Shrinking Core Model for Leaching Kinetics: Worked

Last updated: 2026-10-26

The shrinking core model turns leaching kinetics into one diagnostic question: which resistance — liquid film, ash layer, or surface reaction — sets the pace? In the worked example below, the same 10 mm copper oxide particle reaches 80% conversion in 4.0 h, 10.0 h or 15.0 h depending on the answer, and that 3.7-fold spread flows directly into residence time and equipment sizing.

Key Takeaways

• Three linearized plots — X vs t, 1 − 3(1−X)^(2/3) + 2(1−X) vs t, and 1 − (1−X)^(1/3) vs t — identify the controlling regime.
• The most linear plot over the 20–80% conversion window usually names the rate-controlling step; its slope equals 1/τ.
• Ash-layer diffusion makes τ scale with particle radius squared, so crush size is the strongest single design lever.
• Film control is rare in well-agitated leaching but common in poorly mixed, high-solids pulps and stagnant heap zones.
• Fitted τ values convert directly into tank residence time and heap lift duration — see the wider leaching flowsheet family.

What Is the Shrinking Core Model?

Process engineers who arrive searching for leaching kinetics shrinking core model equations are usually trying to answer two practical questions: how long must ore stay in contact with the lixiviant, and which physical step is throttling the rate? The shrinking core model (SCM), developed by Yagi and Kunii in the 1950s for gas–solid reactions such as roasting and combustion and later adopted wholesale by hydrometallurgy, answers both with a surprisingly compact set of equations.

The shrinking core model is a classical idealization that treats every ore particle as a dense sphere of reactive mineral wrapped in an inert shell of leached residue. As dissolution proceeds, the reaction front — the boundary between fresh mineral and spent residue — advances inward toward the particle center, while the outer diameter stays fixed because the gangue skeleton does not dissolve. Conversion X relates simply to geometry: X = 1 − (r_c/R)³, where R is the particle radius and r_c the radius of the unreacted core.

Three resistances act in series between the bulk solution and that reaction front, and whichever is largest controls the rate. The ash layer is a porous shell of leached, metal-depleted gangue that dissolved reagent must diffuse across before it can reach fresh mineral. Film diffusion control is a kinetic regime in which the slowest step is transport of dissolved reagent through the stagnant liquid boundary layer that clings to the outside of every particle. Surface chemical reaction control, the third regime, means the intrinsic dissolution reaction itself — bond breaking at the mineral surface — is the slowest step.

For a constant-size spherical particle leached under pseudo-steady-state conditions, each regime produces its own conversion–time equation. With X the fractional conversion of the valuable mineral and τ the time for complete conversion:

• Film diffusion control: X = t/τ
• Ash-layer diffusion control: 1 − 3(1−X)^(2/3) + 2(1−X) = t/τ
• Surface chemical reaction control: 1 − (1−X)^(1/3) = t/τ

The parameter τ condenses everything physical about the system. Levenspiel’s standard results for a fluid–solid reaction A(fluid) + bB(solid) → products give:

Here ρ_B is the molar density of the valuable mineral in the particle (mol/m³), k_l the liquid-film mass-transfer coefficient (m/s), D_e the effective diffusivity through the ash layer (m²/s), k_s the surface rate constant (m/s), and C_Ag the reagent concentration in the bulk solution (mol/m³). Note that reagent concentration appears in every denominator: anything that depresses the concentration the particle actually sees — gangue acid consumption, for instance — lengthens τ proportionally. Our note on acid consumption of copper oxide ores shows how to quantify that loss before it corrupts a kinetic fit.

The Three Controlling Regimes

Film diffusion control. The particle consumes reagent faster than the surrounding liquid can deliver it, so concentration falls across the boundary layer and the surface reaction starves. Rates respond strongly to agitation, viscosity and solids loading — the classic signature being bottle-roll tests whose apparent rate changes when stirrer speed changes. In plant terms, film control usually signals a mixing or slurry-transport problem rather than an intrinsic ore property. Because the entire outer surface is fed equally, τ scales linearly with radius, and conversion rises linearly with time: X = t/τ.

