McCabe-Thiele Extraction Isotherm: Stage Count by Hand

Last updated: 2026-10-28

A McCabe-Thiele extraction isotherm converts measured shakeout data into a theoretical stage count: plot the equilibrium curve, draw the operating line from the plant mass balance, and step off stages between the two. For a typical 2.00 g/L copper PLS the construction gives 2 theoretical extraction stages — and any published count still needs an 85–95% per-stage efficiency correction before it sizes real mixers.

This page works through the full graphical and arithmetic construction, with three worked examples: a copper extraction duty, a nickel/cobalt scrub section, and a stripping circuit. It sits inside our broader solvent extraction resource cluster, alongside guides to the underlying chemistry and hardware.

Key Takeaways

• The extraction isotherm is measured, not assumed: six shakeout points across the O/A range define the equilibrium curve.
• The operating line comes from the plant mass balance; its slope is the aqueous-to-organic flow ratio.
• Stage stepping means bouncing between the isotherm and the operating line until the raffinate target is reached.
• Published stage counts assume perfect equilibrium; divide theoretical stages by 0.85–0.95 per-stage efficiency.
• A pinch point means the target is unreachable at that O/A — change flows or chemistry, not the drawing.

What Is a McCabe-Thiele Extraction Isotherm?

A McCabe-Thiele extraction isotherm is the solvent extraction adaptation of the classic McCabe-Thiele diagram from distillation: a single chart that combines an equilibrium curve with an operating line so that the number of contacts in a counter-current cascade can be counted by hand. Several components of that chart deserve precise definitions before any stepping begins.

Mixer-Settler Settler Area and Mixer Volume Calculator: selection rules

An extraction isotherm is a curve that relates the metal concentration in the equilibrium organic phase to the metal concentration in the equilibrium aqueous phase, typically measured over an aqueous metal range of 0.1–5 g/L for copper PLS work. It is generated experimentally — from the shakeout test procedure described in the next section — and it is specific to the reagent concentration, diluent, pH, and temperature at which it was measured.

The operating line is a straight line that links the rich and lean ends of the counter-current cascade, with a slope equal to the aqueous-to-organic flow ratio. In operating line solvent extraction work it represents the mass balance constraint: every point on the line pairs the aqueous stream leaving one position in the cascade with the organic stream passing it. It is a constraint that the plant imposes, whereas the isotherm is a constraint that the chemistry imposes.

A theoretical stage is a mixing-and-settling step that brings its two exit streams to mutual equilibrium, something no industrial mixer fully achieves. Real mixer boxes achieve 85–95% of that equilibrium transfer, which is why every theoretical stage calculation ends with an efficiency correction.

Separation factor is a ratio that compares the distribution coefficients of two metals, D(A)/D(B), and sets how completely one can be separated from the other in a fixed number of stages. In a cobalt/nickel circuit, for example, a separation factor of 250 between cobalt and nickel means a scrub or extraction section needs very few stages to meet a tight impurity specification — while a separation factor near 10 means the same specification may be unreachable at any stage count.

The pinch point is a region that appears where the operating line touches the isotherm: driving force vanishes there, and additional stages stop producing additional transfer. Recognising a pinch during stage stepping is one of the most valuable outputs of the whole exercise, because it tells the engineer to change the O/A ratio or the reagent strength rather than to keep adding stages.

The axes matter. The x-axis is always aqueous-phase metal (g/L), the y-axis is always organic-phase metal (g/L). In extraction, the operating line runs from the raffinate/lean-organic corner up to the feed/loaded-organic corner. In scrubbing and stripping the same construction applies with the roles of the streams reversed.

How Do You Build a McCabe-Thiele Extraction Isotherm From Shakeout Tests?

The isotherm is only as good as the shakeout data behind it, so the laboratory procedure deserves as much care as the stage stepping that follows. A McCabe-Thiele extraction isotherm is built, not downloaded, and a disciplined sequence looks like this:

