Turbines — Definition of Unit Speed (Speed at Unit Head) In turbine performance scaling, the unit speed of a runner is defined as the rotational speed the turbine would have if it operated under a head of 1 m. Select the correct expression for unit speed in terms of actual speed N (rpm) and operating head H (m).

Introduction:
Unit quantities help compare turbines independent of head by normalizing performance to a head of 1 m. Unit speed indicates how fast the runner would turn at unit head, enabling fair comparison across different sites and tests.

Given Data / Assumptions:

• Actual rotational speed = N (rpm).
• Operating head = H (m).
• Geometric similarity and dynamically similar operation are implied when comparing across heads.

Concept / Approach:
Under dynamically similar operation, characteristic velocities scale with sqrt(H). Therefore rotational speed, which is proportional to characteristic velocity divided by a length scale that is held fixed for a given machine, also scales with sqrt(H). To reduce from head H to unit head, divide by sqrt(H).

Step-by-Step Solution:
Velocity scale: V ∝ sqrt(g * H).Speed scale: N ∝ V / D with fixed D ⇒ N ∝ sqrt(H).Unit speed is the speed at H = 1 m: N_u = N / sqrt(H).

Verification / Alternative check:
The related unit relations are N_u = N / sqrt(H), Q_u = Q / sqrt(H), and P_u = P / H^(3/2). All consistently remove the head dependence for similar operation.

Why Other Options Are Wrong:

• N * sqrt(H), N * H: These increase with head; unit speed must be smaller at higher H.
• N / H, N / H^2: Over-correct; rotational speed does not scale linearly or quadratically with head.

Common Pitfalls:
Mixing unit speed with specific speed, or using linear head scaling instead of the square-root law.

Final Answer:
N_u = N / sqrt(H)