Nozzle size for maximum power transmission through a pipe A nozzle is fitted to the end of a straight pipe of length 320 m and diameter 0.10 m. If the Darcy friction factor f = 0.01 for the pipe, determine the nozzle diameter that maximizes the power transmitted by the jet.

Introduction / Context:
When a nozzle is attached to a long pipe, enlarging the nozzle increases jet power (more discharge) but also increases pipe velocity and friction loss, reducing available head at the nozzle. There exists an optimal nozzle size that maximizes the jet power.

Given Data / Assumptions:

• Pipe length L = 320 m, pipe diameter D = 0.10 m.
• Darcy friction factor f = 0.01 (Darcy–Weisbach).
• Steady, incompressible flow; negligible minor losses; jet coefficient ≈ 1.

Concept / Approach:
For a pipe with end nozzle, the optimal area ratio a/A that maximizes jet power satisfies a classical condition obtained by differentiating power with respect to nozzle area, leading to the relation (8 f L / D) * (a/A)^2 = 1, hence d = D / ( (8 f L / D)^(1/4) ).

Step-by-Step Solution:

Verification / Alternative check:
The same result is consistent with the classical criterion that maximum power occurs when head lost due to pipe friction equals one-third of the static head at the pipe entrance for the optimal area ratio.

Why Other Options Are Wrong:
2.4, 2.6, and 2.7 cm correspond to area ratios that depart from the optimal a/A, yielding less jet power.

Common Pitfalls:
Confusing Darcy and Fanning friction factors (off by a factor of 4) or including minor losses without stating them, which would shift the optimum slightly.

Final Answer:
2.5 cm