Flocculation Calculator

Velocity gradient equals square root of power over viscosity times volume

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Velocity Gradient Equation

Calculate the velocity gradient (shear intensity) in a flocculation basin from the power input, dynamic viscosity, and tank volume. Typical G values range from 20 to 80 s⁻¹.

G = √(P / (μ × V))

Power Dissipated (Paddle)

Calculate the power dissipated by paddle mixers from the drag coefficient, paddle area, water density, and relative paddle velocity.

P = Cd × A × ρ × v³ / 2

How It Works

Flocculation gently stirs chemically treated water so that tiny destabilized particles collide and stick together into larger clumps called flocs. The velocity gradient G quantifies the shear intensity -- too low and particles rarely collide, too high and fragile flocs break apart. Designers also calculate the power needed by paddle mixers using fluid density, drag coefficient, and blade velocity. Typical G values for flocculation range from 20 to 80 s⁻¹, with detention times of 20-40 minutes, giving G×t products of roughly 10,000-100,000.

Example Problem

A flocculation tank has 150 W of power input, a volume of 50 m³, and the water viscosity is 0.001 Pa·s. What is the velocity gradient?

  1. Identify the knowns. Power input P = 150 W, dynamic viscosity μ = 0.001 Pa·s (≈ water at 20 °C), tank volume V = 50 m³.
  2. Identify what we're solving for. We want the velocity gradient G — the shear intensity in the flocculation basin, in inverse seconds.
  3. Write the velocity gradient equation: G = √(P / (μ × V)). The units inside the radical work out to s⁻², so taking the square root gives s⁻¹.
  4. Substitute the values: G = √(150 / (0.001 × 50)).
  5. Simplify the arithmetic: the denominator μ × V = 0.001 × 50 = 0.05, then 150 / 0.05 = 3,000, and √3,000 ≈ 54.77.
  6. **The velocity gradient is G ≈ 54.77 s⁻¹** — inside the typical 20-80 s⁻¹ window for flocculation, gentle enough to grow flocs without shearing them apart.

G ≈ 55 s⁻¹ sits within the typical 20-80 s⁻¹ flocculation range, so the basin is correctly sized for gentle floc growth — strong enough to bring particles together, gentle enough to avoid shearing the flocs apart.

When to Use Each Variable

  • Solve for Velocity Gradient (G) — when you know power input, viscosity, and tank volume and need to verify the mixing intensity is within the 20-80 s^-1 range for flocculation.
  • Solve for Power Input — when you have a target G value and need to size the mixer motor for a given tank volume and water viscosity.
  • Solve for Dynamic Viscosity — when you need to back-calculate viscosity from measured power and G — useful for verifying water temperature assumptions.
  • Solve for Tank Volume — when you have a fixed mixer power and target G and need to determine the required basin size.
  • Solve for Power Dissipated — when you know paddle geometry, drag coefficient, density, and velocity and need to calculate the power delivered by paddle mixers.

Key Concepts

Flocculation is the gentle mixing stage in water treatment where chemically destabilized particles collide and aggregate into larger, settleable flocs. The velocity gradient G quantifies shear intensity and is the square root of power input divided by viscosity and volume. Too little shear means insufficient collisions; too much shear breaks fragile flocs apart. The G*t product (gradient times detention time) characterizes total mixing energy, with typical values of 10,000 to 100,000 for water treatment.

Applications

  • Drinking water treatment: designing flocculation basins that precede sedimentation and filtration
  • Wastewater treatment: sizing mixing tanks for chemical phosphorus removal and coagulation
  • Industrial process water: flocculating suspended solids from cooling tower blowdown or mining wastewater
  • Stormwater management: designing detention basins with gentle mixing zones for particulate removal

Common Mistakes

  • Confusing rapid mix (G = 300-1000 s^-1) with flocculation (G = 20-80 s^-1) — the two stages have very different mixing intensities
  • Ignoring temperature effects on viscosity — cold water is more viscous, which changes G for the same power input
  • Using paddle tip speed instead of relative velocity — the effective paddle velocity is about 0.75 times the tip speed because the water moves with the paddle
  • Neglecting the G*t product — achieving the right G without adequate detention time results in poor floc formation

Frequently Asked Questions

What does the velocity gradient G represent in a flocculation basin?

G is the root-mean-square shear rate (s⁻¹) experienced by water inside the basin, calculated from G = √(P / (μV)). It measures how aggressively the fluid is sheared — high enough to bring particles together but low enough to keep delicate flocs intact. Typical design values are 20–80 s⁻¹.

What is the G·t product and why does it matter?

G·t is the dimensionless product of velocity gradient and hydraulic detention time. It captures the total mixing exposure a parcel of water receives. Water treatment flocculation typically targets G·t = 10,000–100,000 — enough cumulative shear for particles to collide many times without breaking apart the flocs that form.

How does paddle speed affect floc formation?

Faster paddles raise G and the collision frequency, accelerating floc growth in the early stages. Past a threshold, however, additional shear shears existing flocs back into small fragments. Paddle tip speeds are usually kept between 0.3 and 0.9 m/s, with relative blade-to-water velocity around 0.75 of the tip speed.

How is rapid mix different from flocculation?

Rapid mix is the high-shear stage where coagulant is dispersed (G = 300–1000 s⁻¹, detention ~30 s) and chemical destabilization happens. Flocculation is the gentle, downstream stage (G = 20–80 s⁻¹, detention 20–40 min) where destabilized particles aggregate. Using flocculation G values in rapid mix — or vice versa — defeats the unit process.

