**1. Introduction**

There is an increasing interest in the use of Organic Rankine Cycles (ORCs) as a suitable way of generating power from low-temperature heat sources such as geothermal, solar thermal, biomass, waste heat, and bottoming cycles. As is well known, a key aspect in the optimal implementation of an ORC for a given heat source is the choice of the working fluid. An appropriate working fluid selection should take into account several criteria such as thermo-economic efficiency, safety, environmental aspects, chemical stability, etc. [1–6]

One of the most relevant aspects in ORC working fluid selection is the analysis of the shape of the liquid–vapor saturation curve in a temperature–molar entropy (*T*-*<sup>s</sup>*) diagram because it has a direct influence both in the thermal efficiency and in the particular design of the cycle. Let us consider a simple, ideal ORC with an evaporation temperature *T*ev and a condensation temperature *T*con, so that *T*con < *T*ev < *Tc* where *Tc* is the critical temperature. In this simple ORC, the isentropic expansion that takes place in the turbine and starts from a saturated vapor state at *T*ev can lead to three different situations depending on the shape of the saturation curve: (1) If the mean slope of the vapor branch of the *T*-*s* saturation curve between *T*con and *T*ev is negative, the working fluid has a *wet fluid behavior* and the isentropic expansion process in the turbine gives rise to condensation, i.e., it ends in the two-phase region of the *T*-*s* diagram. This situation should be avoided (via superheating) since the mixture of vapor with liquid droplets could lead to damage of the turbine blades [7]. (2) If the mean slope is positive, the working fluid has a *dry fluid behavior* and the isentropic expansion leads to superheated vapor. This implies a reduction in the cycle efficiency which can only be partially remediated by resorting to a regenerator. (3) Finally, if the absolute value of the mean slope is very large, the working fluid *behaves as an isentropic fluid* so that the turbine isentropic expansion ends near the saturated vapor state at *T*con. In this case, neither regeneration nor superheating is required.

Fluids such as water, carbon dioxide or ammonia that always have a negative slope in the vapor branch of the *T*-*s* saturation curve (d *<sup>T</sup>*/d*<sup>s</sup>*g < 0) always *behave as wet fluids* and are usually termed as *wet*. On the other hand, fluids such as siloxanes or alkanes with a large number of carbons have a positive slope for most temperatures in the *T*-*s* saturation curve and are called *dry fluids* since they usually lead to a *dry fluid behavior*. Finally, fluids like the refrigerants RE143a, R11, or R116 present a wide range of temperatures for which the saturated vapor curve is almost vertical. These fluids are usually termed as *isentropic*, although *isentropic behavior* can also be obtained with a particular class of *dry fluids* [8–10]. To summarize, the slope of the vapor branch of the *T*-*s* saturation curve gives rise to a basic classification of working fluids into three categories: wet, dry, and isentropic. According to Liu et al. dry or isentropic fluids are preferred for ORC applications since they eliminate the problems related to condensation in the isentropic expansion [7].

Most studies on the shape of the *T*-*s* saturation boundary have focused in the analysis of the slope of the vapor branch d *<sup>T</sup>*/d*<sup>s</sup>*g and its relation with the molar heat capacity of the fluid. In a seminal work, Morrison [11] presented a study in the context of refrigeration cycles, concluding that negative slopes in the vapor branch do arise for fluids with large isochoric molar heat capacity and, consequently, for fluids with large, complex molecules. Liu et al. [7] derived an approximate expression of the (inverse) slope *ξ* = <sup>d</sup>*<sup>s</sup>*g/d*<sup>T</sup>* in terms of the isobaric molar heat capacity of the saturated vapor and the molar enthalpy of vaporization, and related the dry or wet character of the fluid to the sign of *ξ*. Other authors have obtained results for different model equations of state, concluding that a key aspect in the shape of the *T*-*s* saturation curve is the ideal gas contribution to the molar heat capacity of the fluid [12–15].

