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Solution chemistry
Free cyanide ion and dissolved oxygen both have to reach the gold surface.
Slurry and circuit
Agitation thins the diffusion film; grain size and residence time set the recovery.
Model checks
The science
Elsner mixed diffusion control
Gold dissolves as 4 Au + 8 CN- + O2 + 2 H2O → 4 Au(CN)2- + 4 OH-. In a stirred slurry the reaction is limited by diffusion of cyanide and oxygen across a thin film of thickness δ. Equating the two fluxes through the film (Habashi) gives the rate per unit area J = 2 DCN DO2 [CN-][O2] / (δ (DCN[CN-] + 4 DO2[O2])). At high cyanide this tends to 2 DO2[O2]/δ (oxygen-controlled); at low cyanide to DCN[CN-]/(2δ) (cyanide-controlled). The two limits meet at [CN-]/[O2] = 4 DO2/DCN ≈ 6, the textbook value (reported 4 to 7.4).
A spherical grain of radius r shrinks at dr/dt = -J MAu/ρAu, so a grain of diameter d is gone after d/(2 dr/dt) and the dissolved fraction is 1 - (1 - kt/r0)3. The pH matters because only CN- leaches; below the HCN pKa of 9.31 cyanide is lost as HCN: [CN-]/[CN]tot = 1/(1 + 10pKa - pH).
What the simulation does
Grains drawn from a log-normal size distribution (geometric s.d. 1.8) travel through five equal stirred tanks with exponential residence times in each (a cascade of ideal tanks, so total time is Erlang-distributed). The film thickness scales as δ = 30 um / sqrt(agitation) and the diffusivities follow an Arrhenius law with 12.8 kJ/mol. The recovery of the whole cascade is computed both by the grains you see and analytically.
Honest simplifications
- Pure liberated gold spheres; no silver, copper or sulfide cyanicides, no passivation, no ore matrix, no mass-transfer limits from the bulk.
- Reagent concentrations are held constant along the cascade (no cyanide consumption or carbon adsorption), and dissolved oxygen is an input rather than being set by aeration and temperature.
- The film thickness law and the 30 um reference are representative values, not measurements of a specific plant.
What to try
- Load Cyanide starved and watch the rate follow the cyanide limit; double NaCN and the rate doubles. Then load Oxygen starved and raise cyanide: nothing changes.
- Lower the pH to 8.5: more than 80 % of cyanide turns into volatile HCN and recovery collapses.
- Raise the grain size to 80 um: the shrinking-sphere cube law makes a few coarse grains escape the last tank.