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A hybrid programming methodology was proposed to derive a simple empirical formulation for the estimation of the required pipe diameter in the sizing problems (Type 3) of pipe distribution systems. The model was derived based on multiple regression-based analysis using the Richardson's extrapolation approach and the Levenberg–Marquardt algorithm with double precision. The proposed formulation was developed using a total of 300,000 different data points within the framework of MATLAB and DataFit scientific software. The application of the model was explored for a wide range of five fundamental pipeline design variables [absolute roughness of the pipe wall (ε=0–9 mm), water temperature (T=5°C–100°C), pipe length (L=5–500 m), flow rate (Q=0.01–1 m3/s), and head loss (Δh=1–30 m)] and tested against a total of 10,000 additional computational scenarios and the available models reported in the literature. The uncertainty prediction of the proposed formula was quantified and compared with those of existing prediction models. For the new empirical equation, the mean prediction errors between the estimated and the theoretical diameter values (calculated from the numerical solution of the Colebrook–White equation) were significantly smaller than those of existing models. Moreover, the narrowest uncertainty bands, the lowest 95% confidence prediction error intervals, and the lowest amounts of expanded uncertainty (U95) were achieved for the proposed model. Other statistics (e.g., mean absolute relative error, absolute relative error, coefficient of variation of root mean squared error, determination coefficient) also corroborated that the proposed empirical model produced realistic estimations that were superior to those obtained from other well-known explicit models in the literature. The findings of this study concluded that the computational analysis yielded a simple mathematical structure to be easily and accurately used for educational and practical purposes. [ABSTRACT FROM AUTHOR]

The discussers mentioned that "the developed explicit solution of Eq. (7) of the original paper is applicable only to the transitional region and cannot predict the pipe diameter for either the rough ( HT ht ) or the smooth ( HT ht ) flow regime because for these values, Eq. (7) yields HT ht ." These ranges correspond an absolute roughness of the pipe wall ( HT ht ) range of 0.000396-8.999052 mm. [Extracted from the article]