Application of Variational Methods to Transient Flow in Complex Liquid Transmission Systems
Authors: Rachford H.H. , Ramsey E.L. 
Published: December 2007

Abstract
Surge pressures in liquid pipelines are often sufficiently severe either to stress the pipe past its yield point or to produce cavitation. These phenomena are most likely to be serious in complex, phenomena are most likely to be serious in complex, interconnected piping like a water-injection system, an oil-gathering system, or a dense-phase LNG system. In any of these, pressure waves generated by changes in rates at pump stations or terminal points may add in magnitude as they propagate points may add in magnitude as they propagate through the system and result in locally excessive pressures. pressures. The magnitude and time of arrival of surges depends critically on the wave speed in the liquid/pipe system. Examples are given of typical liquids in which this speed is a strong function of pressure because the compressibility changes with pressure because the compressibility changes with pressure. One example illustrates that the transient pressure. One example illustrates that the transient response of models of pipeline networks carrying such fluids must properly reflect the dynamic, point-by-point influence of pressure on wave speed. point-by-point influence of pressure on wave speed. Otherwise, the model will improperly predict surge events and will be unsuitable as a basis for design or control.
The transient flow of liquid through pipe is described by a nonlinear hyperbolic system of equations. The formulation described in this paper treats isothermal flow, taking proper account of the pipe expansibility and dependence of fluid pipe expansibility and dependence of fluid compressibility on pressure, factors that play a major role in determining the magnitude and propagation velocity of disturbances in liquid propagation velocity of disturbances in liquid systems. The solution of the single-pipe hyperbolic equations is approximated numerically using a convergent, discrete-time Galerkin procedure, and is structured to treat the boundary conditions necessary to model an interconnected system with nonlinear, time-dependent elements like pumps, valves, regulators, co other pressure-dependent flow control devices.
The model was used to study the water-hammer pressure surges caused by pump failure in a complex, pressure surges caused by pump failure in a complex, multipump water-injection system. The effectiveness of control systems proposed to deal with surges caused by such changes in operation can be evaluated only by modeling the system response. One such control system was found to induce rather than alleviate pipe-damaging pressures during emergency shutdown.

Introduction
Liquid pipeline systems seldom achieve steady state. Random disturbances or intentional changes in operating conditions almost always cause some degree of pressure surge within a system. Moreover, under upset conditions, surges can become so extreme that equipment may be damaged or pipelines ruptured. Such pulses or surges in pressure have been the subject of much study and are well documented by many unexpected failures in liquid piping systems caused by severe changes in flow piping systems caused by severe changes in flow rate.
Quantitative predictions of pressure surges have long been an integral part of piping-system design. Methods used generally have been based on the assumption of constant liquid compressibility, resulting in a linear or affine relationship for the equation of state. We shall illustrate that many important fluids have nonlinear pressure/density relationships, that the wave speed in such fluids is not constant but is subject to significant variation with pressure, and that such variations can strongly influence the transient response of the pipe/liquid system. For this reason, protective systems designed without correctly considering the effect of dynamic changes of pressure on wave speed may lead to serious operating problems.
The purpose of this work is to develop a general mathematical model of isothermal fluid transmission systems that correctly treats any physically meaningful equation of state and, therefore, properly predicts surges in fluids with properly predicts surges in fluids with pressure-dependent wave speed. pressure-dependent wave speed. SPEJ
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