Hardy Cross Method Research Articles

Hydraulic Modeling of Water Distribution Networks: Hardy-Cross Method Implemented in Python

Distribution network is a part of the water supply system that, formed by pipes and accessory parts, aims to provide potable water to consumers continuously with recommended quantity and pressure (TSUTIYA, 2006). However, due to the exponentially growing demand for potable water, the design of network layouts has become increasingly complex nowadays. In this regard, due to regional parameters such as topography and supply size, the use of meshed networks is feasible, in other words, networks that can supply a point through different paths due to their ring or block shape. Thus, this study aims to evaluate the performance of a meshed distribution network of a fictitious layout developed by undergraduate Civil Engineering students at the Federal University of Alagoas (UFAL) through system modeling in a Python program using the Hardy-Cross method. Furthermore, the effectiveness of the computational program was validated by comparing its results with those provided by the EPANET software. The methodology was based on the bibliographic review of issues related to solving meshed networks using the Hardy-Cross method, where a fictitious problem was modeled with parameters specified in NBR 12218/2017. Thus, this method was implemented in a computational program in Python language, and from this, the results related to the established layout were obtained. Additionally, the same layout was built in the EPANET software, and its results were used to validate the computational program. Therefore, it was possible to conclude that the computational program presented satisfactory results when compared to the EPANET software data, obtained due to recent performance advances in the Python language, which adds greater ease of use, result accuracy, modeling problem speed, and expands the possibilities of evaluating network performance. Finally, it was found that both modeling approaches are in accordance with the reference values established by TSUTIYA (2006) and recommended by NBR 12218/2017.

Multi-objective topology optimization of macro structure and microtubule network structure for self-healing material

Abstract Self-healing materials possess the capability to promptly repair minor damages occurring during service, thereby effectively preventing safety accidents. This paper investigates a multi-objective topology optimization method for the macro structure and microtubule network of self-healing materials around pure epoxy resin materials, aiming to enhance the damage healing capability of the microtubule network while meeting the mechanical performance requirements of the macro structure. By introducing the design variables of macro structure and microtubule network, the corresponding topological description functions are established respectively. And study applies logical operations and post-processing techniques to generate an embedded microtubule network structure description. The objective functions include the flexibility of the macro structure, the along-travel head loss, and the total length of the microtubule network, with material volume serving as a constraint. In order to determine the head loss of the three-dimensional microtubule network structure, a Hardy-Cross method based on flow initialization and loop search is proposed. Multi-objective topology optimization is designed based on moving morphable components algorithm, enumeration method and Pareto principle. Develop iterative termination conditions by assessing the disparity between Pareto solution sets in each generation, thereby ensuring algorithm convergence. The numerical example of the Messerschmitt–Bölkow–Blohm (MBB) beamyields a flexibility of 0.059 without a carrier and 0.0728 with a carrier the macrostructural flexibility without a carrier is 81.0% compared to with a carrier, and the macrostructural profiles and the overall flexibility of the MBB beams with/without a carrier are close to each other. This method serves as a reference for optimizing large-scale self-healing structures.

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

Closed-loop pipe systems allow the possibility of the flow of gas from both directions across each route, ensuring supply continuity in the event of a failure at one point, but their main shortcoming is in the necessity to model them using iterative methods. Two iterative methods of determining the optimal pipe diameter in a gas distribution network with closed loops are described in this paper, offering the advantage of maintaining the gas velocity within specified technical limits, even during peak demand. They are based on the following: (1) a modified Hardy Cross method with the correction of the diameter in each iteration and (2) the node-loop method, which provides a new diameter directly in each iteration. The calculation of the optimal pipe diameter in such gas distribution networks relies on ensuring mass continuity at nodes, following the first Kirchhoff law, and concluding when the pressure drops in all the closed paths are algebraically balanced, adhering to the second Kirchhoff law for energy equilibrium. The presented optimisation is based on principles developed by Hardy Cross in the 1930s for the moment distribution analysis of statically indeterminate structures. The results are for steady-state conditions and for the highest possible estimated demand of gas, while the distributed gas is treated as a noncompressible fluid due to the relatively small drop in pressure in a typical network of pipes. There is no unique solution; instead, an infinite number of potential outcomes exist, alongside infinite combinations of pipe diameters for a given fixed flow pattern that can satisfy the first and second Kirchhoff laws in the given topology of the particular network at hand.

