Centrifugal Pump Affinity Laws — Head Scaling with Speed and Impeller Diameter For dynamically similar operation of a centrifugal pump, how does the developed head H scale with impeller speed N and impeller diameter D?

Introduction:
Selection and scaling of centrifugal pumps use the affinity laws relating flow rate, head, and power to speed and impeller size. Knowing the correct proportionality prevents mis-sizing during variable-speed operation or trim changes.

Given Data / Assumptions:

• Dynamically similar operation (same efficiency region and flow coefficient).
• Impeller diameter = D, rotational speed = N.

Concept / Approach:
The pump affinity laws state: Q ∝ N * D^3H ∝ N^2 * D^2P ∝ N^3 * D^5 Thus head varies with the square of both speed and diameter, not linearly with either one or their simple product.

Step-by-Step Solution:
Compare each option to H ∝ N^2 * D^2.Option A (∝ D) and B (∝ N) omit the square dependence.Option C (∝ N * D) is still linear overall and incorrect.Therefore, none of the listed linear relations match; choose “none of the above”.

Verification / Alternative check:
Dimensionless analysis using head coefficient psi = g * H / (N^2 * D^2) confirms psi remains approximately constant under similarity, leading to H ∝ N^2 D^2.

Why Other Options Are Wrong:

• Linear statements underestimate/overestimate head change when speed or diameter changes.

Common Pitfalls:
Confusing linear speed scaling with the quadratic dependence inherent to head.

Final Answer:
none of the above