Power Transmission Through a Pipeline Condition for Maximum Power: The power delivered at the outlet is maximized when the head lost due to friction equals one-third of the total supply head.

Introduction:
This item examines the condition for maximizing hydraulic power transmitted through a pipeline when head losses occur due to friction. It is a classic optimization result in fluid power systems.

Given Data / Assumptions:

• Total supply head H at the pipeline inlet.
• Head loss due to friction h_f; remaining head at outlet H_d = H - h_f.
• Discharge Q depends on head loss via Darcy–Weisbach or equivalent; proportional reasoning suffices.

Concept / Approach:

Power transmitted P is proportional to Q * H_d. For a given pipeline, Q increases with head loss, but H_d decreases as h_f increases. Maximizing P with respect to h_f yields the optimal split.

Step-by-Step Solution:

Verification / Alternative check:

Substituting h_f = H/3 gives delivery head H_d = 2H/3 and confirms that small deviations reduce P on either side of this optimum.

Why Other Options Are Wrong:

One-fourth, one-half, two-third: These do not satisfy the first-order optimality condition and produce lower transmitted power.

Common Pitfalls:

Assuming maximum power at minimum losses; forgetting that Q depends on h_f; confusing efficiency optimization with power maximization.

Final Answer:

one-third of the total supply head