Turbine Efficiencies — Relationship What is the correct relationship among hydraulic efficiency (eta_h), mechanical efficiency (eta_m), and overall efficiency (eta_o) of a turbine?

Introduction:
Overall efficiency measures how effectively a turbine converts water power at inlet to shaft power at the coupling. It combines hydraulic conversion inside the runner and mechanical transmission to the shaft.

Given Data / Assumptions:

• Definitions: eta_h = runner output / water power at inlet, eta_m = shaft output / runner output, eta_o = shaft output / water power at inlet.
• Steady operation; negligible measurement errors.

Concept / Approach:
Overall efficiency is the product of individual stage efficiencies along the energy path from fluid to shaft. Multiplying eta_h and eta_m cancels the intermediate runner output term, leaving the ratio of final to initial power.

Step-by-Step Solution:
eta_h = P_runner / P_water.eta_m = P_shaft / P_runner.eta_o = (P_shaft / P_runner) * (P_runner / P_water) = eta_m * eta_h.

Verification / Alternative check:
Energy balance across turbine confirms product form when losses in the runner and mechanical train are independent.

Why Other Options Are Wrong:
eta_h = eta_o * eta_m or eta_m = eta_o * eta_h: algebraically incorrect rearrangements.none of these: incorrect because the product relation is standard.

Common Pitfalls:
Confusing volumetric or generator efficiencies with hydraulic and mechanical efficiencies; those would extend the product further.

Final Answer:
eta_o = eta_h * eta_m