Abstract
The 80 year-old empirical Colebrook function ξ, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor λ, with the known Reynolds number Re and the known relative roughness of a pipe inner surface ε*; λ=ξ(Re,ε*,λ). It is based on logarithmic law in the form that captures the unknown flow friction factor λ in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Padé polynomials with only one log-call in total for the whole procedure (expensive log-calls are substituted with Padé polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution.
Highlights
IntroductionThe empirical Colebrook equation [1,2] implicitly relates the unknown flow friction factor λ with the known Reynolds number Re and the know relative roughness of inner pipe surface, ε∗ ; λ =
The empirical Colebrook equation [1,2] implicitly relates the unknown flow friction factor λ with the known Reynolds number Re and the know relative roughness of inner pipe surface, ε∗ ; λ =ξ ( Re, ε∗, λ), where ξ is functional symbol, Equation (1). 2.51 1 ε∗ √ = −2·log10 ·√ + Re λ λ 3.71 (1)In Equation (1) Re is Reynolds number; 4000 < Re < 108, ε∗ is relative roughness of inner pipe surface; 0 < ε∗ < 0.05, and λ is Darcy flow friction factor; 0.0064 < λ < 0.077
For the most pairs of the Reynolds number Re and the relative roughness of inner pipe surfaces ε∗ which are in the domain of applicability, the initial starting point x0 = 7.273124147 requires the least number of iterations
Summary
The empirical Colebrook equation [1,2] implicitly relates the unknown flow friction factor λ with the known Reynolds number Re and the know relative roughness of inner pipe surface, ε∗ ; λ =. In Equation (1) Re is Reynolds number; 4000 < Re < 108 , ε∗ is relative roughness of inner pipe surface; 0 < ε∗ < 0.05, and λ is Darcy flow friction factor; 0.0064 < λ < 0.077 (all three quantities are dimensionless). The Colebrook equation is based on experiments performed by Colebrook and White in 1937 with the flow of air through a set of artificially roughened pipes [2]. Numerous evaluations of flow friction factor such as in the case of complex networks of pipes pose extensive burden for computers, so an accurate and
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