Explicit Solution for Pipe Diameter Problem Using Lambert W -Function

Two Iterative Methods for Sizing Pipe Diameters in Gas Distribution Networks with Loops

“…where D is the inner pipe diameter (m); Q is the gas flow through the pipe in real conditions of pressure and temperature (m 3 /s); note that the Renouard relation, on the contrary, operates with Q st , gas flow at standard conditions; p st is the standard pressure of 10 5 Pa; u is the gas velocity (m/s); here, used for optimisation, u = 15 m/s; p is the real pressure in pipes (Pa); here, p/p st ≈ 4; π is the Ludolpf number ≈ 3.1415. Pipe diameters as part of a gas network with loops are optimised in this article, which is a different problem [88][89][90] compared to determining the diameter of a single pipe (a solution to the problem of a single pipe in particular conditions is given in [91][92][93]103]). In any case, pipes are standardised and therefore they must be chosen from the prescribed list available for sale on the market [94].…”

Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter

“…This paper uses symbolic regression, Lambert-W function, and new suitable dimensionless groups to find simple but still accurate approximations. The other approaches were successfully tested as well, such as iterative methods [7,8], explicit solutions based on special functions [24,25], and neural networks [37]. Although the goodness of fit is a subject of intensive research [38][39][40][41], the goodness of fit methods are not widely used for flow friction modeling.…”

“…The variety of approximate formulas offered as an alternative to the implicitly given original Colebrook equation suitable for solving the unknown ∆p (head loss ∆h) can be expressed mathematically in a wide range of accuracy and complexity of their structure [9][10][11]. Explicit approximations for direct solutions of the unknown diameter D also exist [12,16,[22][23][24][25]. The error caused by the symbolic regression approach also belongs to this type.…”

“…This section gives the developed formulas for solving for the unknown diameter D. Three different approaches were tried in this article to solve the problem of the unknown diameter D while symbolic regression is used here in a threefold way: (1) brute force of computing power (a similar approach for flow friction factor λ is used in [69]), (2) one method based on the Lambert W-function is simplified [24,25] (details about the Lambert W-function can be seen in [70]), and (3) some old approaches associated with nomograph solutions based on suitable dimensionless groups were further developed and improved in a form suitable for the contemporary pipe-design community [14,16,20,21].…”

“…The flow-discharge problem Q can be easily solved because a closed-form solution exists [12] and, also, the unknown ∆p (head loss ∆h) can be solved relatively easily due to the suitable structure of the input parameters of the Colebrook equation. On the other hand, due to the unfavorable structure of the Colebrook equation, the problem of the unknown pipe diameter D, although very important, was not the main focus of researchers, but, nevertheless, many efforts have been made, such as Powell in 1950 [14], Rajaratnam in 1960 [15], Ranga Raju and Garde in 1966 [16], Lai and Lee in 1975 [17], Swamee and Jain in 1976 [12], Debler in 1977 [18], Swamee and Rathie in 2007 [19], Sakkas in 2014 [20], LaViolette in 2017 [21], Medina et al in 2017 [22], Yetilmezsoy et al in 2021 [23], Lamri and Easa in 2022 [24], Brkić and Praks in 2023 [25], etc. Many of them from the early years solve the problem through today's obsolete nomograms [14][15][16][17][18].…”