On the computation of the fundamental derivative of gas dynamics using equations of state

The value of the fundamental derivative of gas dynamics, Γ$\mathrm{\Gamma }$, is a quantitative measure of the variation of the speed of sound with respect to density in isentropic transformations, such as those occurring, for example, in gas-dynamic nozzles. The accurate computation of its value, which is a constant for a perfect gas, is key to the understanding of real-gas flows occurring in a thermodynamic region where the polytropic ideal gas law does not hold. The fundamental derivative of gas dynamics is a secondary thermodynamic property and so far, no experiments have been conducted with the aim of measuring its value. Several studies document the estimation of Γ$\mathrm{\Gamma }$ for fluids composed of complex molecules using mainly simple thermodynamic equations of state, e.g., that of Van der Waals. A review of these studies has revealed that the calculated values of Γ$\mathrm{\Gamma }$ are affected by large uncertainties; these uncertainties are due to the functional form of the adopted equations and because of uncertainties in the available fluid property data on which these equations were fitted. In this work, the fundamental derivative of gas dynamics of molecularly simple fluids is computed with the aid of, among other models, modern reference equations of state. The accuracy of these computations has been assessed. Reference thermodynamic models however, are not available for molecularly complex fluids; some of these molecularly complex fluids are the substances of interest in studies on the so-called nonclassical gas dynamics. Therefore, results of the computation of Γ$\mathrm{\Gamma }$ for few, molecularly simple hydrocarbons, like methane, ethane, etc., are used as a benchmark against which the performance of simpler equations of state, can be assessed. For the selected substances, the Peng–Robinson, Stryjek–Vera modified, cubic equation of state yields good results for Γ$\mathrm{\Gamma }$-predictions, while the modern multiparameter technical equations of state, e.g., the one in the Span–Wagner functional form, are preferable, provided that enough accurate thermodynamic data are available. Another notable result of this study, is that Γ$\mathrm{\Gamma }$ for a fluid composed of complex molecules is less affected by the inaccuracy of _Cv$C_v$-information (_Cv$C_v$ is the isochoric heat capacity), if compared to the estimation of Γ$\mathrm{\Gamma }$ for simple molecules. Inspection of the results of the calculation of Γ$\mathrm{\Gamma }$ in the proximity of the critical point confirms that analytical equations of state fail to predict the correct physical behavior, even if they include terms which allow for the correct estimation of thermodynamic properties.