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.
Mechanical Engineering
Hydraulics and Fluid Mechanics
Difficulty: Medium
Choose an option
-
Ad / D = ( D / (8 f l) )<sup>1/4</sup>
-
Bd / D = ( D / (4 f l) )<sup>1/4</sup>
-
Cd / D = ( 8 f l / D )<sup>1/4</sup>
-
Dd = ( D<sup>3</sup> / (8 f l) )<sup>1/4</sup>
Answer
Correct Answer: d / D = ( D / (8 f l) )<sup>1/4</sup>
Explanation
Given data
- Pipe diameter D, length l; nozzle diameter d; Darcy friction factor f.
- Total head available = H (constant).
Concept/Approach
- Head relation: H = hf + Vj2/(2g) with hf = (4 f l / D) × v2/(2g) and continuity v = (a/A) × Vj = (d2/D2) × Vj.
- Thus H = [1 + (4 f l / D) × (d4/D4)] × Vj2/(2g).
- Power delivered by jet: P ∝ a Vj3 ∝ d2[1 + k d4]-3/2 with k = 4 f l / D5.
Step-by-step optimizationDifferentiate 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 Answerd / D = ( D / (8 f l) )1/4