Water flows through a pipe of 200 mm in diameter 60 m long with a velocity of 2.5 m/s. What is the head loss due to friction using Darcy's formulae? (Assume f = 0.005)
- A1.52 m
- B1.79 m
- C1.85 m
- D1.91 mCorrect
Solution
Concept
Head loss due to friction in a pipe by the Darcy formula, where f is the Fanning-type coefficient (hf = 4fLV²/2gD).
Formula
hf = 4 f L V2 / (2 g D)
Step-by-step solution
1. Data: D = 0.2 m, L = 60 m, V = 2.5 m/s, f = 0.005, g = 9.81 m/s². 2. V2 = 6.25 m²/s². 3. Numerator = 4 × 0.005 × 60 × 6.25 = 7.5 4. Denominator = 2 × 9.81 × 0.2 = 3.924 5. hf = 7.5 / 3.924 = 1.911 m ≈ 1.91 m
Explanation
In Indian textbooks, f = 0.005 is a small coefficient, so Darcy's formula is used as hf = 4fLV²/(2gD). Putting in the values gives a head loss of about 1.91 m.
Why the other options are wrong
A, B and C come from calculation or rounding errors. The correct value is 1.911 m.
Shortcut
4f = 0.02, so hf = 0.02 × (60/0.2) × (6.25/19.62) = 0.02 × 300 × 0.3186 ≈ 1.91 m
Common mistake
Using hf = fLV²/(2gD) without the 4, which gives 0.478 m (not among the options).
Correct answer: D) 1.91 m