Advertisement

Water Hammer Modeling for Water Networks via Hyperbolic PDEs and Switched DAEs

  • Rukhsana Kausar
  • Stephan Trenn
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 237)

Abstract

In water distribution network, instantaneous changes in valve and pump settings introduce jumps and sometimes impulses. In particular, a particular impulsive phenomenon which occurs due to sudden closing of valve is the so-called water hammer. It is classically modeled as a system of hyperbolic partial differential equations (PDEs). We observed that under some suitable assumptions the PDEs usually used to describe water flows can be simplified to differential algebraic equations (DAEs). The idea is to model water hammer phenomenon in the switched DAEs framework due to its special feature of studying such impulsive effects. To compare these two modeling techniques, a system of hyperbolic PDE model and the switched DAE model for a simple setup consisting of two reservoirs, six pipes, and three valves is presented. The aim of this contribution is to present results of both models as motivation for the claim that a switched DAE modeling framework is suitable for describing a water hammer.

Keywords

Water hammer Solution theory Switched system Dirac impulse 

Notes

Acknowledgement

We are thankful to Jochen Kall for fruitful discussions concerning the PDE simulations of earlier versions of this work.

References

  1. 1.
    S. Adami, X.Y. Hu, N.A. Adams, Simulating three-dimensional turbulence with SPH, in Center for Turbulence Research. Proceedings of the Summer Program 2012 (2012), pp. 177–185Google Scholar
  2. 2.
    M.H. Chaudhry, L. Mays, Computer Modeling of Free-Surface and Pressurized Flows (Springer Science & Business Media, 2012)Google Scholar
  3. 3.
    M. Herty, J. Mohring, V. Sachers, A new model for gas flow in pipe networks. Math. Methods Appl. Sci. 33, 845–855 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Izquierdo, R. Pérez, P.L. Iglesias, Mathematical models and methods in the water industry. Math. Comput. Model. 39, 1353–1374 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Jansen, J. Pade, Global unique solvability for a quasi-stationary water network model. Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2013-11, (2013)Google Scholar
  6. 6.
    L. Jansen, C. Tischendorf, A unified (P)DAE modeling approach for flow networks, in Progress in Differential-Algebraic Equations: Deskriptor 2013, ed. by S. Schöps, A. Bartel, M. Günther, W.E.J. ter Maten, C.P. Müller (Springer, Berlin, Heidelberg, 2014), pp. 127–151Google Scholar
  7. 7.
    J. Kall, R. Kausar, S. Trenn, Modeling water hammers via PDEs and switched DAEs with numerical justification, in Proceedings of IFAC World Congress 2017, Toulouse, (2017)CrossRefGoogle Scholar
  8. 8.
    B.E. Larock, R.W. Jeppson, G.Z. Watters, Hydraulics of Pipeline Systems (CRC press, 1999)Google Scholar
  9. 9.
    D. Liberzon, S. Trenn, Switched nonlinear differential algebraic equations: solution theory, Lyapunov functions, and stability. Automatica 48, 954–963 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S. Trenn, Distributional differential algebraic equations. PhD thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Germany, 2009Google Scholar
  11. 11.
    S. Trenn, Regularity of distributional differential algebraic equations. Math. Control Signals Syst. 21, 229–264 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Trenn, Switched differential algebraic equations, in Dynamics and Control of Switched Electronic Systems-Advanced Perspectives for Modeling, Simulation and Control of Power Converters, ch. 6, ed. by F. Vasca, L. Iannelli, (Springer, London, 2012), pp. 189–216Google Scholar
  13. 13.
    E.B. Wylie, V.L. Streeter, Fluid Transients (McGraw-Hill International Book Co., New York, 1978)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTU KaiserslauternKaiserslauternGermany
  2. 2.Bernoulli Institute for Mathematics, Computer Science and Artificial IntelligenceUniversity of GroningenGroningenNetherlands

Personalised recommendations