Determination of the Inside Diameter of Pressure Pipes for Drinking Water Systems Using Artificial Neural Networks

Abstract

The fifth-degree polynomial equation determines the diameter in pressurized drinking water systems. The input variables are Q: flow (m3/s), H: pressure drop (m); L: pipe length (m); ε: roughness (m), ϑ: kinematic viscosity (m2/s), and Ʃk: sum of minor loss coefficients (dimensionless). After applying the energy equation for a hydraulic system composed of two tanks connected to a pipe of constant diameter and accepting the Colebrook-White and the Darcy-Weisbach equations, an undetermined expression is obtained since more unknowns than equations are established. This problem is solved by implementing a nested loop for the coefficient of friction and the diameter. This article proposes an Artificial Neural Network (ANN) implementing the Levenberg-Marquardt backpropagation method to estimate the diameter from the log-sigmoidal transfer function under stationary flow conditions. The training signals set consists of 5,000 random data that follow a normal distribution, calculated in Visual Basic (®Excel). The statistics used for the network evaluation correspond to the mean square error, the regression analysis, and the cross-entropy function. The architecture with the best performance had a hidden layer with 25 neurons (6-25-1) presenting an MSE equal to 5.41E-6 and 9.98E+00 for the Pearson Correlation Coefficient. The cross-validation of the neural scheme was carried out from 1,000 independent input signals from the training set, obtaining an MSE equal to 6.91E-6. The proposed neural network calculates the diameter with a relative error equal to 0.01% concerning the values ​​obtained with ®Epanet, evidencing the generalizability of the optimized system.