Agitation and mixing, power number and impeller types

Stirred-tank geometry, baffles and impeller types, power number versus Reynolds number (laminar and turbulent), and scale-up by constant P/V or tip speed.

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Why it matters

Stirred tanks are the most common reactors and blenders in the chemical, pharmaceutical and food industries. The impeller type decides the flow pattern, and the power number correlation decides the motor size; scale-up from a pilot tank to a plant vessel fails if you keep the wrong quantity constant.

Key ideas

Agitation versus mixing. Agitation is inducing a motion in a material in a specified way, usually a circulatory pattern in a vessel. Mixing is the random distribution of two or more initially separate phases into each other. Purposes include suspending solids, blending miscible liquids, dispersing a gas or an immiscible liquid as small bubbles or drops, and promoting heat transfer to a jacket or coil.

Standard vessel. A vertical cylindrical tank with a dished bottom, liquid depth about equal to the tank diameter, an impeller on a central shaft at about one impeller diameter above the bottom, and (for low-viscosity liquids) four vertical baffles of width about one-tenth (or one-twelfth) of the tank diameter. Without baffles the liquid swirls as a body and forms a central vortex, with little mixing and air entrainment at high speed; baffles convert swirl into top-to-bottom and radial flow. Off-centre or angled mounting is an alternative in small tanks.

Impeller types.

  • Propellers (marine type) and pitched-blade turbines / hydrofoils: axial-flow — they pump liquid parallel to the shaft, giving strong top-to-bottom circulation at low shear. Used for blending and solids suspension in low-viscosity liquids.
  • Flat-blade and disc (Rushton) turbines: radial-flow — they discharge liquid outward toward the wall, with high shear and turbulence. Used for gas dispersion and liquid–liquid dispersion.
  • Paddles: simple, slow, for moderate viscosity.
  • Anchors and helical ribbons: close-clearance, slow impellers for very viscous liquids and pastes, also scraping the wall to improve heat transfer.

Dimensional analysis of power. Power P drawn by an impeller depends on impeller diameter D, speed N, liquid density ρ and viscosity μ, gravity g and the vessel geometry. For geometrically similar systems this reduces to N_P = f(Re, Fr), where Re = ρND²/μ is the impeller Reynolds number and Fr = N²D/g the Froude number. Fr matters only when a vortex forms (unbaffled tanks at high Re); in baffled tanks N_P depends on Re alone.

The power curve.

  • Laminar region, Re < about 10: N_P·Re = K_L, so P = K_L·μ·N²·D³, independent of density.
  • Fully turbulent region in baffled tanks, Re > about 10⁴: N_P = K_T is constant, so P = K_T·ρ·N³·D⁵, independent of viscosity.
  • Transition in between: read N_P from the power curve. K_L and K_T depend on impeller type and geometry; take them from the power-curve table in your data book (for example, a six-blade Rushton disc turbine in a standard baffled tank has K_T of about 5–6 and K_L of about 65; propellers and pitched-blade turbines have much lower K_T).

Scale-up. With geometric similarity, one further criterion must be chosen. Constant power per unit volume (P/V) suits mass transfer and dispersion; in turbulent flow it gives N₂ = N₁(D₁/D₂)^(2/3). Constant tip speed (πND) suits shear-sensitive processes and gives N₂ = N₁(D₁/D₂). Constant Re is rarely used because it demands impossibly low speeds in large tanks. No single rule keeps everything the same.

Formulas

N_P = P / (ρ·N³·D⁵) Re = ρ·N·D² / μ, Fr = N²·D / g Laminar (Re < ~10): P = K_L·μ·N²·D³ Turbulent, baffled (Re > ~10⁴): P = K_T·ρ·N³·D⁵ Scale-up, constant P/V (turbulent): N₂ = N₁·(D₁/D₂)^(2/3); constant tip speed: N₂ = N₁·(D₁/D₂)

  • P: power (W); ρ: liquid density (kg/m³); μ: viscosity (Pa·s); N: impeller speed (rev/s, never rad/s); D: impeller diameter (m); g = 9.81 m/s²; K_L, K_T: constants for the impeller (from tables).

Worked examples

Example 1 (standard): power for a turbine. A Rushton turbine of D = 0.6 m runs at 2 rev/s in a baffled tank of water (ρ = 1000 kg/m³, μ = 10⁻³ Pa·s). Take K_T = 5.0 (from tables). Then repeat for a liquid with μ = 100 Pa·s and ρ = 1200 kg/m³ at 1.5 rev/s, K_L = 65.

  1. Water: Re = 1000 × 2 × 0.6²/10⁻³ = 7.2 × 10⁵ → turbulent.
  2. P = K_T·ρ·N³·D⁵ = 5.0 × 1000 × 8 × 0.07776 = 3110 W.
  3. Viscous liquid: Re = 1200 × 1.5 × 0.36/100 = 6.5 → laminar.
  4. P = K_L·μ·N²·D³ = 65 × 100 × 2.25 × 0.216 = 3160 W. Similar powers, but for completely different reasons: density controls the first, viscosity the second.

Example 2 (GATE level): scale-up at constant P/V. A pilot tank uses a 0.3 m turbine at 3 rev/s in water (N_P = 5.0, turbulent). The plant tank is geometrically similar and 5 times larger in linear scale. Find the plant speed and power for constant P/V.

