Solvent Extraction Mass Balance: A Worked Example

Last updated: 2026-11-04

A solvent extraction mass balance that closes within 2-3% is the difference between a copper SX plant that is controlled and one that is merely observed. In the worked example below, 100 m3/h of PLS at 2.0 g/L Cu delivers 165 kg/h of copper to strip — 3.96 t/d on paper — while an entrainment audit and a reconciliation against the cathode tally explain a measured 4.8% gap.

Key Takeaways

• A hand-built solvent extraction mass balance is the audit tool that catches what simulators smooth over.
• Worked example: 100 m3/h PLS at 2.0 g/L Cu moves 165 kg/h of copper into loaded organic.
• Entrainment, not chemistry, drives organic losses: 30 ppm in raffinate is about 61 kg/d of organic.
• Reconciliation against the cathode tally turns the balance into gross error detection.
• Flow meter bias and sampler placement, not stage chemistry, explain most multi-percent gaps.

Why Build a Solvent Extraction Mass Balance by Hand?

Simulators can converge a hundred streams in seconds, so why do it with a calculator? Because the value of solvent extraction circuit accounting lies less in the answer than in what the act of building it reveals. A solvent extraction mass balance is a copper accounting ledger that tracks every kilogram of metal into, through, and out of the circuit, closing the books at each stage boundary. Build it by hand once and you know which number depends on which measurement — knowledge no converged simulator output will hand you.

Reconciliation is a structured comparison that pits the calculated balance against an independent production tally, most often the electrowinning tankhouse cathode count. If the ledger says 3.96 t/d of copper left the SX section and the tankhouse plated 3.77 t/d, one of two things is true: copper is going somewhere you have not modeled, or one of your inputs is wrong. Either way, you have learned something the simulator would have buried inside a converged solution.

Gross error detection is a discipline that hunts for single mistakes large enough to explain the entire residual — a mis-scaled flow, a stale assay, a flow meter reading in actual versus standard cubic meters. Hand calculation is brutal about gross errors because every number passes through your fingers twice: once as a measurement, once as arithmetic. A simulator will happily converge a circuit fed by a PLS flow meter that over-reads 2%; the hand balance simply will not close, and the 4.8% residual in the worked example below was chased down exactly this way.

There is also a quieter reason: the copper SX circuit balance is the document you defend in metallurgical accounting reviews, and auditors trust a balance whose lineage they can check line by line — flow, grade, mass, multiplication. When a plant reconciliation disputes three months of metal accounting, the argument is won with a stream table and transparent arithmetic, not software screenshots. Finally, the hand balance is a training ground for judgment: it teaches you that raffinate copper, entrainment, and electrolyte advance are the three numbers that actually move cathode production, which is precisely where your sampling budget belongs.

The Circuit and the Stream Inventory

Everything below is a worked example with synthetic flowsheet numbers — realistic for a copper heap-leach SX-EW operation, but invented for internal consistency, not measured from any plant. The circuit is a 2E-1W-2S configuration: two counter-current extraction stages, one wash stage, and two counter-current strip stages, the workhorse arrangement for oxide copper leach liquor. The same bookkeeping extends to a four-stage mixer settler cascade or any other configuration; only the stage count changes.

The basis: PLS (pregnant leach solution) enters at 100 m3/h carrying 2.0 g/L Cu, so copper arrives at 100 × 2.0 = 200 kg/h. Extraction runs at an O/A ratio of 1:1, so organic flow is 100 m3/h; the reasoning behind setting that ratio — and why density ruins naive volume ratios at high acid — is covered in the O/A ratio solvent extraction calculation guide and is not repeated here. Loaded organic carries 2.15 g/L Cu, within the 2.0-2.4 g/L loading strength band typical of fresh aldoxime/ketoxime reagents (indicative); stripped organic returns at 0.50 g/L Cu, so the organic picks up 1.65 g/L Cu per pass. Wash liquor flows at 0.1 volumes per volume of organic — 10 m3/h against 100 m3/h — at 10 g/L H2SO4 and essentially zero copper. Strip receives spent electrolyte at 35 g/L Cu and advances it to 45 g/L; to move 165 kg/h at a 10 g/L lift, the electrolyte flow must be 165,000 g/h ÷ 10 g/L = 16.5 m3/h.

