The Colebrook Equation and the Moody Chart: Where Pipe Friction Factors Come From

The Problem Laminar Flow Never Had

For laminar pipe flow the friction factor is a solved problem: f = 64/Re, derived exactly from a force balance, with no dependence on how rough the pipe wall is. Turbulent flow offers no such luck. The chaotic eddies near the wall interact with every bump and pit in the surface, and nobody has ever derived the resulting friction factor from first principles. What engineers use instead is a correlation built on a few decades of very careful experiments, and nearly all of it traces back to two people working in the 1930s.

The first was Johann Nikuradse, working in Prandtl's laboratory in Göttingen. In 1933 he glued sand grains of carefully graded, uniform size to the inside of smooth pipes and measured pressure drop across a huge span of Reynolds numbers. His data showed something important: at high enough Reynolds number, a rough pipe's friction factor stops changing with Re at all and depends only on the relative roughness, ε/D. That plateau is now called the fully rough regime, and it exists because the roughness elements poke right through the thin viscous layer next to the wall, so viscosity stops mattering.

Colebrook's Blend of Two Laws

Nikuradse's uniform sand was tidy, but real commercial pipes are not. Their roughness is a random mix of sizes, and in the transition between smooth and fully rough behaviour they did not show the dip that sand-grain pipes did. Cyril Colebrook, working with Cedric White at Imperial College, tackled exactly this gap. In his 1939 paper he combined the known smooth-pipe law and the fully rough law into a single expression: 1/√f = −2 log₁₀(ε/(3.7D) + 2.51/(Re√f)), with f the Darcy friction factor.

The form is clever. At very high Re the second term inside the logarithm vanishes and you recover the fully rough law; for a perfectly smooth pipe the first term vanishes and you recover the smooth law. In between, the two blend smoothly, which matched commercial pipe data far better than Nikuradse's curves did. The price is that f appears on both sides of the equation, so it has to be solved by iteration. A starting guess of f = 0.02 usually converges in three or four passes.

Moody's Chart, and How Accurate It Really Is

Iterating a logarithmic equation by hand was tedious in the 1940s, so in 1944 Lewis Moody plotted Colebrook's equation as a family of curves, friction factor against Reynolds number, one curve per relative roughness, with the laminar line 64/Re on the left. That chart has been reprinted in essentially every fluid mechanics textbook since. It is worth remembering that the Moody chart is not separate data. It is a picture of the Colebrook equation.

It is also not as precise as its neat curves suggest. The roughness values tabulated for commercial pipe, about 0.045 mm for new commercial steel and about 0.0015 mm for drawn tubing, are themselves averages, and corrosion or scale can raise them several-fold over a pipe's life. White's widely used textbook puts the practical accuracy of a friction factor read from the chart at roughly ±15%. For design that is fine, which is why engineers add margin to pump head, but quoting a friction factor to four significant figures is false precision.

Explicit Approximations and the Fanning Trap

Because iteration is awkward inside spreadsheets and hand calculations, several explicit approximations have been proposed. Haaland's 1983 formula, 1/√f = −1.8 log₁₀[(ε/D/3.7)^1.11 + 6.9/Re], agrees with Colebrook to within about two percent over the turbulent range, well inside the uncertainty of the roughness value itself. Swamee and Jain published a similar explicit form in 1976. For any practical calculation, either is as good as the iterated original.

The single most common error with all of this has nothing to do with turbulence. There are two friction factors in use. The Darcy friction factor, used by Moody and by most mechanical and civil engineers, gives 64/Re in laminar flow. The Fanning friction factor, common in chemical engineering texts, is exactly one quarter of it and gives 16/Re. Both are correct; mixing them is not. Before using any chart or correlation, check which one it uses, because a factor-of-four error in pressure drop is not something a design margin will hide.

References

1. Nikuradse, J. (1933). Strömungsgesetze in rauhen Rohren. VDI-Forschungsheft 361. English translation: Laws of flow in rough pipes, NACA Technical Memorandum 1292 (1950).
2. Colebrook, C. F. (1939). Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers, 11(4), 133–156. doi:10.1680/ijoti.1939.13150
3. Moody, L. F. (1944). Friction factors for pipe flow. Transactions of the ASME, 66(8), 671–678. doi:10.1115/1.4018140
4. Haaland, S. E. (1983). Simple and explicit formulas for the friction factor in turbulent pipe flow. Journal of Fluids Engineering, 105(1), 89–90. doi:10.1115/1.3240948
5. White, F. M. (2011). Fluid Mechanics (7th ed.), Chapter 6. New York: McGraw-Hill.