Maximum Entropy Analysis of Hydraulic Pipe Flow Networks

Abstract

A maximum entropy (MaxEnt) method is developed to predict mean external and internal flow rates and mean pressure gradients (potential differences) in hydraulic pipe networks, without or with sufficient constraints to enable a closed-form solution. This substantially extends existing methods for the analysis of flow networks (e.g., Hardy Cross), applicable only to deterministic networks. The analysis is based on a continuous relative entropy defined on the space of flow rates, requiring one dimension for each edge and node within the network. This entropy is maximized subject to observable constraints on the mean values of certain flow rates and/or potential differences, and also physical constraints arising from Kirchhoff’s first and second laws (node and loop constraints) and the frictional properties of each pipe. A semianalytical and numerical algorithm is developed to solve the equation system, which involves implicit solution of multidimensional integrals and root finding with quasi-Newton and Newton methods. The analysis is demonstrated by application to (1) a simple three-node network, and (2) a 1123-node, 1140-pipe water distribution network in the suburb of Torrens, Australian Capital Territory, Australia.