Raw water carries colloidal particles too small to settle on their own — electrostatic charge on their surfaces keeps them apart (double-layer repulsion). A coagulant dose neutralizes that charge; too little leaves particles repelling, too much re-charges them the other way (restabilization), so there is a real optimum dose. Once neutralized, collisions stick: Brownian motion drives collisions between the smallest particles (perikinetic aggregation), while the mixing paddle's velocity gradient G sweeps larger flocs into each other (orthokinetic aggregation) — together this is Smoluchowski's collision model. Each successful collision conserves volume, so flocs grow as the cube-root sum of their radii, and because settling velocity scales with radius² (Stokes' law), bigger flocs fall out of suspension fast.
α(dose) = collision efficiency, peaks at optimum dose
J_peri = (4kT / 3μ) · (r₁+r₂)²/(r₁r₂) — Brownian collisions
J_ortho = (4/3)·G·(r₁+r₂)³ — shear-driven collisions
r_new = (r₁³ + r₂³)^(1/3) — volume-conserving merge
v_settle ∝ r² — Stokes settling
- Coagulant dose — too low: particles stay charged and bounce apart. Near-optimal: charge is neutralized and every collision sticks. Overdosed: charge reverses sign and particles restabilize, hurting removal again.
- Mixing intensity G — the velocity gradient in the tank. Higher G means more orthokinetic collisions and faster floc growth, but excess G can shear fragile flocs apart once they get large.
- Raw turbidity — starting particle load; denser suspensions collide more often per second at any dose or G.
Real-world relevance: this coagulation → flocculation → sedimentation sequence is the first treatment stage at almost every municipal water plant, run as a bench "jar test" before full-scale dosing is set.