Advances in Civil Engineering

Volume 2018, Article ID 5451034, 18 pages

https://doi.org/10.1155/2018/5451034

## Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

^{1}European Commission, Joint Research Centre (JRC), Directorate C: Energy, Transport and Climate, Unit C3: Energy Security, Distribution and Markets, Via Enrico Fermi 2749, 21027 Ispra (VA), Italy^{2}IT4Innovations National Supercomputing Center, VŠB-Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic^{3}Alfatec, Bulevar Nikole Tesle 63/5, 18000 Niš, Serbia

Correspondence should be addressed to Pavel Praks; moc.liamg@skarp.levap and Dejan Brkić; sr.ca.gb.bucr.alset@fgrnajed

Received 31 March 2018; Accepted 13 August 2018; Published 10 December 2018

Academic Editor: Giuseppe Oliveto

Copyright © 2018 Pavel Praks and Dejan Brkić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 10^{8}) through pipes with roughness between negligible relative roughness (*ε*/*D *⟶* *0) to very rough (up to *ε*/*D* = 0.05). The Colebrook equation includes the flow friction factor *λ* in an implicit logarithmic form, *λ* being a function of the Reynolds number Re and the relative roughness of inner pipe surface *ε*/*D*: *λ* = *f*(*λ*, Re, *ε*/*D*). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, *λ* ≈ *f*(Re, *ε*/*D*), it is necessary to determinate the value of the friction factor *λ* from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

#### 1. Introduction

To evaluate flow resistance in turbulent flow through rough or smooth pipes, the empirical Colebrook equation is in common use [1]:

In the Colebrook equation, *λ* represents Darcy flow friction factor, Re Reynolds number, and *ε*/*D* relative roughness of inner pipe surfaces (all three quantities are dimensionless).

The experiment performed by Colebrook and White [2] dealt with flow of air through a pipe, diameter *D* = 53.5 mm, and length *L* = 6 m, with six different roughness of inner surface of the pipe artificially simulated with various mixtures of two sizes of sand grain (0.035 mm and 0.35 mm diameter) to simulate conditions of inner pipe surface from almost smooth to very rough. The sand grains were fixed using a sort of bituminous adhesive waterproof insulating compound to form five types of relatively uniform roughness of inner pipe surfaces while the sixth one was without sand, that is, it was left smooth. The experiment revealed, contrary to the previous, that the flow friction, *λ*, does not have a sharp transition from the smooth to the fully rough law of turbulence. This evidence Colebrook [1] later captured in today famous and widely used empirical equation, Equation (1).

The Colebrook function relates the unknown flow friction factor *λ* as function of itself, the Reynolds number Re, and the relative roughness of inner pipe surface *ε*/*D*, *λ* = *f*(*λ*, Re, *ε*/*D*). It is valid for 4000 < Re < 10^{8} and for 0 < *ε*/*D* < 0.05. The Colebrook equation is transcendent and thus cannot be solved in terms of elementary functions [3–6]. Although empirical, and therefore with questionable accuracy, its precise solution is sometimes essential in order to repeat or to evaluate the previous findings in a concise way [7–9].

Few approaches are available today for solving the Colebrook equation:(1)*Graphical solution—Moody diagram*: To represent the Colebrook equation graphically, Rouse in 1942 had developed an appropriate diagram which Moody later adapted in 1944 in the famous diagram widely used in the past in engineering practice [10, 11]. The diagram was preferred because the Colebrook equation is implicitly given. Today, graphical solution has only value for educational purposes.(2)*Iterative solution of the Colebrook equation*:(a)*Simple fixed-point iterative method.* The simple fixed-point iterative method [12] is in common use for solving accurately the Colebrook equation (special case of the Colebrook equation for Re ⟶ ∞ gives explicit form valid only for the fully turbulent flow in rough pipes [13–16] but which can be used as initial starting point for all cases covered by the Colebrook equation *λ*_{0} = *f*(*ε*/*D*)* *⟶* *Equation (2); now using the Colebrook equation, new value can be calculated *λ*_{1} = *f*(*λ*_{0}; Re; *ε*/*D*); starting from *i* = 1, the procedure *λ*_{i+1} = *f*(*λ*_{i}; Re; *ε*/*D*); *i* = *i* + 1 goes until *λ*_{i} ≈ *λ*_{i+1}, where we set *λ*_{i+1} − *λ*_{i} ≤ 10^{−8}). It usually reaches the satisfied accuracy after no more than 10 iterations.(b)*Householder’s iterative methods.* On the other hand, Newton’s method (also known as the Newton–Raphson method [17–19]) needs few iterations less compared to the fixed-point method to reach the same level of accuracy. A shortcoming of Newton’s method is that it additionally requires the first derivative of the Colebrook function (here we show analytical form of the first derivative including the symbolic form generated in MATLAB [20]). Also knowing that the Newton–Raphson method is the 1st order of Householder’s method [21, 22], here we also analyze the 2nd order, which is known as the Halley [23] and the Schröder [24, 25] method, and also the 3rd order. The third-order methods use the third, the second, and the first derivative, the 2nd order use the second and the first, while the 1st order use only the first derivative. Today, all mentioned types of iterative solutions can easily be implemented in software codes and they are accepted as the most accurate way for solving the Colebrook equation, and hence, they are preferred compared to the graphical solution.(c)*Three-point iterative methods.* Three-point iterative methods need only 1 to 3 iterations in three points *x*_{0}, *y*_{0}, and *z*_{0} (three internal iterations) to achieve the high level of accuracy [26–28]. *x*_{0} is initial starting point, *y*_{0} is auxiliary step, while *z*_{0} is the solution. Three-point methods are very accurate and can reach high accuracy in some cases even after 1 to 2 iterations. Also slightly less fast two-point methods in terms of required number of iterations to reach the demanded accuracy do exist.(3)*Approximations of the Colebrook equation*: Colebrook’s equation can be expressed in the explicit form only in an approximate way: *λ *≈* f*(Re, *ε*/*D*) [29–36]. Numerous explicit approximations to the Colebrook equation are available in the literature [29–32, 34–36]. The iterative solutions as the most accurate methods are used for evaluation of accuracy of such approximations. Also, based on our findings, we provide an approximation, Equation (28), with the error of no more than 0.69% and 0.0617%. The Colebrook equation can also be approximately simulated using Artificial Neural Networks [37–39].(4)*Lambert W-function*: Until now, the only one known way to express the Colebrook equation exactly in explicit way is through the Lambert W-function, *λ* = *W*(Re, *ε*/*D*) [3, 8, 40–43], where further evaluation of the Lambert W-function can be only approximated [44–48]. Here, we show the procedure how to solve the Lambert W-function using the Householder iterative procedure (2nd order: Halley’s method and 1st order: Newton–Raphson). Also approach with the shifted Lambert W-function in terms of the Wright Ω-function exists [40, 43].

