Darcy–Weisbach friction loss in a circular pipe: Select the correct head loss expression (f = Darcy's friction coefficient, l = pipe length, d = diameter, v = mean velocity).

Introduction / Context:
The Darcy–Weisbach equation gives a unified way to compute head loss due to wall friction in pipes, valid across laminar and turbulent regimes when the appropriate friction factor is known. Many textbooks define f as Darcy’s coefficient and include a factor of 4 compared with the Fanning factor. Choosing the correct constant and velocity power is essential.

Given Data / Assumptions:

• Steady, incompressible, fully developed internal flow in a straight circular pipe.
• f is the Darcy (not Fanning) friction factor.
• v is mean velocity; elevation changes are negligible for the segment considered.

Concept / Approach:
The Darcy–Weisbach head loss is h_f = f_D * (l/d) * (v^2/(2g)) when f_D denotes the Darcy factor. Some Indian texts denote “coefficient of friction f” such that the formula appears as h_f = 4 f (l/d) (v^2/(2g)). Here the options map to that convention, making option A correct.

Step-by-Step Solution:

Verification / Alternative check:
Using Moody chart values (Darcy factor), numerical results align with the A-form; converting to Fanning requires dividing by 4.

Why Other Options Are Wrong:

• B: Missing the factor 4 under this notation.
• C: Uses v to the first power, which is dimensionally inconsistent for energy loss.
• D: Same as A but doubled overall—incorrect constant.

Common Pitfalls:
Mixing Darcy and Fanning definitions; using centerline velocity instead of mean; ignoring minor losses (separate from friction loss).

Final Answer:
h_f = (4 * f * l * v^2) / (2 * g * d)