Non ideal compressible fluid dynamics

For dilute thermodynamic conditions, the ideal-gas equation of state (EoS) provides sufficiently accurate results in modelling the fluid thermodynamics. This occurs in general for low values of reduced pressure and high values of reduced temperature, where the term reduced refers to the ratio of a certain thermodynamic quantity and its critical value. For some fluids such as air, the assumption of considering ideal conditions is perfectly reasonable and it is widely used. defined as : Z = \frac{Pv}{RT} where • P is the pressure [Pa]; • v is the specific volume [m3/kg]; • R is the specific gas constant [J/(kg K)], namely the universal gas constant divided by the fluid's molecular mass; • T is the absolute temperature [K]. The compressibility factor is a dimensionless quantity which is equal to 1 for ideal gases and deviates from unity for increasing levels of non-ideality. Several non-ideal models exist, from the simplest cubic equations of state (such as the Van der Waals and the Peng-Robinson models) up to complex multi-parameter ones, including the Span-Wagner equation of state. State-of-the-art equations of state are easily accessible through thermodynamic libraries, such as FluidProp or the open-source software CoolProp. == Non-ideal gasdynamic regimes ==

The dynamic behavior of compressible flows is governed by the dimensionless thermodynamic quantity \Gamma , which is known as the Landau derivative or fundamental derivative of gas dynamics and is defined as : \Gamma = \frac{v^3}{2c^2} \left(\frac{\partial^2 P}{\partial v^2} \right)_s = 1+ \frac{c}{v} \left(\frac{\partial c}{\partial P} \right)_s where • c is the speed of sound [m/s]; • s is the specific entropy per unit mass [J/(kg K)]. From a mathematical point of view, the Landau derivative is a non-dimensional measure of the curvature of isentropes in the pressure-volume thermodynamic plane. From a physical point of view, the definition of \Gamma tells that the speed of sound increases with pressure in isentropic transformations for values of \Gamma > 1 , while, by contrast, it decreases with pressure for \Gamma . Based on the value of \Gamma , three gas dynamic regimes can be defined: Typically, for single-phase fluids made of simple molecules, only the ideal gasdynamic regime can be reached, even for thermodynamic conditions very close to saturation. It is for example the case of diatomic or triatomic molecules, such as nitrogen or carbon dioxide, which can only experience small departure from the ideal behavior. Such fluids belong to different classes of chemical compounds, including hydrocarbons, siloxanes and refrigerants. Indeed, for an ideal gas expanding isentropically in a converging-diverging nozzle, the Mach number increases monotonically as the density decreases. By contrast, in oblique shock waves, the post-shock Mach number can be larger than the pre-shock one. Non-classical gas-dynamic regime Finally, fluids with an even higher molecular complexity can exhibit non-classical behavior in the single-phase vapor region near saturation. They are called Bethe-Zel'dovich-Thompson (BZT) fluids, from the name of physicists Hans Bethe, Yakov Zel'dovich, and Philip Thompson, who first worked on these kinds of fluids. For thermodynamic conditions lying in the non-classical regime, the non-monotone evolution of the Mach number in isentropic expansions can be found even in subsonic conditions. In fact, for values of \Gamma , positive values of J can be reached also in subsonic flows ( M ). In other words, the non-monotone Mach number evolution is also possible in the convergent section of an isentropic nozzle. Composite waves, instead, are referred to as phenomena in which two elementary waves propagate as a single entity. Experimental evidence of a non-classical gas-dynamic regime is not available yet. The main reasons are the complexity of performing experiments in such challenging thermodynamic conditions and the thermal stability of these very complex molecules. == Applications ==

Compressible flows in non-ideal conditions are encountered in several industrial and aerospace applications. They are employed for example in Organic Rankine Cycles (ORC) and supercritical carbon dioxide (sCO2) systems for power production. In the aerospace field, fluids in conditions close to saturation can be used as oxiders in hybrid rocket motors or for surface cooling of rocket nozzles. Gases made of molecules of high molecular mass can be used in supersonic wind tunnels instead of air to obtain higher Reynolds numbers. Finally, non-ideal flows find application in fuels transportation at high-speed and in Rapid Expansion of Supercritical Solutions (RESS) of CO2 for particles generation or extraction of chemicals. Organic Rankine cycles at the LUT University in Lappeenranta, Finland.Usual Rankine cycles are thermodynamic cycles that employ water as a working fluid to produce electric power from thermal sources. In Organic Rankine cycles, by contrast, water is substituted by molecularly complex organic compounds. Since the vaporization temperature of these kinds of fluids is lower than that of water at atmospheric pressure, low-to-medium temperature sources can be exploited allowing for heat recovery, for example, from biomass combustion, industrial waste heat, or geothermal heat. For these reasons, ORC technology belongs to the class of renewable energies. For the design of mechanical components, such as turbines, working in ORC plants, it is fundamental to take into account typical non-ideal gas-dynamic phenomena. In fact, the single-phase vapor at the inlet of an ORC turbine stator usually evolves in the non-ideal thermodynamic region close to the liquid-vapor saturation curve and critical point. Moreover, due to the high molecular mass of the complex organic compounds employed, the speed of sound in these fluids is low compared to that of air and other simple gases. Therefore, turbine stators are very likely to involve supersonic flows even if rather low flow velocities are reached. High supersonic flows can produce large losses and mechanical stresses in the turbine blades due to the occurrence of shock waves, which cause a strong pressure raise. However, when working fluids of the BZT class are employed, expander performances could be improved by exploiting some non-classical phenomena. Supercritical carbon dioxide cycles When carbon dioxide is held above its critical pressure (73.773 bar) and temperature (30.9780 °C), For example, it is employed in domestic water heat pumps, which can reach high efficiencies. By contrast, mechanical components within sCO2 Brayton cycles, especially turbomachinery and heat exchangers, suffer from corrosion. == See also ==