Explicit pipe friction factor equations: evaluation, classification, and proposal

Abstract

The Colebrook equation has been used to estimate the friction factor (f) in turbulent fluids. In this regard, many equations have been proposed to eliminate the iterative process of the Colebrook equation. The goal of this article was to perform an evaluation, classification, and proposal of the friction factor for better development of hydraulic projects. In this study, Gene Expression Programming (GEP), Newton-Raphson, and Python algorithms were applied. The accuracy and model selection were performed with the Maximum Relative Error (∆f/f), Percentage Standard Deviation (PSD), Model Selection Criterion (MSC), and Akaike Information Criterion (AIC). Of the 30 equations evaluated, the Vatankhah equation was the most accurate and simplest to obtain the friction factor with a classification of very high, reaching a value of ∆f/f<0.5% and 1.5<PSD<1.6. A new equation was formulated to obtain the explicit f with fast convergence and accuracy. It was concluded that the combination of GEP, error theory, and selection criteria provides a more reliable and strengthened model.

Keywords

• Colebrook equation
• turbulent fluid
• relative roughness
• Reynolds number