Mineral Processing and Ore Handling Systems Formula Sheet

This formula sheet collects first-pass relationships used in mineral processing and ore handling systems. Use it to keep dry solids, wet tonnes, component grade, recovery, water balance, circulating load, screening, crushing, slurry flow, pump power, conveyor capacity, tailings mass, and reconciliation on a consistent basis.

The formulas are not a substitute for metallurgical test work, equipment vendor curves, plant sampling standards, geometallurgical models, site safety rules, tailings design criteria, or environmental permits. They are a calculation framework for plant review, shift troubleshooting, production accounting, and engineering decisions.

Basis and Notation

State the basis before calculating. A plant number may refer to dry solids, wet ore, slurry, contained metal, volume flow, or stream assay.

Symbol	Meaning	Typical unit
F	dry feed solids rate	\text{t/h}
C	dry concentrate or product solids rate	\text{t/h}
T	dry tailings or reject solids rate	\text{t/h}
R	recycle or circulating solids rate	\text{t/h}
f,c,t	component grade in feed, concentrate, tailings	mass fraction or percent
\dot{m}	mass flow rate	\text{t/h} or \text{kg/s}
Q	volumetric flow rate	\text{m}^3/\text{s} or \text{m}^3/\text{h}
\rho	density	\text{t/m}^3 or \text{kg/m}^3
w_s	solids mass fraction in slurry	dimensionless
P	power	\text{kW}
\eta	efficiency	dimensionless

Use grades consistently. Percent grades can be used in ratios only when all grades use the same percent basis.

Dry, Wet, and Slurry Basis

Dry solids from wet mass:

Water mass from wet mass:

where w_m is moisture mass fraction.

Slurry mass from dry solids and solids mass fraction:

Water in slurry:

Volumetric slurry flow:

Engineering Comment

Do not combine wet belt-scale tonnes with dry assay grades unless the moisture correction is known. Many reconciliation errors start as a hidden basis mismatch.

Metallurgical Balance

Total dry solids balance:

Component balance:

Concentrate mass fraction from a two-product balance:

Tailings mass fraction:

Metal or useful component recovery:

Using the two-product balance:

Enrichment ratio:

Validity

These formulas assume a steady two-product circuit with representative assays and no unmeasured inventory change. Recycle streams, middlings, stockpile changes, moisture, sampling bias, delayed assays, and unmeasured losses require a wider balance.

Circulating Load and Internal Flow

Circulating load ratio:

Percent circulating load:

Total mill or screen feed:

Net product fraction of total internal flow:

Engineering Comment

High circulating load is not automatically bad. It becomes a problem when it coincides with coarse product, high energy, roping, screen overload, crusher instability, or downstream recovery loss.

Screening and Classification

Recovery of true undersize to the undersize product:

where U_p is true undersize in the undersize product and U_f is true undersize in screen feed.

Misplaced oversize fraction in undersize product:

where O_p is oversize mass reporting to undersize product and P_u is total undersize product mass.

Classifier cut-size notation:

Partition value:

Engineering Comment

Screen and cyclone performance depend on feed size distribution, density, water addition, pressure, aperture, near-size material, blinding, wear, viscosity, rheology, and sampling method. A single efficiency number is weak unless the size fractions and sampling basis are documented.

Crushing and Grinding

Size reduction ratio:

Specific energy:

where P is power in \text{kW} and \dot{m}_{product} is dry product rate in \text{t/h}, giving \text{kWh/t}.

Net throughput shortfall:

Relative shortfall:

Engineering Comment

Reduction ratio and specific energy are screening metrics. Crusher chamber design, closed-side setting, liner wear, feed gradation, ore hardness, moisture, screen condition, and tramp protection decide whether the circuit is actually operating inside its envelope.

Conveyor and Ore Handling Capacity

Volumetric capacity:

Mass capacity:

where A is loaded cross-sectional area, v is belt speed, \rho_b is bulk density in \text{t/m}^3, and k_L is a loading or availability factor.

Capacity utilization:

Stockpile residence time:

Engineering Comment

Conveyor capacity is not only belt speed. Chute flow, moisture, fines, segregation, belt mistracking, spillage, dust control, take-up, drive power, skirt friction, guarding, and maintenance access can reduce practical capacity.

Slurry Flow and Pump Power

Pipe velocity:

Hydraulic head from pressure difference:

Hydraulic power:

Pump input power:

Pressure rise:

Engineering Comment

Slurry pump calculations must be checked against pump curves, solids concentration, particle size, settling velocity, wear, viscosity, air entrainment, suction conditions, cavitation margin, and shutdown flushing procedure.

