Power transmission with an end nozzle: for a pipe of diameter D, length L, and Darcy–Weisbach friction factor f, the optimum nozzle diameter d (for maximum power at the jet) is:

Introduction / Context:
When a long pipeline ends in a nozzle driving a jet (e.g., impulse turbines, firefighting), the nozzle size controls both discharge and head lost to pipe friction. There exists an optimum nozzle diameter that maximizes power delivered by the jet.

Given Data / Assumptions:

• Total available head at pipe entrance is H (symbolic).
• Pipe: length L, diameter D, friction factor f (Darcy–Weisbach).
• Nozzle of diameter d discharging to atmosphere; single, smooth, frictionless nozzle.

Concept / Approach:

Head loss in pipe: h_f = 4 * f * (L / D) * V^2 / (2 * g), where V is mean pipe velocity. Jet velocity v_j = sqrt( 2 * g * (H − h_f) ). Continuity gives V = v_j * (d^2 / D^2). Jet power P = (rho / 2) * A_n * v_j^3. Maximizing P with respect to d yields the classical condition (4 * f * L / D) * (d^4 / D^4) = 1/2, which rearranges to the formula below.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Dimensional check confirms d has dimension of length; the ratio inside the fourth-root is dimensionless.

Why Other Options Are Wrong:

Options (b)-(e) contradict the fourth-root dependence or invert the friction/length effect. Larger f or L should reduce optimum d, which (a) captures.

Common Pitfalls (misconceptions, mistakes):

Using the Hazen–Williams form directly in optimization; forgetting that nozzle losses are neglected in this ideal result.

Final Answer:

d = D * ( D / (8 * f * L) )^(1/4)