A General Solution of the Equations Describing Steady State Turbulent Compressible Flow in Circular Conduits

Abstract

Abstract Although a number of solutions to the equations describing steady compressible flow in pipes exist, even the most rigorous of these make simplifying assumptions concerning thermal effects; i.e., either that the flowis isothermal or that the gas assumes identically the temperature of the surroundings. There are, however, instances where these assumptions may not be warranted. This paper examines the differential equations which characterize momentum and heat transfer, rearranges them so as to make them amenable to numericalsolution and proposes a computational technique suitable for use on digital computers. The approach represents an improved and generalized method ofhandling gas well calculations as well as pipeline design problems. Introduction The flow of fluids, in general can be described by a set of partial differential equations – comprising the equation of continuity, aforce-momentum balance for each of the three dimensions and a total energy balance. A general solution of this set of five partial differential equations involving four independent variables – three geometric, one time – is currently beyond our capabilities. Fortunately, in many cases of gas flow, the flow ishighly turbulent, steady and occurs in circular conduits. In such cases, the system may be described by two ordinary differential equations, namely, aforce-momentum and a total energy balance. These two equations may be writtenin terms of one independent variable, length, and two dependent variables, which may be two fluid properties. Although any two independent properties willdefine the state of a fluid, temperature and pressure are the obvious choices, as they are the ones usually measured in practice. Until recently, the solution of simultaneous nonlinear differential equations has been too time-consuming to find wide engineering acceptance. Consequently, further simplifying assumptions have been made. It has been assumed that the gas temperature is either constant or an explicit function of length, thereby reducing the number of differential equations to one, viz., theforce-momentum balance. It should be pointed out, however, that the assumption of isothermal flow is normally not defensible on physical grounds. With regardto the alternate proposed assumption, while the temperature of the surroundings may be known, the assumption that the fluid assumes identically the temperature of the surroundings implies that the overall heat transfer coefficient between the fluid and the surroundings is infinite - a questionable hypothesis.