Ash-layer diffusion control. Dissolution at the reaction front is fast, and the bottleneck becomes the tortuous migration of reagent through the growing porous residue shell. Because the shell thickens as leaching proceeds, the rate declines with conversion and the conversion–time curve flattens markedly at high X. The quadratic dependence of τ on radius — τ ∝ R² — is the single most consequential relation in leaching kinetics: halving the particle size cuts ash-diffusion leach time by a factor of four, which is why diffusion-limited ores repay every increment of grinding while reaction-limited ores repay only linearly. Temperature has a weak effect because diffusion activation energies (typically 5–20 kJ/mol) are far below chemical ones (40–80 kJ/mol).

Surface chemical reaction control. The intrinsic dissolution reaction is slowest, the reagent concentration at the reaction front equals the bulk value, and agitation is irrelevant. Conversion time follows 1 − (1−X)^(1/3) = t/τ, τ ∝ R, and the rate shows strong Arrhenius temperature dependence. Fine-grained, porous ores leached in strong, well-stirred lixiviant frequently sit in or near this regime.

Because the three equations differ in shape, a single set of conversion–time data can discriminate between them.

There is also a shape-only diagnostic that needs no fitting at all: the ratio of time to 90% conversion over time to 50% conversion. Film control gives t₉₀/t₅₀ = 0.9/0.5 = 1.8; reaction control gives 0.536/0.206 = 2.6; ash-layer diffusion gives 0.554/0.110 = 5.0. A leach curve whose late tail stretches badly is announcing diffusion control before any regression is run.

How Do You Fit Leaching Data to the Shrinking Core Model?

Fitting is arithmetic on ordinary bottle-roll or column test data: timed residue assays, converted to fractional extraction, tested against the three linearized forms. The procedure below takes one afternoon with a spreadsheet.

1. Convert the assays to conversion. For each timed sample, compute X = 1 − (metal remaining in residue) / (metal in head assay), on a dry basis, correcting for the residue mass yield when residue masses are measured. This is the same extraction arithmetic used in the leaching recovery calculation — the only difference is that here you need it as a function of time, not just at the end point.
2. Compute the three linearized terms at every time point. For each (t, X) pair evaluate y₁ = X, y₂ = 1 − 3(1−X)^(2/3) + 2(1−X), and y₃ = 1 − (1−X)^(1/3). Tabulate them side by side; each column corresponds to one candidate controlling regime.
3. Plot all three columns against time and test linearity. The regime whose plot is straight over roughly the 20–80% conversion window is the controlling one. Inspect visually first, then confirm with least-squares R²; a fit that bends systematically is telling you the regime is wrong or changing.
4. Extract τ from the slope. Every linearized form has slope 1/τ, so τ = 1/slope, with units inherited from the time axis (a slope of 0.10 h⁻¹ gives τ = 10 h). Report τ together with the particle size fraction and conditions it was fitted at — a τ without a size is meaningless, because τ depends on R and possibly R².
5. Check the tail before trusting the fit. Curvature above about 90% conversion is normal for ash diffusion (the last few percent genuinely take longest) but pathological for reaction control. A fit that is linear early and steeply curved late often means the regime transitions from reaction to diffusion control as the residue shell thickens; fit the two windows separately and say so in the report.
6. Confirm the regime with independent levers. Repeat at a second crush size: τ dropping by half with a halved radius indicates film or reaction control (τ ∝ R); dropping to a quarter indicates ash diffusion (τ ∝ R²). Repeat at two temperatures: an Arrhenius-type rate acceleration confirms chemical control, while near-insensitivity confirms diffusion control. Two levers agreeing is far stronger evidence than one good R².

A well-executed fit yields more than a curve: it yields τ (and its size scaling) as a transferable parameter that survives into design.

