Minerallum Technical Lab

Mineral-processing recovery calculation: complete two-product balance example

A complete worked example covering mass yield, recovery, product and reject flows, and independent mass-balance verification.

5-minute read Published on 14 October 2026 Written by Minerallum
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Read the derivation and balance conditions

Open the Minerallum calculator

This example shows how to manually verify the Minerallum calculator for a simple system with one feed, one product and one reject stream.

All data below are fictional and are used only for educational demonstration.

Starting data

Consider a steady-state separator.

Dry-solids feed rate:

F=100t/hF=100\ t/h

Tracked-component assay:

Stream Assay
Feed f 10%
Product p 40%
Reject t 2%

All three values use the same unit and analytical basis.

The component is enriched in product and depleted in reject:

p>f>tp>f>t

Step 1 - calculate product mass yield

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

Substitute:

Y=10-240-2Y=\frac{10-2}{40-2}
Y=838Y=\frac{8}{38}
Y=0.210526315789...Y=0.210526315789...

Therefore:

Y=21.0526315789%Y=21.0526315789\%

About 21.05% of the feed dry-solids mass reports to product.

Step 2 - calculate product dry-solids flow

P=F×YP=F\times Y
P=100×0.210526315789P=100\times0.210526315789
P=21.0526315789t/hP=21.0526315789\ t/h

Step 3 - calculate reject dry-solids flow

T=F-PT=F-P
T=100-21.0526315789T=100-21.0526315789
T=78.9473684211t/hT=78.9473684211\ t/h

Total-mass check:

P+T=21.0526315789+78.9473684211=100t/hP+T=21.0526315789+78.9473684211=100\ t/h

The total mass balance closes.

Step 4 - calculate component recovery to product

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

Since P/F = Y:

RP=YpfR_P=Y\frac{p}{f}

Substitute:

RP=0.210526315789×4010R_P=0.210526315789\times\frac{40}{10}
RP=0.842105263158R_P=0.842105263158

Therefore:

RP=84.2105263158%R_P=84.2105263158\%

The calculation indicates that about 84.21% of the tracked component entering with the feed reports to product.

Step 5 - component distribution to reject

Assuming conservation of the tracked component:

RT=1-RPR_T=1-R_P
RT=1-0.842105263158R_T=1-0.842105263158
RT=0.157894736842R_T=0.157894736842

Therefore:

RT=15.7894736842%R_T=15.7894736842\%

Step 6 - verify the component balance in tonnes per hour

Because the assays in this example are mass percentages, we can directly check contained-component mass.

Feed

100t/h×10%=10t/h100\ t/h\times10\%=10\ t/h

Product

21.0526315789t/h×40%21.0526315789\ t/h\times40\%
=8.42105263156t/h=8.42105263156\ t/h

Reject

78.9473684211t/h×2%78.9473684211\ t/h\times2\%
=1.57894736842t/h=1.57894736842\ t/h

Closure

8.42105263156+1.578947368428.42105263156+1.57894736842
=10t/h=10\ t/h

For this idealized dataset, the component mass leaving the separator exactly equals the component mass entering it.

Step 7 - feed-to-product mass ratio

K=FP=1YK=\frac{F}{P}=\frac{1}{Y}
K=10.210526315789K=\frac{1}{0.210526315789}
K=4.75K=4.75

In this fictional example, 4.75 mass units of feed correspond to one mass unit of product.

Step 8 - product/feed assay ratio

E=pfE=\frac{p}{f}
E=4010=4E=\frac{40}{10}=4

The tracked-component assay in product is four times the feed assay.

Step 9 - reconstruct the feed assay

Another useful check is to reconstruct the feed assay from the two outputs:

f^=Yp+(1-Y)t\hat f=Yp+(1-Y)t
f^=(0.210526315789×40)+(0.789473684211×2)\hat f=(0.210526315789\times40) +(0.789473684211\times2)
f^=8.42105263156+1.57894736842\hat f=8.42105263156+1.57894736842
f^=10\hat f=10

This exactly reproduces the entered feed value:

f=10%f=10\%

It is an excellent internal calculator invariant.

Results summary

Metric Result
Product mass yield 21.0526%
Reject mass yield 78.9474%
Product dry-solids flow 21.0526 t/h
Reject dry-solids flow 78.9474 t/h
Component distribution to product 84.2105%
Component distribution to reject 15.7895%
Feed-to-product mass ratio 4.75
Product/feed assay ratio 4.00

Why the calculator should expose each step

A standalone result such as “recovery = 84.21%” requires trust in the software.

A result that shows:

Y=10-240-2Y=\frac{10-2}{40-2}

followed by:

RP=4010×YR_P=\frac{40}{10}\times Y

can immediately be checked with a normal calculator or spreadsheet.

That is the purpose of the Minerallum Technical Lab approach.

A case where the tracked component is depleted in product

The same balance also works when the tracked component is an impurity or another species preferentially reporting to reject.

For example:

  • f = 20
  • p = 5
  • t = 30

Then:

Y=20-305-30=0.4Y=\frac{20-30}{5-30}=0.4

The product represents 40% of feed mass, while only:

RP=0.4×520=0.1R_P=0.4\times\frac{5}{20}=0.1

or 10% of the tracked component reports to product.

The calculation therefore does not require the tracked component to be economically valuable. It simply calculates distribution from the entered values.

When should this example not be used as a model?

The simple calculation is not appropriate without modification when:

  • the process has more than two significant outputs;
  • the system is not sufficiently steady;
  • assays do not represent the same time period;
  • analytical units or bases differ;
  • the tracked component is not conserved;
  • reaction or transformation changes the quantity being tracked.

Real measurements contain uncertainty

The numbers in this example were constructed to close perfectly.

Real measurements may fail to close exactly because of sampling, assay uncertainty and process variation.

Published research in Minerals Engineering also shows that the two-product formula can amplify assay uncertainty in some configurations.

The Minerallum calculator should therefore describe its output as calculated from the entered values, not as a certified measurement of true plant performance.

Technical 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. Wills’ Mineral Processing Technology, 8th ed., Chapter 3. 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

Use the calculator

Enter the assays for the three streams and, if available, the dry-solids feed rate.

The calculator will show the result and every step used to obtain it.

The data in this article are fictional. Always confirm that real process data satisfy the model assumptions before interpreting the result.

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

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

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