Power Transmission Through a Pipe — Efficiency Expression If the total supply head is H and the head lost due to friction is h_f, what is the efficiency of power transmission through the pipe?

Introduction / Context:
When conveying water through long pipelines, frictional losses reduce the available head and thus the useful power at the delivery end. Efficiency quantifies how much of the supply head remains to do useful work (for example, to drive a turbine or supply pressure at the outlet).

Given Data / Assumptions:

• Total head at inlet: H (measured above the delivery datum).
• Head loss due to pipe friction: h_f.
• Neglect minor losses for this expression or include them within h_f.

Concept / Approach:
Power in a steady incompressible flow is proportional to ρ g Q times the available head. Since discharge Q is the same throughout a single pipe, the ratio of output power to input power reduces to the ratio of available heads: (H − h_f) / H.

Step-by-Step Derivation:

Verification / Alternative check:
If h_f = 0 (ideal frictionless pipe), η = 1; if h_f approaches H, η → 0, both consistent with physical expectations.

Why Other Options Are Wrong:
h_f / H measures loss fraction, not efficiency; squaring terms has no basis; square-root forms are unrelated to power proportionality.

Common Pitfalls:
Confusing head efficiency with mechanical efficiency of pumps; forgetting to include all losses in h_f when comparing to field data.

Final Answer:
η = (H − h_f) / H.