scholarly journals Hardy Cross Method for Pipe Networks

2019 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Flow Resistance ◽  
Water Pipe ◽  
Pipe Network ◽  
Pipe Networks ◽  
Hardy Cross ◽  
Hydraulic Networks ◽  
Newton Raphson ◽  
Balanced Solution ◽  
Number Of Iterations ◽  
Looped Network

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 the flow resistance versus the electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties and it still remains an important tool for analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method which threats the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from the Russian practice published during 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the life of Hardy Cross are also shown. Finally, an improved version of the Hardy Cross method, which significantly reduces number of iterations, is presented and discussed. Also we tested multi-point iterative methods which can be used as substitution for the Newton-Raphson approach used by Hardy Cross, but this approach didn’t reduce number of required iterations to reach the final balanced solution. Although, many new models have been developed since the time of Hardy Cross, main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks or ventilation systems.

scholarly journals Improved Hardy Cross Method for Pipe Networks

2018 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Electrical Resistance ◽  
Flow Resistance ◽  
Engineering Method ◽  
Pipe Network ◽  
Electrical Analog ◽  
Pipe Networks ◽  
Hardy Cross ◽  
Hydraulic Networks ◽  
Number Of Iterations ◽  
Looped Network

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 the flow resistance versus the electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties and it still remains an important tool for analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method which threats the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from the Russian practice published during 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the life of Hardy Cross are also shown. Finally, an improved version of the Hardy Cross method, which significantly reduces number of iterations, is presented and discussed.

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

2019 ◽  
Vol 9 (10) ◽  
pp. 2019 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Flow Resistance ◽  
Flow Distribution ◽  
Water Pipe ◽  
Pipe Network ◽  
Iterative Solver ◽  
Pipe Networks ◽  
Hardy Cross ◽  
Newton Raphson ◽  
Cross Type ◽  
Number Of Iterations

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 that 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 modeling 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.

scholarly journals Application of Third-Order Schemes to Improve the Convergence of the Hardy Cross Method in Pipe Network Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Majid Niazkar ◽  
Gökçen Eryılmaz Türkkan
Keyword(s):  
Network Analysis ◽  
Lower Number ◽  
Pipe Network ◽  
Third Order ◽  
New Methods ◽  
Pipe Networks ◽  
Hardy Cross ◽  
One Step ◽  
The One ◽  
Number Of Iterations

In this study, twenty-two new mathematical schemes with third-order of convergence are gathered from the literature and applied to pipe network analysis. The presented methods were classified into one-step, two-step, and three-step schemes based on the number of hypothetical discharges utilized in solving pipe networks. The performances of these new methods and Hardy Cross method were compared by solving a sample pipe network considering four different scenarios (92 cases). The results show that the one-step methods improve the rate of convergence of the Hardy Cross method in 10 out of 24 cases (41%), while this improvement was found to be 39 out of 56 cases (69.64%) and 5 out of 8 cases (62.5%) for the two-step and three-step methods, respectively. This obviously indicates that the modified schemes, particularly the three-step methods, improve the performance of the original loop corrector method by taking lower number of iterations with the compensation of relatively more computational efforts.

scholarly journals An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks

2017 ◽  
Author(s):  
Dejan Brkic
Keyword(s):  
Gas Flow ◽  
Solution Method ◽  
Distribution Networks ◽  
Original Method ◽  
Gas Distribution ◽  
Gas Network ◽  
Hardy Cross ◽  
Successive Substitution ◽  
Newton Raphson ◽  
Looped Network

Hardy Cross method is common for calculation of loops-like gas distribution networks with known node gas consumptions. This method is given in two forms: original Hardy Cross method-successive substitution methods and improved-simultaneous solution method (Newton-Raphson group of methods). Problem of gas flow in looped network is nonlinear problem; i.e. relation between flow and pressure drop is not linear while relation between electric current and voltage is. Improvement of original method is done by introduction of influence of adjacent contours in Yacobian matrix which is used in calculation and which is in original method strictly diagonal with all zeros in non-diagonal terms. In that way necessary number of iteration in calculations is decreased. If during the design of gas network with loops is anticipated that some of conduits are crossing each other without connection, this sort of network became, so there has to be introduced corrections of third or higher order.

scholarly journals Object-oriented integrated algorithms for efficient water pipe network by modified Hardy Cross technique

2020 ◽  
Vol 7 (1) ◽  
pp. 56-64
Author(s):  
Kailash Jha ◽  
Manish Kumar Mishra
Keyword(s):  
Flow Analysis ◽  
Object Oriented ◽  
Water Pipe ◽  
Pipe Network ◽  
Breadth First Search ◽  
Third Order ◽  
Flow Adjustment ◽  
Hardy Cross ◽  
Nodal Pressure ◽  
Water Pipe Network

