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21`eme Congr`es Fran ̧cais de M ́ecanique Bordeaux, 26 au 30 aouˆt 2013 Runge-Kutta scheme. Local time stepping, implicit residual smoothing and multi-grid acceleration are used in order to drive the solution to the steady state. The accuracy of the numerical solver, already demonstrated in previous works [1, 2], will be not investigated further. The geometrical configurations under investigations are axial reaction turbines with a variable number of stages according to the working fluid in use. The reaction degree is fixed to 0.5. The rotational speed is fixed at 3000 RPM to allow direct coupling with the alternator. The number of stages and their relative average radius are calculated in the pre-design phase by considering the operating conditions (inlet total conditions, mass flow and pressure ratio), as well as cost and system complexity considerations. A three-stage turbine working with refrigerant R245fa is designed to work at subcritical nominal conditions, and four-stage turbines, designed for supercritical nominal conditions, are used with fluids R134a and CO2. The blades use the same airfoils for both stator and rotor wheels. Both the subcritical and supercritical turbines were tested by using both R134a and R245fa as the working fluid, to check the impact of different working fluids on the overall performance of a given turbine. The different turbine configurations under investigation are described in Tab. 1. Blade vanes are discretized by C-meshes made of 272x32 cells. Total temperature and pressure are imposed at inlet, static pressure at the outlet, periodicity at upper and lower boundaries, and no-slip at the wall. A mixing plane condition is used at stator/rotor interfaces. 3 Numerical simulation results For each turbine configuration and each stage, we computed the isentropic efficiency defined as the real-to-ideal static enthalpy jump ratio. In the present 2D inviscid calculations, only losses due to shock waves can be taken into account. Numerical results show that shock waves are generated, at upper surfaces of rotor blades, for some turbine configurations and operating conditions. However, these shocks remain relatively weak, and the associated entropy losses small. Even in the case the flow field is entirely shock-free, the computed efficiencies are slightly lower than unity, because of errors introduced by the numerical approximation of governing equations and boundary conditions. Fig. 1 shows distributions of relative Mach number, Fundamental Derivative of Gas Dynamics Γ, and sound speed for case SUPR134a. Weak shock waves are created on the suction side of the rotor blades for all stages. Going from the first to the last stage, Γ becomes lower than unity, and the sound speed increases as the expansion proceeds. Since flow velocity grows more slowly than the sound speed, stronger shocks are generated in the first stages where Mach is higher; this is reflected by stage isentropic efficiencies, which increase as shocks weaken (see Tab. 2). For case SUPR245fa shown in Fig. 2, a similar behavior is observed. Nevertheless, entropy losses are higher than for R134a. In this case, Γ is lower than unity through the entire turbine : as a consequence, the speed of sound increases during the expansion. However, its value in the upstream stages is much lower than in SUPR134a, whereas flow speed is almost the same, so that Mach number is higher and shocks stronger. This results in efficiencies roughly 5% lower than in the SUPR134a case. For subcritical configurations SUBR134a and SUBR245fa (not shown for brevity), the overall perfor- mance is lower. Lower isentropic efficiencies are due to the fact that the computed Γ value is nearly constant and close to unity : the sound speed growth is negligible with respect to the flow speed one, so that Mach number and shock strength increase moving downstream. Performance loss is partly due to the fact that the subcritical turbine is designed for 3 stages instead of 4; as a consequence, each stage processes a higher pressure ratio with respect to the supercritical turbine (see Tab. 1). Subcritical turbine efficiencies about 1.5% lower w.r.t. supercritical configurations. Finally, case SUPCO2 is shown in Fig. 3. In this case the flow remains always subsonic and no shocks are formed, since carbon dioxide has a sound speed about double than R134a and R245fa. Thus, even if Γ is higher than unity and the speed of sound decreases with fluid expansion, it is still high enough to prevent the flow from becoming supersonic. In the absence of shocks and viscous effects, isentropic efficiencies differ from unity only because of numerical errors. According to these numerical results, CO2 seems to be the working fluid that offers the highest adiabatic efficiency. Nevertheless, the choice of an optimal working fluid for ORC turbines also involve other kinds of consideration, like economical and safety considerations. For instance, supercritical CO2 cycles involve pressures of the 3PDF Image | Numerical investigation of dense gas flows through transcritical multistage axial Organic Rankine Cycle turbines
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