Internal flow – How frictional resistance scales with speed For turbulent flow in a typical engineering pipe, the frictional resistance (formulated as head loss or drag) varies approximately with which function of mean velocity?

Introduction:
Estimating frictional losses in pipes is central to sizing pumps, predicting energy consumption, and avoiding excessive pressure drops. While the exact dependence involves Reynolds number and roughness via the Darcy–Weisbach equation, a common rule of thumb is that head loss is roughly proportional to the square of the velocity in turbulent regimes.

Given Data / Assumptions:

• Single-phase liquid flow in a prismatic circular pipe.
• Fully developed turbulent regime (Re well above transition).
• Friction factor variation with Re and roughness treated as weak over a narrow operating range.

Concept / Approach:

The Darcy–Weisbach head loss is h_f = f * (L/D) * (v^2 / (2 * g)). For a given line and modest Re changes, f varies slowly compared with v^2, so h_f ∝ v^2 is a good engineering approximation. Equivalently, the required pump power ∝ Q * h_f ∝ v * v^2 ∝ v^3 for fixed diameter, which emphasizes why small increases in speed can significantly raise power demand.

Step-by-Step Solution:

Verification / Alternative check:

On a Moody chart, moving along a line of roughly constant f shows h_f changing with v^2; pump curves and system characteristic curves also reflect this square-law behavior.

Why Other Options Are Wrong:

Velocity or square root: Underpredict losses in turbulent pipe flow.Cube: Relates to power demand scaling, not directly to head loss under constant diameter.Pressure: Not a functional dependence; pressure drop is the outcome, not the input variable here.

Common Pitfalls:

Applying laminar flow relation (h_f ∝ v) to turbulent conditions; ensure regime is identified first.

Final Answer:

square of velocity