Recently [16], we proposed a semiempirical method to obtain the *T*-*s* saturation curve by means of: (1) a modified rectilinear diameter law for the two branches of the saturated entropy; and (2) an appropriate expression for the entropy of vaporization Δ*s*. The method only requires the knowledge of three parameters: *Tc*, the acentric factor *ω*, and the slope of the modified rectilinear diameter <sup>−</sup>*b*. Two approximations were considered for *b*. The simplest one requires the knowledge of the critical molar volume *νc*, which together with *Tc* and *ω* can be obtained in any Thermophysical Properties database. The most accurate approximation requires the calculation of the slope *ξ* at a certain temperature and therefore, the use of programs like RefProp [17] or CoolProp [18]. The goal of the present work is to study the relation between the modified rectilinear diameter law and the ideal gas molar heat capacity considered in earlier works [12–15]. More concretely, in this work, we propose a new approximation for *b* in terms of *ω* and the ideal-gas isobaric molar heat capacity at an appropriate reduced temperature. The new approximation has the same application range as the previous ones, with the advantage that it does not require the calculation of *ξ* and yields better results for very dry fluids.

This paper is structured as follows. In Section 2, we outline the main features of the semiempirical method in [16]. In Section 3, we introduce the ideal gas contribution to the entropy and investigate its relevance in the dry or wet behavior of a working fluid. From the results of the preceding sections, a new approximation for the slope of the modified rectilinear diameter is proposed in Section 4, in terms of the ideal gas molar heat capacity of the fluid. In Section 5, we compare the results of the new approximation with previous ones. Finally, we conclude in Section 6 with a brief summary.

#### **2. A Semiempirical Method for the** *T***-***<sup>s</sup>* **Saturation Curve**

In [16], we proposed a semiempirical method to obtain approximate expressions for the liquid–vapor phase boundary in a temperature–entropy diagram. Here, we present the basic equations of the method that are employed in connection with the ideal gas molar heat capacity of the fluid. We find it convenient to work in reduced coordinates so that, without loss of generality, we consider a *Tr* − *s*<sup>∗</sup> diagram where *Tr* = *T*/*Tc* is the reduced temperature and *s*<sup>∗</sup> = (*s* − *sc*)/*<sup>R</sup>*

is a dimensionless molar entropy, where *sc* is the molar critical entropy and *R* is the gas constant. In this context, the entropies of the gas and liquid branches of the phase boundary are denoted as *s*∗g and *s*∗l , respectively, and are functions of *Tr*. Furthermore, with these definitions, for all fluids, one has *<sup>s</sup>*<sup>∗</sup>g(*Tr* → 1) = *s*∗l (*Tr* → 1) = 0.

To determine *<sup>s</sup>*<sup>∗</sup>g(*Tr*) and *s*∗l (*Tr*), two relations are considered in the semiempirical method. The first one is the well-known relation between the dimensionless molar enthalpy of vaporization Δv*hr* = Δv*h*/*RTc* and the molar entropy of vaporization Δv*s*<sup>∗</sup> = *s*∗g − *s*∗l:

$$
\Delta\_{\mathbf{v}} h\_{\mathbf{r}} = T\_{\mathbf{r}} \Delta\_{\mathbf{v}} \mathbf{s}^\* \,. \tag{1}
$$

Several approximate expressions for Δv*hr* are available in the literature [19]. For simplicity, we consider the following corresponding states version [20] of the Watson equation [21]:

$$
\Delta\_{\rm v} h\_{\rm r} = K(\omega)(1 - T\_{\rm r})^{0.38},\tag{2}
$$

where

$$K(\omega) = 7.2729 + 10.4962\,\omega + 0.6061\,\omega^2 \tag{3}$$

is a function of the acentric factor of the fluid *ω*.

The second relation in the semiempirical method is based on the observation [16] that the line of constant quality *q* in the two-phase region of the *Tr* − *s*<sup>∗</sup> diagram,

$$s\_q^\*(T\_r) = q s\_\mathcal{g}^\*(T\_r) + (1 - q) s\_1^\*(T\_r) \,. \tag{4}$$

has an approximately linear behavior in the range 0.6 < *Tr*<0.99 for an appropriate value of *q*. The optimal value of *q* varies slightly with the fluid, with a mean value *q*¯ = 0.385 for the 121 fluids of the RefProp 9.1 program [17]. Therefore, one can write the following *modified rectilinear diameter* relation for the entropies of the saturated liquid and vapor [16]:

$$s\_{\overline{q}}^{\*}(T\_{\overline{r}}) \approx b(1 - T\_{\overline{r}}) \qquad \left(0.6 < T\_{\overline{r}} < 0.99\right),\tag{5}$$

where −*b* is the slope of the modified rectilinear diameter. We note here that this linear relation is similar to the well known rectilinear diameter law of Cailletet and Mathias [22] for the saturation densities but, instead of using *q* = 0.5, one has to consider a different value of *q* for the entropy.

