In mixing, how does impeller diameter affect power draw? (Assume fixed speed, turbulent regime, and constant fluid properties.)

Introduction / Context:
Agitator power scaling is fundamental in mixer selection and scale-up. For geometrically similar systems under turbulent flow at fixed rotational speed, the dependence of power on impeller diameter is strong and guides both energy estimates and motor sizing.

Given Data / Assumptions:

• Newtonian liquid, fully turbulent regime.
• Geometric similarity (constant impeller type, tank/impeller ratios).
• Fixed rotational speed N and constant density rho.

Concept / Approach:
The power number correlation gives P = Np * rho * N^3 * D^5 for turbulent mixing (where Np ≈ constant for a given impeller in fully turbulent flow). With speed fixed, P scales as D^5, showing why seemingly small diameter increases drastically raise power requirements.

Step-by-Step Solution:
Start from P ∝ rho * N^3 * D^5 (Np constant).Hold rho and N constant → P ∝ D^5.Compare with options: D^5 is the correct dependence.

Verification / Alternative check:
Pilot data commonly collapse onto P ∝ N^3 D^5 in turbulent regions; deviations occur in laminar/transition regimes where P ∝ mu * N^2 * D^3.

Why Other Options Are Wrong:
D, D^2: underpredict power; applicable to no standard turbulent scaling.D^9: vastly overpredicts; not supported by dimensional analysis.

Common Pitfalls:
Scaling diameter without adjusting speed or checking regime; assuming Np is constant outside fully turbulent conditions.

Final Answer:
Proportional to D^5