Questions: Friction Factor and the Darcy-Weisbach Equation

Question 1

An engineer doubles the flow velocity in a smooth pipe with laminar flow. What happens to the friction factor and head loss?

• A. Friction factor doubles, head loss quadruples
• B. Friction factor halves (since Re doubles), but head loss still doubles because the velocity-squared term dominates
• C. Friction factor halves and head loss also halves
• D. Friction factor is unchanged since it only depends on pipe roughness

Answer: (B)

In laminar flow, f = 64/Re. Doubling velocity doubles Re, so f halves. Head loss h_f = f(L/D)(V²/2g): with f halving and V² doubling, h_f = (f/2)(4V²/2g)... wait — f halves while V² doubles, so h_f doubles overall. This is counterintuitive: faster flow halves the friction factor, but head loss still increases because the velocity-squared term grows faster than f shrinks. Option D is wrong: in laminar flow f depends on Re (and thus on velocity), not roughness.

Question 2

At very high Reynolds numbers in a rough pipe, what primarily determines the friction factor?

• A. Reynolds number alone — faster flow always reduces friction factor
• B. Relative roughness ε/D alone — the viscous sublayer is thinner than roughness elements so Re no longer matters
• C. The Colebrook equation reduces to f = 64/Re at high Re
• D. Friction factor goes to zero at very high Reynolds numbers because turbulence fully develops

Answer: (B)

This is the 'hydraulically rough' or 'fully rough' regime. At very high Re, the viscous sublayer at the pipe wall becomes thinner than the roughness elements. Turbulent eddies interact directly with roughness protrusions, and pressure drag from those elements dominates over viscous wall stress. The Colebrook equation reduces to 1/√f = −2 log₁₀(ε/3.7D), which is independent of Re — the flat rightward portion of the Moody diagram. Option A is true for laminar flow but fails at very high Re in rough pipes.

Question 3

In turbulent pipe flow, increasing the flow velocity generally decreases the friction factor.

• T. True
• F. False

Answer: False

This is true in laminar flow (f = 64/Re) and in the smooth-pipe turbulent regime, but NOT in the hydraulically rough regime. Once Re is high enough that the viscous sublayer is thinner than the roughness elements, f becomes constant — independent of velocity. The Moody diagram shows a clear transition: f decreases with Re in the smooth-pipe region, then levels off to a horizontal asymptote in the fully rough regime. Applying the laminar intuition to turbulent rough-pipe flow is a common error.

Question 4

The Colebrook equation is preferred over f = 64/Re for turbulent flow partly because it is explicit in f, making it easy to solve directly.

• T. True
• F. False

Answer: False

The Colebrook equation is preferred for turbulent flow because it accounts for both viscous effects (Re term) and roughness (ε/D term). However, it is *implicit* in f — f appears on both sides: 1/√f = −2 log₁₀(ε/3.7D + 2.51/Re√f). This requires iterative solution or an explicit approximation like the Swamee-Jain formula. The implicitness is precisely why engineers use the Moody diagram graphically or rely on explicit approximations in practice.

Question 5

Explain the physical reason why the Darcy friction factor for turbulent flow depends on pipe roughness (ε/D), while for laminar flow it does not.

Model answer: In laminar flow, the velocity profile is smooth and parabolic, with a thick viscous sublayer near the wall that completely submerges roughness elements — the flow does not 'see' wall irregularities. Friction comes only from viscous shear in the sublayer, giving f = 64/Re regardless of roughness. In turbulent flow, chaotic eddies extend toward the wall. When the viscous sublayer is thinner than the roughness protrusions (hydraulically rough regime), turbulent bursts strike the protrusions directly, creating pressure drag ('form drag') on each bump. This form drag depends on the size and density of roughness elements (captured by ε/D) and dominates over viscous drag.

This explains why pipe material matters in turbulent applications: commercial steel (ε ≈ 0.046 mm) behaves much like a smooth pipe at moderate Re but diverges significantly from cast iron (ε ≈ 0.26 mm) at high Re. Selecting pipe material based on expected flow regime is an engineering decision informed directly by this physics.