Worked Example: 10 mm Copper Oxide Particle

The example below is illustrative — the numbers are constructed to demonstrate the method and are internally consistent, but they are not measurements from any specific ore or client. Suppose bottle-roll tests on a narrowly sized fraction of 10 mm malachite-bearing particles (R = 5.0 mm) in 0.5 mol/L sulfuric acid at 25 °C, 600 rpm, have been fitted by the procedure above and returned complete-conversion times of τ_film = 5 h, τ_reaction = 24 h and τ_ash = 40 h. These are coherent with physical expectations for a well-agitated test: film resistance is small, ash diffusion through a porous gangue shell is slowest, and surface kinetics sit in between.

Step 1 — film diffusion. X = t/τ, so at X = 0.80: t = τ·X = 5 h × 0.80 = 4.0 h.

Step 2 — ash-layer diffusion. The linearized form is 1 − 3(1−X)^(2/3) + 2(1−X) = t/τ. At X = 0.80 the unreacted fraction is 1 − X = 0.20. First, (0.20)^(1/3) = 0.5848, so (0.20)^(2/3) = 0.5848² = 0.3420. Then:

t/τ = 1 − 3(0.3420) + 2(0.20) = 1 − 1.0260 + 0.40 = 0.374

t = 0.374 × 40 h = 14.96 h ≈ 15.0 h

Step 3 — surface chemical reaction. The linearized form is 1 − (1−X)^(1/3) = t/τ. With (0.20)^(1/3) = 0.5848:

t/τ = 1 − 0.5848 = 0.415

t = 0.415 × 24 h = 9.97 h ≈ 10.0 h

So the same particle reaches 80% conversion in 4.0 h under film control, 10.0 h under chemical control and 15.0 h under ash-layer control — the diffusion-controlled case takes 3.7 times longer than the film-controlled case. Extending the same arithmetic across the conversion range:

The table exposes two features that every leaching test exhibits in some form. First, the order of difficulty flips: at low conversion the reaction-controlled curve is slowest (2.19 h at X = 0.25 versus 0.94 h for ash diffusion), because the ash shell is still thin, but by X = 0.80 diffusion has become the most punishing regime. Second, the tails differ enormously: the last 19% of conversion (0.80 → 0.99) costs 20.3 h under ash diffusion against 8.9 h under chemical control, which is why diffusion-limited circuits chase a long, expensive tail while reaction-limited ones finish more cleanly.

Size leverage. Suppose the same ore is retested at 5 mm (R = 2.5 mm). Under ash diffusion, τ ∝ R², so τ falls from 40 h to 10 h and the 80% conversion time becomes 0.374 × 10 = 3.7 h — a four-fold improvement from halving the diameter. Under chemical control, τ ∝ R, so τ falls to 12 h and the 80% time becomes 0.415 × 12 = 5.0 h — only a two-fold gain. This asymmetry is the quantitative heart of crush-size economics in the SCM.

When Does the Shrinking Core Model Break Down?

The SCM earns its keep on dense, single-mineral, roughly spherical particles — and leaching feeds are often none of those. Knowing where the idealization fails is what separates a useful fit from a misleading one.

Non-spherical and fractured particles. Crushed fragments are angular, flaky or cracked, not spheres. The reaction front then advances unevenly and shape factors deviate from the cubic relations above, typically by tens of percent at high conversion. Fits still land near the textbook equations at low-to-moderate X, but quote τ as an apparent value for that size fraction and shape, not as a material constant.

Changing diffusivity. The model assumes the ash layer keeps a constant effective diffusivity D_e. Real residue shells evolve: gypsum, jarosite or ferric hydroxide precipitating inside the pores throttles D_e with time, so the fitted τ drifts and the linearized plot bends. Where this is suspected, fit conversion windows separately and treat the drift itself as diagnostic of solution-chemistry problems.

