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 (e/D) of inner pipe surface; i.e. λ0=f(λ0, Re, e/D). The paper presents a problem that requires iterative methods for the solution. In particular, the implicit method used for calculating the friction factor λ0 is an application of fixed-point 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, e/D) compared with the iterative solution of implicit equation which can be treated as accurate.