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Chapter 8 general interface to several thermodynamic libraries [73]. An unstructured mesh refinement tech- nique is adopted to increase the accuracy in regions where the solutions exhibit the largest gradients. Figure 8.6 shows the flow field isobars from flow simulations for different mixtures of MDM/MD6 M, whereby the mole fraction of MDM varies from xMDM = 1 in figure 8.6a to xMDM = 0 in figure 8.6f, under the assumption of negligible fluid viscosity and thermal conductivity. Figure 8.6a displays the supersonic flow of a pure MDM vapour. As expected, since Γ > 0, a classical isentropic expansion fan is observed in this case. It is remarkable that differently from supersonic expansions of a constant specific heats ideal gas, the Mach number M variation across the expansion is non-monotone, as it can be appreciated from figure 8.7a, where the Mach number is depicted for a representative streamline across the expansion wave. The present non monotone behaviour is consistent with the value of the parameter J, namely 1 J(s,ρ,M)=1−Γ(s,ρ)− M2, (8.2) across the expansion wave. Indeed, for isentropic processes from a given state A one has, see Ref. [82], dM M dρ =J(sA,ρ,M)ρ. (8.3) As shown in figure 8.7b, in the expansion of pure MDM vapour depicted in figure 8.6a, J can have both negative and positive values and therefore M is non-monotone. Note that for a constant specific heat ideal gas, J = (1 − γ)/2 − 1/M2 < 0, with γ ratio of the isobaric and isochoric specific heats, and therefore M always increases monotonically during a supersonic expansion. A reversed, nonclassical, behaviour is observed for the supersonic expansion of pure MD6M, shown in figure 8.6f. An oblique nonclassical rarefaction shock wave, which forms an angle of 60◦ with respect to the upstream flow direction, is observed. Intermediate situations are observed in the case of mixtures of MDM/MD6M—shown in fig- ures 8.6b, 8.6c, 8.6d and 8.6e—where the supersonic expansions of mixtures of increasing concen- tration of the more complex component MD6M are depicted. In particular, in figure 8.6b, where the flow of a MDM(0.75)/MD6 M(0.25) mixture is shown, a classical rarefaction fan is observed since Γ > 0. The angular sector encompassed by the fan is larger thanthatobservedinfigure8.6aforpurefluidMDM,althoughthefinalturningangleθB =−13.169◦ is the same in both conditions. Therefore, since the slope of the limiting characteristic line at the right boundary of the fan is λ(θB ) = tan (θB + μ(θB )) where sin μB = 1/MB , one can conclude that the Mach number at the end of the expansion is larger in this case, as it is confirmed also by the values in table 8.3 and 8.4. The above can be explained by recalling the dependence of the Mach number on the local velocity angle θ given by the Prandtl-Meyer relation, see Ref. [7], namely, √M2 −1 1 − (Γ − 1) M2 Indeed, despite the larger average molecular weight, along the considered isentrope the value of J for the mixture is always lower than that computed for pure MDM, see figure 8.7b, and the Mach number difference between the upstream and downstream state is larger. In the conditions depicted in 8.6c and 8.6d, Γmin > 0, see figure 8.3 and table 8.1, and therefore a classical flow is observed in both cases. In case 8.6c, the Mach number in state B is larger than that observed in case 8.6a, consistently with the Γ and J profiles across the expansion, see figures 8.7c and 8.7b, respectively. As a consequence, the angle encompassed by the rarefaction fan is larger than that observed for pure fluid MDM. The opposite situation is found in the case in 8.6d and the fan is narrower than its pure fluid counterpart. dθ = dM. (8.4) 214PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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