Explicit power formula for the Darcy-Weisbach pipe flow equation: Application in optimal pipeline design

Although the Darcy-Weisbach equation combined with the Colebrook-White semitheoretical formula for calculating the friction coefficient is a highly accurate generalized pipe-water flow resistance equation, most users prefer the use of simple, explicit power law form formulas. Because of their simplicity (despite their limitations) the purely empirical power formulas of Hazen-Williams and Manning remain the most popular pipe flow resistance equations used in routine hydraulic engineering applications. In this paper, a new simple power law form formula is derived to approximate the generalized Darcy-Weisbach combined with the Colebrook-White equation. The two main pipe flow parameters, such as the discharge (or velocity) and the diameter, appeared explicitly in the proposed formula. The suggested power-form formula compared with the Darcy-Weisbach and Coolbrook-White equation yields a maximum relative error of about +/- 4.5%. The power-form suggested formula is dimensionally homogeneous and its accuracy is sufficient for practical engineering applications. A correction factor is introduced for the variation of kinematic viscosity with temperature. The usefulness of the formula is demonstrated in an application concerning the optimal design of a delivery pipeline with pumping. The power form of the friction formula facilitates the formulation of the problem leading to the derivation of a simple equation from which the economic diameter is explicitly calculated.