Shrinking Core Model
Also known as: Shrinking Unreacted Core Model, SCM, Leaching Kinetics Model
The Shrinking Core Model, formalized by Szekely, Evans, and Sohn in 1976, describes the kinetics of chemical reactions between solid ore particles and surrounding fluids (leaching solutions, roasting gases). As the reaction proceeds from the particle surface inward, an unreacted core shrinks while products accumulate in a product layer. The model enables prediction of leaching times and optimization of hydrometallurgical processes.
Key highlights
- Simple analytical model with physically interpretable rate parameters
- Identifies limiting steps, guiding process optimization
- Robust: predicts conversion-time relationships across wide ranges of temperature and concentration
- Easy to fit to laboratory data and scale to industrial operations
- Widely used in hydrometallurgy and pyrometallurgy; abundant reference literature
Intuition
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How it works
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When to use it
Use the Shrinking Core Model when designing hydrometallurgical processes where ore particles react with leaching solutions (acid leaching, heap leaching, bioleaching) or roasting with gases. Assume spherical particles and well-mixed, isothermal bulk fluid. Works best for single ore minerals or simple ore assemblies. For complex multicomponent ores or non-spherical particles, use numerical simulations.
Strengths & limitations
- Simple analytical model with physically interpretable rate parameters
- Identifies limiting steps, guiding process optimization
- Robust: predicts conversion-time relationships across wide ranges of temperature and concentration
- Easy to fit to laboratory data and scale to industrial operations
- Widely used in hydrometallurgy and pyrometallurgy; abundant reference literature
- Assumes spherical particles; non-spherical ore breaks assumption
- Model does not account for cracking or fragmentation of the product layer, which accelerates reaction in real systems
- Product layer is assumed solid and impermeable; some products are permeable or dissolve, invalidating assumption
- Bulk concentration of reactant is assumed constant; in closed systems or batch leaching, concentration drops significantly
- Assumes no interactions between particles; in dense slurries, particle-particle effects (osmotic pressure, concentration gradients) matter
Common pitfalls
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Applications
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Frequently asked
How do I know which limiting step (diffusion vs. reaction) applies to my leaching process?
Plot the linearized data. For diffusion control: plot [1 - 3(1-X)^(2/3) + 2(1-X)] versus time—should be linear. For reaction control: plot 1-(1-X)^(1/3) versus time—should be linear. If neither is perfect, your process may have mixed control or involve other steps.
What is a typical effective diffusivity (D_e) for ion diffusion in ore particle layers?
Typical values range 10^-6 to 10^-8 cm²/s at room temperature, decreasing with temperature. Real measured values depend on pore geometry, tortuosity, and fluid properties. Measure or look up literature values for your specific ore type.
How do I apply shrinking core to non-spherical ore particles?
The model assumes spheres. For other shapes, use shape factors or numerically solve the diffusion equation. Alternatively, measure the relationship between time and mean particle size empirically and ignore the analytical solution.
If my ore fragments during leaching, does the shrinking core model break?
Yes, fragmentation introduces new surface area and breaks the constant-size assumption. Empirically, apply corrections: adjust the model to account for surface area increase (multiply rate by fragmentation factor >1). Fragmentation factors are ore-specific and best determined experimentally.
Can I use shrinking core for heap leaching where particle sizes are not uniform?
Use population balance models that track the shrinking core model for each size class. Average the results over the size distribution. This is more complex but captures reality. Alternatively, apply the shrinking core model to the average particle size as a rough estimate.
Sources
- 1.Szekely, J., Evans, J. W., & Sohn, H. Y. (1976). Gas-solid reactions. Academic Press, New York.
- 2.Levenspiel, O. (1999). Chemical reaction engineering (3rd ed.). John Wiley & Sons.
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Cite this page
ScholarGate. (2026, June 3). Shrinking Core Model. ScholarGate. https://scholargate.app/mining-engineering/shrinking-core-model