ANALISIS JARINGAN PIPA DISTRIBUSI AIR BERSIH DI DESA SRIMULYO KECAMATAN AIR SALEK KABUPATEN BANYUASIN

Srimulyo Village, Air Salek District has a water connection with PDAM Tirta Betuah as a clean water service provider. The problem that occurs is the lack of water pressure, so it cannot meet the need for clean water for 24 hours. Apart from loss of head, it is projected that there will be changes in population which will increase the burden on clean water capacity at the research location. This research was conducted to determine the existing clean water needs, population projections, and determine head loss. This calculation uses geometric methods to calculate population projections, as well as the Hardy Cross method to calculate pressure loss. In the drainage area in Srimulyo Village, the total need for clean water in 2022 is 371,760 lt/day, the projected population for the next 5 years in 2027 is 5,164 people, and in the next 10 years 2032 it is 6,435 people. And the loss of pressure height occurred in block 3, namely 6.03 m and the remaining pressure was 33.34 m. This is caused by a change in pipe diameter from 6 inches to 4 inches.

Use of EPANET Software in Network Planning for Drinking Water Supply Systems in Balikpapan Tengah District and Balikpapan City District

Clean water is a primary need for everyone. The need for clean water will increase over time as well as the increase in population, this makes it necessary to carry out an assessment related to water demand in the future up to 10 years in the future. Therefore, considering the raw water source whose availability cannot be predicted so that it can be calculated the need for clean water for the District of Balikpapan Tengah and Balikpapan City, the water demand value is obtained, namely the debit in the District of Central Balikpapan is 352 liters/second and the District of Balikpapan Kota is 295 liters. /second. After that, an analysis of the drinking water distribution routing network was carried out using the Hardy-Cross method and the use of EPANET 2.0 software. Then calculations are carried out by entering data in the form of needs data such as maps of the main pipeline network, 24-hour flow patterns, and data related to contours or elevation. In terms of speed and accuracy of analysis using EPANET 2.0 software for drinking water supply systems is considered more fulfilling.

Optimisation of the choice of the sizing method for drinking water networks in the developing countries: A case of Cameroon

In developing countries (DC), particularly in Cameroon, the demand for drinking water is increasing. To date, the methods of design, sizing and extension of existing drinking water networks are done through the EPANET application which is a simulation software based on the Hardy-Cross method. The main objective of this study is to set up a local tool adapted to the context of D.C. allowing the design, sizing and extension of networks based on the rough model method (RMM). Thus, on the basis of a previously modelled existing network, a comparison was made between the Hardy-Cross method and the RMM (according to the criteria: costs, speed of execution, aptitude for automation, fidelity and precision) with the multi-criteria analysis tool PROMETHEE II. The results of this analysis show that the best adapted method for the design, sizing and extension of drinking water networks is the RMM. It appears that the setting up of water supply networks in DC must be done by the RMM for affordable costs, rapid execution, easy reproducibility and reliable precision.

Two-layer hydrodynamic network model for redox flow battery stack with flow field design

A two-layer hydrodynamic network model is proposed to analyze the electrolyte distribution within a redox flow battery (RFB). The proposed model consists of a channel, a porous electrode, and an interconnection part through which the electrolyte flows through the channel and porous electrode. The flow rate and pressure drop of each network component are calculated using the Hardy-Cross method. The predicted pressure drop is consistent with experimental and three-dimensional computational fluid dynamics (CFD) results. The velocity and pressure distributions are also in good agreement with CFD results. Furthermore, the proposed model analyzes the flow distribution of cells in RFB stacks with different flow field designs. The results show that the flow rate difference between the first and last cells of the RFB stack with an interdigitated flow field is 20 times higher than that of a flow-through-type.

Numerical and Theoretical Investigation to Estimate Darcy Friction Factor in Water Network Problem Based on Modified Chun-Hui He’s Algorithm and Applications

In this work, we applied the modified Chun-Hui He’s algorithm to evaluate for estimation of flow friction factor f for value of friction factor by using Colebrook–White relation. The speedy, precise, and consistent evaluation of flow friction factor f are essential for evaluation of pressure dips and streams in complex network prototypes at distinct values of diameters of pipes. Friction factor estimated outcomes are applied in everyday engineering routine. Numerous computational systems tested for distinguishing of water pipe networks resolution, such as Hardy Cross method (HCM), Newton method (NM), and modified Newton method (MNM), are presented. As a novelty, a modified Newton method tabulated data, graphical results, and comparisons that are presented with different numerical schemes.

