Short overview of early developments of the Hardy Cross type methods for computation of flow distribution in pipe networks

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.Web of Science910art. no. 201

An efficient iterative method for looped pipe network hydraulics free of flow-corrections

The original and improved versions of the Hardy Cross iterative method with related modifications are today widely used for the calculation of fluid flow through conduits in loop-like distribution networks of pipes with known node fluid consumptions. Fluid in these networks is usually natural gas for distribution in municipalities, water in waterworks or hot water in district heating systems, air in ventilation systems in buildings and mines, etc. Since the resistances in these networks depend on flow, the problem is not linear like in electrical circuits, and an iterative procedure must be used. In both versions of the Hardy Cross method, in the original and in the improved one, the initial result of calculations in the iteration procedure is not flow, but rather a correction of flow. Unfortunately, these corrections should be added to or subtracted from flow calculated in the previous iteration according to complicated algebraic rules. Unlike the Hardy Cross method, which requires complicated formulas for flow corrections, the new Node-loop method does not need these corrections, as flow is computed directly. This is the main advantage of the new Node-loop method, as the number of iterations is the same as in the modified Hardy Cross method. Consequently, a complex algebraic scheme for the sign of the flow correction is avoided, while the final results remain accurate.Web of Science42art. no. 7

Spreadsheet-based pipe networks analysis for teaching and learning purpose

An example of hydraulic design project for teaching purpose is presented. Students’ task is to develop a looped distribution network for water (i.e. to determinate node consumptions, disposal of pipes, and finally to calculate flow rates in the network’s pipes and their optimal diameters). This can be accomplished by using the original Hardy Cross method, the improved Hardy Cross method, the node-loop method, etc. For the improved Hardy Cross method and the node-loop method, use of matrix calculation is mandatory. Because the analysis of water distribution networks is an essential component of civil engineering water resources curricula, the adequate technique better than the hand-oriented one is desired in order to increase students’ understanding of this kind of engineering systems and of relevant design issues in more concise and effective way. The described use of spreadsheet solvers is more than suitable for the purpose, especially knowing that spreadsheet solvers are much more matrix friendly compared with the hand-orientated calculation. Although matrix calculation is not mandatory for the original Hardy Cross method, even in that case it is preferred for better understanding of the problem. The application of commonly available spreadsheet software (Microsoft Excel) including two real classroom tasks is presented

Evaluation of Flow Rate Correction in Water Pipeline Distribution Network by Two Numerical Methods of Solution

This study evaluates flow rate correction and approximate flow rates in  loops for three different case studies of closed looped pipe distribution network systems using Hardy Cross and Newton Raphson. Darcy Weisbach head loss equation was also used to account for major losses. Manual calculation was initially done for each case study followed by a C-Sharp programming software which was developed to affirm the manual  calculation. For one looped network, head loss around the loop converged from 25.60 m to 0.13 m at the third iteration. The two looped network head loss around each loop converged from 170.97 m and 8.92 m to 0.05 and 0.06 m for Hardy Cross at the sixth iteration while the head loss are 0.88 m and 0.24 m at the fourth iteration for both Hardy Cross and Newton Raphson method while for the three looped network, it has head losses around the three loops converged after the fourth iteration from 0.26, 1.36 and 18.32 m to 0.13, 0.11 and 0.10 m respectively for Hardy Cross at thirditeration while the head losses are 0.03, 0.00 and 0.05 m for Newton  Raphson method. Newton Raphson method was found to have a better convergence pattern because it convergences in a uniform manner unlike Hardy Cross method. Also, the program developed gave almost but more accurate results as compared to that of manual calculations with the agreement between them rated at 98%. Some slight differences encountered in the mathematical terms calculated were as a result of some accumulated approximation errors.Keywords: Pipe Distribution Network, Head loss, Convergence, Iteratio

Optimization methods applied to the design of eco-industrial parks: a literature review

With the growing environmental concern, there is evidence that increasing symbiotic relationship between plants in the same industrial area, highly contributes to a more sustainable development of industrial activities. The concept of industrial ecology extended to the terms of eco-industrial park (or ecopark) or industrial symbioses is the topic of extensive research since the five last years. More particularly, even if a lot of ecopark examples and realizations already exist throughout the world, a lot of ecopark proposals are in progress but not achieved. Recently, this vision leads the research community to focus on works proposing methods to optimize the exchanges of an ecopark prior to its design and construction. We find it especially interesting for the scientific community to propose a detailed paper review focused on optimization works devoted to the design of eco industrial parks. This paper is based on a comprehensive literature search in Web of Science database for publications that listed ‘industrial symbiosis’ (or ‘eco industrial park’, or ‘inter plant integration’) and ‘optimization’. This study is segmented into different sections with first, a description of the different concepts evoked in the literature. Then, the several types of networking in an eco-industrial park are detailed in association with the optimization methods employed to solve each problem. The following sections reviews the different objective functions that are formulated to optimally design an eco-industrial park. The last part of the paper is devoted to a critical analysis of the state of the art by proposing several routes to improve the methodologies found in the literature. Another aim of this paper review consists in finding the gaps existing in previous studies. These major gaps are found to be: the lack of multiobjective optimization studies, the absence of social/societal objectives formulation also needs to be addressed and the lack of works taking into account flexibility of ecoparks in an operational point of view