Unified Matrix Frameworks for Water Hammer Analysis in Pipe Networks
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RESEARCH PAPER
Unified Matrix Frameworks for Water Hammer Analysis in Pipe Networks Naser Moosavian1 • Barbara Lence1 Received: 7 May 2019 / Revised: 27 May 2020 / Accepted: 6 July 2020 Ó Iran University of Science and Technology 2020
Abstract Several numerical methods have been developed to analyze water hammer dynamics, and among these, the method of characteristics (MOC) is the most widely applied, establishing the time-dependent characteristic equations that account for the components of the unsteady flow. This paper introduces a matrix formulation of the characteristic equations, which is based on an assumed initial incidence matrix that describes the topology of the network, including loops. Unlike traditional MOC, the matrix equations are used to simultaneously solve for all pressures and flows in the network, at each time step, as a linear function of the pressure and flow at all locations in the network in the previous time step. Two solution procedures are proposed, both of which solve for pressures by decomposing the linear systems of equations into a reduced linear system of equations that is on the order of the number of nodes, and use the resulting pressures to update the flow vector, that is on the order of the number of pipes or pipe reaches, at each time step. The proposed solution procedures are not dependent to the network topologies. For different shapes of pipe network, the formulation remains unchanged and the user need only enters different input data in the form of vectors to find the solution using matrix–vector multiplications at each time step. Fast linear solvers can also be implemented to speed up the process, because the linear system of equations at the core of these algorithms is a Stieltjes matrix. These solution procedures are applied for matrix formulations of two numerical examples, and the resulting nodal pressure and pipe flow at each time step are nearly identical. Keywords Matrix calculations Method of characteristics Networks
1 Introduction Water hammer in pipe networks, resulting from abrupt changes in the flow or velocity, due to, e.g., sudden changes in demand, or irregular valve and pump operations, causes high positive or negative pressures, and may lead to mechanical failures. This paper proposes a matrix formulation of the water hammer equations for analyzing designs and operations of pipe networks undergoing fully compressible unsteady flow. The matrix formulation advances the applicability of the method of characteristics & Naser Moosavian [email protected] Barbara Lence [email protected] 1
Department of Civil Engineering, University of British Columbia, 6250 Applied Science Lane, Vancouver, BC V2T1Z4, Canada
(MOC) for analyzing such events, may be applied for looped pipe networks, and is computationally efficient. The MOC is applicable for fully compressible unsteady flow, has the desirable attributes of accuracy, simplicity, and numerical efficiency [1], and is one of the most comprehensive and wide
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