Stirred tank power scaling at high Reynolds number: if the power number is constant, how does power input vary with impeller speed N?

Introduction / Context:
In turbulent mixing (high Reynolds number), the power number Np becomes approximately constant for a given impeller–tank geometry. Power scaling with speed is fundamental for mixer design and scale-up.

Given Data / Assumptions:

• Np = P / (rho * N^3 * D^5) is constant at high Re.
• Fluid density rho and impeller diameter D are fixed.
• We vary only the rotational speed N.

Concept / Approach:
Rearranging the definition of Np: P = Np * rho * N^3 * D^5. With Np, rho, D fixed, power P scales as N^3. This cubic dependence underpins motor sizing and energy-cost estimates during process optimization.

Step-by-Step Solution:
Start with Np = P / (rho * N^3 * D^5).Solve for P: P = Np * rho * D^5 * N^3.Hence, P ∝ N^3.

Verification / Alternative check:
Log–log plots of power versus speed show slope near 3 for fully turbulent regimes with constant Np.

Why Other Options Are Wrong:
N^0, N^1, N^2: underestimate the steep increase of power with speed in turbulent mixing.

Common Pitfalls:
Applying viscous regime scaling (P ∝ N^2 or N^1) outside of their valid Reynolds number ranges.

Final Answer:
N^3