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42 3 Numerical methods Buijtenen et al., 2003; van Buijtenen, 2009). However, detailed experimental results on the turbine flow field and performance are lacking in the literature and thus, no validation of the accuracy of the solver has been carried out. Finflo was selected as a flow solver since it has been extended to allow for real-gas simulations, by using a real gas models based on accurate polynomial fittings. Finflo employs the finite-volume method for spatial discretization, and Roe’s flux differ- ence splitting (Roe, 1981) was used as an approximate Riemann solver. The second order upwind-method was used as a spatial discretization method in the presented simulations. The turbine simulations were performed by using the Chien’s k − ε (Chien, 1982) and k − ω SST-turbulence model (Menter, 1993), and the results obtained by the different turbulence models were compared to discuss the effect of the turbulence model on the predicted flow field and losses. The turbine stator flow channel and the turbine rotor were modelled separately, and the used boundary conditions and the methods are presented in more detail in the section Stator and nozzle simulations as well as in the section Rotor simulations. 3.3.1 Real Gas Model The accurate real gas model was implemented in a CFD-flow solver to predict the flow field in the designed turbine. The real gas model is based on the polynomial fitting of the fluid properties in a superheated gas region, and the method has been described pre- viously by (Tang, 2006; Turunen-Saaresti et al., 2006). In the real-gas model used in this study, temperature and pressure are independent variables, and the model is based on the polynomial fitting of the fluid properties in the superheated gas region at subcritical pressure levels. Density, internal energy, dynamic viscosity, speed of sound, and thermal conductivity are functions of temperature and pressure. The applied polynomial is of fourth order, and an example of fourth order polynomial for density, ρ, as a function of temperature, T , and pressure, p, is defined as ρ(T,p) = a1 +a2T +a3p+a4T2 +a5Tp+a6p2 +a7T3 +a8T2p+a9Tp2 +a10p3 + a11T4 + a12T3p + a13T2p2 + a14Tp3 + a15p4. Multivariable non-linear regression was used in determining the coefficients a1-a15 by using nonlinear least square fitting. The minimum sum of relative values was aimed in the regression instead of minimum sum of absolute values since the minimum sum of the relative difference provides more accurate polynomials in the studied superheated gas region. The objective function of minimizing the sum of relative difference between the values calculated by the polynomial fitting and the values calculated by Refprop can be defined as The used polynomial reconstruction allows to implement accurate thermodynamic prop- erties of the fluid for the solver. The relative difference between the results for speed min(f ) = minΣ( ρpolynomial − ρRefprop ). (3.28) ρRefprop (3.27)PDF Image | WORKING FLUID SELECTION AND DESIGN OF SMALL-SCALE WASTE HEAT RECOVERY SYSTEMS BASED ON ORGANIC RANKINE CYCLES
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