Why does water temperature change the required power input?

Cold water has higher dynamic viscosity (μ at 5 °C is about 1.5× that at 20 °C). G = √(P/(μV)) means more viscous water requires more power to achieve the same G. Plants in northern climates often install variable-speed drives to compensate seasonally.

What is the drag coefficient in the paddle power equation?

C_d in P = C_d × A × ρ × v³ / 2 captures how efficiently a paddle transfers shaft power to fluid shear. Flat paddles run at C_d ≈ 1.5–1.8, while streamlined paddles or impellers have lower values. The drag coefficient is geometry-dependent and usually taken from manufacturer or hydraulic-handbook tables.

Can the velocity gradient model be used for jet or hydraulic flocculators?

Yes — G = √(P / (μV)) is a general energy-dissipation formulation. P can be supplied by paddles, in-line static mixers, baffled hydraulic flocculators, or jet diffusers. The model assumes uniform energy dissipation; real basins have dead zones and short-circuiting, which CFD or tracer testing can reveal.

Worked Examples

Drinking Water — Rapid Mix

What G value does a 25 m³ rapid-mix tank achieve with a 1.5 kW mixer?

Rapid-mix (flash-mix) tanks disperse coagulants like alum within 1–2 seconds at very high G values. For a 25 m³ basin powered by a 1.5 kW mixer in water at 20 °C (μ = 1.0×10⁻³ Pa·s), compute the velocity gradient G to confirm it falls in the textbook rapid-mix range of 600–1000 s⁻¹ ... or if it under-mixes.

  • Knowns: P = 1500 W, μ = 0.001 Pa·s, V = 25 m³
  • G = √(P / (μ × V))
  • G = √(1500 / (0.001 × 25))
  • G = √(1500 / 0.025)
  • G = √60,000

G ≈ 245 s⁻¹

245 s⁻¹ is well below the typical rapid-mix design target of G ≥ 600 s⁻¹ — either the mixer is undersized for this basin or the basin is too large for true flash-mix. Operators would tighten residence time (smaller mixing chamber upstream) or upgrade the impeller.

Drinking Water — Slow Mix (Flocculation)

Does a 600 m³ flocculation basin hit the 30 s⁻¹ G target with 75 W mixing power?

Slow-mix flocculation basins use much lower G values (20–80 s⁻¹) than rapid-mix to grow floc without shearing it apart. For a 600 m³ basin with 75 W input and water viscosity 0.001 Pa·s, compute G and compare against the typical 30 s⁻¹ design target.

  • Knowns: P = 75 W, μ = 0.001 Pa·s, V = 600 m³
  • G = √(P / (μ × V))
  • G = √(75 / (0.001 × 600))
  • G = √(75 / 0.6)
  • G = √125

G ≈ 11.2 s⁻¹

11.2 s⁻¹ is below the typical 20–80 s⁻¹ flocculation range — floc particles will form but mixing intensity is on the gentle end. Tapered flocculation basins decrease G in stages (e.g., 50 → 30 → 15 s⁻¹) to grow large, settleable floc without breaking it.

Paddle Flocculator Power

How much power does a 5 m² paddle blade dissipate at 0.3 m/s in water?

Paddle flocculators rotate slowly through water; the power dissipated drives flocculation but also limits how fast you can spin them before floc shear. For paddles with total projected area A = 5 m², drag coefficient C_D = 1.8 (flat plates), water density ρ = 1000 kg/m³, and relative paddle velocity v = 0.3 m/s, compute the dissipated power.

  • Knowns: C_D = 1.8, A = 5 m², ρ = 1000 kg/m³, v = 0.3 m/s
  • P = C_D × A × ρ × v³ / 2
  • P = 1.8 × 5 × 1000 × (0.3)³ / 2
  • P = 1.8 × 5 × 1000 × 0.027 / 2
  • P = 243 / 2

P = 121.5 W

Paddle tip speeds above ~0.6 m/s shear floc apart — designers stay in the 0.15–0.45 m/s window. The relative velocity v is paddle tip speed × (1 − slip), where slip accounts for water rotating with the paddle (often 0.25 for un-baffled tanks).

Flocculation Formulas

Two complementary equations describe flocculation-basin energy and paddle mixer hydraulics:

G = √(P / (μ × V))Velocity gradient — basin shear intensity
P = Cd × A × ρ × v³ / 2Power dissipated by a paddle mixer

Where:

  • G — velocity gradient (s⁻¹); design target 20–80 s⁻¹ for flocculation, 300–1000 s⁻¹ for rapid mix
  • P — net power input to the water (W)
  • μ (mu) — dynamic viscosity of water (Pa·s); ~0.0010 at 20 °C, rising as water cools
  • V — flocculation basin volume (m³)
  • Cd — paddle drag coefficient (dimensionless); ~1.5–1.8 for flat paddles
  • A — paddle area perpendicular to motion (m²)
  • ρ (rho) — water density (kg/m³); ~1000 for fresh water
  • v — relative paddle-to-water velocity (m/s); ≈ 0.75 × paddle tip speed

The velocity gradient equation, originally developed by Camp and Stein, derives G from the root-mean-square fluid shear that a uniform power input produces in a basin. The paddle-power relation is the drag-force equation P = F × v with drag force F = ½ × Cd × ρ × A × v². Together they let designers match mixer geometry and motor sizing to the target G and G·t product.

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