Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

Abstract

P.P.) Dejan Brkić: https://orcid.org/0000-0002-2502-0601, Pavel Praks https://orcid.org/0000-0002-3913-7800 #Dejan Brkić was on short visit funded by IT4Innovations, VŠB -Technical Abstract: Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method which considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton-Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modelling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton-Raphson iterative method.

What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

“…The proposed exercises are suitable for students of hydraulics (civil, mechanical, chemical, or petroleum engineering curriculum, including the exploitation of oil and gas), but can be used as practical examples in teaching of numerical methods for students of mathematics. The proposed exercises can be extended for the teaching of pipe network analysis [31][32][33][34][35][36]. These methods can be combined with recent experimental works [63].…”

“…Students can learn how to avoid using these time-consuming functions [27][28][29][30]. This simplification process can have a large effect on the students' future jobs, regarding the design of complex pipe networks, which involve extensive computations [31][32][33][34][35][36]. For example, in Section 2.2.1, students can reuse the value of the logarithm, which is computed only in the first iteration, to all subsequent iterations [30].…”

Special Issue on Mathematical Modeling Using Differential Equations and Network Theory

“…In [4], the authors provide an overview of early developments of the Hardy-Cross-type methods for computation of flow distribution in pipe networks.…”

“…In addition, in [4] a method from a Russian practice published during the 1930s, which is similar to the Hardy-Cross method, is also described. Some notes from the work of Hardy-Cross are also presented.…”

Optimization Based on Product and Desirability Functions for Flow Distribution in Multi-Channel Cooling Systems of Power Inverters in Electric Vehicles

“…Therefore, analytical methods have been used in several studies on channel networks to efficiently optimize flow rate distribution. Brkić and Praks proposed the hardy cross method for calculating channel network flow distribution [8,9]. Also, Kim, Yoo, and Cho analyzed incompressible looped flow networks using constitutive topologic matrix equations [10].…”