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.