Laminar flow friction factor (Darcy) and Reynolds number In steady, fully developed laminar flow through a circular pipe, if Re denotes the Reynolds number based on pipe diameter, what is the Darcy–Weisbach friction factor f expressed in terms of Re?

Introduction / Context:Friction factor correlations are fundamental in internal pipe-flow calculations for head loss, pump sizing, and energy auditing. In the laminar regime, viscous forces dominate, and the Darcy–Weisbach friction factor takes a simple inverse relation with Reynolds number. This question checks your ability to recall and correctly identify that canonical formula and distinguish it from turbulent-flow correlations.

Given Data / Assumptions:

• Incompressible, Newtonian fluid.
• Steady, fully developed laminar flow in a straight, circular pipe.
• Reynolds number Re = (ρ V D) / μ is less than the laminar limit (typically Re < 2100).
• Darcy–Weisbach convention for friction factor (not Fanning).

Concept / Approach:For laminar flow in a round pipe, the velocity profile is parabolic and can be derived from a balance between pressure gradient and viscous shear. Integrating the Navier–Stokes equations leads to a linear relationship between wall shear stress and mean velocity. Substitution into the Darcy–Weisbach definition yields a friction factor inversely proportional to Reynolds number.

Step-by-Step Solution:

Verification / Alternative check:The Fanning friction factor is f_F = 16 / Re. Since the Darcy factor is four times the Fanning factor for round pipes, f = 4 f_F = 64 / Re, confirming the result.

Why Other Options Are Wrong:

• 16 / Re: This is the Fanning friction factor, not the Darcy factor requested here.
• 0.316 / Re^0.25 and 0.079 / Re^0.25: These are Blasius-type turbulent correlations, not valid for laminar flow.
• 4 / Re: Incorrect scaling; arises from mixing definitions.

Common Pitfalls:Confusing Darcy with Fanning friction factor; applying turbulent correlations to laminar conditions; using entrance-region or noncircular-duct results.

Final Answer:f = 64 / Re