Advanced Analytical Modeling of Polytropic Gas Flow in Pipelines: Unifying Flow Regimes for Efficient Energy Transport

In the present work, a new analytical model of polytropic flow in constant-diameter pipelines is developed to accurately describe the flow of compressible gases, including natural gas and hydrogen, explicitly accounting for heat exchange between the fluid and the environment. In contrast to conventional models that assume isothermal or adiabatic conditions, the proposed model simultaneously accounts for variations in pressure, temperature, density, and entropy, i.e., it is based on a realistic polytropic gas flow formulation. A system of differential equations is established, incorporating the momentum, continuity, energy, and state equations of the gas. An implicit closed-form solution for the specific volume along the pipeline axis is then derived. The model is universal and allows the derivation of special cases such as adiabatic, isothermal, and isentropic flows. Numerical simulations demonstrate the influence of heat flow on the variation in specific volume, highlighting the critical role of heat exchange under real conditions for the optimization and design of energy systems. It is shown that achieving isentropic flow would require the continuous removal of frictional heat, which is not practically feasible. The proposed model therefore provides a clear, reproducible, and easily visualized framework for analyzing gas flows in pipelines, offering valuable support for engineering design and education. In addition, a unified sensitivity analysis of the analytical solutions has been developed, enabling systematic evaluation of parameter influence across the subsonic, near-critical, and heated flow regimes.