Turbine Specific Speed — Definition Which expression correctly defines the specific speed (Ns) of a turbine in terms of rotational speed N (rpm), power P, and head H (m)?

Introduction:
Specific speed is a dimensionless (or quasi-dimensional in rpm-based forms) index used to classify turbines and select appropriate types for a site. It reflects how speed and power scale with head for dynamically similar machines.

Given Data / Assumptions:

• Conventional metric definition with N in rpm, P in kW (or W with consistent constants), and H in meters.
• Geometric similarity and dynamic similarity assumptions.

Concept / Approach:
By definition, specific speed is the speed at which a geometrically similar turbine would run to develop unit power under unit head. Rearranging similarity relations yields Ns ∝ N * P^0.5 / H^1.25, the widely used selection formula.

Step-by-Step Solution:
Start from condition: H = 1 m and P = 1 unit for the similar machine.Apply similarity: N ∝ H^0.5 and P ∝ H^1.5 for turbines.Derive Ns = N * sqrt(P) / H^(5/4).

Verification / Alternative check:
Type ranges: Pelton low Ns, Francis medium, Kaplan high; computed Ns sorts sites appropriately.

Why Other Options Are Wrong:
N * P / H and N / H^(3/2): wrong exponents; do not match similarity scaling.N * P^(3/2) / H^(1/2): incorrect power exponent; not dimensionally consistent.

Common Pitfalls:
Confusing pump specific speed formulas with turbine formulas; although similar, constants and exponents differ.

Final Answer:
Ns = N * sqrt(P) / H^(5/4)