**5. Results**

Figure 3 compares the results of RefProp 9.1 with those of approximations A1 and A3 for the liquid–vapor saturation boundary in a *Tr* − *s*<sup>∗</sup> diagram for: (a) ammonia; (b) benzene; (c) methyl palmitate;and (d) D6. In the case of ammonia, approximation A3 gave better agreemen<sup>t</sup> with RefProp 9.1 than A1 for the saturated vapor branch, whereas we obtained the opposite behavior for the saturated liquid branch. This can be ascribed to the fact that the slope of the modified rectilinear diameter is slightly larger for A3 (−*b*A3 = 5.0275, obtained from Equation (25) with *c*ig<sup>∗</sup> *p* (*<sup>T</sup>*M*r*) = 4.3795) than for A1 (−*b*A1 = 4.4225). Figure 3b shows that A1 and A3 yield similar results for the saturation entropies of benzene, with excellent agreemen<sup>t</sup> with RefProp 9.1 results except for *s*∗g for *Tr* < 0.6. In this case, the slope −*b*A1 = 18.5302 is very similar to −*b*A3 = 18.4997 (obtained from *c*ig<sup>∗</sup> *p* (*<sup>T</sup>*M*r*) = 15.4544). In Figure 3c, the approximation A3 gives a better agreemen<sup>t</sup> with RefProp 9.1 than A1. This indicates that the slope −*b*A3 = 107.9614 (obtained from *c*ig<sup>∗</sup> *p* (*<sup>T</sup>*M*r*) = 84.2017) yields better results than that of approximation A1 (−*b*A1 = 110.949). Finally, Figure 3d shows that A3 yields much better agreemen<sup>t</sup> with RefProp 9.1 than A1. We note that D6 gives rise to the maximum deviation between A1 and RefProp 9.1, as we shown below. This is clearly due to the fact that the slope −*b*A1 = 128.0031 for D6 is much larger than −*b*A3 = 120.7889.

**Figure 3.** Liquid–vapor saturation boundary in a *T*r-*s*<sup>∗</sup> diagram for: (**a**) ammonia; (**b**) benzene; (**c**) methyl palmitate; and (**d**) D6. The solid lines correspond to the results of approximation A3 for the entropies of the saturated vapor *<sup>s</sup>*<sup>∗</sup>g,A3 (solid red line) and the saturated liquid *<sup>s</sup>*<sup>∗</sup>l,A3 (solid blue line). The dashed lines correspond to the results of approximation A1 for the entropies of the saturated vapor *<sup>s</sup>*<sup>∗</sup>g,A1 (dashed red line) and the saturated liquid *<sup>s</sup>*<sup>∗</sup>l,A1 (dashed blue line). The symbols are RefProp 9.1 results for *s*∗g (circles) and *s*∗l(triangles).

The only difference between approximations A1 and A3 comes from the value of the parameter *b* that is calculated from Equation (11) in the case A1 and from Equation (25) for A3. In Figure 4, we present a plot of *b*A3 vs. *b*A1 for the 121 fluids of RefProp 9.1 [17], which are also listed in Table A1. In all cases, we obtained negative values of *b* in the range −130 < *b* < −1 (notice that the modified rectilinear diameter relation is given by *b*(1 − *Tr*) and thus the slope of the straight line, <sup>−</sup>*b*, is always positive). Overall, the agreemen<sup>t</sup> between both approaches is very good except for very dry fluids (with large values of −*b*) where approximation A3 yields results for −*b* slightly smaller than those of approximation A1, as one can see in Figure 4a. Wet fluids are those for which *ξ*<sup>∗</sup>M < 0 and thus they present smaller values of <sup>−</sup>*b*. As shown in Figure 4b, the agreemen<sup>t</sup> between the results of A1 and A3 for wet fluids is still good but presents more spread. The transition from dry to wet fluids occurs for *b* ≈ −14 but it is not sharp, mainly due to the effect of the acentric factor in Equation (11).

**Figure 4.** The parameter *b*A3 vs. *b*A1. The symbols represent the values of *b*A3 and *b*A1 obtained from Equations (25) and (11), respectively, using RefProp 9.1 data [17]. The solid blue line corresponds to the identity *b*A3 = *b*A1. Dry fluids (*ξ*<sup>∗</sup>M > 0) are plotted with red circles and wet fluids (*ξ*<sup>∗</sup>M < 0) correspond to black squares. (**b**) A zoom of (**a**). The data for *b*A3 and *b*A1 are listed in Table A1.

To provide a quantitative measurement of the performance of approximations A1 and A3, we consider the following expression for the percent relative deviation Δ*r*<sup>i</sup> of approximation Ai (i = 1, 3), which was introduced in [16]:

$$\Delta\_{\rm ri} = 100 \frac{\int\_{0.6}^{1} |s\_{\rm g}^\*(T\_r) - s\_{\rm g, \rm Ai}^\*(T\_r)| \mathrm{d}T\_r + \int\_{0.6}^{1} |s\_1^\*(T\_r) - s\_{\rm l, \rm Ai}^\*(T\_r)| \mathrm{d}T\_r}{\int\_{0.6}^{1} |s\_{\rm g}^\*(T\_r) - s\_{\rm l}^\*(T\_r)| \mathrm{d}T\_r} \, \mathrm{} \tag{29}$$

where *s*∗g and *s*∗l are RefProp 9.1 results. We obtained Δ*r*<sup>1</sup> = 4.82% and Δ*r*<sup>3</sup> = 3.42% for ammonia, Δ*r*<sup>1</sup> = 2.11% and Δ*r*<sup>3</sup> = 2.28% for benzene, and Δ*r*<sup>1</sup> = 24.68% and Δ*r*<sup>3</sup> = 14.12% for methyl palmitate.

