Why can't the Colebrook equation be solved algebraically?

Because the friction factor f appears inside both a logarithm and implicitly on the left side of the equation (as 1/√f), the equation can't be rearranged into a form with f isolated on one side — this is the definition of an "implicit" equation, and it requires numerical iteration (or an explicit approximation like Swamee-Jain) rather than direct algebraic solution.

Does the Moody chart apply to non-circular ducts?

The standard Moody chart and Colebrook/Swamee-Jain correlations are formulated for circular pipes. For non-circular ducts, engineers commonly use a hydraulic diameter (4 × cross-sectional area ÷ wetted perimeter) as a substitute for D in the same correlations — an approximation that works reasonably well for many practical duct shapes but introduces additional uncertainty compared to a true circular pipe.

Engineering·7 min read·August 14, 2026

📉 The Moody Chart and Colebrook Equation: Reading Turbulent Friction Factor

Q: How accurate is the Swamee-Jain approximation compared to Colebrook?

Swamee-Jain agrees with Colebrook to within about 1% across the normal turbulent-flow range (Reynolds number roughly 5,000 to 10⁸, relative roughness up to about 0.05) — accurate enough for essentially all practical engineering design work, avoiding the need for iterative solution.

How the Moody chart organizes friction factor data by Reynolds number and relative roughness, what the implicit Colebrook equation actually solves, and why explicit approximations like Swamee-Jain exist.

By EngineersUniverse Editorial Team

Engineering Editorial & Technical Review Team

Published August 14, 2026

Why Turbulent Friction Factor Isn't a Simple Formula

In laminar flow, the friction factor has an exact closed-form solution (f = 64/Re) derived directly from the governing fluid mechanics equations. Turbulent flow has no equivalent exact analytical solution — the friction factor in turbulent flow depends on both the Reynolds number and the pipe's relative roughness (ε/D, the ratio of absolute surface roughness to pipe diameter) in a way that can only be captured through empirical correlation, not derived from first principles alone.

What the Moody Chart Shows

The Moody chart (or Moody diagram) is the standard graphical tool for reading turbulent friction factor — it plots the Darcy friction factor (vertical axis, typically log scale) against Reynolds number (horizontal axis, log scale), with a family of curves, each representing a different relative roughness value. Reading the chart means locating the calculated Reynolds number on the horizontal axis, following it up to the curve matching the pipe's relative roughness, and reading the friction factor off the vertical axis. The chart also shows the laminar region (Re < 2300, a single straight line since f = 64/Re there), a critical/transition zone, and the fully turbulent region.

The Colebrook Equation — The Data Behind the Chart

The curves on the Moody chart are themselves generated from the Colebrook equation (sometimes called Colebrook-White), an empirical correlation fit to experimental pipe-friction data: 1/√f = -2·log₁₀(ε/(3.7D) + 2.51/(Re·√f)). The defining characteristic of this equation — and the reason it's called "implicit" — is that the friction factor f appears on both sides of the equation, including inside a square root and a logarithm, so it can't be algebraically rearranged to solve for f directly. Solving it requires an iterative numerical method (starting with a guess for f, computing the right side, comparing to the left, and refining the guess until it converges).

Why Explicit Approximations Exist

Because iterating the Colebrook equation by hand is tedious and impractical for routine engineering calculations (and was a genuine obstacle before computers and calculators made iteration trivial), several explicit approximations have been developed that closely match Colebrook's results without requiring iteration. The Swamee-Jain equation — used by this site's Pressure Drop Calculator — is one of the most widely used: f = 0.25 / [log₁₀(ε/(3.7D) + 5.74/Re^0.9)]². It's accurate to within about 1% of the Colebrook equation across the normal turbulent range (Reynolds numbers roughly 5,000 to 10⁸, relative roughness up to about 0.05), which is well within the accuracy needed for real engineering design work.

How Relative Roughness Shapes the Result

Relative roughness (ε/D) — not absolute roughness alone — is what actually matters for the friction factor, since the same absolute roughness has a much larger relative effect in a small-diameter pipe than in a large one. This is why the Moody chart plots curves by relative roughness rather than by pipe material or absolute roughness directly — the same commercial steel pipe with the same absolute roughness value will sit on a different relative-roughness curve depending on its diameter, and therefore show a different friction factor even at the identical Reynolds number.

Reading the Chart at High Reynolds Number

At sufficiently high Reynolds number for a given relative roughness, the Moody chart curves flatten out into the "fully rough" or "fully turbulent" zone, where friction factor becomes essentially independent of Reynolds number and depends only on relative roughness. This makes physical sense: at very high flow velocity, the roughness elements on the pipe wall dominate the friction behavior so completely that further increases in flow speed don't meaningfully change the dimensionless friction factor, even though the absolute pressure drop keeps increasing (since pressure drop also scales directly with v² independent of f).

Topics covered

Moody chartColebrook equationrelative roughness pipeturbulent friction factor