Power Requirements for Rushton Turbine Impellers in Bioreactors

Rushton turbine impellers are widely used in bioreactors for mixing
Newtonian fluids. In the turbulent regime
under unaerated conditions, the power required scales with
the cube of the stirring speed.
The correct answer is option (C).

Correct Answer

Option (C): Proportional to the cube of the stirring speed of the impeller

Power Relationship in Turbulent Flow

The power consumption for an impeller is given by:

P = N_p ρ N³ D⁵

where:

• P = Power consumed
• N_p = Power number
• ρ = Fluid density
• N = Stirring speed
• D = Impeller diameter

In turbulent flow (Re > 10,000), the power number
N_p becomes constant. For fixed geometry and fluid
properties, this simplifies to:

P ∝ N³

This relationship holds for Rushton turbines operating in
unaerated Newtonian fluids.

Option Analysis

• (A) Proportional to stirring speed:
  Applies to the laminar regime (low Reynolds number), where
  N_p ∝ 1/Re and power does not vary linearly with N.
• (B) Proportional to square of stirring speed:
  Represents transitional flow behavior, not fully turbulent conditions
  where inertial forces dominate.
• (C) Proportional to cube of stirring speed:
  Correct. In turbulent flow, constant N_p leads to
  P ∝ N³.
• (D) Inversely proportional to stirring speed:
  Incorrect. Power consumption always increases with stirring speed in
  mechanical agitation.

Key Implications for Bioreactor Design

Rushton turbines require relatively high power due to their six flat blades,
which generate strong radial flow and intense turbulence.

During scale-up with geometric similarity, power per unit volume follows:

P_g = P/V ∝ N³ / D²

Because power increases cubically with speed, even small increases in
agitation can lead to large energy demands—making this relationship critical
in bioreactor design and operation.