  1. Pilot power: P₁ = 5.0 × 1000 × 3³ × 0.3⁵ = 5.0 × 1000 × 27 × 0.00243 = 328 W. (Re₁ = 2.7 × 10⁵, turbulent.)
  2. N₂ = N₁(D₁/D₂)^(2/3) = 3 × (1/5)^(2/3) = 3 × 0.342 = 1.03 rev/s.
  3. P₂ = 5.0 × 1000 × 1.026³ × 1.5⁵ = 5.0 × 1000 × 1.080 × 7.594 = 41.0 kW.
  4. Check: V₂/V₁ = 125 and P₂/P₁ = 41 006/328 = 125, so P/V is indeed constant. Tip speed rises from 2.83 to 4.83 m/s.

Common mistakes

  • Using N in rad/s or rpm in N_P and Re. N must be in revolutions per second.
  • Applying the turbulent constant K_T to a viscous liquid in the laminar region.
  • Forgetting D⁵: doubling impeller diameter at the same speed raises power 32 times.
  • Expecting an unbaffled tank to follow the baffled power curve at high Re; vortexing brings in the Froude number.
  • Assuming all quantities can be kept constant in scale-up.

For GATE CH

Expect: power from N_P in the turbulent or laminar region; impeller Re to choose the region; effect of changing speed or diameter on power; scale-up by constant P/V or tip speed; impeller selection (axial vs radial, anchors for viscous fluids) and the role of baffles. Practise unit discipline with N in rev/s.

Quick check

  1. In fully turbulent baffled flow, how does power change if speed doubles?
  2. In the laminar region, does power depend on liquid density?
  3. Which impeller type is preferred for gas dispersion?
  4. What is the purpose of baffles? Answers: 1. It increases 8 times. 2. No; P = K_L·μ·N²·D³. 3. A radial-flow (Rushton disc) turbine. 4. To break the swirl and prevent vortex formation, converting rotation into useful circulation.

Interview questions

All Mechanical Operations interview questions

Try answering each one aloud before you open it.

  1. 1.What is agitation in the context of chemical engineering?Concept

    Agitation refers to the process of stirring or mixing fluids to enhance mass and heat transfer. It is commonly used in chemical engineering to ensure uniformity in chemical reactions, prevent sedimentation, and improve the contact between reactants.

  2. 2.Explain the role of an impeller in a mixing process.Concept

    An impeller is a rotating component of a mixing system that transfers energy to the fluid, causing it to move and mix. It is designed to create flow patterns that enhance mixing efficiency, and its design can vary depending on the type of mixing required, such as axial or radial flow.

  3. 3.What is the power number in agitation and mixing, and why is it important?Concept

    The power number is N_P = P/(ρN³D⁵), the impeller power divided by density times the cube of speed (rev/s) times the fifth power of impeller diameter; it is a dimensionless drag coefficient for the impeller. For geometrically similar baffled tanks it depends only on the impeller Reynolds number ρND²/μ: N_P·Re is constant in laminar flow and N_P itself is constant (about 5–6 for a Rushton turbine) in fully turbulent flow. That single correlation lets you size the motor for any speed and scale, and compare impeller types.

  4. 4.Why are axial flow impellers preferred in certain mixing applications?Application

    Axial flow impellers are preferred in applications where top-to-bottom circulation is needed, such as in blending and solid suspension. They create a flow pattern that moves fluid along the axis of the impeller, which is effective for mixing large volumes and ensuring uniform distribution of components.

  5. 5.What happens if the impeller speed is increased significantly in a mixing tank?Application

    Increasing the impeller speed increases the shear rate and turbulence in the tank, which can enhance mixing efficiency. However, it can also lead to higher power consumption, potential damage to sensitive materials, and increased wear on the equipment. It is important to balance speed with the desired mixing outcome.

  6. 6.How does the viscosity of a fluid affect the choice of impeller type?Application

    For low-viscosity fluids, high-speed impellers like turbines are often used to create sufficient turbulence. For high-viscosity fluids, low-speed impellers like paddles or helical ribbons are preferred to ensure effective mixing without excessive energy consumption. The impeller type must match the fluid's properties to achieve efficient mixing.

  7. 7.Calculate the power number if the power input is 500 W, the impeller diameter 0.5 m, the fluid density 1000 kg/m³ and the rotational speed 10 rad/s.Numerical

    N in the power number must be in revolutions per second: N = 10/(2π) = 1.592 rev/s. Then N_P = P/(ρN³D⁵) = 500/(1000 × 1.592³ × 0.5⁵) = 500/(1000 × 4.031 × 0.03125) ≈ 3.97. Substituting 10 rad/s directly would give 0.016, which is wrong by a factor of (2π)³ ≈ 248.

  8. 8.What is the effect of baffles in a mixing tank?Application

    Baffles are vertical strips attached to the inner wall of a mixing tank. They prevent the formation of a vortex and improve mixing by creating additional turbulence. This results in more efficient mixing and better distribution of materials throughout the tank.

  9. 9.Explain why radial flow impellers are used in gas-liquid mixing applications.Application

    Radial flow impellers are used in gas-liquid mixing because they create high shear and turbulence, which enhances gas dispersion into the liquid. This is crucial for processes like aeration, where maximizing the gas-liquid contact area is important for efficient mass transfer.

  10. 10.Determine the impeller speed required to draw 200 W if the power number is 0.02, the impeller diameter is 0.4 m and the fluid density is 950 kg/m³.Numerical

    From P = N_P·ρ·N³·D⁵, N = [P/(N_P·ρ·D⁵)]^(1/3) = [200/(0.02 × 950 × 0.01024)]^(1/3) = (1028)^(1/3) ≈ 10.1 rev/s, about 606 rpm. The result is in revolutions per second because the power number is defined with N in rev/s. (A power number of 0.02 is unusually low for a real impeller, so in practice you would check it against the impeller's power curve.)

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