Porous, fine-grained ores and the grain model. Many industrial ores are not dense particles at all but porous aggregates of micron-scale grains. The grain model is a companion formulation that pictures such a particle as thousands of tiny mineral grains, each dissolving as its own miniature shrinking core. The practical consequence is profound: the diffusion path length becomes the grain radius, not the particle radius, so the R² penalty of ash diffusion largely collapses and coarse crushes can leach with near reaction-controlled kinetics. Applying the dense-particle SCM to such an ore overpredicts the benefit of fine grinding — an expensive error to discover in a plant.

Shrinking particles. If the particle matrix itself dissolves (no inert residue), the constant-size assumption fails and the shrinking-particle variant of the equations applies. Partially soluble gangue sits between the two cases and is a common source of fits that match neither set of equations cleanly.

Bed-scale effects. Finally, the SCM describes a single particle in fresh solution. A heap or column adds solution bypass, channeling, wetting hysteresis and concentration profiles along the flow path. A particle-level fit is still the right starting point, but scale-up needs bed-level hydrology layered on top — exactly why column tests, not bottle rolls alone, underpin heap design.

From Kinetics to Equipment Sizing

A fitted τ is only useful once it becomes hours of residence in a vessel or days on a pad, and the translation differs sharply by flowsheet.

Agitated tank leaching. Take the reaction-controlled fit from the worked example (τ = 24 h at R = 5 mm) and target 90% extraction. The SCM gives t = τ·[1 − (1−X)^(1/3)] = 24 × [1 − (0.10)^(1/3)] = 24 × 0.5358 = 12.9 h of ideal batch time. Continuous stirred-tank trains never achieve ideal plug-flow kinetics — short-circuiting costs roughly 25–35% of the theoretical kinetics at typical stage numbers — so a design at 70% stage efficiency needs about 12.9/0.70 ≈ 18.4 h of total residence, conventionally split across three agitated tanks of roughly 6 h each. Two cautions from practice: fit τ at the design pulp density, because high solids loadings thicken the slurry and can push a reaction-controlled system toward film control; and re-fit after any crush-size change, since τ ∝ R for reaction control means coarser feed silently stretches every tank. Equipment implications — impeller choice, solids suspension, tank volume — are covered on our agitated leaching tank page.

Heap leaching. For percolation systems the SCM parameters arrive from column tests, and the fitted regime dictates the design response. An ash-layer-controlled column fit (τ ∝ R²) argues for finer crushing and longer lifts with extended leach cycles; a reaction-controlled fit argues that crush size matters less than solution contact and reagent supply. The fitted conversion–time curve also sets the economic cutoff: with the illustrative numbers above, reaching X = 0.95 instead of 0.80 under diffusion control costs 12.8 extra days per lift — a trade against pad throughput that must be made explicitly, not by default. Column-to-heap translation, lift design and irrigation strategy are treated in our heap leaching guide.

In both cases the discipline is the same: carry the fitted τ, its particle size, and the regime identification into design documents, so that every downstream sizing decision traces back to measured kinetics rather than a rule of thumb.

Frequently Asked Questions

What assumptions does the SCM make?

It assumes spherical particles of constant outer size with an inert gangue skeleton, a sharp reaction front advancing inward, pseudo-steady-state diffusion, and one rate-controlling step among film diffusion, ash-layer diffusion and surface reaction. Reagent concentration in the bulk solution is treated as uniform. Departures from any assumption — angular particles, collapsing porosity, mixed control — show up as curvature in the linearized plots.

Which regime usually controls industrial leaching?

In well-agitated tank leaching of moderately fine ore, surface chemical reaction or ash-layer diffusion control dominate; film control is uncommon because good mixing keeps the boundary layer thin. In heap and coarse-ore leaching, diffusion — both through ash layers and through the bed itself — usually dominates. The regime can also shift with conversion, typically from reaction control early to diffusion control late as the residue shell thickens.

How can I tell film control from ash-layer diffusion experimentally?

