Questions: Darcy-Weisbach Equation: Major Head Loss Calculation

Question 1

Water flows through a commercial steel pipe at Re = 100,000 (turbulent). An engineer doubles the pipe's wall roughness ε by switching to a rougher material while keeping the same pipe diameter, length, and flow velocity. What happens to the friction factor f and the head loss?

• A. f is unchanged because Reynolds number is unchanged, and head loss stays the same
• B. f increases because higher relative roughness ε/D raises f in turbulent flow on the Moody diagram, so head loss also increases
• C. f decreases because a rougher surface disrupts the turbulent boundary layer and actually reduces skin friction
• D. f is not defined for rough commercial pipes; Darcy-Weisbach only applies to hydraulically smooth surfaces

Answer: (B)

In turbulent flow, the friction factor depends on both Reynolds number AND relative roughness ε/D. On the Moody diagram, for a given Re, moving to a higher ε/D curve gives a higher f. Since head loss H = f(L/D)(V²/2g) and f increases, head loss increases proportionally. This contrasts with laminar flow, where f = 64/Re depends only on Re and roughness is irrelevant — the viscous sublayer completely covers the roughness peaks in laminar flow, so they produce no additional resistance.

Question 2

Two geometrically identical pipes carry fluid in laminar flow at the same Reynolds number. One pipe has smooth walls; the other has significant wall roughness ε/D = 0.05. Which has the higher Darcy friction factor?

• A. The rough pipe, because roughness always increases friction regardless of flow regime
• B. Both have identical friction factors — in laminar flow, f = 64/Re regardless of wall roughness
• C. The smooth pipe, because turbulent eddies cannot form on smooth walls, reducing overall flow resistance
• D. The rough pipe at low Re, but both are identical only at very high laminar Reynolds numbers

Answer: (B)

This is a counterintuitive result that students who apply turbulent reasoning to laminar flow consistently get wrong. In laminar flow (Re < 2300), the velocity profile is dominated by viscosity; the fluid moves in orderly layers, and the viscous sublayer completely submerges any roughness features. The theoretical result f = 64/Re is exact for laminar pipe flow and has no roughness term. Roughness only matters when the flow is turbulent and roughness elements protrude through the viscous sublayer to disturb the turbulent eddies.

Question 3

At very high Reynolds numbers in rough pipes (the 'fully rough' turbulent regime), the Darcy friction factor f becomes independent of Reynolds number and is determined solely by the relative roughness ε/D.

• T. True
• F. False

Answer: True

This is visible on the Moody diagram as the horizontal 'plateaus' at high Re for each roughness curve. The physical reason: at very high Re, the viscous sublayer becomes thinner than the roughness height ε. The roughness peaks protrude fully into the turbulent flow field and dominate friction, while viscous effects become negligible. In this regime, each constant-roughness curve on the Moody diagram flattens to a constant f value given by the fully rough formula (1/√f = −2 log(ε/3.7D) from the Colebrook equation at Re → ∞).

Question 4

Doubling the length of a pipe in a Darcy-Weisbach calculation doubles the friction factor f.

• T. True
• F. False

Answer: False

This confuses two different quantities. The friction factor f is a dimensionless property of the flow regime and pipe roughness — it depends on Re and ε/D, not on pipe length. What doubles when you double pipe length is the head loss H_loss = f(L/D)(V²/2g). The factor (L/D) doubles, so H_loss doubles, but f itself is unchanged. f is an intensive property of the flow; H_loss is the extensive result of applying f over a given pipe geometry.

Question 5

Why does expressing head loss in meters of fluid column — rather than in pascals — make the Darcy-Weisbach equation useful across different fluids without modification?

Model answer: Head loss in meters of fluid (H_loss = f(L/D)(V²/2g)) is a measure of energy per unit weight of fluid, and it is independent of fluid density. To convert to pressure drop, you multiply by ρg: ΔP = ρg·H_loss. Since different fluids have different densities, expressing the result in meters keeps the equation form identical regardless of fluid — you simply multiply by the appropriate ρg at the end. This means the same friction factor and the same equation apply whether you are pumping water, crude oil, or air through a pipe.

This is the practical engineering power of the Darcy-Weisbach formulation over the Fanning friction factor or direct pressure-drop equations. In Bernoulli-form energy equations, all terms (velocity head V²/2g, elevation head z, pressure head P/ρg) share the same units of meters. Head loss drops in directly without needing to know the fluid density. The Moody diagram gives f as a function of Re and ε/D for any fluid; the density enters only when converting back to pressure for pump sizing.