Recycled paper is re-pulped, and small ink/toner flecks detach from the fibers as free particles. In a flotation deinking cell, air is sparged into the pulp slurry from the bottom as fine bubbles; a bubble captures a particle only if it collides with it and the particle is hydrophobic enough to attach and survive the froth journey to the top, where the ink-laden froth is skimmed off (reject) while clean fibers stay in the pulp (accept). This 2D view is a side cross-section through the cell: the lower band is the pulp slurry, the upper band is the froth layer that gets skimmed.
Collision probability: P_c = min(1, 3 · r_particle / r_bubble)
Attachment probability: P_a = sin²(θ / 2)
Capture probability: P_capture = P_c · P_a
- Contact angle θ — set by fatty-acid/soap collector dosage on the ink surface. Higher θ → more hydrophobic ink → higher P_a. Cellulose fiber stays near θ ≈ 8° (hydrophilic) regardless of dosage, so it is almost never captured.
- Bubble diameter — smaller bubbles raise the collision probability P_c (more surface area per unit air volume) but also rise more slowly and carry less lift; the sweet spot for deinking cells is typically 0.3–1 mm.
- Airflow rate — how often new bubbles nucleate at the sparger; more bubbles mean more collision events per second but can also over-froth and drag some fiber out mechanically (entrainment).
- Pulp consistency — the fiber mass fraction of the slurry. Higher consistency raises viscosity, slowing bubble rise and particle diffusion, which lowers both ink removal and throughput.
Brightness gain is reported in ISO brightness points, an illustrative linear proxy (~8.5 pts per 100% ink removed) for the international paper-brightness scale mills use to grade deinked pulp. The lower probability panel plots P_a(θ) and marks the live operating point for both ink and fiber.