Impeller scaling at constant power input (turbulent regime): If impeller diameter Di is increased by 20%, by what factor must the speed N be decreased to keep power constant?

Introduction:
In fully turbulent mixing with constant power number (Np), the impeller power draw scales as P ∝ ρ * N^3 * Di^5. Designers frequently adjust diameter and speed to meet power constraints or to retrofit vessels. This problem asks how to adjust speed when diameter increases by 20% while keeping P constant.

Given Data / Assumptions:

• Fully turbulent regime; Np is constant.
• Fluid density ρ constant; no aeration effects.
• Initial power P held constant after diameter change.

Concept / Approach:
With P ∝ N^3 * Di^5, holding P constant implies N^3 * Di^5 = constant. If Di,new = 1.2 * Di,old, then N_new^3 * (1.2^5) * Di,old^5 = N_old^3 * Di,old^5. Canceling common terms gives N_new = N_old / (1.2)^(5/3). Thus, speed must be decreased by a factor (1.2)^(5/3).

Step-by-Step Solution:
Start with P ∝ N^3 * Di^5.Set P_new = P_old ⇒ N_new^3 * Di_new^5 = N_old^3 * Di_old^5.Substitute Di_new = 1.2 * Di_old.Solve: N_new = N_old / (1.2)^(5/3).

Verification / Alternative check:
Dimensional consistency and known scale-up rules (constant tip speed or constant power per volume) align with this result; here it is constant power with constant Np.

Why Other Options Are Wrong:

• (1.2)^(2/3) or (3/5) mismatch the required exponent 5/3 from P-scaling.
• (1.2)^(3/2) or (1.2)^(1/2) are unrelated to the N^3 * Di^5 relationship.

Common Pitfalls:
Using constant tip-speed (N * Di) instead of constant power scaling; they lead to different exponents and design outcomes.

Final Answer:
(1.2)^(5/3)