A Hardy Cross Approach for Hydraulic Modelling of Water Pipe Networks

Whenever there are substantial variations in the quantity of demands within a metropolitan water network, it is necessary to assess the pipe network to aid the water utilities in decision making. Variability in demand exists every time new industries or residences are connected to the network. In cases where no analyses are done prior to making new connections, unnecessarily huge funds are incurred and use of unreasonably bigger pipes is inevitable, some of which may stay redundant. The present study aims at developing a user-friendly numerical hydraulics model for analysing compound pipe networks. The model was developed using the V-Model approach, written in visual basic language to resolve the elementary pipe system equations using the improved Hardy Cross method. This program examines steady-state flows, head losses, flow velocities, and pressures for single, two, three, and four loop water distribution networks. The four-loop example represents the entire network of the case study area in consideration. The comparative study conducted on results from the program and EPANET indicated consistency in the results as coefficient of determinant, R^2, for all the computed variables was approximately unity (1). The Root Mean Square Error (RMSE) and Mean Bias Error (MBE) were found to be reasonably so small. Therefore, it can be concluded from the statistical analysis that the model is reliable for the analysis of a water network consisting of 1, 2, 3, and 4 closed loops.

Analysis of water distribution systems and head losses using the hardy-cross method

Distribution services have not been running optimally at PDAM Tirta Meulaboh, that This study aims to determine the level of pressure loss in the PDAM’s clean water pipe network. The method used in this research is a quantitative descriptive survey method supported by primary data and secondary data. Calculation of the amount of pressure loss using the hardy cross method. The results showed that the observed value of discharge with the highest water consumption was at 18:00 WIB at 0.172 m3 / person, while for the lowest use was at 14:00 WIB at 0.119 m3 / person. It can be concluded that this time is the peak time for water consumption. High-pressure water where the maximum average is 0.00000599 m, because high pressure is not adequate. The maximum loss of high pressure is found in the loop (VII) of 2.57025 m or 0.257025 atm, for the minimum pressure there is in the loop (VI) of - 0.00024 m or -0.000024 atm. It is necessary to re-evaluate the distribution network system by PDAM such as surveys to find out the causes of high water pressure losses to customers such as surveys for water leaks in distribution pipes, replacing old pipes.

Effect of Asymmetry in the Wing Vein Network

A wing vein network of a fore wing of a Drosophila fly has asymmetry. One vein breaking symmetry of the wing vein network was found via application of our topological transformation. This asymmetry-creating wing vein can be a strategy to improve functionality of the wing veins. As one of multiple functions of the wing veins, they are pipes to transport blood according to the entomological knowledge. However, there have been no engineering attention to the function of vein network as pipe network for blood transportation and are no understandings of how blood circulation is achieved in such a network consisting of extremely narrow pipes based on physics. In this study, we revealed the effect of the asymmetric vein network structure on the blood circulation via comparison with symmetric vein network without the asymmetry-creating vein. A numerical simulation based on the Hardy-Cross method allowed us to demonstrate how addition of this vein influences blood circulation in the wing. The result of that showed the asymmetrical structure improves blood flow distribution in the posterior region of the wing and energy consumption by pumping organ for circulation. Our results interpreted importance of the asymmetry caused by one irregularly positioned vein for the efficient maintenance of the wing via circulation.

Functional Asymmetry In Fly’s Wing Veins as Pipe Network

The wing veins of insects are blood-transporting pipes that form a network. The wing vein network of a Drosophila fly has asymmetry created by one vein named V14. The advantage of the asymmetry in the fly’s wing veins was discussed via pipe network calculation, the Hardy-Cross method. Our numerical simulation revealed V14 influences blood flow in the veins on the posterior region of the wing, and it contributes to a decrease in the pressure difference between the inlet and outlet of the network required to flow the blood. This result suggests that the asymmetric wing vein network is a strategy for less energy consumption in pumping blood into the wing veins of the fly.

Fastest Modified Model of Hardy Cross Method for Ventilation Network Analysis of Mines (Second Conflation Model)

Ventilation network design is done in manual and computerized methods. Computerized method is based on mathematical approximate methods. Several algorithms were presented in mathematical approximate methods for analyzing of water distribution and ventilation networks. Hardy Cross method is the most commonly model of mathematical approximate method for analyzing of ventilation networks in mine. For faster convergence to at the final result of Hardy Cross method were presented other models such as Wang model, conflation model and Newtonian models (First, third and sixteenth). In this paper is performed an initial review of Hardy Cross method and its modified models. Then first, third, and sixteenth modified models of Newtonian are presented for more accurate analysis of ventilation networks in mines. Finally, second conflation model will be presented as the fastest modified model of Hardy Cross method to achieve at the final result.

