A power-law approximation of the turbulent flow friction factor useful for the design and simulation of urban water networks

Abstract

An approximation of the friction factor of the Colebrook–White equation is proposed, which is expressed as a power-law function of the pipe diameter and the energy gradient and is combined with the Darcy–Weisbach equation, thus yielding an overall power-law equation for turbulent pressurised pipe flow. This is a generalised Manning equation, whose exponents are not unique but depend on the pipe roughness. The parameters of this equation are determined by minimising the approximation error and are given either in tabulated form or as mathematical expressions of roughness. The maximum approximation errors are much smaller than other errors resulting from uncertainty and misspecification of design and simulation quantities and also much smaller than the errors in the original Manning and the Hazen–Willians equations. Both these can be obtained as special cases of the proposed generalised equation by setting the exponent parameters constant. However, for large roughness the original Manning equation improves in performance and becomes practically equivalent with the proposed generalised equation. Thus its use, particularly when the networks operate with free surface flow is absolutely justified. In pressurised conditions the proposed generalised Manning equation can be a valid alternative to the combination of the Colebrook–White and Darcy–Weisbach equations, having the advantage of simplicity and speed of calculation both in manual and computer mode.