1. Fix the aqueous feed and reagent system. Decide the feed metal tenor, pH, reagent concentration, diluent, and temperature. The isotherm is valid only at those conditions, and every later design decision inherits them.
2. Run the shakeout matrix. Contact feed liquor and organic across a range of O/A ratios — for a 2 g/L copper PLS, roughly 0.15:1 to 4:1 — in sealed flasks. Agitate for 3–5 minutes at controlled temperature, allow full phase disengagement, and run each point in duplicate.
3. Analyse both phases. Measure metal in the raffinate and in the loaded organic. Never back-calculate one from the other by difference alone; independent analyses are the internal check on the whole dataset.
4. Plot the isotherm. Plot aqueous metal out on the x-axis against organic loading on the y-axis, one point per shakeout, and draw a smooth curve through them. The curve should be monotonically increasing and should flatten toward the reagent’s loading capacity.
5. Draw the operating line. From the intended plant flows, compute the two endpoints: the rich point (feed aqueous tenor, loaded organic tenor) and the lean point (raffinate target, stripped organic tenor). The straight line through them is the operating line solvent extraction design relies on.
6. Step off stages and correct for efficiency. Walk between the isotherm and the operating line as shown in the worked examples below, then divide the theoretical count by the expected per-stage efficiency.

A worked dataset makes the idea concrete. Six shakeout points on a synthetic 2.00 g/L copper PLS with a typical oxime reagent give the following table (illustrative values, consistent with a typical copper PLS, to be confirmed by testwork on your own liquor):

Each row closes its own mass balance: the metal removed from the aqueous phase, (2.00 − aqueous out), must equal O/A multiplied by the organic loading. Row two, for instance, removes 1.70 g/L of copper, and 1.89 × 0.90 ≈ 1.70 g/L appears on the organic side. The extraction percentage is simply the removal divided by the feed tenor. Note how the curve is steep at low aqueous tenor — the reagent pulls copper down hard — and flattens above about 1.9 g/L of loading, where the reagent approaches saturation. That flattening is the region where pinch points form, which is why the choice of O/A ratio in SX flows deserves deliberate attention before the operating line is drawn at all.

Worked Example 1: Copper SX Stage Count

The duty: treat a 2.00 g/L Cu PLS, hit a raffinate target of 0.30 g/L Cu (85% overall extraction), with organic that has been stripped down to a lean tenor of 0.20 g/L Cu. Set the O/A ratio at 1.0:1 on a basis of 1 L/min of PLS.

Mass balance first. Copper removed from the aqueous phase must equal copper picked up by the organic:

F × (x_in − x_out) = O × (y_out − y_in)

1.00 L/min × (2.00 − 0.30) g/L = 1.00 L/min × (y_out − 0.20 g/L)

1.70 g/min = 1.00 L/min × (y_out − 0.20) → y_out = 1.90 g/L Cu

The loaded organic tenor is 1.90 g/L — comfortably inside the isotherm’s measured range. The operating line has slope F/O = 1.00 and passes through the lean endpoint (x = 0.30, y = 0.20) and the rich endpoint (x = 2.00, y = 1.90). Written out: y = x − 0.10.

Stage stepping. Start at the rich corner (2.00, 1.90). Step horizontally to the isotherm: the curve passes through (1.00, 1.90) — an actual measured point, so no interpolation is needed. That horizontal step is Stage 1 complete: aqueous drops from 2.00 to 1.00 g/L. Step vertically down to the operating line at x = 1.00: the line gives y = 0.90. Step horizontally again: the isotherm passes through (0.30, 0.90), another measured point. That is Stage 2, and the aqueous tenor has landed exactly on the 0.30 g/L target. Two steps: 2 theoretical stages.

The table must satisfy two checks, and it does. Equilibrium: Stage 1 pairs (1.00, 1.90) and Stage 2 pairs (0.30, 0.90), both points on the measured isotherm. Mass balance: Stage 1 transfers 1.00 g/L of copper from aqueous (2.00 → 1.00) and the same 1.00 g/L into organic (0.90 → 1.90 at O/A 1.0). Stage 2 transfers 0.70 g/L each way (1.00 → 0.30 aqueous; 0.20 → 0.90 organic). The overall balance closes: 1.70 g/L removed from the PLS equals 1.70 g/L gained by the organic.

Efficiency correction. The count of 2 is theoretical. Applying N_actual = N_theoretical / E with a realistic per-stage efficiency of 90% gives 2 / 0.90 = 2.2, which rounds up to 3 mixer boxes — or justifies 2 stages only if pilot work verifies efficiency at the top of the range. This is the single most common omission when flowsheet stage counts are quoted from a textbook: the numbers assume perfect equilibrium contacts that no stage efficiency in mixer-settlers discussion would accept uncorrected. On real industrial mixer-settler equipment, entrainment and disengagement behaviour reinforce the case for the correction.

