One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Pade Polynomials

TL;DR

This paper introduces a novel iterative method using Pade polynomials to solve the Colebrook equation efficiently with only one logarithmic evaluation, reducing computational cost without sacrificing accuracy.

Contribution

It presents a new Pade polynomial-based iterative algorithm for solving the Colebrook equation with minimal log evaluations, improving computational efficiency.

Findings

Requires only one logarithmic call in the entire iterative process

Maintains accuracy comparable to standard iterative methods

Reduces computational effort compared to existing approaches

Abstract

The 80 year-old empirical Colebrook function, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor, with the known Reynolds number and the known relative roughness of a pipe inner surface. It is based on logarithmic law in the form that captures the unknown flow friction factor in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Pade polynomials with only one log-call in total for the whole procedure (expensive log-calls are substituted with Pade polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution.