Accurate and Efficient Explicit Approximations of the Colebrook Flow Friction Equation Based on the Wright ω-Function: Reply to the Discussion by Majid Niazkar

Pavel Praks,Dejan Brkić

doi:10.3390/math8050796

Pavel Praks, Dejan Brkić

Open Access

https://doi.org/10.3390/math8050796

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Journal: Mathematics	Publication Date: May 14, 2020
Citations: 1	License type: CC BY 4.0

Affiliation: Technical University of Ostrava, University of Nis

Abstract

In this reply, we present updated approximations to the Colebrook equation for flow friction. The equations are equally computational simple, but with increased accuracy thanks to the optimization procedure, which was proposed by the discusser, Dr. Majid Niazkar. Our large-scale quasi-Monte Carlo verifications confirm that the here presented novel optimized numerical parameters further significantly increase accuracy of the estimated flow friction factor.

Highlights

We thank discusser Dr Majid Niazkar for his valuable discussion of our article [1]. He proposed to optimize numerical parameters of our explicit approximations of the Colebrook’s equation, in order to increase their accuracy in comparison to the implicitly given Colebrook’s flow friction factor [2]
The original Colebrook equation for flow friction [3] is of implicit nature, and our goal is to develop its computational simple but accurate explicit approximations
Approximations given in Equation (3) of our discussed article [1] are based on the approximation of the Wright ω-function as given in [12], while Equations (5) and (6) of [1] are found using symbolic regression software Eureqa [13,14] and they are based on the approximation of the same branch of the Wright ω-function as in Equation 3

Summary

Introduction

We thank discusser Dr Majid Niazkar for his valuable discussion of our article [1] In his discussion, he proposed to optimize numerical parameters of our explicit approximations of the Colebrook’s equation, in order to increase their accuracy in comparison to the implicitly given Colebrook’s flow friction factor [2]. Using the methodology proposed by Niazkar [8], our approximation can be classified in the group of one-step relations which clearly confirm their simplicity In this reply we present updated numerical parameters for these models, which are computational simple, but with further increased accuracy. Comparable approaches can be found in the scientific literature [9,10,11], such as optimization of the other already available approximations of the Colebrook equations using genetics algorithms, etc

Updated Approximations

Findings

Conclusions