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Nonclassical Gasdynamics of Vapour Mixtures 1.05 1 0.95 0.9 0.85 0.8 1.45 1.05 1.45 1.05 1.45 1.4 1 1.4 1 1.4 P rP r P r Mach 1.35 1.3 1.25 1.2 1.15 0.95 0.9 0.85 0.8 1.35 1.3 1.25 1.2 1.15 0.95 1.35 0.9 1.3 0.85 1.25 Mach 0.8 1.2 0.75 1.15 0 0.05 0.1 0.15 0.2 Flow direction [rad] From table 8.1 and figure 8.4, for small concentrations of MDM (xMDM < 0.15), Γ assumes negative values in the vapour phase and nonclassical behaviour is admissible. This is the case of the flow of mixture MDM(0.05)/MD6 M(0.95) in figure 8.6e, where the expansion occurs through a nonclassical composite wave made of a continuous fan that is terminated by a nonclassical rarefac- tion shock wave, with close-to-sonic downstream state. The expansion is depicted in the thermody- namic plane in figure 8.5b, which displays the pre- and post-expansion states A and B in the reduced P-v plane. While for the case of the pure fluid MD6M a single rarefaction shock wave connects the two states, in the case of the mixture MDM(0.05)/MD6M(0.95), an initial isentropic expansion is observed from state A to state A’, which lies on the double sonic line. Then, a rarefaction shock wave expands the fluid from A’ to B. Results are summarized in table 8.4, which displays the downstream Mach number M and the variations of pressure P, the temperature T, the densityρ and the velocity u across the waves for all the considered different mixture compositions. The entropy difference across composite waves and the RSW is very small, due to both the relatively small pressure difference across the shock and the small value of Γ. Indeed, Landau and Lifshitz showed that a Taylor-series expansion of the Rankine-Hugoniot jump conditions delivers the following relation P [-] r Mach [-] P [-] r Mach [-] P [-] r Mach [-] Mach 0.75 0 0.05 0.1 0.15 0.2 Flow direction [rad] 0.75 0 0.05 0.1 0.15 0.2 Flow direction [rad] (b) MDM(0.75)/MD6M(0.25) differential equation (8.4) and numerical simulation (—). (c) MDM(0.40)/MD6M(0.60) (a) MDM Figure 8.8: Comparison between the numerical integration (- - -) of the Prandtl–Meyer ordinary 2 3 4 ΓAcA∆v ∆v ∆s ≃ − + o , 6TA vA vA where ∆(·) = (·)B − (·)A , which is valid for weak shock waves [83]. In the present study, the largest entropy difference was computed by solving the non-linear Rankine-Hugoniot jump conditions for theRSWinMD6Manditisassmallas ∆s ∼5×10−6. sA To conclude, figure 8.8 reports the comparison of the numerical integration of the Prandtl– Meyer ordinary differential equation (8.4) and the numerical simulations for the three classical ex- pansions of pure fluid MDM and of the mixtures MDM(0.75)/MD6 M(0.25) and MDM(0.40)/MD6 M(0.60). The very good agreement obtained by these two different approaches increases the authors’ confi- dence on the correctness of the presented results. Furthermore, the values of the upstream and 217PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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