Circuits built around copper solvent extraction chemistry almost always converge on this kind of answer — two extraction stages plus the correction — because the oxime isotherm is steep at the raffinate end and nearly flat at the loaded end. The same two-stage pattern carries through the copper SX-EW flowsheet, where the stripped organic leaving the strip section closes the loop back to Stage 2.

Worked Example 2: Nickel/Cobalt Scrub Section

The duty: a loaded organic from a Cyanex 272 cobalt circuit carries 10.0 g/L Co and 0.40 g/L Ni. Before stripping, the organic is scrubbed with dilute acid at an organic-rich O/A of 4:1 to knock the nickel down to ≤0.05 g/L. The scrub liquor enters essentially nickel-free.

The chemistry is summarised by a separation factor, and it is the one input a McCabe-Thiele extraction isotherm cannot supply on its own. At scrub acidity the distribution coefficients (organic/aqueous) are roughly D_Co = 25 and D_Ni = 0.10, so the separation factor β = D_Co/D_Ni ≈ 250. That large number is why the scrub section is short: nickel transfers readily into the scrub liquor while cobalt stays on the organic, losing only about 10.0/25 × 0.25 ≈ 0.10 g/L Co per litre of organic — roughly 1% of the cobalt inventory. Circuits compared in D2EHPA vs Cyanex 272 differ sharply here, and the comparison is really a statement about scrub stage counts. The same logic governs full nickel-cobalt refining flowsheets.

The two lines. Equilibrium: with D_Ni = 0.10, the isotherm is the straight line y = 0.10x (y = Ni on organic, x = Ni in scrub liquor). Operating line: on a basis of 1 L/min of loaded organic, scrub flow is 0.25 L/min, so the line has slope A/O = 0.25. The overall balance fixes its position: organic loses (0.40 − y_out) g/min and the scrub liquor gains 0.25 × x_1. Stepping from the organic-feed end, the operating line works out to y = 0.25x + 0.04, running from the point (x = 1.44, y = 0.40) down to (x = 0, y = 0.04).

Stage stepping. Start at the top of the operating line at (1.44, 0.40): the scrub liquor leaving Stage 1 carries x_1 = (0.40 − 0.04)/0.25 = 1.44 g/L Ni. Step horizontally to the equilibrium line: y_1 = 0.10 × 1.44 = 0.144. Step vertically to the operating line: x_2 = (0.144 − 0.04)/0.25 = 0.416. Step horizontally again: y_2 = 0.10 × 0.416 = 0.0416. Step vertically once more: the operating line gives x_3 = (0.0416 − 0.04)/0.25 ≈ 0.006, essentially the fresh scrub liquor. Two steps reach the lean end: 2 theoretical scrub stages.

Both stages close their balances. Stage 1: the organic loses 0.40 − 0.14 = 0.26 g/L of nickel, and the scrub liquor gains (1.44 − 0.41) × 0.25 = 0.26 g/L per litre of organic. Stage 2: the organic loses 0.10 and the scrub liquor gains 0.41 × 0.25 ≈ 0.10. Overall: 0.36 g/L removed from the organic, 1.44 × 0.25 = 0.36 g/L carried out in the advance scrub liquor. The scrubbed organic at 0.04 g/L Ni meets the 0.05 specification with margin.

The sensitivity that matters. If scrub acidity control slips and D_Ni rises to 0.30, the same two stages leave 0.16 g/L of nickel on the organic — and no stage count fixes it. With D_Ni = 0.30 the driving force is so weak that the count approaches an asymptote near 0.067 g/L no matter how many stages are installed; the 0.05 specification becomes unreachable by construction. The correct response is chemical (stronger scrub acid, more stages of acid washing upstream), not graphical. That is the separation factor speaking: it, not the drawing, sets what stage stepping can achieve.

Worked Example 3: Stripping Circuit Operating Line

The strip section closes the copper loop: loaded organic at 1.90 g/L Cu — exactly the tenor produced in Worked Example 1 — must be stripped to ≤0.20 g/L with spent electrolyte entering at 32.0 g/L Cu and roughly 180 g/L free acid.

Strip shakeouts at that acidity produce a flat, unfavourable isotherm, and it flattens for the same reason the extraction isotherm in the McCabe-Thiele extraction isotherm construction above was steep: the reagent’s affinity reverses with acidity. Illustrative points read (aqueous Cu, organic Cu) = (32, 0.14), (36, 0.22), (40, 0.32), (44, 0.45), (48, 0.60). Compare those organic tenors with the extraction isotherm of Example 1 — at strip acidity the reagent holds almost nothing, which is of course the point.

