Head loss in fully rough turbulent pipe flow: how does the frictional head loss (for a given length) vary with average fluid velocity?

Introduction:
Pump sizing and pipeline design rely on knowing how head loss changes with flow rate. This question probes the classic Darcy–Weisbach relation for turbulent flow in pipes and the dominant velocity-squared dependence of frictional head loss.

Given Data / Assumptions:

• Internal pipe flow, single-phase, steady.
• Turbulent regime.
• Length and diameter fixed for the comparison.

Concept / Approach:
The Darcy–Weisbach equation is h_f = f * (L/D) * (v^2/(2g)), where f is the friction factor. Over many practical turbulent conditions, head loss varies approximately with v^2; f depends weakly on Reynolds number and relative roughness, but the dominant scaling with velocity is quadratic.

Step-by-Step Solution:
Start with h_f = f (L/D) v^2/(2g).For a given pipe (fixed L and D), the term (L/D)/(2g) is constant.Although f can vary with Re and roughness, for a velocity change the leading dependence is with v^2.Therefore, head loss ∝ v^2.

Verification / Alternative check:
Moody chart and the Blasius correlation (for smooth pipes, f ≈ 0.316 Re^−0.25) both leave the v^2 dependence intact in h_f, since f varies sublinearly with v, while the kinetic head term scales as v^2.

Why Other Options Are Wrong:

• Linear with v: underestimates losses in turbulent flow.
• Inverse with diameter squared: dimensional form is incorrect; for fixed v, h_f ∝ 1/D (not 1/D^2).
• Inverse with velocity or independent: contradicts Darcy–Weisbach.

Common Pitfalls:
Confusing laminar (Δp ∝ v) with turbulent (h_f ∝ v^2) behavior or misapplying Hazen–Williams empiricism beyond its range.

Final Answer:
Proportional to (velocity)^2