Change agitation and temperature. Film control responds strongly to stirrer speed or solution velocity and weakly to temperature, because mass-transfer activation energies are low. Ash-layer diffusion also shows weak temperature sensitivity, but it is insensitive to agitation and its τ scales with the square of particle size. A two-size, two-temperature test matrix separates the regimes with just four runs.

Can the SCM be applied to heap leaching?

As a particle-level starting point, yes — column test data fitted to the three linearized forms reveal whether diffusion or reaction controls the crush size being tested. But the heap adds bed-scale physics: channeling, incomplete wetting and concentration profiles down the flow path. Treat the SCM fit as one input to a heap model that also includes hydrology, and validate against columns at the design crush size before committing lift heights.

What is the grain model and when should I use it instead?

The grain model treats a porous ore particle as an assembly of micron-scale grains, each behaving as its own shrinking core. Use it when mineralogy shows the ore is a porous aggregate of fine grains — common in oxidized and some secondary sulfide ores. Its key prediction is that leach time scales with grain size rather than particle size, so coarse crushes can perform far better than the dense-particle shrinking core model would predict.

How does temperature affect leaching kinetics in the model?

Through the constants inside τ. Surface reaction rates follow Arrhenius behavior with activation energies typically 40–80 kJ/mol, so a 10 °C rise can roughly double the rate. Diffusion coefficients through films and ash layers have much smaller activation energies, around 5–20 kJ/mol, so diffusion-controlled leaching barely accelerates with heat. This contrast makes temperature a powerful diagnostic as well as a process lever.

What data do I need to fit the SCM?

Timed extraction data — residue assays at five or more points spanning roughly 20–95% conversion — plus the particle size fraction, lixiviant concentration, temperature and agitation conditions. Narrowly sized fractions are strongly preferred, since mixing sizes smears the conversion–time curve. End-point head and residue assays on a dry basis anchor the conversion calculation. With that, the fitting procedure in this article needs only a spreadsheet.

Conclusion

The shrinking core model survives in modern hydrometallurgy because it converts a messy question — why is this ore leaching slowly? — into three testable equations and one number, τ. The three linearized plots identify the controlling regime; the regime’s size scaling tells you whether grinding, agitation or temperature buys recovery; and the fitted τ converts directly into tank residence hours or heap lift days. When the model strains against the data, the strain is information: porous ores point to the grain model, drifting fits point to evolving residue chemistry, and curvature at high conversion points to the diffusion tail that every design must price honestly. Treat leaching kinetics as measurement, not folklore. If you are characterizing a new orebody or troubleshooting an underperforming circuit, send your column or bottle-roll data to Lexin-tech through our pilot testing services and we will fit the kinetics, identify the controlling regime and translate the result into equipment sizing. More worked calculations and technical guides are available in our resource library of hydrometallurgy technical guides.

References

1. Levenspiel, O. (1999). Chemical Reaction Engineering, 3rd edition, Chapter 25 “Fluid-Particle Reactions: Kinetics”. John Wiley & Sons, New York.
2. Habashi, F. (1993). A Textbook of Hydrometallurgy. Métallurgie Extractive Québec, Québec City.
3. Weiss, N. L. (ed.) (1985). SME Mineral Processing Handbook. Society for Mining, Metallurgy & Exploration (SME), Littleton, CO.
4. Wadsworth, M. E. (1979). “Leaching kinetics.” In: Sohn, H. Y. & Wadsworth, M. E. (eds.), Rate Processes of Extractive Metallurgy. Plenum Press, New York.
5. Szekely, J. & Evans, J. W. (1971). “A structural model for gas-solid reactions with a moving boundary — II.” Metallurgical Transactions, Vol. 2, pp. 1699–1710.
6. Dixon, D. G. & Petersen, J. (2003). “Comprehensive modelling study of chalcocite column and heap bioleaching.” Canadian Metallurgical Quarterly, Vol. 42, No. 4, pp. 497–508 (Metallurgy and Materials Society of CIM).

Changelog

2026-10-26 — first published.