For fully turbulent agitation in a geometrically similar stirred tank, how is the impeller power number (Np) correlated with the impeller Reynolds number (Rei)?

Introduction / Context:
The power number Np characterizes energy draw by an impeller and is central to scale-up. Its dependence on Reynolds number Rei changes across flow regimes. Recognizing the turbulent asymptote simplifies power estimation at industrial scales where Rei is large.

Given Data / Assumptions:

• Newtonian fluid, geometric similarity, standard baffling.
• Turbulent regime with Rei ≫ 10^4.
• Same impeller type across scales.

Concept / Approach:
At low Rei (laminar), Np ∝ 1/Rei. In transition, Np varies with Rei. In fully turbulent flow, viscous effects are negligible and Np approaches a constant that depends on impeller and tank geometry. Thus P = Np * ρ * N^3 * Di^5 with Np ≈ constant is used for design and scale-up.

Step-by-Step Solution:
Identify regime: Rei > 10^4 ⇒ fully turbulent for typical impellers.Apply asymptote: Np → constant K (geometry-dependent).Use power law: P = Np * ρ * N^3 * Di^5 with Np ≈ K.Therefore, Np does not depend on Rei in this regime.

Verification / Alternative check:
Manufacturer curves show flat Np vs Rei at high Rei for Rushton or pitched-blade impellers, validating constancy.

Why Other Options Are Wrong:
Np = K * (Rei)^-1 applies to laminar, not turbulent, conditions.Including N^2 * Di with Rei^-1 mixes variables redundantly; not a standard correlation.“Cannot be correlated” is incorrect; the constant-Np correlation is standard.

Common Pitfalls:
Applying laminar correlations to turbulent scale-up; ignoring the need for geometric similarity and consistent baffling when using constant Np.

Final Answer:
Np = K (approximately constant, geometry-dependent)