Some explicit formulations of Colebrook–White friction factor considering accuracy vs. computational speed

Abstract

The Colebrook-White formulation of the friction factor is implicit and requires some iterations to be solved given a correct initial search value and a target accuracy. Some new explicit formulations to efficiently calculate the Colebrook-White friction factor are presented herein. The aim of this investigation is twofold: (i) to preserve the accuracy of estimates while (ii) reducing the computational burden (i.e. speed). On the one hand, the computational effectiveness is important when the intensive calculation of the friction factor (e.g. large-size water distribution networks (WDN) in optimization problems, flooding software, etc.) is required together with its derivative. On the other hand, the accuracy of the developing formula should be realistically chosen considering the remaining uncertainties surrounding the model where the friction factor is used. In the following, three strategies for friction factor mapping are proposed which were achieved by using the Evolutionary Polynomial Regression (EPR). The result is the encapsulation of some pieces of the friction factor implicit formulae within pseudo-polynomial structures.Key words 9 9 9 9 Colebrook-White formula, computational speed, evolutionary polynomial regression, friction factor, pipe flow

Intelligent Flow Friction Estimation

“…It can produce outputs for inputs and its inner complexity is not crucial [47,48]. Users would easily apply the ANN without any difficulty due to its structure complexity, in contrast to use of the approximate formulas [38]. The same circumstances of comfort can be experienced by users applying the prepared computer codes for the approximate formulas.…”

“…In fact, the complexity of explicit approximations (e.g., approach with the Lambert W-function [8,49]) was considered as a serious issue few decades ago when pocket calculators were widely used [38]. Nowadays, even the very complex approximations can be easily used in computer codes.…”

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

“…number of logarithmic and power expression stays unchanged [9,18]. Only change of integer power to non-integer power in some approximation can increase computational burden, but even than not significantly.…”

“…The Colebrook equation (1) relates hydraulic flow friction (λ) through Reynolds number (R) and relative roughness (ε/D) of inner pipe surface but in implicit way; λ=f(λ, R, ε/D) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. On the other hand, to express flow friction (λ) in implicit way a number of approximations can be used .…”

Discussion of “Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model” by Angus Simpson and Sylvan Elhay

“…The procedure shown in the discussed paper is more accurate mathematically and also can be more easily implemented in a computer code. This increased computing burden of the improved procedure has to be Text DB Click here to download Manuscript: Discussion DB (Simpson and Elhay).doc carefully investigated where the ratio between complexity and accuracy will decide whether this method will be used in every day engineering practice (Giustolisi et al 2011).…”