In this paper, we show the three-point and the Householder iterative procedures (the 3rd order, the 2nd order: Halley’s [49] and Schröder’s method, and the 1st order: Newton–Raphson) with the additional recommendations in order to solve the empirical Colebrook equation implicitly given in respect of the flow friction factor *λ*. The goal of this paper is to show the improved iterative solutions which can obtain the value of the unknown friction factor *λ* accurately after the least possible number of iterations. Additionally, we developed a strategy how to choose the best starting point [50] for the iterative procedure in the domain of interest of the Colebrook equation, how to generate required symbolic derivatives to the Colebrook equation in MATLAB, and how to avoid use of the derivatives (secant method). Finally, we use findings from our paper to present a novel explicit approximation of the Colebrook equation, which would be interesting for engineering practice. We also present distribution of the relative error in respect of the presented approximation over the applicability domain of the Colebrook equation.

To evaluate the efficiency of the presented methods, the unknown flow friction factor *λ* is calculated for two pairs of the Reynolds number Re and relative roughness of inner pipe surfaces *ε*/*D*: (1) (Re = 5·10^{6}, *ε*/*D* = 2.5·10^{−5})* *⟶* λ* = 0.010279663295529 and (2) (Re = 3·10^{4}, *ε*/*D* = 9·10^{−3})*⟶ λ = 0.038630738574792.*

*2. Initial Estimate of Starting Point for the Iterative Procedures*

*The starting point is a significant factor in convergence speed in the three-point and the Householder methods [50], and there are the different methods to choose a good start, but here we examine (1) starting point as function of the input parameters and (2) initial starting point with the fixed value.*

*One of the essential issues in every iterative procedure is to choose the good starting point [51, 52]. Here, we try to find the fixed starting point (the initial value of the flow friction factor λ_{0} or the related transmission factor ) valid for all cases from the practical domains of applicability of the Colebrook equation which is for the Reynolds number Re, 4000 < Re < 10^{8}, and for the relative roughness ε/D, 0 < ε/D < 0.05. In the cases when this approach does not work efficiently, we show how to choose the starting value in function of the Reynolds number Re and the relative roughness ε/D, that is, using some kind of the rough approximations to the Colebrook equation which can be relatively inaccurate but simply and which put the initial value reasonable close to the final and accurate solution. This initial guess then needs to be plugged into the shown numerical methods and iterated recursively few times (usually two or three times and up to ten in the worst case) to converge upon the final solution. In any case, a sample of size 65536 was considered for analysis of the iteration methods. The input sample was generated according to the uniform density function of each input variable. The low-discrepancy Sobol sequences were employed [53]. These so-called quasirandom sequences have useful properties. In contrary to the random numbers, quasirandom numbers cover the space more quickly and evenly. Thus, they leave very few holes.*

*The Colebrook equation can also be expressed in terms of the Lambert W-function analytically, λ = f(λ, Re, ε/D) ⟶ λ = W(Re, ε/D) [41, 42, 54]. The Lambert W-function further can be evaluated only approximately through the Householder iterative procedures which also require the appropriate initial starting point. The analysis of this initial starting point has wider applicability, because the Lambert W-function has extensive use in many branches of physics and technology [55, 56].*

*2.1. Starting Point as Function of Input Parameters*

*2.1.1. Starting Point as Function of the Relative Roughness **ε*/*D* (When Re* *⟶* *∞)

*ε*/

*D*(When Re

*⟶*

*∞)*

*The special case of the Colebrook equation when Re ⟶ ∞ physically means that the flow friction factor λ in that case depends only on ε/D, for Re ⟶ ∞, λ = f(ε/D), that is, the flow friction factor λ is not implicitly given [14]. In that way, the starting point can be calculated using the explicit equation which has only one variable, λ_{0} = f(ε/D) (Equation (2)). The results obtained in that way are accurate only for the case Re ⟶ ∞ but for the smaller values of the Reynolds number Re which corresponds to the smooth turbulent flow, the error can goes up to 80% [13, 57]. Anyway, in that way, calculated value can be efficiently used as an initial starting guess for the iterative procedures for the whole domain of applicability of the Colebrook equation.*

*The initial starting point obtained using the previous equation is referred as “traditional,” and it introduces the maximal relative error of 80% over the domain of applicability of the Colebrook equation where the error can be neglected in case of fully developed turbulent flow through the pipes with very rough inner surface. To reach the accuracy of λ_{i+1} − λ_{i} ≤ 10^{−8}, usually 6 steps are enough regarding the Newton–Raphson method (Figure 1).*