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?
-
Af = 64 / Re
-
Bf = 16 / Re
-
Cf = 0.316 / Re^0.25
-
Df = 4 / Re
-
Ef = 0.079 / Re^0.25
Answer
Correct Answer: f = 64 / Re
Explanation
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:
Start from laminar solution: τ_w = (Δp * D) / (4 L).Mean velocity V relates to Δp via Poiseuille law: Δp = 32 μ V L / D^2.Hence τ_w = (32 μ V L / D^2) * (D / 4 L) = 8 μ V / D.Darcy friction factor f = 8 τ_w / (ρ V^2) = 8 * (8 μ V / D) / (ρ V^2) = 64 μ / (ρ V D) = 64 / Re.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