Comparison of the Lambert W-function based solutions to the Colebrook equation
Purpose - The Colebrook equation for determination of hydraulic resistances is implicit in fluid flow friction factor and hence it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors. The purpose of this paper is to compare different Lambert W based solutions of the Colebrook equation and to make comparisons among them and identify some constraints in applicability of certain solutions. Design/methodology/approach - Alternate mathematically equivalents to the implicit Colebrook equation in explicit form with no approximation involved actually exist. Findings - These alternate equations were developed using Lambert W-function. The paper compares various implementations of the Lambert W methodology and shows that some of these are less able than others to yield solutions using modern computer hardware. This is because the functions require the evaluation of terms with numerical values outside the ranges that can be expressed on most computers. Research limitations/implications - Some of existed transformations cannot be applied for high values of relative roughness of inner pipe surface and the Reynolds number. This limitation applied only for computer computations. Other presented transformations do not sufferer of this limitation. Practical implications - Presented procedures can be easily implemented in a computer code. Recommended solution can be used in all cases that can occur in engineering practice. Originality/value - The paper shows some possible practical procedures for solution of the transformed Colebrook equation. Accuracy analysis and comparisons of presented formulas are also performed and recommendation for use is shown.
lambert w-function colebrook equation hydraulic resistances fluid flow flow friction friction factor mathematics turbulent flow