Centrifugal Pumps — Affinity Laws The power required to drive a centrifugal pump varies how with rotational speed N (other factors geometrically similar)?

Introduction:
Pump affinity laws predict how head, flow, and power scale with speed for a given impeller and fluid. These relations are essential for estimating motor requirements and evaluating speed-control strategies.

Given Data / Assumptions:

• Geometrically similar operation; same impeller and casing geometry.
• Same fluid, negligible change in efficiency over the considered range.
• Rotational speed N is varied.

Concept / Approach:
The affinity laws state: Q ∝ N, H ∝ N^2, and P ∝ N^3 for similar conditions. Intuitively, power combines flow (N) and head (N^2), giving N^3 overall. Thus, modest changes in speed can have large power implications.

Step-by-Step Solution:
Flow scaling: Q2 / Q1 = N2 / N1.Head scaling: H2 / H1 = (N2 / N1)^2.Power scaling: P2 / P1 = (Q2 / Q1) * (H2 / H1) ≈ (N2 / N1) * (N2 / N1)^2 = (N2 / N1)^3.

Verification / Alternative check:
Manufacturer performance curves shifted by speed align when scaled by the affinity laws.

Why Other Options Are Wrong:
inversely proportional to N^3: physically unreasonable; higher speed increases power draw.proportional to N or N^2: underestimates the cubic effect.

Common Pitfalls:
Ignoring efficiency changes at off-design, which can slightly alter the exact scaling but not the cubic trend.

Final Answer:
directly proportional to N^3