Thermal Simulation of Pipeline Flow

Abstract

A new numerical method for studying one-dimensional fluid flow through pipelines is presented and analyzed. This work extends in a certain direction a collocation method described by Luskin (SIAM J. Numer. Anal., 16 (1979), pp. 145--164). The pressure and velocity of an isothermal fluid in a pipeline can be described by a coupled pair of nonlinear first-order hyperbolic partial differential equations. When thermal effects are important a third equation for temperature is added. While Luskin's method works well for the isothermal situation he discussed, it does not apply in certain common cases when thermal effects are modeled. The analysis of this new method shows how the difficulties that come from the application of standard collocation can be overcome. Experiments indicate that this method is a substantial improvement over standard collocation. The analysis also illustrates an approach to analyzing nonlinear evolution equations with smooth solutions that produces convergence theorems about the nonlinear system analogous to the corresponding linear theorems with relatively little extra work. This analysis also yields an H1 estimate in the isothermal case. 1. Introduction. Thermal modeling of fluid flow through pipelines is increas- ingly necessary as practical engineering applications demand greater fidelity between model and reality. When pipeline simulators are used for leak detection, for instance, real-time thermal modeling may be required to distinguish leaks from pressure changes due to heat exchange with the environment. Thermal modeling may also be important in attempting to optimize pipeline usage for maximum economic utility. Isothermal fluid flow in pipelines is described by a pair of coupled first-order nonlinear hyperbolic partial differential equations for pressure and velocity. In 1979 Mitchell Luskin (4) described and analyzed a numerical method applicable to a large class of systems, including the isothermal pipeline case. The method relies upon the fact that the eigenvalues of the matrix that determines the characteristic directions are both bounded far away from zero. In its simplest version, the method uses piece- wise linear approximations in space and time and can be described as box-centered collocation. Thermal effects in pipeline flow introduce a third coupled equation for tempera-