Spodumene Flotation - 2D Cell Cascade Recovery

First-order flotation kinetics

The floatable mineral is removed at a rate proportional to its concentration: dC/dt = -k C. In a batch cell recovery is R = R_∞ (1 - e^-kt). For N perfectly mixed cells in series, each with residence time τ/N, the steady-state recovery is R = R_∞ [1 - (1 + kτ/N)^-N], which approaches the batch curve as N grows. The page integrates the cell balances dC_i/dt = (C_i-1 - C_i)N/τ - kC_i to steady state and compares with this formula.

Where k comes from

A simple collision model gives k = ¼ S_b E_c E_a E_s, with bubble surface-area flux S_b = 6 J_g/d_b (J_g superficial gas velocity, d_b bubble diameter), collision efficiency E_c = 1.5 (d_p/d_b)² (interception regime), attachment efficiency E_a and a stability factor E_s = exp[-(d_p/d_max)²] that models detachment of coarse particles. Fine particles collide rarely, very coarse ones detach, so recovery peaks at an intermediate size.

Honest simplifications: the collision efficiency is the Gaudin interception limit, entrainment and froth drop-back are ignored, every cell has the same k and R_∞, and the reagent chemistry that makes spodumene hydrophobic is only represented by E_a. It illustrates trends in a mineral-processing circuit, not a plant model.