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of separation bubbles will not be predicted accurately. Nevertheless, it is expected that the model will roughly be able to predict the main trends and qualitative features of the flow field. 2.2. Thermodynamic models 2.2.1. Equations of state The perfect gas equation of state is not valid at temperatures and pressures of the order of magnitude of those of thermodynamic critical point. Then other more complex equations of state should be used. The van der Waals equation of state is the earliest attempt to correct the perfect gas law in order to take into account co-volume effects and attractive intermolecular forces. It only satisfies two thermodynamic constraints: the horizontal slope and inflection of the critical isotherm at the critical point. However, these are sufficient conditions that make the van der Waals equation capable to model BZT fluid behavior. It has the following general form: point. In the limit cv / R → ∞ , the isentropes and isotherms coincide. Therefore, a van der Waals gas with sufficiently high cv / R → ∞ ratio is expected to exhibit reversed isentrope concavity above the upper saturation curve: hence it possesses BZT properties. It is possible to show that, taking the specific heat ratio γ in the range 1 < γ < 1.06 , a region of negative values of the Fundamental Derivative appears. The van der Waals thermodynamic model is computationally inexpensive (because it’s possible to write an explicit relation between pressure and internal energy), and has been often utilized to provide a qualitative description of BZT fluid flows. On the other hand, this model is not very accurate for thermodynamic conditions close to saturation, i.e. the region of interest in the present study, and largely over-predicts the extent of the inversion zone [6]. The Amagat (p-v) diagrams reported in Figure 1a give an idea of the behavior of a van der Waals gas with γ= 34 p = RT − α vdW V−β V2 (Eq. 6) where αvdW, βvdW, are some coefficients obtained by imposing that ∂v2 = 0 at the critical vdW ∂2 pPDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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