Jacobian Matrix for Solving Water Distribution System Equations with the Darcy-Weisbach Head-Loss Model
This article has been corrected.
VIEW CORRECTIONPublication: Journal of Hydraulic Engineering
Volume 137, Issue 6
Abstract
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.
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Acknowledgments
The writers wish to acknowledge the assistance of Matt Gibbs.
References
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Rossman, L. (2000). EPANET 2 Users Manual, Environmental Protection Agency, Washington, DC, 〈www.epa.gov/nrmrl/wswrd/dw/epanet.html#support〉 (Apr. 17, 2011).
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Todini, E., and Pilati, S. (1988). A gradient algorithm for the analysis of pipe networks, B. Coulbeck and O. Chun-Hou, eds., Wiley, London, 1–20.
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Published In
Journal of Hydraulic Engineering
Volume 137 • Issue 6 • June 2011
Pages: 696 - 700
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Jul 8, 2009
Accepted: Oct 7, 2010
Published online: Oct 12, 2010
Published in print: Jun 1, 2011
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