Nozzle optimization at end of pipeline: A pipe of diameter D and length L feeds a nozzle of diameter d. Neglecting minor losses, what d maximizes the power carried by the jet (friction factor f by Darcy–Weisbach)?

Introduction / Context:

Nozzles at the end of pipelines convert pressure head into kinetic energy. For a given supply head H and line length L with friction factor f, there exists an optimal nozzle size that maximizes jet power delivered to a turbine or impact plate downstream.

Given Data / Assumptions:

• Pipe diameter D, length L; nozzle diameter d at the end.
• Single straight run; Darcy–Weisbach friction with constant f in the pipe (minor losses neglected).
• Steady incompressible flow; nozzle discharges to atmosphere at the same elevation.

Concept / Approach:

Energy: H = hf + v^2/(2g), where v is jet velocity, hf = f * (L/D) * V^2/(2g), and V is pipe velocity. Continuity gives V = v * (d^2 / D^2). Jet power P = ρ g Q * v^2/(2g) = 0.5 * ρ * A_n * v^3 (A_n = π d^2/4). Maximizing P with respect to d (or y = d^2) yields the optimum.

Step-by-Step Solution:

Verification / Alternative check:

The equivalent form d/D = ( D / (2 f L) )^(1/4) follows by dividing both sides by D^4. Both representations are standard results.

Why Other Options Are Wrong:

• Forms with inverted ratios (2 f L / D)^(1/4) or d^4 = 2 f L / D^5 reverse the dependence and contradict optimization.
• Exponents like 1/5 lack derivational basis.

Common Pitfalls:

• Maximizing discharge instead of jet power; these are not identical under frictional losses.

Final Answer:

d^4 = D^5 / (2 f L)