The O/A is set by electrolyte advance requirements, not by the extraction duty. Electrowinning needs the advance electrolyte roughly 10 g/L richer in copper than spent, and the aqueous flow per unit of organic flow is correspondingly small: here O/A = 6:1 organic-rich, an aqueous-to-organic ratio of just 0.17 — far below the 1:1 of extraction. The overall balance follows the same equation as before, with the streams swapped: on 1 L/min of organic, electrolyte flow is 0.167 L/min, and 0.167 × (x_advance − 32.0) = 1.90 − y_stripped.

The operating line runs from the spent corner up to the advance corner. With two stages in view, stepping gives the following:

Read the stepping from the top. Stage 1: organic leaves in equilibrium with the advance electrolyte at x_1 = 42.4 g/L; the strip isotherm gives y_1 ≈ 0.40 g/L. Vertical to the operating line, whose slope is A/O = 0.167: x_2 = 42.4 − 6 × (1.90 − 0.40) = 33.4. Stage 2: the isotherm at 33.4 g/L gives y_2 ≈ 0.17 g/L. One more vertical step lands at 42.4 − 6 × (1.90 − 0.17) = 32.0 g/L — exactly the spent electrolyte. The count closes at 2 theoretical strip stages, and the balances close too: Stage 1 moves 1.50 g/L from organic to electrolyte (42.4 − 33.4 = 9.0 g/L rise on 1/6 the flow = 1.50), Stage 2 moves 0.23 g/L ((33.4 − 32.0)/6 ≈ 0.23), overall 1.73 g/L each way.

Why not one stage? Stepping a single stage against the same operating line lands at about 0.36 g/L on the stripped organic — nearly double the specification — because the last stage equilibrates organic with electrolyte that is already 32 g/L in copper, and the flat isotherm offers little pull below that. Hence the rule of thumb that strip sections need 2–3 stages at a much lower aqueous-to-organic ratio than extraction, with the third mixer justified by the same efficiency correction as before: 2 / 0.90 ≈ 2.2 rounds up to 3 when stripped-organic tenor is critical to the raffinate guarantee one stage downstream.

What Makes a Stage Count Wrong?

The construction is simple; the ways it fails are numerous. Six pitfalls account for most bad stage counts, each with a recognisable symptom, cause, and correction.

1. Symptom: the drawn operating line never intersects the design endpoints. Cause: the O/A used for the line is an assumed flowsheet value that the plant’s actual flows will not reproduce — pump rates drift, and phase disengagement changes hold-up. Correction: derive the O/A from measured plant flows, recompute both endpoints, and redraw; the mixer-settler design flowsheet, not a textbook default, supplies the numbers.
2. Symptom: the loaded organic endpoint sits above the measured isotherm. Cause: the mass balance demands more loading than the reagent can deliver at the feed tenor — often an O/A chosen too high for the metal tenor. Correction: lower the O/A, or accept a higher raffinate; the isotherm is physics and will not be negotiated with.
3. Symptom: plant raffinate is consistently poorer than the diagram promised. Cause: theoretical stages were installed without the efficiency correction. Correction: apply the explicit formula N_actual = N_theoretical / E, with E the per-stage efficiency (0.85–0.95 for well-designed copper extract duties), then round up — 2 theoretical stages at E = 0.90 means 3 mixers, not 2.
4. Symptom: the isotherm from the lab looks better than plant stage-by-stage samples. Cause: shakeouts were run at a fixed pH while plant stages drift through a pH profile, particularly with chelating reagents whose loading releases acid. Correction: run the shakeout test procedure at the pH each stage will actually see, or add partial-neutralisation make-up to the laboratory contacts.
5. Symptom: the last step of the staircase will not land on the lean corner. Cause: a genuine pinch — the operating line has approached the isotherm so closely that driving force has vanished near the raffinate or stripped-organic end. Correction: change the O/A ratio, raise reagent strength, or relax the specification; adding mixer boxes at a pinch adds cost, not transfer.
6. Symptom: stepping gives fractional and shifting answers between runs. Cause: short-equilibration shakeouts bias the isotherm — the “equilibrium” curve is really a kinetic snapshot, so no two stepping exercises reproduce. Correction: extend shakeout contact time until replicate analyses agree, then rebuild the isotherm; a theoretical stage calculation is only as reproducible as the curve it steps against.

Every one of these pitfalls is cheaper to find on paper than in concrete. The McCabe-Thiele extraction isotherm’s real value is not the stage number itself but the discipline it forces: a mass balance that closes, an isotherm that replicates, and an efficiency assumption stated out loud.

