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Power equals specific weight times flowrate times headloss

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Static Mixer Power

Static mixers dissipate energy through headloss as water flows through baffles or orifices. Power depends on the specific weight of the fluid, flow rate, and headloss through the mixer.

P = γ × Q × h

Laminar Impeller Power

In the laminar regime, impeller power depends on the mixing constant, fluid viscosity, rotational speed, and impeller diameter. Used when the Reynolds number is below about 10.

P = k × μ × n² × D³

Turbulent Impeller Power

In the turbulent regime, impeller power depends on the mixing constant, fluid density, rotational speed, and impeller diameter. Used when the Reynolds number exceeds about 10,000.

P = k × ρ × n³ × D⁵

How It Works

Mixing injects energy into water to blend chemicals, suspend solids, or transfer oxygen. This calculator covers six equation families used in wastewater treatment mixing design: static mixer power, laminar and turbulent impeller power, Reynolds number for flow regime classification, and two pneumatic mixing power equations (volume-based and flow-based). Static mixers dissipate energy through headloss as water flows through baffles or orifices. Mechanical impeller mixers operate in either laminar or turbulent regimes, and their power draw depends on rotational speed, impeller diameter, and fluid properties. Pneumatic mixers use compressed air to agitate the liquid.

Example Problem

A static mixer treats 0.2 m³/s of water with a specific weight of 9,810 N/m³ and a headloss of 0.5 m. What power is dissipated?

  1. Identify the knowns. Specific weight γ = 9,810 N/m³ (water at 20 °C), volumetric flow rate Q = 0.2 m³/s, headloss across the mixer h = 0.5 m.
  2. Identify what we're solving for. We want the power P dissipated by the static mixer — the energy per unit time extracted from the flow to blend the chemicals.
  3. Write the static mixer power equation: P = γ × Q × h. Dimensionally (N/m³)(m³/s)(m) = N·m/s = W.
  4. Substitute the values: P = 9,810 N/m³ × 0.2 m³/s × 0.5 m.
  5. Simplify the arithmetic: 9,810 × 0.2 = 1,962, then 1,962 × 0.5 = 981.
  6. **The dissipated power is P = 981 W** — the energy the flow loses to friction through the mixer baffles, which is the power available for chemical blending.

When to Use Each Variable

  • Solve for Static Mixer Power — when you know the flow rate, specific weight, and headloss through the mixer and need the power dissipated.
  • Solve for Laminar Impeller Power — when the impeller Reynolds number is below about 10 and you need the power draw from viscosity, speed, and diameter.
  • Solve for Turbulent Impeller Power — when the impeller Reynolds number exceeds about 10,000 and you need the power draw from density, speed, and diameter.

Key Concepts

Mixing design determines the energy input required to blend chemicals, suspend solids, or transfer gases in treatment basins. Static mixers use flow energy (headloss) with no moving parts. Mechanical impeller mixers operate in either laminar or turbulent regimes, and the power number (mixing constant k) varies by impeller type and Reynolds number. The velocity gradient G = sqrt(P / (mu x V)) is the key design parameter — rapid mix requires G > 300 s^-1 while slow mix uses G = 20-80 s^-1.

Applications

  • Water treatment: designing rapid-mix chambers for coagulant injection and slow-mix basins for flocculation
  • Wastewater aeration: sizing mechanical aerators and diffused air systems for activated sludge basins
  • Chemical processing: scaling up mixing operations from bench to pilot to full-scale reactors
  • Industrial blending: designing mixers for paint, pharmaceutical, and food processing applications

Common Mistakes

  • Using the laminar power formula in turbulent flow (or vice versa) — the formulas have different exponents and use viscosity versus density; check the Reynolds number first
  • Ignoring impeller type when selecting the mixing constant k — Rushton turbines (k around 5) draw far more power than marine propellers (k around 0.3) at the same speed and diameter
  • Sizing mixers using only power without checking the velocity gradient G — adequate power at the wrong G value produces poor mixing quality

Frequently Asked Questions

How are rapid mix and slow mix different in design intent?

Rapid mix uses high shear (G > 300 s⁻¹) for 10–30 s to uniformly disperse coagulant before particles destabilize. Slow mix (flocculation) applies gentle shear (G = 20–80 s⁻¹) for 20–40 min so destabilized particles collide and build flocs without being torn back apart.

Which mixing constant k applies to my impeller?

k (the impeller power number) depends on impeller geometry and Reynolds regime. A 6-blade Rushton turbine in turbulent flow runs at k ≈ 5.0; marine propellers and hydrofoil impellers are k ≈ 0.3–0.4. Manufacturers publish curves of k vs. NRe so designers can read off the value for their selected mixer and flow regime.

When should I pick a static mixer over a mechanical mixer?

Static (in-line) mixers shine in pressurized pipelines where you want fast chemical blending, no moving parts, and minimal maintenance. Mechanical impeller mixers are better for open basins with longer detention times, where you need adjustable speed and can tolerate routine motor and seal maintenance.

How do I know if mixing is laminar or turbulent?

Compute the mixing Reynolds number NRe = D² × n × ρ / μ, where D is impeller diameter, n is rotational speed (rev/s), and ρ and μ are water density and viscosity. NRe < 10 is laminar (use P = k × μ × n² × D³), NRe > 10,000 is turbulent (P = k × ρ × n³ × D⁵), and the band between is transitional.

How does pneumatic (air) mixing power scale with depth?

Diffused-air mixing power per unit air volume scales with the natural log of the compression ratio. Deeper basins produce more mixing power per unit airflow because air is compressed against a larger hydrostatic head — P = p_a × V_a × ln(p_c / p_a) captures this, where p_a is atmospheric and p_c is total pressure at the diffuser.

