Real Gas Thermodynamics

and the isentropic behavior of substances

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Abstract

A generalized isentropic gas model is derived following earlier work by Kouremenos et al. in the 1980s, by replacing the traditional adiabatic exponent γ by the real exponents γPv, γTv, and γPT, describing the isentropic pressure-volume, temperature-volume, and pressure-temperature relations respectively. The real adiabatic exponents are expressed as functions of state variables to take into account compressibility effects on the isentropic behavior of substances. Due to the implicit analytical nature of the real exponents, any equation of state or thermodynamic library can be used for their evaluation. The theoretical limits and overall behavior of the real isentropic gas model are explored for a Van der Waals substance. In the two opposing physical limits, the model is shown to reduce to the incompressible substance model for liquid densities and the ideal gas model as the temperature increases or the pressure goes to zero.

The relation of the generalized isentropic gas model with other thermodynamic properties is explored, leading to the development of specific heat relations and other thermodynamic properties in terms of the real exponents γPv, γTv, and γPT. Besides providing alternative schemes for their evaluation, special features of thermodynamic properties such as the state of maximum density and inversion temperature may be related to the value of the isentropic exponents determined by the local compressibility of the substance. Due to the exact definitions of the real adiabatic exponents at a state point, the relations between properties is thermodynamically consistent – another physical requirement.

The generalized isentropic gas model is then applied to isentropic flows to derive traditional gas dynamic relations such as speed of sound, stagnation properties, and choked flow conditions for non-ideal compressible fluid flows. Exact solutions are provided for Prandtl-Meyer expansion fans, and approximate Rankine-Hugoniot jump conditions are explored for real gases. Finally, attributes of the fundamental derivative of gas dynamics are explored under the generalized isentropic gas model to gain new insights into its mathematical properties. Under the generalized isentropic model, the fundamental derivative is shown to satisfy both liquid and gaseous physical limits. Non-classical behavior is attributed to higher-order derivatives of the real exponents.

The application of the generalized isentropic gas model is demonstrated and validated for use in non-ideal compressible fluid dynamic (NICFD) codes by simulation of the one-dimensional Euler equations for a standard shock tube problem. Several numerical schemes for the evaluation of thermodynamic properties of varying levels of accuracy are presented for evaluation of the isentropic gas model. In the application of the shock tube problem, the general equation for the speed of sound is demonstrated to be equivalent to the speed of sound of a Van der Waals gas, proving the validity of the model.