FAQ

How many shakeout points do I need for a reliable extraction isotherm?
Six to eight points spread across the O/A range is usually sufficient, provided they bracket both design endpoints. The critical points are near the raffinate end, where the curve is steep and stage crowding occurs, and near the loading plateau, where pinch risk lives. A curve built from only three or four points forces long interpolations exactly where accuracy matters most, and the resulting stage count inherits that error.

What O/A range should shakeout tests cover?
Cover at least the range the operating line could plausibly occupy, plus margin. For a 2 g/L copper PLS that typically means O/A from about 0.15:1 to 4:1. Points below the design O/A map the crowded top of the isotherm; points above it map the steep raffinate end. If a flowsheet optimisation may push O/A in either direction, extend the matrix accordingly before the laboratory campaign closes.

How do I convert theoretical stages to actual mixer boxes?
Divide by per-stage efficiency and round up: N_actual = N_theoretical / E, with E typically 0.85–0.95. A count of 2 theoretical stages at E = 0.90 gives 2.2, so three mixer boxes are installed — or two are accepted with a documented raffinate penalty. Never quote a theoretical count to a construction team without this correction; the difference is concrete, tankage, and organic inventory.

Does the operating line ever bend?
It stays straight only while flows and phase volumes are constant. In scrub and strip sections, and wherever metal transfer is dense, phase volumes shift and the O/A differs stage to stage, so the “line” becomes a piecewise curve. For extraction duties on dilute PLS the straight-line assumption is normally fine; for concentrated feeds, construct the line segment by segment between stage-wise flow estimates.

Does the McCabe-Thiele extraction isotherm method work for chelating extractants like those in nickel-cobalt circuits?
Yes — the construction is generic. What changes is the isotherm’s shape and the presence of competing metals, which the single-metal isotherm hides. For nickel-cobalt systems, build separate isotherms or use separation factors to track the impurity alongside the product metal, as the scrub example above does. The stepping arithmetic is identical; the data workload is heavier.

What is a pinch point and how do I fix one?
A pinch is where the operating line touches the isotherm, driving force falls to zero, and added stages transfer nothing. Fix it by moving the line away from the curve: change the O/A ratio, strengthen the reagent, alter the strip or scrub liquor tenor, or relax the specification. Adding mixers at a pinch is the classic error — the diagram is telling you the flows or chemistry must change, not the box count.

How does temperature affect the isotherm and the stage count?
Strongly. Distribution coefficients for most chelating extraction systems fall as temperature rises, so an isotherm measured at 20 °C overstates performance in a plant stream at 40 °C. Run shakeouts at the expected operating temperature, and if the circuit sees a range — an Australian PLS pond in summer versus winter — measure both and step stages against the pessimistic curve.

Conclusion

The McCabe-Thiele extraction isotherm remains the fastest defensible way to turn a page of shakeout numbers into a mixer count: a measured curve, a mass-balance line, a staircase, and one honest efficiency correction. Worked by hand, it exposes pinch points and bad O/A choices long before they are poured in concrete, and it scales from a two-stage copper extraction duty to scrub and strip sections of a full refining flowsheet. If you want a stage count reviewed against your own liquor assay and reagent system, request a quote and the Lexin-tech process team will size the circuit with you.

References

• Lo, T.C., Baird, M.H.I. and Hanson, C. (eds.), Handbook of Solvent Extraction, John Wiley & Sons, New York, 1983.
• Wills, B.A. and Finch, J.A., Wills’ Mineral Processing Technology, 8th ed., Elsevier, 2016.
• Treybal, R.E., Mass-Transfer Operations, 3rd ed., McGraw-Hill, New York, 1980.
• Green, D.W. and Southard, M.Z. (eds.), Perry’s Chemical Engineers’ Handbook, 9th ed., McGraw-Hill, New York, 2019.
• Coulson, J.M. and Richardson, J.F., Chemical Engineering, Volume 2: Particle Technology and Separation Processes, 5th ed., Butterworth-Heinemann, Oxford, 2002.
• Mular, A.L., Halbe, D.N. and Barratt, D.J. (eds.), Mineral Processing Plant Design, Practice, and Control, SME, Littleton, Colorado, 2002.
• Solvent Extraction and Ion Exchange, Taylor & Francis — peer-reviewed journal covering distribution isotherms, separation factors, and solvent extraction circuit design.

Changelog

2026-10-28 — first published.