Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model

The widely used Todini and Pilati method for solving the equations that model water distribution systems was originally developed for pipes in which the head loss is modeled by the Hazen-Williams formula. The friction factors in this formula are independent of flow. Rossman’s popular program EPANET implements elements of the Todini and Pilati algorithm, but when the Darcy-Weisbach head-loss formula is used, it does not take into account the dependence of the friction factors on the Reynolds number, and therefore flow, in computing the Jacobian. We present the correct Jacobian matrix formulas, which must be used in order to fully account for the friction factor’s dependence on flow when the Todini and Pilati method is applied with the Darcy-Weisbach head-loss formula. With the correct Jacobian matrix the Todini and Pilati implementation of Newton’s method has its normally quadratic convergence restored. The new formulas are demonstrated with an illustrative example.