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

Abstract

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 methodsuccessive 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 inYacobian 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. 2. Note your eq (10) and (16) Answer: English expression is now probably improved and the text is shorter for about 1000 words, one table is deleted, and one figure is deleted. Shorter abstract is included. Readers will have access to online base e.g. science direct, so there will be available version in colour. * Detailed Response to ReviewersEditor:1. The length of the manuscript is over the required 10 printing pages by Applied Energy.Authors shall shorten to make it much concise. Suggestion accepted in full: Text is shorter for about 1000 words, one table is deleted, and one figure is deleted. Many equations are deleted, only necessary equations are now in the text. References are checked, and only essential references are still in the text. In the text is something above 5000 words including title, abstract, references… 2. Authors shall discuss how to apply the results in real applications to give readers who might be interesting to use the results from this paper.

One-Log Call Iterative Solution of the Colebrook Equation for Flow Friction Based on Padé Polynomials

“…The procedure proposed in this paper can significantly reduce the computational burden for evaluating complex distribution networks with various applications (water, gas) (Brkić 2009, Brkić 2011a, Praks et al 2015, Praks et al 2017, Brkić 2016. For example, a probabilistic approach using time dependent modeling of distribution or transmission networks requires many millions of evaluations of Colebrook's equation, which means that it is not a computationally cheap task at all.…”

“…Numerous evaluations of flow friction factor such as in the case of complex networks of pipes pose extensive burden for computers, so not only an accurate but also a simplified solution is required. Calculation of complex water or gas distribution networks (Brkić 2009, Brkić 2011ab, Praks et al 2015, Praks et al 2017 which requires few evaluations of logarithmic function for each pipe, presents a significant and extensive burden which available computer resources hardly can easily manage (Clamond 2009, Giustolisi et al 2011, Danish et al 2011, Winning and Coole 2013, Vatankhah 2018. The Colebrook equation is based on logarithmic law where the unknown flow friction factor is given implicitly, i.e., it appears on both sides of Equation (1) in form , from which it cannot be extracted analytically; an exception is through the Lambert -function (Keady 1998, Sonnad and Goudar 2004, Brkić 2011cd, Brkić 2012ab, Biberg 2017, Brkić 2017a.…”

Discussion of “Method to Cope with Zero Flows in Newton Solvers for Water Distribution Systems” by Nikolai B. Gorev, Inna F. Kodzhespirov, Yuriy Kovalenko, Eugenio Prokhorov, and Gerardo Trapaga

“…In other words, the lowest pressure of water is somewhere between the two nodes (Brkić 2009). This situation is not allowed and cannot be calculated using any of the Hardy Cross type methods of s for calculation of looped (Brkić 2011b).…”

“…Because the Darcy-Weisbach model with the Colebrook formula for the friction factor is theoretically more sound (Brkić 2011c(Brkić , 2012b, the use of the Hazen-Williams equation is strongly discouraged. Finally, the Darcy-Weisbach model can also be used for calculation of gas distribution networks, whereas the Hazen-Williams model cannot be used in any circumstances (Brkić 2009(Brkić , 2011a. …”

Evolutionary Optimization of Colebrook’s Turbulent Flow Friction Approximations

“…Looped network of pipes: The use of the accurate explicit approximations should be prioritized over the use of the iterative solution in the calculation of looped networks of pipes, since in that way, double iterative procedures, one for the Colebrook equation and one for the solution of the whole looped system of pipes, can be avoided [63][64][65][66][67].…”