Novel high-precision explicit equations for pipe diameter in turbulent flow via modified rough model method (MRMM)

Abstract

This study addresses key hydraulic engineering challenges in turbulent pipe flow - computing flow rate (Q), hydraulic energy slope Sf , and pipe diameter (D) - by introducing the Modified Rough Model Method (MRMM). We propose novel, high-precision explicit equations for D (Eqs. 56 and 60). These achieve maximum relative errors of 0.017 % and 0.0085 %, respectively. We also introduce an innovative friction factor equation (54) with 0.086 % error. Validated across the entire Moody diagram (ε/D = 0 to 0.05, and 2300 ≤ R ≤ 108) using a brute-force approach with over 7 million data points, these non-iterative solutions outperform existing models. A comprehensive set of statistical metrics including Mean Absolute Error (MAE), Root Mean Square Error (RMSE), correlation coefficients (R² and Pearson’s R), Bias, Mean Relative Error (MRE), Standard Deviation (SD), Coefficient of Variation (CV), and maximum relative error were employed to assess the accuracy and reliability of the proposed and existing formulas; the results of the Statistical metrics confirm their robustness, establishing a new benchmark for accuracy in pipeline design. This advancement enhances efficiency and reliability in water, oil, and gas transport systems.