Centrifugal Pump — Specific Speed (Ns) Which formula correctly gives the specific speed of a centrifugal pump in terms of rotational speed N (rpm), discharge Q, and head H (m)?

Introduction:
Specific speed groups pumps of different sizes by hydraulic similarity. It indicates the impeller style best suited to a duty point, varying from radial (low Ns) to axial (high Ns) designs.

Given Data / Assumptions:

• N in rpm, Q in m^3/s (or consistent units), H in m.
• Geometric similarity and similar efficiency near the best efficiency point.

Concept / Approach:
For pumps, specific speed is based on discharge, not power. From affinity/similarity, the widely used empirical form is Ns = N * Q^0.5 / H^0.75 (up to a constant depending on units). This correlates duty with impeller type: low Ns for radial, medium for mixed flow, high for axial-flow pumps.

Step-by-Step Solution:
Start with model law: Q ∝ N * D^3 and H ∝ N^2 * D^2.Eliminate D to obtain dimensionless grouping linking N, Q, H.Arrive at Ns ∝ N * Q^0.5 / H^0.75 (rpm-based form).

Verification / Alternative check:
Catalog charts mapping Ns to impeller geometry confirm the correlation trends.

Why Other Options Are Wrong:
N * Q / H: wrong exponents; not similarity-consistent.N * sqrt(P) / H^(5/4): that is a turbine-specific-speed form using power.N / H^(3/2): omits discharge term entirely.

Common Pitfalls:
Confusing pump Ns (uses Q) with turbine Ns (uses P).

Final Answer:
Ns = N * sqrt(Q) / H^(3/4)