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21`eme Congr`es Fran ̧cais de M ́ecanique Bordeaux, 26 au 30 aouˆt 2013 is required to model the fluids. Hereafter, we first briefly recall the governing equations and thermo- dynamic models in use, then we describe the ORC turbines under investigation, compare the different flow behaviors obtained with different working fluids, and comment on entropy losses associated to shock wave formation as a function of fluid complexity. 2 Governing Equations and Flow solver Since in this work we are mainly concerned with the impact of thermodynamic properties of fluids on their inviscid flow behavior, we restrict our attention to the Euler equations, written in integral form for a control volume Ω with boundary ∂Ω : d∫∫ dt w dΩ + f · n dS = 0 (1) Ω ∂Ω where w = (ρ,ρv,ρE)T is the conservative variable vector, n is the outer normal to ∂Ω and f = (ρv, p ̄I + ρvv, ρvH)T is the flux density ; v is the velocity vector, E the specific total energy, H = E+p/ρ the specific total enthalpy, p the pressure and ̄I the unit tensor. These equations are completed by a thermal and a caloric equation of state : p = p(ρ(w), T (w)) and e = e(ρ(w), T (w)) (2) where e is the specific internal energy and T the absolute temperature. Dynamic behavior of dense gases is governed by the fundamental derivative of Gas Dynamics Γ [13] : (3) Γ=1+a ∂ρ s ρ (∂a) being ρ the density and a = [−v2(∂p/∂v)s]12 the speed of sound. Γ represents a measure of the rate of change of the sound speed. According to whether Γ is higher or lower than unity, the flow exhibits different sound-speed variation in isentropic perturbations : when Γ > 1, a drops in isentropic expansions and grows in isentropic compressions, which is what happens in common fluids ; an opposite behavior is obtained when Γ < 1. Thermodynamic regions with Γ lower than unity are observed for molecularly complex fluids at temperatures and pressures close to saturation conditions. The gas response is modeled through equations of state (EOS) based on Helmholtz free energy Φ, which represent the most accurate available models for the fluids of interest in thermodynamic regions close to the critical point and the saturation curves [7]. EOS are written in the form proposed by Setzmann and Wagner [10] : Φ(δ, τ ) = Φ0 (δ, τ ) + Φr (δ, τ ). They use a reduced form (variables normalized with critical-point properties) and are composed by an ideal-gas part, Φ0, function of the ideal-gas isobaric heat capacity, and by a residual term, Φr taking into account real-gas corrections. M1 M2 ∑∑ Φr(δ, τ) = m=1 + where δ = ρ/ρc is the reduced density and τ = Tc/T is the inverse reduced temperature. The number of polynomial and exponential terms and the values of coefficients and exponents are calibrated from experimental data using an optimization algorithm described in [10] . A in-house flow solver has been equipped with reference EOS available for R134a [14] and CO2 [12]. For R245fa, no reference EOS is available, thus we adopt the short technical multiparameter EOS proposed by Span and Wagner [7]. The governing equations are discretized using a cell-centered finite volume scheme for structured multi- block meshes of third-order accuracy. The equations are then integrated in time using a four-stage amδim τjm + amδim τjm exp(−δkm ) m=M1 +1 m=M2 +1 M3 ∑imjm 22 amδ τ exp[−αm(δ−εm) −βm(τ−γm) ] (4) 2PDF Image | Numerical investigation of dense gas flows through transcritical multistage axial Organic Rankine Cycle turbines
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