Maximum efficiency of power transmission through a pipeline If the total head available at the pipe inlet is H and the head lost due to friction is h_f, what is the maximum possible efficiency of power transmission through the pipe?

Introduction / Context:
When transmitting power by water through a long pipe to a turbine/nozzle, frictional losses reduce the head at the outlet. The efficiency depends on how much head is lost en route versus delivered for useful work.

Given Data / Assumptions:

• Total head at inlet = H.
• Head loss due to friction along pipe = h_f.
• No other losses or shaft work until the outlet.

Concept / Approach:

Hydraulic power delivered is proportional to the product of discharge and available head at outlet. Optimization (with velocity related to head) gives a specific ratio of loss to available head at maximum efficiency: h_f = H/3, yielding η_max = 2/3.

Step-by-Step Solution:

Verification / Alternative check:

Classical derivation in fluid mechanics texts confirms η_max = 66.7% for power transmission by a pipeline without other losses.

Why Other Options Are Wrong:

Other percentages do not correspond to the optimal friction–head trade-off; they arise if h_f differs from H/3 and are suboptimal.

Common Pitfalls:

Maximizing outlet head alone (min V) or flow alone (max V) does not maximize power; the product must be optimized.

Final Answer:

2/3 (66.7%)