Before any arithmetic, write the stream inventory. A stream inventory is a single table that lists every flow, its copper grade, its acid strength, and its measurement method — the contract between the flowsheet and the instruments. Every later disagreement traces back to one of these rows.

Two inventory habits separate a useful table from a decoration. First, record the measurement method in the notes column: a flow measured by magnetic meter, a tank dip, and a pump curve are not equally trustworthy, and the reconciliation section needs that hierarchy. Second, mark assumed values explicitly — the acid figures above are indicative until assayed, organic copper grades come from laboratory shake-outs, and organic flows are almost always inferred from pump settings rather than metered, which is acceptable only if you accept the density correction that comes with it.

What Does the Stage-by-Stage Copper Arithmetic Look Like?

Now carry the numbers stage by stage, in plain arithmetic a reader can redo. The split of the 1.65 g/L organic lift between the two extraction stages is not arbitrary — it comes from the extraction isotherm and the operating line, drawn exactly as in the McCabe-Thiele extraction isotherm method; stage counting is not repeated here. Each stage achieves roughly 90-95% of its equilibrium approach, the band discussed in the mixer settler stage efficiency guide, and one self-consistent split that honors the endpoints is the one below.

Extraction stage 1 (E1). Aqueous enters at 2.00 g/L and leaves at 1.10 g/L, a drop of 0.90 g/L. With 1:1 flows, the organic must rise by the same mass: it enters from E2 at 1.25 g/L and leaves loaded at 2.15 g/L, a rise of 0.90 g/L. Copper transferred: 0.90 g/L × 100 m3/h × 1000 L/m3 = 90 kg/h. Check: 90 kg/h removed from 200 kg/h of aqueous copper input.

Extraction stage 2 (E2). Aqueous continues from 1.10 g/L down to the 0.35 g/L raffinate target, a drop of 0.75 g/L, or 0.75 × 100 × 1000 = 75 kg/h. Organic enters lean at 0.50 g/L and leaves at 1.25 g/L, a rise of 0.75 g/L — the books balance stage by stage.

Overall extraction check. Aqueous copper: 200 kg/h in, 0.35 × 100 = 35 kg/h out in raffinate, so 165 kg/h extracted — an 82.5% extraction from PLS. Organic copper: rises 1.65 g/L × 100 m3/h = 165 kg/h. The two sides agree, which is the whole point of a solvent extraction mass balance: every copper atom is accounted for twice, from both the aqueous and the organic view.

Strip stage 2 (S2, organic side). Organic enters from S1 at 1.325 g/L and leaves lean at 0.50 g/L, a drop of 0.825 g/L, releasing 0.825 × 100 × 1000 = 82.5 kg/h. Electrolyte at 16.5 m3/h rises from 35 g/L to 40 g/L, absorbing 5 g/L × 16.5 × 1000 = 82.5 kg/h.

Strip stage 1 (S1, organic side). Loaded organic enters at 2.15 g/L and leaves at 1.325 g/L, a drop of 0.825 g/L, releasing another 82.5 kg/h. Electrolyte continues from 40 g/L to the 45 g/L advance grade, absorbing 82.5 kg/h.

Strip liquor copper balance overall. Organic gives up 1.65 g/L × 100 m3/h = 165 kg/h; electrolyte gains 10 g/L × 16.5 m3/h = 165 kg/h. The advance electrolyte therefore carries 165 kg/h toward electrowinning: 165 × 24 = 3,960 kg/d = 3.96 t/d of copper. This is the number the tankhouse tally will be asked to match, and the number the entrainment audit is about to whittle down.

Where Does the Metal Leak, and How Do You Build the Balance From Plant Data?

