Colebrook White Equation Research Articles

Constraints for Using Lambert W Function-Based Explicit Colebrook–White Equation

We analyze the general applicability of a recent explicit expression of the Colebrook.White equation for turbulent flow friction factor calculation. This explicit expression, which is based on the Lambert \IW\N function, is characterized by an exponential term which imposes restrictions on its use. These constraints have been expressed in terms of pipe roughness (ϵ/\ID\N) and the Reynolds number R that are required for friction factor calculation. These constraints were determined as 8.0666 ln(R) + (ϵ/\ID\N)R<721.97 and 8.0666 ln(R) + (ϵ/\ID\N)R<5731.83, respectively, for machines using single precision and double precision computations. Using the Lambert W function, an explicit equation relating Rand ϵ/\ID\N was derived at the limiting case which allowed for a graphical representation of the applicability of the explicit form of the Colebrook.White equation in the Rversus ϵ/\ID\N space. Before computing friction factors using the explicit Colebrook-White equation, a quick check must be performed to see if the desired combination of Rand ϵ/\ID\N values satisfies the applicable constraint mentioned above.

Symbolic and numerical regression: experiments and applications

This paper describes a new method for creating polynomial regression models. The new method is compared with stepwise regression and symbolic regression using three example problems. The first example is a polynomial equation. The two examples that follow are real-world problems, approximating the Colebrook–White equation and rainfall-runoff modelling. The three example problems illustrate the advantages of the new method.

Improved explicit equations for estimation of the friction factor in rough and smooth pipes

The most common correlations for calculating the friction factor in rough and smooth pipes are reviewed in this paper. From these correlations, a series of more general equations has been developed making possible a very accurate estimation of the friction factor without carrying out iterative calculus. The calculation of the parameters of the new equations has been done through non-linear multivariable regression. The better predictions are achieved with those equations obtained from two or three internal iterations of the Colebrook–White equation. Of these, the best results are obtained with the following equation: 1 f =−2.0 log ε/D 3.7065 − 5.0272 Re log ε/D 3.827 − 4.567 Re log ε/D 7.7918 0.9924+ 5.3326 208.815+Re 0.9345 .

A ROBUST INTEGRATED COMPUTER-AIDED DESIGN PACKAGE FOR URBAN DRAINAGE NETWORKS

This paper presents the development and verification of a computer aided design and drafting package for medium sized municipal storm water drainage systems (DRAINAGE). The numerical model, which is designed for use on microcomputers, is written in PASCAL Language and is compiled by PC software TURBO PASCAL version 6.0. The computer package for flow prediction and drainage design applies the Colebrook White Equation and the Rational Method to route pipe flows through tree-type drainage networks, automatically adjusting drainage pipe diameters to fulfil flow requirements and backwater effects. The progrmn outputs are written as DXF files which can be read and displayed readily as drawings of drainage layout plml ,md longitudinal profiles in an AutoCAD environment. DRAINAGE replaces the timeconsuming conventional method for designing stormwater drainage networks. Although the prognun is tailored for application and use in Hong Kong, it can be easily adapted to other situations.

ANÁLISE DE EQUAÇÕES EXPLICITAS PARA O CÁLCULO DO COEFICIENTE "f" DA FÓRMULA UNIVERSAL DE PERDA DE CARGA

Neste estudo foram comparadas três equações explícitas para o cálculo do coeficiente "f da fórmula universal de perda de carga com a equação de Cole-brook-White. A equação obtida pela substituição do segundo termo (&lt;img border=0 width=26 height=21 src="../../../../../../img/revistas/cr/v22n2/a06img01.gif"&gt;) do argumento do logaritmo da equação de Colebrook-White, pela aproximação de Konakov (NEKRASOV, 1968), foi a que apresentou o melhor ajuste (r² = 0.9996), quando se correlacionou os resultados obtidos pelas três equações explícitas aos valores fornecidos pela fórmula de Colebrook-White. O erro relativo máximo foi da ordem de 2.72 %.

“Negative roughness” and polymer drag reduction

Based on an analogy to the Colebrook-White equation, a technique has been developed to allow polymer-solution extrapolation or “scaling” from one pipe size to another at constant values of ΔB. Each experimental data point can be transferred to a new pipe size by a simple, pocket-calculator method which preserves the experimental value of ΔB exactly. Thus scaling can be easily accomplished, without resorting to iteration or graphical techniques. The “negative-roughness” idea can also explain the loss of ΔB or drag reduction with increasing flow velocity.

Applicability of the Colebrook-White Formula to Represent Frictional Losses in Partially Filled Unsteady Pipeflow.

The use of Manning's n as a friction factor is shown to be unsuitable in the case of small bore (less than about one meter diameter) partially filled pipeflow, particularly for relatively smooth materials such as glass and cast-iron. The Colebrook-White equation with the roughness coefficient k is presented in a form suitable for inclusion in a computer program to solve the partially filled unsteady pipeflow equations by means of the method of characteristics. Results are presented which show that the Colebrook-White equation provides substantially improved predictions of the wave velocity along the pipe. It provides slightly improved predictions for the maximum depth of flow along the pipe.

Explicit Equations for Pipe-Flow Problems

Direct solutions of pipe flow problems are not possible because of the implicit form of Colebrook-White equation which expresses the hydraulic resistancee of commercial pipes. The three basic and major problems encountered in hydraulic engineering practice are the determination of pipe diameter, the discharge and the head loss. The solution of these problems on conventional lines involves many trials and tedious computations. Some research workers have proposed graphical solutions which have their own inherent limitations. Reported herein are explicit and accurate equations for pipe diameter and head loss and a closed form solution for the discharge through the pipe, based on Colebrook-White equation. These explicit equations can also be utilized with advantage in optimization studies of pipelines and water distribution systems.