Fast and Accurate Approximations for the Colebrook Equation

The standard expression for pipe friction calculations, the Colebrook equation, is in an implicit form. Here, we present two accurate approximate solutions, given by replacing the numerically unstable term in Keady's exact Lambert function solution with a truncated series expansion. The resulting expressions have a higher accuracy than most advanced approximations and a lower computational cost than basic engineering formulas. The simplest expression, given by retaining only three terms in the series expansion, has a maximum error of less than 0.153% for Re >= 4000. The slightly more involved expression, based on five terms, has a maximum error of 0.0061%.