The stage arithmetic assumed perfect phase separation. Real mixer settlers leak droplets both ways. Entrainment is a physical carry-over that sends droplets of one phase along with the other at every phase disengagement, and it is audited with the same seriousness as the copper itself — the mechanisms and countermeasures are detailed in the entrainment in mixer settlers guide. The indicative bands below are typical for well-run copper SX; run your own audit before quoting any of them.

Work through the entrainment loss calculation line by line. Organic entrained in raffinate: 30 ppm of 100 m3/h is 100,000 L/h × 30 × 10⁻⁶ = 3.0 L/h. At an organic density of 0.85 kg/L (stated assumption for a kerosene-diluent reagent), that is 3.0 × 0.85 = 2.55 kg/h, or 61 kg/d of organic inventory walking out with the raffinate — the single largest consumable line item in most copper SX operating budgets. The copper riding in those droplets, 3.0 × 2.15 = 6.45 g/h, is trivially small against 165 kg/h; the loss that hurts is the diluent and extractant, plus the crud it drags along.

Aqueous entrained in organic is a recycle, not a loss, but it moves copper and acid in directions worth checking. Loaded organic leaving E2 carries 100 ppm aqueous — 10 L/h of raffinate-grade liquor at 0.35 g/L Cu — into the wash. The wash stage, fed with 10 m3/h of copper-free liquor at 10 g/L H2SO4, exchanges that entrained raffinate for clean wash; the displaced copper, 10 L/h × 0.35 g/L = 3.5 g/h, leaves in the wash effluent back toward the raffinate pond. Acid direction check number one: wash liquor enters at 10 g/L and leaves at roughly 10 g/L plus traces — the wash consumes no acid, it only ferries the entrained raffinate’s acid out.

Leaving the strip section, the organic carries 10 L/h of spent electrolyte at 35 g/L Cu and roughly 180 g/L H2SO4 (indicative). That is 0.35 kg/h of copper and 1.8 kg/h of acid cycling back into extraction — acid direction check number two, and the important one: acid travels from strip to extraction via aqueous entrainment in organic, never the reverse, which is why raffinate acid creeps up in closed PLS loops by 1.8 × 24 = 43 kg/d until a bleed or neutralization catches it. The 0.35 kg/h of recycled copper quietly re-loads the organic, shaving about 0.004 g/L off the effective lift — negligible here, but directionally real.

How do you build this balance from plant data rather than from a textbook? Follow the same sequence every time:

1. Fix the time basis and the boundary. Choose a 24 h composite period and draw the battery limit around SX only, with PLS in and advance electrolyte out; every stream crossing that line needs a number.
2. Inventory streams and sample points. Walk the circuit and mark each stream’s sampling location; a stream with no sample point gets flagged as an estimate before the arithmetic starts.
3. Verify flows with two independent methods. Compare the magnetic meter against a tank dip or pump curve over the composite period; if the two disagree by more than 3%, calibrate before computing anything.
4. Correct volumes for density and temperature. Convert tank dips to volume with the actual liquor density — aqueous SX liquors run 1.05-1.25 kg/L depending on acid — and record the temperature at which every volume was taken.
5. Convert all assays to a common basis. Copper in g/L of liquor for solutions; for any solids-bearing stream, convert to dry basis explicitly and state the moisture used.
6. Compute stage by stage, then close the overall balance. Do the E1, E2, S1, S2 arithmetic exactly as above; if any stage is off by more than the measurement precision, stop and find it before proceeding.
7. Reconcile against production and chase the residual. Compare Cu to EW against the cathode tally, then work the discrepancy section below until the residual is inside sampling precision.

If you are commissioning a new circuit, this sequence is run daily for a week on a continuous solvent extraction mini plant before it is trusted at full scale — the mini plant is where sample point mistakes are cheap.

Worked Reconciliation: Chasing a 4.8 Percent Gap

Here is the reconciliation that closes the article’s loop, using the synthetic worked-example numbers throughout. The balance says 165 kg/h × 24 = 3.96 t/d of copper delivered to electrowinning. The tankhouse tally for the same 24 h period, weighed on the cathode crane, says 3.77 t/d. The gap is 190 kg/d, or (3.96 − 3.77) / 3.96 = 4.8% — far outside the 1-2% closure a healthy copper SX circuit balance should show.

