Pipe friction — For steady, fully developed flow in a pipe, the Darcy–Weisbach friction factor depends on Reynolds number and relative roughness. Which parameter below does the friction factor <em>not</em> directly depend upon?

Introduction:
The Darcy–Weisbach friction factor f is a dimensionless quantity used to calculate pressure losses due to wall friction in internal flow. Correctly identifying which variables it truly depends on is essential for pipe sizing and pump calculations. This question focuses on eliminating a common misconception: that f depends on pipe length.

Given Data / Assumptions:

• Steady, fully developed, single-phase flow in a straight, prismatic pipe.
• Darcy–Weisbach equation: Δp = f * (L/D) * (ρ * V^2 / 2).
• f is a function of Reynolds number Re and relative roughness ε/D.

Concept / Approach:
In laminar flow, f = 64/Re (independent of roughness). In turbulent flow, f is obtained from the Colebrook–White equation or Moody chart as a function of Re and ε/D. Pipe length L appears linearly in the pressure-drop equation, but it does not affect the value of f itself. Likewise, density, viscosity, and mass flow rate influence f indirectly via Re, not as separate independent variables.

Step-by-Step Solution:
Write the head-loss relation: h_f = f * (L/D) * (V^2 / (2g)).Note: f = f(Re, ε/D) only; L multiplies f but does not determine f.Therefore, among the choices, “pipe length” is not a direct determinant of f.

Verification / Alternative check:
Using the Moody chart: for a given Re and ε/D, f is fixed irrespective of L. Changing L scales Δp but leaves the chosen f unchanged.

Why Other Options Are Wrong:

• Pipe roughness: affects f in turbulent and transitional regimes through ε/D.
• Fluid density & viscosity: set Re with velocity and diameter; thus they influence f via Re.
• Mass flow rate: sets velocity V (for fixed D), hence changes Re and f.
• Relative roughness: fundamental determinant in turbulent regime.

Common Pitfalls:
Confusing parameters that scale pressure drop (like L) with parameters that set f; forgetting that fittings/entrances introduce additional loss coefficients separate from f.

Final Answer:
Pipe length