Moody Friction Factor (Colebrook)
Solve the Colebrook-White equation exactly by iteration — the friction factor behind the Moody chart, without squinting at log-log axes.
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The engineering
The Colebrook-White equation is the definition underneath every Moody chart — it blends the smooth-pipe law and the fully-rough law into one implicit relation covering the whole turbulent range. This card solves it exactly: seed from the Swamee-Jain explicit fit, then fixed-point iterate on 1/√f until it converges to machine precision. Feed the result straight into Darcy-Weisbach (ΔP = f·L/D·½ρV²) or divide by 4 for the Fanning convention common in chemical-engineering texts.
The classic gotcha is exactly that factor of 4: if a handbook value looks a quarter of yours, it's Fanning, not wrong. Sanity checks: commercial steel water lines almost always land between 0.015 and 0.035; at high Re the answer should approach the fully-rough asymptote shown on the card (roughness dominates, Re stops mattering); and anything in the 2300–4000 transitional band is a guess no matter whose correlation you use.
Relative roughness is ε divided by inside diameter — for new commercial steel ε ≈ 0.045 mm, so a 2-in Sch 40 line (52.5 mm ID) runs ε/D ≈ 0.00086. Crane TP-410 tabulates ε for common pipe materials if you need more than steel.
Where this math comes from
Johann Nikuradse glued graded sand grains inside pipes at Göttingen in 1933 and mapped how roughness bends the friction curve — beautiful data, but real commercial pipe doesn't behave like uniform sand in the transition zone. Cyril Colebrook, working with Cedric White at Imperial College on tests of actual galvanized and tar-coated pipes, published the interpolation formula in 1939 that smoothly joins the smooth-pipe and fully-rough limits. It was implicit in f, which meant iteration by hand — a real cost in 1939.
Lewis Moody's contribution was packaging: his 1944 ASME paper plotted Colebrook's equation as one chart with relative-roughness contours, and it became the single most-reproduced figure in fluids engineering. Crane's Technical Paper 410, first issued in 1942 and revised ever since, carried the chart and the roughness tables onto every process engineer's shelf. A computer doesn't need the chart — it just iterates, as this card does.
- 1933Johann NikuradseSand-grain roughness experiments at Göttingen map the turbulent friction regimes.
- 1937Colebrook & WhiteTests on commercial pipe show real roughness transitions gradually, unlike uniform sand.
- 1939Cyril ColebrookPublishes the implicit transition formula this card solves.
- 1942Crane Co.Technical Paper 410 puts the correlation and roughness tables in every process engineer's toolkit.
- 1944Lewis MoodyPlots Colebrook as the famous Moody chart in Transactions of the ASME.
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