Batch froth flotation of fluorite (CaF₂) from calcite and quartz: first-order kinetics for each mineral, the grade–recovery trade-off and a live cell.
The science
First-order flotation kinetics. For each mineral i, the fraction still in the cell decays as dCi/dt = −ki Ci, so the recovered fraction is Ri(t) = R∞,i(1 − e−kit) (García-Zuñiga form), where R∞ is the ultimate recovery of the floatable fraction. Rate constants are in 1/min.
Reagents. The collector adsorbs on fluorite and calcite surfaces (Langmuir-type saturation): kf = kmax θ(col) A, θ = col/(col + Kc), with A proportional to the air rate (more bubble surface). Calcite is similar to fluorite chemically, so it floats too (kc = 0.85 kf without depressant); a silicate depressant coats calcite and quartz: kc = 0.85 kf / (1 + dep/D50). Quartz hardly floats; it enters the froth mainly by entrainment, proportional to the water recovery: Rent = W(1 − e−t/τ).
Grade and recovery. Concentrate mass is the sum of component masses; grade = mCaF₂ / mconc, recovery = mCaF₂,conc / mCaF₂,feed. Scraping longer raises recovery but the slower-floating impurities catch up, so grade falls: the grade–recovery curve is the trade-off. Commercial classes: acidspar ≥ 97 % CaF₂, ceramic about 85–96 %, metallurgical 60–85 %.
Numerical checks. (1) An RK4 integration of dC/dt = −kC (step 0.01 min) reproduces e−kt for fluorite and calcite. (2) A seeded Monte Carlo gives each of 200 000 particles an exponential flotation time −ln U / k; the fraction floated by the stop time must match the closed-form recovery within 4 standard errors. (3) The grade recomputed from RK4-integrated recoveries must equal the closed-form grade.
Honest simplifications. A single rate constant per mineral ignores the particle-size distribution (a real ore has a distribution of k); constants (kmax = 1.1/min, Kc = 250 g/t, D50 = 450 g/t, entrainment time constant 2 min) are illustrative values, not plant data. Froth stability, pulp density, pH, temperature and bubble-size effects are folded into those few numbers. Real fluorspar plants use several cleaning stages to reach acid grade.