Pipe head loss terminology: what is the ratio between the Darcy–Weisbach friction factor and the Fanning (coefficient of) friction used in head-loss relations?

Introduction / Context:
Two friction parameters are common in hydraulics: the Darcy–Weisbach friction factor f (used in head loss h_f = f * (L/D) * V^2/(2g)) and the Fanning friction factor f_F (often called the coefficient of friction in some texts). Consistency between them avoids errors in design calculations.

Given Data / Assumptions:

• Steady internal flow in a pipe of diameter D and length L.
• Mean velocity V; gravitational acceleration g.
• Standard definitions of Darcy and Fanning friction factors.

Concept / Approach:

The relationship is f (Darcy) = 4 * f_F (Fanning). This arises because the Darcy form lumps wall shear on all sides, whereas the Fanning form is based directly on wall shear stress relative to dynamic pressure. Using the wrong conversion multiplies head loss or pressure drop by a factor of four in error.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis and standard Moody chart annotations confirm the 4:1 relationship.

Why Other Options Are Wrong:

(a), (b), (c), and (e) contradict the established equivalence; using them would mispredict losses.

Common Pitfalls:

Reading a Moody chart labeled with the Darcy factor but plugging into a Fanning-based formula (or vice versa) without conversion.

Final Answer:

4