The chase, in order. First, tankhouse losses were audited and found under 0.3% of production, so most of the gap sits upstream. Second, the PLS magnetic flow meter was field-verified against a reservoir draw-down over 2 h and found reading 2.0% high: true flow 98 m3/h, not 100. Third, the PLS composite sampler was cross-checked with a series of manual grabs taken after a full line purge, and the composite read 2.00 g/L against a grab mean of 1.96 g/L — a +0.04 g/L bias traced to the sampler cutting preferentially from the pipe’s faster centerline (more on dead-legs below). Recomputing with corrected inputs: extracted copper = 98 m3/h × (1.96 − 0.35) g/L = 98 × 1.61 = 157.8 kg/h, or 157.8 × 24 / 1000 = 3.79 t/d. The residual against the 3.77 t/d tally is now (3.79 − 3.77) / 3.79 = 0.5% — inside sampling precision, and the books close.

The lesson generalizes: a 4.8% SX reconciliation gap resolved into two mundane instrument errors, each worth roughly 2%, with zero chemistry involved. Most multi-percent gaps in real plants are exactly this — and both of these errors would have silently inflated the simulator’s answer too.

When the Solvent Extraction Mass Balance Disagrees With Production, What Then?

When the residual exceeds 2% and survives a first-pass check, work through the standard suspects in order of probability. Gross error detection is a discipline that hunts for single mistakes large enough to explain the whole residual before blaming distributed effects: a flow transcribed in the wrong units, an assay entered from the wrong date, a stream assigned to the wrong boundary. Test one gross error at a time by deleting it and recomputing; if the residual collapses, you have your culprit, and no statistical smoothing is needed.

Instrument drift comes next. Magnetic flow meters drift with liner wear and electrode coating; a meter reading 2% high for months is invisible until compared to a tank dip, which is why step 3 demands two independent flow measurements on every balance campaign. Density meters and pressure-based level instruments drift with lining fouling. Organic flow inferred from pump strokes drifts whenever wear clears the pump internals — a 5% slip change on a 100 m3/h organic flow is 5 m3/h of phantom copper capacity.

The subtlest class is basis mix-ups, and density sits at the center of them. At high acid strength, aqueous density rises toward 1.25 kg/L, so a volumetric O/A ratio set by pump settings is silently a different mass ratio than the one the chemistry assumes — the O/A ratio solvent extraction calculation guide quantifies this, and the practical symptom is an O/A error that hides inside a density number: the pump strokes never changed, but the 10 g/L acid drift in the PLS moved the effective ratio anyway. Assay basis mix-ups are equally common: copper reported as g/L of liquor versus mg/kg of slurry, wet versus dry basis on any solids-bearing stream, or a laboratory reporting on a density the sample never had.

One judgment only a commissioning engineer would know: sample point dead-legs corrupt PLS copper assays more than sampler design does. A sampling nozzle on a stagnant stub of pipe holds liquor from hours ago; the first liter drawn is yesterday’s PLS at yesterday’s grade, and on a 2 g/L liquor a stale dead-leg can bias a composite by 0.05-0.15 g/L Cu (indicative) — several times larger than the centerline-vs-average error any well-designed cross-stream sampler makes. Purge three to five line volumes before every sample, and put sample points on flowing tees, never on instrument taps at the end of forgotten spools. It costs nothing at design time and saves a reconciliation argument every month afterward.

FAQ

How accurate should a hand-built SX mass balance be?

A daily copper SX circuit balance built from good instruments should close to within 1-2% between copper into extraction and copper into electrowinning; weekly and monthly balances tighten further as sampler noise averages out. A persistent 3%-plus miss in the same direction is an instrument problem, not a process one — chase flows and assays before questioning the stage chemistry.

Why does my raffinate copper assay drift day to day?

Raffinate copper responds to PLS grade, flow, organic-to-aqueous ratio, organic condition, and temperature, so daily drift of 0.02-0.05 g/L is normal within sampler precision. Sustained drift usually traces to extractant degradation, crud buildup, or a drifting PLS flow meter. Compare only assays drawn by the same method at the same point; mixing composite and grab data creates artificial trends.