Why doesn't impeller power scale linearly with speed?

In turbulent flow P ∝ n³, so doubling the speed multiplies the power draw by eight. This cubic relationship is why slowing a mixer modestly saves a disproportionate amount of energy and is the driving force behind variable-frequency drives on biological treatment basins.

What is the velocity gradient G and how does it relate to mixing power?

G = √(P / (μ × V)) is the root-mean-square fluid shear rate in a basin (s⁻¹). It links mixing power to mixing quality: doubling power increases G by only √2, so achieving the next G band often requires substantially more energy. G is the primary specification for rapid-mix and flocculation basins.

Worked Examples

Chemical Process — Rushton Turbine

How much power does a 0.5 m Rushton turbine draw at 2 rev/s in water?

A 0.5 m diameter Rushton (six-blade flat disk) turbine in fully baffled water at 2 rev/s runs in the turbulent regime (Re > 10⁴) with power number k ≈ 5. Compute the shaft power using the turbulent impeller form P = k × ρ × n³ × D⁵.

  • Knowns: k = 5, ρ = 1000 kg/m³, n = 2 rev/s, D = 0.5 m
  • P = k × ρ × n³ × D⁵
  • P = 5 × 1000 × (2)³ × (0.5)⁵
  • P = 5 × 1000 × 8 × 0.03125

P = 1,250 W ≈ 1.25 kW

Power scales with D⁵ — doubling the impeller diameter increases the motor load 32×. The dimensionless power number k depends only on impeller geometry above Re ≈ 10⁴: Rushton ≈ 5, pitched-blade turbine ≈ 1.3, marine propeller ≈ 0.3.

Reynolds Regime — CSTR Sizing

Is the flow turbulent in a 0.3 m impeller spinning at 4 rev/s in 998 kg/m³ water?

An engineer sizing a 1 m³ stirred-tank reactor with a 0.3 m pitched-blade impeller at 4 rev/s needs to know whether the power-number correlation should use the turbulent or transitional form. Compute the impeller Reynolds number Re = D² × n × ρ / μ for water (μ = 0.001 Pa·s).

  • Knowns: D = 0.3 m, n = 4 rev/s, ρ = 998 kg/m³, μ = 0.001 Pa·s
  • Re = D² × n × ρ / μ
  • Re = (0.3)² × 4 × 998 / 0.001
  • Re = 0.09 × 4 × 998 / 0.001
  • Re = 359.28 / 0.001

Re ≈ 359,000

Re > 10⁴ confirms fully turbulent mixing — use the turbulent power number (k ≈ 1.3 for pitched-blade) directly without a Reynolds correction. Transitional (10 < Re < 10⁴) requires a manufacturer chart; laminar (Re < 10) uses the P = k × μ × n² × D³ form.

Water Treatment — Static Mixer

How much power does a static mixer dissipate on 50 L/s of water with 0.2 m of head loss?

An in-line static mixer just downstream of a coagulant injection point passes 50 L/s of water with a measured pressure drop of 0.2 m of head. Compute the power dissipated using the static-mix form P = γ × Q × h, with water specific weight γ = 9810 N/m³.

  • Knowns: γ = 9810 N/m³, Q = 0.05 m³/s, h = 0.2 m
  • P = γ × Q × h
  • P = 9810 × 0.05 × 0.2

P = 98.1 W

Static mixers achieve their mixing energy from the head loss across the unit — no motor, no maintenance. Typical drinking-water static mixers operate at 0.1–0.5 m head loss; higher head loss gives better coagulant distribution but pumps must overcome it.

Mixing Design Formulas

Several equation families cover the major mixing technologies in water and wastewater treatment — static, mechanical (laminar and turbulent), and pneumatic — plus the Reynolds number that selects between laminar and turbulent regimes:

P = γ × Q × hStatic (in-line) mixer power from headloss
P = k × μ × n² × D³Mechanical impeller power — laminar regime (NRe < 10)
P = k × ρ × n³ × D⁵Mechanical impeller power — turbulent regime (NRe > 10,000)
NRe = D² × n × ρ / μMixing Reynolds number — selects flow regime
P = pa × Va × ln(pc / pa)Pneumatic mixer power — volume-based form
P = 35.28 × Qa × ln((h + 33.9) / 33.9)Pneumatic mixer power — US flow-based form (h in ft)

Where:

  • P — mixing power (W, ft·lbf/s, or hp depending on form)
  • γ (gamma) — specific weight of water (N/m³); ~9810 at 20 °C
  • Q — volumetric flow rate through the mixer (m³/s)
  • h — headloss across the static mixer, or diffuser depth for pneumatic (m or ft)
  • k — impeller power number (dimensionless); 5.0 for Rushton turbine, 0.3–0.4 for marine propeller
  • μ (mu) — dynamic viscosity (Pa·s)
  • ρ (rho) — fluid density (kg/m³)
  • n — impeller rotational speed (rev/s)
  • D — impeller diameter (m)
  • NRe — mixing Reynolds number (dimensionless)
  • pa, pc — atmospheric and diffuser-depth absolute pressure (kPa)
  • Va, Qa — air volume / air flow rate at standard conditions

These equations cover the dominant mixing technologies but assume uniform energy dissipation. Real basins have spatial gradients; critical applications often validate the design with computational fluid dynamics or tracer studies. The velocity-gradient framework (G = √(P / (μV))) ties any of these power inputs back to a basin shear intensity that matches design guidelines for rapid mix or flocculation.

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