Hydraulic Models for Calculating Head Loss in Water Distribution System: a case study in Konya

The management of the water supply system becomes crucial day by day because of limited natural water sources and rapidly increasing population all over the world. However, the water distribution systems have various technical problems and the energy loss in the system can be counted one of these problems. In this study, the water distribution system of a designated area in the Yaka district of Konya was examined using Hardy-Cross method and, the hydraulic simulation softwares EPANET, and WaterCAD which are used in hydraulic modeling of the water distribution systems frequently. The same data of designated area were used for EPANET and WaterCAD in the study. The calculations demonstrated that the head loss calculated in EPANET is 3.5% and %16.2 lower than that calculated by WaterCAD and Hardy-Cross, repectively when PVC pipes were used in the water network. When polyethelene pipes were used in the water network, the head loss calculated in EPANET is 3.8% and %26.6 lower than that calculated by WaterCAD and Hardy-Cross, repectively. In the case of choosing WaterCAD and Hardy-Cross as the model for water distribution system, the project cost would increase slightly due to increasing the diameter of the pipes for reducing the head loss. It can be concluded that by comparing the head losses, EPANET is more efficient than WaterCAD and Hardy-Cross.

Numerical and simulation analysis comparison of hydraulic network problem base on higher-order efficiency approach

The hydraulic network study is a real-life engineering problem extensively base on applied and computational approaches. Several numerical schemes are tested for defining of hydraulic pipe networks solution, such as, Hardy Cross Method, Newton Method and Modified Newton method are presented in this paper. Convergence analysis are also compared and deliberated by using a multi-loop hydraulic network. As a novelty, a Modified Newton method tabulated data, graphical results and comparisons are presented with the Hardy-Cross method and 2nd -order Newton–Raphson method. On other hand, we applied Ancient Chinese Algorithm and its modification for value of friction factor by using Colebrook-White relation. Numerical efficiency and comparison of Ancient Chinese Algorithm with Newton method and Bubbfil Algorithm I and Algorthm-II is presented and discussed quantitative approach to demonstrate the feasible result.

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

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.

The Hardy Cross method and its implementation in Spain

In May 1930, Hardy Cross (1885-1959) published an article called ‘Analysis of continuous frames by distributing fixed-end moments’ in the American Society of Civil Engineers (ASCE). This article proposed a new approach to Structural Theory, and its relevance could be compared to that of the Three Moments Theorem (also known as the Clapeyron Theorem). The Cross method, as this calculation methodology has been often called, had remarkable significance from the moment it came out until the 70s, when new calculation methods became popular.
 In the present article, we will be trying to evaluate its impact in locations far from its origins; in particular, how it was understood and formulated in Spain. As will be demonstrated, the importance of this method was extremely relevant for theconstruction of new buildings and the implementation of new industries, which started to appear in a decisive moment for the development of the country. Even though the Hardy Cross method was the most widely used methodology at the time, two other procedures were also available; namely, the Kani and the Takabeya methods, methods that would also appear in the technical bibliography of the time. Despite the infrequent implementation of these other methods, we have briefly referred to both of them in the present paper. This article aims to show the relevance of the Cross method as well as its early implementation in Spain, by using academic bibliography of that time.

Analysis of Flowrate in Loop Gas Pipeline Networks Using Pipe 2016 and Hardy Cross Method

Analysis of Flowrate in Loop Gas Pipeline Networks Using Pipe 2016 and Hardy Cross Method

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

AbstractIn 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.

A visual application for teaching pipe flow optimization in engineering curricula

AbstractThe main purpose of this study is to investigate the importance of programming in the learning and understanding of water distribution networks. This paper presents the implementation of a programming language C# for applications for teaching water distribution networks calculations at both undergraduate and graduate levels. A user's guide is also prepared for the software which includes the theory of hydraulic and optimization calculations. There are many ways to design water distribution networks. Hardy Cross method is very often used for optimization of networks. Thus, Hardy Cross optimization method was preferred in this study. It is a perfect study to implement as an educational tool at undergraduate and graduate levels of education.