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

• Novel Explicit Equations: Development of highly accurate and explicit formulas for pipeline diameter ( (56) , (60) ) and friction factor ( Eq. (54) ) using the Modified Rough Model Method (MRMM), improving hydraulic calculations in pipeline networks. • Exceptional Precision: Achieves maximum relative errors as low as 0.0172 % and 0.0086 % for pipeline diameter calculations and 0.086 % for friction factor, outperforming existing models used in pipeline design, optimization, and performance assessment. • Comprehensive Applicability: Proposed formulas are valid across the entire range of the Moody diagram ( 0 ≤ ε / D ≤ 5 × 10 − 2 , 2300 ≤ R ≤ 10 8 ), covering all turbulent flow regimes encountered in pipelines, including water supply systems, oil and gas transmission, and industrial fluid transport. • Robust Statistical Validation: Results evaluated using statistical metrics including MAE, RMSE, R², Bias, CV, and maximum relative error, ensuring reliability and accuracy for practical pipeline engineering applications. • Improved Practicality: Explicit equations eliminate iterative calculations, simplifying their application in pipeline hydraulics, reducing computational time, and facilitating quick assessments for pipeline engineers. • Enhanced Methodological Foundation: Introduces the Modified Rough Model Method (MRMM) as an improvement over the Rough Model Method, addressing limitations in traditional pipeline flow models and broadening applicability to complex pipeline networks. • Significant Engineering Contribution: Establishes a new benchmark for precision in explicit friction factor and pipe diameter calculations, making it highly relevant for the design, efficiency, and operational reliability of pipelines in various industries. This study addresses key hydraulic engineering challenges in turbulent pipe flow - computing flow rate (Q), hydraulic energy slope S f , 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 ≤ 10 8 ) 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.