Water Balance and Tailings

Water to solids ratio:

Water recovery:

Tailings dry solids:

Tailings slurry mass:

Tailings water:

Engineering Comment

Tailings are an engineering output, not a leftover. Water balance, solids concentration, rheology, permeability, chemistry, deposition method, storage capacity, seepage, and closure constraints should be reviewed when plant operating conditions change.

Reconciliation and Measurement Error

Inventory-corrected production:

Relative reconciliation error:

Measurement margin:

Uncertainty ratio:

Engineering Comment

If the uncertainty is larger than the decision margin, the plant should not treat the number as a release decision. Sampling, belt scales, density meters, flow meters, assays, moisture measurements, and stockpile surveys all need calibration and bias checks.

Worked Example 1: Two-Product Metallurgical Balance

A concentrator treats:

Feed copper grade:

Concentrate grade:

Tailings grade:

Concentrate mass fraction:

Concentrate mass:

Tailings mass:

Metal recovery:

Engineering Comment

The calculation is internally consistent, but it is only as good as the assays and mass measurements. A biased tailings sample can move the recovery estimate enough to trigger the wrong operating response.

Worked Example 2: Slurry Flow and Pump Power

A cyclone feed pump handles dry solids:

Solids concentration by mass:

Slurry density:

Pump head:

Pump efficiency:

Slurry mass flow:

Volumetric flow:

Convert:

Pump input power:

Engineering Comment

This is hydraulic screening power, not motor selection. Slurry correction, pump curve, wear, suction pressure, cavitation margin, density measurement, and control-valve position must be checked.

Worked Example 3: Screen Circulating Load and Undersize Recovery

A closed crusher-screen circuit has fresh feed:

Screen oversize recycle:

Screen undersize product:

Circulating load:

Total screen feed:

If size analysis shows true undersize in screen feed:

and true undersize reporting to undersize product:

then undersize recovery is:

Engineering Comment

The circuit is producing net product, but 44 percent circulating load and 86.2 percent undersize recovery suggest avoidable internal work. Screen blinding, near-size material, aperture wear, moisture, and crusher setting should be checked before increasing feed rate.

Worked Example 4: Conveyor Capacity

A crushed-ore conveyor has loaded cross-sectional area:

Belt speed:

Bulk density:

Loading factor:

Mass capacity:

If required capacity is:

utilization is:

Engineering Comment

The belt has nominal capacity margin, but this does not prove the transfer system. Chute plugging, skirt friction, mistracking, moisture, lump size, drive power, belt scale accuracy, and spillage controls remain part of the review.

Worked Example 5: Tailings Water Balance

Dry tailings solids:

Current tailings slurry is:

Current tailings slurry mass:

Current water to tailings:

If thickener control increases underflow to:

then:

New water to tailings:

Water reduction:

Engineering Comment

The water saving is operationally important, but higher solids concentration can increase yield stress, pumping pressure, line plugging risk, and deposition constraints. Tailings rheology and pump limits must be checked.

Worked Example 6: Crusher Reduction and Specific Energy

A crusher-screen circuit has feed:

Product:

Crusher power:

Net product:

Reduction ratio:

Specific energy:

If the stable reference value is:

relative increase is:

Engineering Comment

Higher specific energy with lower net product is a warning sign. The likely investigation path is crusher level, closed-side setting, liner condition, screen oversize, screen blinding, feed gradation, and ore hardness before assuming the motor is simply underpowered.

Common Mistakes

1. Mixing wet tonnes with dry assay grades.
2. Reporting recovery without checking concentrate and tailings assays against a mass balance.
3. Treating circulating load as product throughput.
4. Using screen efficiency without a size-by-size basis.
5. Ignoring water balance when slurry density or cyclone pressure changes.
6. Calculating pump power without checking slurry correction, NPSH, wear, and settling.
7. Reporting conveyor capacity without chute, transfer, spillage, and belt-scale evidence.
8. Treating tailings solids concentration as only a process variable, not a storage and closure variable.
9. Trusting reconciliation when stockpile inventory and moisture are unknown.
10. Changing control limits without validating sampling and instrument calibration.

Validation Evidence

A mineral processing calculation package should be tied to evidence such as:

• dry and wet mass basis for every stream;
• representative assays and moisture measurements;
• belt-scale, density-meter, flow-meter, and sampler calibration records;
• size distributions for feed, product, recycle, cyclone overflow, and screen products;
• water balance and slurry-density checks;
• pump curves, pressure trends, and motor power;
• crusher setting, liner condition, screen condition, and conveyor inspection records;
• tailings solids, water chemistry, rheology, and deposition constraints;
• reconciliation between plant data, laboratory assays, and inventory surveys;
• operating limits, alarm setpoints, interlocks, and shift handover evidence.

The formulas are useful only when they preserve the measurement basis. Mineral processing decisions should be based on reconciled material flow, not isolated numbers from one instrument or one sample.