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 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 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 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 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 A New Class of Halley’s Method with Third-Order Convergence for Solving Nonlinear Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Mohammed Barrada ◽  
Mariya Ouaissa ◽  
Yassine Rhazali ◽  
Mariyam Ouaissa
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Order Convergence ◽  
Eighth Order ◽  
Third Order ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
New Family ◽  
Number Of Iterations

In this paper, we present a new family of methods for finding simple roots of nonlinear equations. The convergence analysis shows that the order of convergence of all these methods is three. The originality of this family lies in the fact that these sequences are defined by an explicit expression which depends on a parameter p where p is a nonnegative integer. A first study on the global convergence of these methods is performed. The power of this family is illustrated analytically by justifying that, under certain conditions, the method convergence’s speed increases with the parameter p. This family’s efficiency is tested on a number of numerical examples. It is observed that our new methods take less number of iterations than many other third-order methods. In comparison with the methods of the sixth and eighth order, the new ones behave similarly in the examples considered.

A Study on the Rearrangement of Dialkyl 1-Aryl-1-hydroxymethylphosphonates to Benzyl Phosphates

2020 ◽  
Vol 24 (4) ◽  
pp. 465-471 ◽  
Author(s):  
Zita Rádai ◽  
Réka Szabó ◽  
Áron Szigetvári ◽  
Nóra Zsuzsa Kiss ◽  
Zoltán Mucsi ◽  
...  
Keyword(s):  
Quantum Chemical ◽  
Room Temperature ◽  
Phase Transfer ◽  
One Pot ◽  
Substrate Dependence ◽  
Triethylbenzylammonium Chloride ◽  
One Step ◽  
The Pudovik Reaction ◽  
The One ◽  
Solid Liquid

The phospha-Brook rearrangement of dialkyl 1-aryl-1-hydroxymethylphosphonates (HPs) to the corresponding benzyl phosphates (BPs) has been elaborated under solid-liquid phase transfer catalytic conditions. The best procedure involved the use of triethylbenzylammonium chloride as the catalyst and Cs2CO3 as the base in acetonitrile as the solvent at room temperature. The substrate dependence of the rearrangement has been studied, and the mechanism of the transformation under discussion was explored by quantum chemical calculations. The key intermediate is an oxaphosphirane. The one-pot version starting with the Pudovik reaction has also been developed. The conditions of this tandem transformation were the same, as those for the one-step HP→BP conversion.

scholarly journals One-Year Clinical Evaluation of the Bonding Effectiveness of a One-Step, Self-Etch Adhesive in Noncarious Cervical Lesion Therapy

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Babacar Faye ◽  
Mouhamed Sarr ◽  
Khaly Bane ◽  
Adjaratou Wakha Aidara ◽  
Seydina Ousmane Niang ◽  
...  
Keyword(s):  
Clinical Performance ◽  
Retention Rate ◽  
Recall Rate ◽  
Acid Etching ◽  
Significant Difference ◽  
Usphs Criteria ◽  
One Step ◽  
One Year ◽  
The One ◽  
Self Etch

This study evaluated the one-year clinical performance of a one-step, self-etch adhesive (Optibond All-in-One, Kerr, CA, USA) combined with a composite (Herculite XRV Ultra, Kerr Hawe, CA, USA) to restore NCCLs with or without prior acid etching. Restorations performed by the same practitioner were evaluated at baseline and after 3, 6, and 12 months using modified USPHS criteria. At 6 months, the recall rate was 100%. The retention rate was 84.2% for restorations with prior acid etching, but statistically significant differences were observed between baseline and 6 months. Without acid etching, the retention rate was 77%, and no statistically significant difference was noted between 3 and 6 months. Marginal integrity (93.7% with and 87.7% without acid etching) and discoloration (95.3% with and 92.9% without acid etching) were scored as Alpha or Bravo, with better results after acid etching. After one year, the recall rate was 58.06%. Loss of pulp vitality, postoperative sensitivity, or secondary caries were not observed. After one year retention rate was of 90.6% and 76.9% with and without acid conditioning. Optibond All-in-One performs at a satisfactory clinical performance level for restoration of NCCLs after 12 months especially after acid etching.

scholarly journals Building up cyclodextrins from scratch – templated enzymatic synthesis of cyclodextrins directly from maltose

2021 ◽  
Author(s):  
Dennis Larsen ◽  
Sophie R. Beeren
Keyword(s):  
Enzymatic Synthesis ◽  
Combinatorial Libraries ◽  
One Step ◽  
The One ◽  
Kinetic Trapping

Template-induced kinetic trapping of specific cyclodextrins in enzyme-mediated dynamic combinatorial libraries of linear and cyclic α-glucans enables the one-step synthesis of cyclodextrins from maltose in water.

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