According to common engineering formulas, friction head loss is sometimes proportional to velocity rather than velocity squared in which case?

According to common engineering formulas, friction head loss is sometimes proportional to velocity rather than velocity squared in which case?

• A. Yes, in laminar flow
• B. Yes, for non-Newtonian fluids
• C. Yes, in smooth pipes
• D. No, never

Answer: (A)

In pipe flow, how friction head loss scales with velocity depends on the flow regime. In turbulent flow, the friction factor f is roughly constant for a given pipe, so the head loss h_f ∝ V^2. But in laminar flow, f is not constant; it scales with 1/Re (with Re = ρVD/μ), specifically f = 64/Re for fully developed laminar flow in a circular pipe. Plugging this into the Darcy–Weisbach form h_f = f (L/D) (V^2 / 2g) gives h_f ∝ V. So friction head loss becomes proportional to velocity in laminar conditions, especially in smooth pipes where the flow remains laminar. The other scenarios don’t produce this linear relationship as reliably: non-Newtonian fluids can complicate the dependence, and in many practical turbulent cases the dependence is quadratic.

In pipe flow, how friction head loss scales with velocity depends on the flow regime. In turbulent flow, the friction factor f is roughly constant for a given pipe, so the head loss h_f ∝ V^2. But in laminar flow, f is not constant; it scales with 1/Re (with Re = ρVD/μ), specifically f = 64/Re for fully developed laminar flow in a circular pipe. Plugging this into the Darcy–Weisbach form h_f = f (L/D) (V^2 / 2g) gives h_f ∝ V. So friction head loss becomes proportional to velocity in laminar conditions, especially in smooth pipes where the flow remains laminar.

The other scenarios don’t produce this linear relationship as reliably: non-Newtonian fluids can complicate the dependence, and in many practical turbulent cases the dependence is quadratic.