Solution of the Implicit Colebrook Equation for Flow Friction Using Excel

Abstract

Empirical Colebrook equation implicit in unknown flow friction factor (λ) is an accepted standard for calculation of hydraulic resistance in hydraulically smooth and rough pipes. The Colebrook equation gives friction factor (λ) implicitly as a function of the Reynolds number (Re) and relative roughness (ε/D) of inner pipe surface; i.e. λ 0 =f(λ 0 , Re, ε/D). The paper presents a problem that requires iterative methods for the solution. In particular, the implicit method used for calculating the friction factor λ 0 is an application of fixedpoint iterations. The type of problem discussed in this "in the classroom paper" is commonly encountered in fluid dynamics, and this paper provides readers with the tools necessary to solve similar problems. Students' task is to solve the equation using Excel where the procedure for that is explained in this "in the classroom" paper. Also, up to date numerous explicit approximations of the Colebrook equation are available where as an additional task for students can be evaluation of the error introduced by these explicit approximations λ≈f(Re, ε/D) compared with the iterative solution of implicit equation which can be treated as accurate.

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials

“…The Colebrook equation is based on logarithmic law where the unknown flow friction factor is given implicitly, i.e., it appears on both sides of Equation (1) in form , from which it cannot be extracted analytically; an exception is through the Lambert -function (Keady 1998, Sonnad and Goudar 2004, Brkić 2011cd, Brkić 2012ab, Biberg 2017, Brkić 2017a. The common way to solve it is to guess an initial value for friction factor and then to try to balance it using the iterative algorithm (Brkić 2017b) which needs to be terminated after the certain number of iterations when the final balanced value is reached. As an alternative to the iterative procedure, numerous approximate formulas are available (Gregory and Fogarasi 1985, Zigrang and Sylvester 1985, Brkić 2011e, Brkić 2012c, Brkić and Ćojbašić 2017, Pimenta et al 2018.…”

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation – Numerical Validation

“…The Colebrook equation for flow friction in its domain of applicability is fast converging. The fixed-point iterative methods are in common use but they demand up to seven iterations to reach the final satisfied and balanced accuracy [30,31]. On the other hand, numerous explicit approximations with different degree of accuracy are available [12][13][14].…”

“…In Equation (2), In Section 3, the presented iterative methods are listed in general from the simplest to the more complex [63]; 3.1 One log-call per iteration: 3.1.1) Fixed-point [30,31] [62].…”

Symbolic Regression-Based Genetic Approximations of the Colebrook Equation for Flow Friction

“…(3)(4)(5)(6). Adding one additional logarithmic form for acceleration using one additional fixed-point iterative step [9,45]; Eq. (2a), accuracy of the approximations increase significantly (about 10 times); Eqs.…”