DIRECT PRESSURIZED PIPE DIAMETER DETERMINATION FROM DARCY-WEISBACH AND COLEBROOK-WHITE RELATIONSHIPS

Abstract

The pipe-diameter design problem constitutes a strongly coupled nonlinear engineering system whose governing equations exhibit multiple nonlinear interactions. The proposed analytical reformulation provides an efficient nonlinear solution framework applicable to engineering systems governed by coupled nonlinear equations.

The determination of the diameter of pressurized circular pipes from the Darcy-Weisbach and Colebrook-White equations remains one of the most challenging problems in hydraulic engineering because the unknown diameter simultaneously influences the Reynolds number, the relative roughness, the Darcy friction factor, and the energy equation. This multiple nonlinear coupling has traditionally required nested iterative procedures involving successive evaluations of both the friction factor and the pipe diameter.

The present work develops a rigorous analytical framework for the direct determination of the pipe diameter without modifying either the Darcy-Weisbach or the Colebrook-White relationships. By introducing an appropriate change of variable, the original coupled hydraulic problem is transformed into a single nonlinear scalar equation involving only one unknown. The transformed equation is shown to be continuous and strictly monotonic over the physically admissible domain, thereby proving the existence and uniqueness of the positive solution.

An efficient computational strategy is then developed by combining the Rough Model Method (RMM) with Halley's third-order iterative scheme. The RMM provides a physically meaningful analytical initial estimate located in the immediate vicinity of the exact solution, while a single Halley correction is generally sufficient to recover the exact diameter with near machine precision. Numerical examples demonstrate that the analytical approximation supplied by the RMM alone already yields exceptionally accurate pipe diameters, whereas the one-step Halley refinement reduces the remaining error to practically negligible levels.

The analytical formulation is further extended to the fully rough and hydraulically smooth turbulent-flow limits. Exact asymptotic expressions for the pipe diameter are derived in closed form using the principal branch of the Lambert W function. To avoid iterative evaluation of this special function, a highly accurate analytical approximation of W₀ is employed, leading to explicit diameter formulas exhibiting extremely small relative deviations from the exact Lambert-based solutions.

Finally, the study demonstrates that the general transformed Colebrook-White equation admits a rigorous series-based analytical treatment in which the exact Lambert W-based asymptotic solutions constitute the leading-order kernels, while the neglected resistance mechanism is progressively restored through convergent logarithmic expansions. This development establishes a unified analytical connection between the general turbulent-flow solution and its two asymptotic limits, thereby providing new insight into the mathematical structure of the Darcy-Weisbach-Colebrook-White diameter-design problem.

This final section represents the cornerstone of the study, providing the definitive methodology for the safe hydraulic design of pressurized circular conduits.