Fractal Analysis of the Optimal Hydraulic Gradient Surface in Water Distribution Networks

Abstract

This study aimed to examine the fractal properties of the optimal hydraulic gradient surface (OHGS), a geometrical body that describes the way in which the available energy should be spent within a water distribution network to ensure the calculation of a minimum capital cost design. For this purpose, multiple benchmark and Colombian systems were optimized and then analyzed to compute the fractal dimension of the OHGS and of the underlying structure of each network, which included the examination of randomly generated nonoptimal designs to recognize the differences in the fractal behavior of a least-cost and a more expensive solution. The results showed a dependency between the fractal properties of the OHGS and those of the topological structure, flow, and energy distribution inside the corresponding optimized network. Moreover, it was found that the degree of irregularity of the OHGS tended to be higher compared to a nonoptimal energy dissipation pattern. This suggests the applicability of the fractal analysis in optimization and operational improvement procedures.

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

“…Explanation of the first improvements to the Hardy Cross method [28,29,33,34,38] (useful are also [30][31][32])-accelerated Hardy Cross and versions of loop-versus nodeoriented approach; • Flow pattern in already existing pipe network with loops [36]; • Very illustrative but simple example of application of Hardy Cross and node-loop methods for water distribution [37]; • Approach involving virtual loop that connects two nodes with the same pressure in order to ensure a linear independent matrix needed for calculation (also application of the methods to ventilation systems of underground mines) [39]; • Various versions of methods for ventilation [39,[49][50][51][52][53], oil [54] (compare also similarities with [2]), water distribution [40][41][42][43][44][45][46][47][48] and district heating systems [57][58][59];…”