Head–power relationship for turbines If H is the net head on a turbine of fixed geometry operating at its best efficiency point, the developed power varies how with H?

Introduction / Context:
Turbine similarity laws link discharge, runner speed, and power to the operating head. For a given turbine design running near its best efficiency point, these relations provide quick scaling rules useful for preliminary sizing and performance prediction.

Given Data / Assumptions:

• Turbine geometry fixed; operating near best efficiency.
• Fluid is water, density approximately constant.
• Losses scale comparably so similarity holds.

Concept / Approach:
From similarity: specific speed and unit quantities give Q ∝ H^(1/2) and power P ∝ ρ g Q H (times efficiency). With efficiency roughly constant at BEP, this yields P ∝ H * H^(1/2) = H^(3/2).

Step-by-Step Solution:
Discharge scaling: Q ∝ H^(1/2).Hydraulic power: P_h = ρ * g * Q * H.Assume η ≈ constant near BEP; then P ≈ η * P_h ∝ Q * H.Hence P ∝ H^(1/2) * H = H^(3/2).

Verification / Alternative check:
Unit power P_u = P / H^(3/2) is approximately constant for dynamically similar operation, commonly used to compare turbines of different sizes.

Why Other Options Are Wrong:
H^(1/2) scaling applies to discharge or speed, not power. Inverse relations contradict energy scaling. Independence from H is physically untenable for hydraulic machines.

Common Pitfalls:
Mixing up unit speed/discharge with power; assuming efficiency changes negate the fundamental scaling (they modify constants but not the basic exponent near BEP).

Final Answer:
Directly proportional to H^(3/2)