How much organic do I lose to entrainment?

In this worked example, 30 ppm organic in 100 m3/h of raffinate is 3.0 L/h, or about 61 kg/d at 0.85 kg/L — roughly 1.5 kg of organic per tonne of cathode. Industrial circuits typically run 20-100 ppm in raffinate (indicative), so the band is wide. Track organic inventory purchases against the entrainment audit; if purchases exceed the audit’s prediction, look for crud recovery losses and spills not being captured.

Do I need isotherm data to build a mass balance?

For the accounting balance itself, no — flows and assays on every stream are sufficient, as this article shows. Isotherm data become necessary when you want to explain the numbers: why the intermediate stage sits at 1.25 g/L, what raffinate grade is achievable, or how many stages a new PLS grade requires. Balance first, isotherm second; the McCabe-Thiele construction then explains the split the plant data reveal.

What flow meters work best for SX streams?

Magnetic flow meters suit PLS, raffinate, and electrolyte because these are conductive aqueous streams; verify them annually against tank draw-down. Organic cannot be measured magnetically — use coriolis meters where the budget allows, or infer from pump curves with a documented density assumption and periodic tank-dip checks. Never mix meter types in a ratio without correcting for their different error behaviors; a 2% meter pairing error looks exactly like a chemistry problem.

How do density corrections change the balance?

Every volume-based number — flows, tank dips, g/L assays — is tied to a density, and SX aqueous streams range from about 1.05 to 1.25 kg/L with acid strength. A g/L assay is mass of copper per liter of liquor; if you convert to mass fractions or scale tank dips, use the actual measured density, not 1.0. A 5% density error shifts every aqueous copper tonnage by 5%, which alone can explain a typical reconciliation gap.

How often should the mass balance be rebuilt?

Rebuild the full hand balance at least monthly, and daily during commissioning or any period with metal accounting disputes. Between rebuilds, track three sentinel numbers — raffinate copper, electrolyte advance rate, and cathode tally — which reveal most developing problems. Rebuild immediately after any flowsheet change: a new wash rate, a changed strip mixer, or a PLS grade shift invalidates the previous balance’s assumptions.

Conclusion

The solvent extraction mass balance remains the metallurgist’s most honest instrument: a stream inventory, stage-by-stage arithmetic you can defend line by line, an entrainment audit that prices the organic you are losing, and a reconciliation that turns a 4.8% gap into two corrected instruments. In the worked example, 100 m3/h of PLS at 2.0 g/L Cu became 165 kg/h of copper per hour and 3.96 t/d on paper — and the chase to 3.79 t/d taught more than any converged simulation would have. Build the balance by hand once per month, keep the stream table current, and your SX circuit will tell you the truth before the tankhouse does. If you want this discipline applied to your circuit — stream inventory design, sampling audit, and a reconciliation framework you can defend — request a quote for a pilot SX program, or start with the rest of our calculation guides.

References

• Ritcey, G.M., Solvent Extraction: Principles and Applications to Process Metallurgy, Revised 2nd ed., Ritcey and Associates.
• Lo, T.C., Baird, M.H.I. and Hanson, C., Handbook of Solvent Extraction, Wiley.
• Kordosky, G.A., “Copper Recovery Using Leach/Solvent Extraction/Electrowinning Technology,” JOM.
• Free, M., Hydrometallurgy: Fundamentals and Applications, Wiley.
• Wills, B.A. and Finch, J.A., Wills’ Mineral Processing Technology, 8th ed., Butterworth-Heinemann.
• SME Mineral Processing and Extractive Metallurgy Handbook, Society for Mining, Metallurgy and Exploration, Englewood, CO.
• Habashi, F., A Textbook of Hydrometallurgy, Métallurgie Extractive Québec.
• Davenport, W.G. et al., Extractive Metallurgy of Copper, 5th ed., Elsevier.

Changelog

• 2026-11-04: Initial publication.