  • replace full statistical data reconciliation;
  • determine whether a material is suitable for an industrial application.

The output is a deterministic calculation from user-entered data.

Quick example

For:

  • feed f = 10%;
  • product p = 40%;
  • reject t = 2%;

mass yield is:

Y=10-240-2=838=0.210526...Y=\frac{10-2}{40-2} =\frac{8}{38} =0.210526...

or:

21.0526%21.0526\%

Product recovery of the tracked component is:

RP=4010×0.210526...=0.842105...R_P=\frac{40}{10}\times0.210526... =0.842105...

or:

84.2105%84.2105\%

The companion worked-example article closes the full mass and component balance.

References

  1. Iveson, S. M.; Galvin, K. P. (2024). Amplification in the Uncertainty of the Yield and Recovery for a Steady-State Mineral Separator calculated using the Two-Product Formula. Minerals Engineering, 212, 108729. https://doi.org/10.1016/j.mineng.2024.108729
  2. Wills, B. A.; Finch, J. (2015/2016). Wills’ Mineral Processing Technology, 8th ed., Chapter 3: Sampling, Control, and Mass Balancing. https://doi.org/10.1016/C2010-0-65478-2
  3. Koermer, S.; Noble, A. (2021). The utility of Bayesian data reconciliation for separations. Minerals Engineering, 169, 106837. https://doi.org/10.1016/j.mineng.2021.106837
  4. Starrett, J. B.; Iveson, S. M.; Galvin, K. P. (2025). Empirical evidence for Fermi-Dirac classification of minerals by particle size. Minerals Engineering, 227, 109277. https://doi.org/10.1016/j.mineng.2025.109277

Key point

The two-product balance is powerful because it is simple: two conservation equations provide the mass split and the distribution of a tracked component.

Its value still depends entirely on the quality of the input data and on respecting its assumptions.

That is why the Minerallum calculator shows the calculation instead of asking users to trust a black box.

Documentation

Sources and references

The technical information presented in this article is based on the following publications and resources.

  1. 01
    Minerals Engineering

    Iveson & Galvin (2024), Two-Product Formula

    View source
  2. 02
    Elsevier

    Wills’ Mineral Processing Technology, Chapter 3

    View source
  3. 03
    Minerals Engineering

    Koermer & Noble (2021), Bayesian data reconciliation

    View source
  4. 04
    Minerals Engineering

    Starrett, Iveson & Galvin (2025), Particle-size classification

    View source

Technical requirements may vary depending on the installation, applicable standards, process and system-designer recommendations.

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