Power transmission through a pipe with nozzle: nozzle diameter for maximum power A pipe of diameter D and length l carries water to a nozzle of diameter d at its end. The Darcy friction factor for the pipe is f (Darcy–Weisbach form). Find the nozzle diameter that maximizes the power transmitted.

Given data

• Pipe diameter D, length l; nozzle diameter d; Darcy friction factor f.
• Total head available = H (constant).

Concept/Approach

• Head relation: H = h_f + V_j²/(2g) with h_f = (4 f l / D) × v²/(2g) and continuity v = (a/A) × V_j = (d²/D²) × V_j.
• Thus H = [1 + (4 f l / D) × (d⁴/D⁴)] × V_j²/(2g).
• Power delivered by jet: P ∝ a V_j³ ∝ d²[1 + k d⁴]^-3/2 with k = 4 f l / D⁵.

Step-by-step optimization
Differentiate ln P: 2/d − (3/2) × (4k d³)/(1 + k d⁴) = 0→ 2(1 + k d⁴) = 6k d⁴ → d⁴ = 1/(2k) = D⁵/(8 f l)→ (d/D)⁴ = D/(8 f l) ⇒ d/D = ( D / (8 f l) )^1/4.

Final Answer
d / D = ( D / (8 f l) )^1/4