Questions: Colebrook-White Friction Factor Correlation

Question 1

An engineer calculates the friction factor for fully turbulent flow in a rough pipe and finds that ignoring the Reynolds number term in the Colebrook-White equation introduces negligible error. When is this simplification valid?

• A. Only when the pipe is hydraulically smooth (ε/D → 0)
• B. When Re is very high, so the viscous sublayer is thinner than the roughness elements
• C. When the flow is laminar, so viscous effects dominate
• D. This simplification is never valid—both terms must always be included

Answer: (B)

At very high Reynolds numbers, the viscous sublayer shrinks to a thickness smaller than the roughness elements, so rough-wall form drag dominates. The term 2.51/(Re√f) becomes negligible, and f depends only on ε/D—the 'fully rough' or 'complete turbulence' regime. This corresponds to the horizontal lines at the far right of the Moody diagram, where curves for different ε/D are flat, indicating Re-independence.

Question 2

Why must the Colebrook-White equation be solved iteratively rather than by direct algebraic manipulation?

• A. The equation involves transcendental functions that have no closed-form solutions under any circumstances
• B. f appears inside the logarithm on the right-hand side as well as on the left, making direct isolation impossible
• C. The equation is only valid for specific ranges of ε/D and Re, making algebra unreliable outside those ranges
• D. Because the Moody diagram was derived empirically without algebraic structure

Answer: (B)

The equation has the form 1/√f = [expression containing √f in the denominator], so any attempt to algebraically isolate f leads in circles. Iteration—starting from an initial guess (often the fully turbulent limit) and repeatedly substituting—converges in 3–5 steps. Explicit approximations like Swamee-Jain sidestep this by sacrificing exact agreement for algebraic tractability, at a cost of less than 3% error.

Question 3

Using the Swamee-Jain or Haaland explicit approximation instead of iterating the Colebrook-White equation is acceptable for most engineering pipe flow calculations.

• T. True
• F. False

Answer: True

Explicit approximations have errors below 2–3% relative to the Colebrook-White equation, which is well within the uncertainty introduced by pipe roughness itself (which varies by surface preparation, aging, and corrosion). Engineering decisions based on friction factor rarely require more than ~5% accuracy; the explicit formulas provide this while allowing direct calculation without iteration.

Question 4

In the Colebrook-White equation, increasing Reynolds number generally decreases the friction factor, regardless of pipe roughness.

• T. True
• F. False

Answer: False

In the fully turbulent (completely rough) regime at high Reynolds numbers, friction factor becomes independent of Re—it plateaus at a value determined only by ε/D. Increasing Re further has no effect because the viscous sublayer is already too thin to matter. The friction factor decreases with Re only in the transitionally rough regime where both terms in the equation matter. This Re-independence of f at high Re is visible as the flat lines on the right side of the Moody diagram.

Question 5

What physical phenomena do the two terms inside the logarithm in the Colebrook-White equation represent, and why does their relative importance shift with Reynolds number?

Model answer: The first term, (ε/D)/3.7, represents pipe wall roughness: when roughness elements protrude through the viscous sublayer, they add form drag and dominate friction loss. The second term, 2.51/(Re√f), represents the viscous sublayer's smoothing effect: at lower Re, the sublayer is thick enough to submerge the roughness elements, making the pipe behave hydraulically smooth. As Re increases, the sublayer thins; when it disappears entirely, only roughness matters and f becomes Re-independent. At intermediate Re, both effects coexist.

This physical duality is why the Colebrook-White equation has its specific form—it unifies the smooth-pipe law (Prandtl) and the rough-pipe law (von Kármán) into a single interpolating formula describing the transition between them.