Analytical study of steady-state compressible flow of perfect gas with constant heat flux and friction in constant area ducts

Exact analytical solutions of one-dimensional gas dynamics are intensively applied in engineering practice as a tool in modelling and simulating the piping systems that utilize a compressible medium as their working fluids. Well-known exact analytical solutions for simple types of flows, i.e. for flow processes in which only a single effect is taken into account (e.g. such limiting cases of flows as isentropic, adiabatic or isothermal), are classics of modern one-dimensional gas dynamics theory formed in the first half of last century. At present, gas dynamics does not possess an exact analytical solution for more than a single factor bringing about changes in fluid properties. In this paper the possibility of obtaining general and particular solutions of a non-linear ordinary differential equation (ODE) system describing one-dimensional steady-state flow of compressible ideal gas in constant area ducts with a constant heat flux and friction factor is discussed. It shows that ODE system variables can be separated, and integrals can be taken in terms of elementary functions. Since an analytical solution is the most important result of the paper, its detailed derivation is presented. The mathematical properties of the solution are analyzed in order to calculate this type of compressible flow. For the purpose of this analysis, the functions of the solution and duct performance characteristics of the flow model are demonstrated for various flows and heat flux values, as well as distributions of flow parameters along the duct for both subsonic and supersonic flows. The thermodynamic constraints of the solution are also studied. The analytical solution formulas obtained in this paper may serve as a definition of heat flux and friction factor in ducts from a viewpoint of one-dimensional gas dynamics.