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Colebrook equation

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The Colebrook Equation is an implicit equation which combines experimental results of studies of laminar and turbulent flow in pipes. It was developed in 1939 by C. F. Colebrook.

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It is defined as:

$\frac{1}{\sqrt{f}}=-2.0 \log \left( \frac { \varepsilon/D} {3.7} + \frac {2.51} {\mathrm{Re} \sqrt{f}} \right)$

where:

$f$ is the Darcy friction factor
$\varepsilon/D$ is the relative roughness
$Re$ is the Reynolds number

Due to the implicit nature of the Colebrook equation, determination of a friction factor requires some iteration or a numerical solving method. Therefore, an approximate explicit relation for $f$ was determined by S. E. Haaland in 1983.

This equation is known as the Haaland equation, and is defined as:

$\frac{1}{\sqrt {f}} = -1.8 \log \left[ \left( \frac{\varepsilon/D}{3.7} \right)^{1.11} + \frac{6.9}{\mathrm{Re}} \right]$