Figure 5 compares the percent relative deviations of A1 and A3 for the 121 fluids of RefProp 9.1. To relate the performance of A1 and A3 to the dry or wet character of the fluid, the percent relative deviations are plotted as a function of the parameter *ξ*<sup>∗</sup>M. We note that, for the sake of clarity, the results are plotted with lines instead of symbols. As one can observe in Figure 5 (see also Table A1) the new approximation A3 fares better than A1 for fluids with *ξ*<sup>∗</sup>M>28, i.e., for very dry fluids. For the remaining fluids, both approximations yield similar results, in most cases with relative deviations less than 5% (below the dotted line in Figure 5). More concretely, in the case of the new approximation A3, we obtained an average percent relative deviation Δ*r*<sup>3</sup> = 4.19% for all fluids of Table A1, with a maximum percent relative deviation max(<sup>Δ</sup>*r*<sup>3</sup>) = 15.56% for methyl stearate (the only fluid with Δ*r*<sup>3</sup> > 15%). In the case of A1, one has [16] Δ*r*<sup>1</sup> = 5.00% and max(<sup>Δ</sup>*r*<sup>1</sup>) = 34.10% for D6, with nine fluids of a total of 121 with Δ*r*<sup>1</sup> > 15%. The reason for obtaining Δ*r*<sup>1</sup> > Δ*r*<sup>3</sup> can be clearly ascribed to the poorer behavior of A1 for very dry fluids. We note, however, that, in the two approximate treatments, 96 fluids have relative deviations less than 5%. Among the wet fluids (*ξ*<sup>∗</sup>M < 0), the largest relative deviations were obtained for helium (Δ*r*<sup>1</sup> = 13.86% and Δ*r*<sup>3</sup> = 14.38%) and R40 (Δ*r*<sup>1</sup> = 12.65% and Δ*r*<sup>3</sup> = 12.83%). Finally, we would like to note that, even with large deviations, e.g. Δ*r*<sup>1</sup> = 34.10% for D6, one can obtain fairly good results, as shown in Figure 3d.

**Figure 5.** The percent relative deviations Δ*r*<sup>1</sup> and Δ*r*<sup>3</sup> vs. *ξ*<sup>∗</sup>M. For clarity, the deviations are plotted with lines: the dashed red line corresponds to Δ*r*<sup>1</sup> and the solid blue line represents Δ*r*3. The dotted line indicates a deviation level of 5%. The data for Δ*r*1, Δ*r*3, and *ξ*<sup>∗</sup>M are listed in Table A1.

## **6. Summary**

In a previous work [16], a semiempirical method was developed for obtaining the liquid–vapor boundary of a fluid in a *Tr* − *s*<sup>∗</sup> diagram. The method assumes that, for a certain value of the quality *q* ¯ = 0.385, the line of constant *q*¯ in the *Tr* − *s*<sup>∗</sup> diagram (*s*<sup>∗</sup>*q*¯(*Tr*)) is very close to the straight line *<sup>s</sup>*<sup>∗</sup>*q*¯(*Tr*) ≈ *b*(1 − *Tr*), i.e., one has a modified rectilinear diameter relation for the saturation entropies. From this assumption, the semiempirical method only requires two ingredients: an appropriate expression for the enthalpy of vaporization and an approximate estimation of the parameter *b*. In what refers to the enthalpy of vaporization, perhaps the simplest choice is the extended corresponding states version [20] of the Watson equation [21] considered in [16] and in the present work. Two approximations, A1 and A2, where developed for *b* in [16]. The more accurate is approximation A1 with *b*A1 given by Equation (11) and therefore depending on the acentric factor *ω*, the parameter *ξ*<sup>∗</sup>M, and the reduced temperature *T*M*<sup>r</sup>*. The main drawback of A1 is that, while *ω* is available for most fluids, the other two parameters must be calculated for each fluid. Approximation A2 is less accurate but it has the advantage that, in addition to *ω*, it only requires the critical molar volume *νc* that can be accessed in most databases.

In the present work, we developed a new approximation A3 where the parameter *b*A3 is given by Equation (25) and only depends on *ω* and on the ideal-gas isobaric molar heat capacity of the fluid *c*ig<sup>∗</sup> *p* at a reduced temperature *T*M*r* = 0.81. We want to recall that excellent approximations for *c*ig<sup>∗</sup> *p* are available for a large variety of fluids in most Thermophysical Properties databases. The new approximation has the virtues of approximations A1 and A2 without their problems: it has an accuracy similar to or even better than that of A1 and only needs the value of easily accessible parameters.

To conclude we would like to comment that the present work has also served to clarify the role played by the ideal-gas isobaric molar heat capacity in the shape of the liquid–vapor saturation boundary in a temperature–entropy diagram, in agreemen<sup>t</sup> with the results of other authors [11–15].

**Author Contributions:** Conceptualization, J.A.W. and S.V.; Formal analysis, J.A.W. and S.V.; and Writing— Original draft, J.A.W. and S.V.

**Funding:** This research was funded by Junta de Castilla y León of Spain gran<sup>t</sup> number SA017P17.

**Conflicts of Interest:** The authors declare no conflict of interest.
