A STUDY OF THE DYNAMIC RESPONSE OF THE FLOW OF LIQUIDS IN PIPE LINE SYSTEMS
This study has shown that a large number of physical effects are involved in dynamic fluid flow in pipe line systems. These effects are governed by coupled nonlinear equations and only a few solutions are possible. The linearized equations apply to some special physical cases that have been completely analysed in this study. A comparison of the theoretical and the experimental results shows good correlation. The flow of liquids in a pipe line when subjected to large disturbances are governed by quasilinear partial differential equations. If the internal diameter of the pipe line is much smaller compared to its length, the liquid flow can be approximated by one dimensional form of the quasilinear partial differential equations. These equations give rise to nonlinear simultaneous partial differential equations which govern the general wave propagation characteristics in one dimension. To develop methods of estimating the effects of the nonlinearities caused by finite disturbances in the flow of liquids in pipe lines, a few nonlinear cases of analysis have been chosen. By considering the quasilinear partial differential equations for fluid flow in one dimension, which form a hyperbolic system of equations, the equations of the two characteristics and the canonical equations are obtained. These canonical equations are four in number and contain derivatives only in one direction, along one of the two characteristic directions. By solving the four canonical equations further, integral equations representing the solutions of the original system of hyperbolic equations are obtained. For analysing the nonlinear effects of the finite disturbances, a simplified nonlinear form of the general wave equation is considered and solutions, which provide a basis for estimating the effects of high pressure level, the effects of high rate of pressure application and the effects of high pressure amplitude on the velocity of the wave propagation and wave front form are presented. Also by using another simplified nonlinear form of the general wave equation, the existence of the subcritical and the supercritical waves is shown. Several cases of practical pipe line systems, including the loaded pipe line systems, are theoretically analysed for small distutbances by considering the linear forms of the fluid flow equations and the wave transmission equations. The existing method of linear analysis using the linearized resistance coefficient approach is improved by developing'a method for an exact estimation of the linearized resistance coefficient. This method and the exact linear analysis, which takes the radial gradients in the axial flow velocity into consideration, in one dimension are compared and are shown to be the same. Simple equations are developed to estimate the effects of the frequency dependent viscous damping and the frequency dependent linearized resistance coefficient on the effective phase velocity. To estimate the pipe line effects, as caused by the radial and the longitudinal tube wall oscillations and their transmission along the pipe line, methods are developed by using perturbations in the flow velocities for estimating the additional resonant characteristics introduced into the frequency response behaviour of the pressure and the axial flow velocity. Also a method is developed for an.approximate estimation of the additional attenuation effects caused by the radial oscillations of the tube wall. Several experimental test setups are used to test the frequency and the transient response behaviour of the different pipe line systems. The experimental results compare with the theoretical results to some extent and show the practical limitations of the theoretical analysis as applicable to the experimental results. By using a theoretical linear model of the load system, the unusual behaviour of a particular configuration of the experimental test setup is analysed with reference to the variations of the load coulomb friction characteristics. In general, the correlation of the theoretical and the experimental results for the frequency and the transient response cases is good within the limitations of the experimental error and the practical limitations of the analysis. Some of the dynamic response-, experimental results and the results from the test setup for measuring the bulk modulus of the hydraulic oil have indicated the unusual and the unpredictable behaviour of the compressible nature of the commercial hydraulic oil used. This property has introduced an element of uncertainty in the accurate prediction of the performance of the systems using this type of hydraulic oil.