Abstract In this work, object-oriented integrated algorithms for an efficient flow analysis of the water pipe network are developed. This is achieved by treating the pipe network as a graph data structure with its nodes as the graph’s nodes and the pipes as the edges. The algorithm for cycle (real cycle or pseudo-cycle) extraction has been developed using nested breadth-first search that gives ordered cycles. Pseudo-loops are found using the shortest path algorithm between the nodes. Pipes are initialized loop by loop using conservation of mass at nodes. A modified Hardy Cross method is used in the proposed work with third-order convergence. The friction factor is updated for every change in discharges. The pressure calculation has been done by the graph traversal algorithm between the reference nodes and node where the pressure is to be calculated using the energy equation. The pressure at all intermediate nodes is obtained in the course of the traversal. Balanced discharges and nodal pressure in the pipe network are compared with the simultaneous loop flow adjustment method and EPANET software. The proposed work gives more efficient flow analysis than the traditional Newton–Raphson-based techniques for complex networks.

scholarly journals Data-driven asset management in urban water pipe networks: a proposed conceptual framework

2021 ◽  
Author(s):  
E. Okwori ◽  
Y. Pericault ◽  
R. Ugarelli ◽  
M. Viklander ◽  
A. Hedström
Keyword(s):  
Data Management ◽  
Conceptual Framework ◽  
Asset Management ◽  
Water Utilities ◽  
Data Driven ◽  
Urban Water ◽  
Water Pipe ◽  
Pipe Network ◽  
Pipe Networks ◽  
Analytical Tools

Abstract Analytical tools used in infrastructure asset management of urban water pipe networks are reliant on asset data. Traditionally, data required by analytical tools has not been collected by most water utilities because it has not been needed. The data that is collected might be characterised by low availability, integrity and consistency. A process is required to support water utilities in assessing the accuracy and completeness of their current data management approach and defining improvement pathways in relation to their objectives. This study proposes a framework to enable increased data-driven asset management in pipe networks. The theoretical basis of the framework was a literature review of data management for pipe network asset management and its link to the coherence of set objectives. A survey to identify the current state of data management practice and challenges of asset management implementation in five Swedish water utilities and three focus group workshops with the same utilities was carried out. The main findings of this research were that the quality of pipe network datasets and lack of interoperability between asset management tools was a driver for creating data silos between asset management levels, which may hinder the implementation of data-driven asset management. Furthermore, these findings formed the basis for the proposed conceptual framework. The suggested framework aims to support the selection, development and adoption of improvement pathways to enable increased data-driven asset management in municipal pipe networks. Results from a preliminary application of the proposed framework are also presented.

Flow Analysis on Piping Networks Using the Finite Element Method

2013 ◽  
Author(s):  
Carlos Luis Moreno
Keyword(s):  
Finite Element ◽  
Local Community ◽  
Computational Simulation ◽  
Flow Analysis ◽  
Working Fluid ◽  
Network Systems ◽  
Hardy Cross ◽  
Newton Raphson ◽  
Number Of Iterations ◽  
Finite Element Methodology

The objective of this work is to apply the Finite Element Methodology (F.E.M.) to several piping systems, using an incompressible working fluid, in order to calculate the volumetric flow on each element and the piezometric load on each node of the network. To accomplish this goal a computational code was designed using Fortran Computational Language. Such a code consists of a main program and six subroutines. The input variables are general data of the network including the number of pipes, the number of nodes, the piezometric load values on nodes where they are constant (tanks for example), demanding flows in those nodes where the fluid is removed from the system, a connectivity table indicating the assumed flow direction in each pipe, and the number of pumps with respective parabolic curve coefficients. Program data also included both the maximum number of iterations and tolerance allowed. Fluid properties such as kinematic viscosity, density and pipe features such as length, diameter and absolute rugosity are also required. The output data include pipe volumetric flows and piezometric load on variable static pressure nodes. In this work, three different network systems were analyzed: 51-, 63- and 65-element networks. All were examples taken from the bibliography. The Finite Element Methodology results were first validated with real data, and then compared with the other results coming from the Hardy-Cross, Newton-Raphson and Linear Methods. The comparison was based on convergence speed and numerical stability. It is concluded that the methodology called Finite Element Methodology requires a smaller number of iterations than the Hardy-Cross, Linear and Newton-Raphson Methods. Another advantage of the Finite Element Methodology is that there is no need to assign the flow initial values that satisfy the Continuity Equation on each node of the piping network before running the program. Also, no loops establishing is needed. In addition, the designed code permits calculations for networks that present both booster and feed pumps. The importance of this work rests on the fact that nowadays it is necessary for piping network flow analysis to use computational simulation in order to design systems more efficiently and economically. Furthermore, this work is important for network construction as well as the satisfaction of consumer demand on a local community level, taking into account prevailing normative requirements. This paper, consequently, aims to contribute to progress in these areas.

scholarly journals A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1306
Author(s):  
Elsayed Badr ◽  
Sultan Almotairi ◽  
Abdallah El Ghamry
Keyword(s):  
Comparative Study ◽  
Numerical Results ◽  
Traditional Methods ◽  
Running Time ◽  
Root Finding ◽  
Average Running Time ◽  
False Position ◽  
Newton Raphson ◽  
Number Of Iterations

In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time.

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