Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines

A finite volume scheme for the numerical solution of a mathematical model for non-isothermal non-adiabatic compressible flow of a real gas in a pipeline is introduced. In order to make an upwind discretization of the flux, the Q-scheme of van Leer is used. Unlike standard Euler equations, the model takes into account wall friction, variable height and heat transfer between the pipe and the environment. Since all these terms are sources, in order to get a well-balanced scheme they are discretized by making a similar upwinding to the one in the flux term. The performance of the overall method has been shown for some usual numerical tests. The final goal, which is beyond the scope of this paper, is to consider a network including several pipelines connected at junctions, as those employed for natural gas transport.