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# Comparison of Explicit Relations for Calculating Colebrook Friction Factor in Pipe Network Analysis Using h-based Methods

## Abstract

Although many explicit correlations have already been presented as alternatives to implicit Colebrook–White (C–W) formula, performances of C–W-based relations in pipe network analysis have not been investigated. In this study, 56 explicit relations available in the literature were implemented in the analysis of four water distribution networks while the benchmark solution is computed considering the implicit C–W formula. In the numerical experiment, these pipe networks were solved using three different h-based methods including h-based Newton–Raphson method, finite element method, and the gradient algorithm. In each scenario, one of these explicit relations was considered in the process of analyzing water networks. According to the obtained results, 15 explicit relations face the convergence problems which were identified as unreliable equations. Moreover, 15 explicit equations, which were successfully performed in analyzing all sample networks with the closest results to that of the benchmark solution, were introduced as the most accurate ones. Moreover, as many scenarios outperform those of the outdated explicit equation used for the same purpose in professional hydraulic solvers such as EPANET and WaterGEMS, it was recommended they be replaced with one of the explicit equations with higher accuracy. Finally, the achieved results demonstrate that the equation selected for computing Darcy–Weisbach friction factor has an inevitable impact not only on the accuracy but also on the convergence of pipe network analysis.

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## Abbreviations

Ai and ai for i = 1, 2, …, 9:

Coefficients

Bi and bi for i = 1, 2, …, 8:

Coefficients

Ci and ci for i = 1, 2, …, 8:

Coefficients

di for i = 1, 2, …, 5:

Coefficients

C–W:

Colebrook–White

D :

Pipe diameter

D–W:

Darcy–Weisbach

$$f$$ :

Darcy–Weisbach friction factor

ii for i = 1, 2, 3, 4:

Coefficients

$$g$$ :

The gravitational acceleration

$$K_{i}$$ :

Pipe coefficient of the ith pipe

$$L$$ :

Pipe length

$$m$$ :

The number of pipes in a typical water network

$$n$$ :

An exponent determined based on the resistance equation used

$$N$$ :

The number of pipes in the loop

Re:

Reynolds number

$$Q_{i}$$ :

Discharge flowing through the ith pipe

$$Q_{\text{in}}$$ :

Discharge entering a node

$$Q_{\text{out}}$$ :

Discharge exiting from a node

$$q_{i}$$ :

Water demand at the ith node

SSqRE:

Sum of squared relative error

xi for i = 1, 2, …, 5:

Coefficients

yi for i = 1, 2, …, 5:

Coefficients

zi for i = 1, 2, …, 5:

Coefficients

ɛ :

Absolute pipe roughness

υ :

Water kinematic viscosity

$$\pi$$ :

The pi number

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