Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks

Abstract

This study proposes hybrid models to solve the Colebrook–White equation by combining explicit equations available in the literature to solve the Colebrook–White equation with an error function. The hybrid model is in the form of fH=fo−eA. fH is the friction factor value f predicted by the hybrid model, fo is the value of f calculated using several explicit formulas for the Colebrook–White equation, and eA is the error function determined using the neural network procedures. The hybrid equation consists of a series of hyperbolic tangent functions whose number corresponds to the number of neurons in the hidden layer. The simulation results showed that the hybrid models using five hyperbolic tangent functions could produce reasonable predictions of friction factors, with the maximum absolute relative error (MAXRE) around one tenth, or ten times lower than that produced by the corresponding existing formula. The simplified hybrid models are also given using four and three tangent hyperbolic functions. These simplified models still provide accurate results with MAXRE of less than 0.1%.

Keywords

• Colebrook–White equation
• friction factor
• explicit approximation
• artificial neural networks
• hybrid model