Equations (1) and (4) allow expressing the saturation entropies in terms of Δv*hr* and *s*∗*q* (*Tr*):

$$s\_{\mathfrak{E}}^{\*}(T\_r) = s\_q^{\*}(T\_r) + (1 - q)\frac{\Delta\_{\mathrm{v}}h\_r}{T\_r} \,. \tag{6}$$

$$s\_1^\*(T\_I) = s\_q^\*(T\_I) - q \frac{\Delta\_\mathrm{v} h\_r}{T\_I} \, , \tag{7}$$

that are exact relations valid for any value of the quality in the range 0 ≤ *q* ≤ 1. Substituting the approximations Equations (2) and (5) with *q* = *q*¯ = 0.385, it is direct to obtain the approximate expressions

$$s\_{\mathfrak{g}}^{\*}(T\_r) = b(1 - T\_r) + (1 - \mathfrak{q})K(\omega)\frac{(1 - T\_r)^{0.38}}{T\_r},\tag{8}$$

$$s\_1^\*(T\_I) = b(1 - T\_I) - \bar{q}K(\omega) \frac{(1 - T\_r)^{0.38}}{T\_r} \,\mathrm{},\tag{9}$$

which are expected to yield good results for the entropies of the saturated vapor and liquid in the range 0.6 < *Tr*<0.99. Equations (8) and (9) require the parameter *b* as an input. Differentiating Equation (8) with respect to *Tr* one has

$$\frac{\mathrm{d}s\_{\mathrm{g}}^{\*}}{\mathrm{d}T\_{\mathrm{I}}} = -b - (1 - \eta)K(\omega)\frac{(1 - 0.62T\_{\mathrm{r}})}{T\_{\mathrm{r}}^{2}(1 - T\_{\mathrm{r}})^{0.62}}.\tag{10}$$

From this equation, two approximations were considered in [16] for *b*. The first one, referred to as approximation A1, is given by the following expression:

$$b = -\zeta\_{\rm M}^{\*} - (1 - \eta)K(\omega)\frac{(1 - 0.62T\_{\rm Mr})}{T\_{\rm Mr}^{2}(1 - T\_{\rm Mr})^{0.62}},\tag{11}$$

where *T*M*r* is the reduced temperature at which the derivative <sup>d</sup>*s*<sup>∗</sup>g/d*Tr* attains its maximum value *ξ*<sup>∗</sup>M (see [16,23] for details). The values of *T*M*r* and *ξ*<sup>∗</sup>M have been calculated in [23] for the 121 pure fluids of RefProp 9.1. From now on, all results obtained from this approximation are labeled with the subscript A1, i.e., we refer to *b*A1, *<sup>s</sup>*<sup>∗</sup>g,A1, *<sup>s</sup>*<sup>∗</sup>l,A1, and *<sup>s</sup>*<sup>∗</sup>*q*¯,A1.

To avoid the dependence of the parameter *b* on *T*M*r* and *ξ*<sup>∗</sup>M, a further approximation A2 was presented in [16], in which *T*M*r* was replaced by its mean value *T*M*r* = 0.81 and a correlation in terms of the critical molar volume *νc* was used for *ξ*<sup>∗</sup>M. Approximation A1 is more accurate, but approximation A2 only depends on *Tc*, *ω*, and *νc* that are easily accessible for several fluids.

#### **3. The Ideal Gas Contribution to the Entropy**

The dimensionless molar entropy of a fluid can be expressed as

$$\mathbf{s}^\*(T\_{r\prime}\boldsymbol{\nu}\_r) = \mathbf{s}^{\rm ig} \mathbf{s}^\*(T\_{r\prime}\boldsymbol{\nu}\_r) + \mathbf{s}^{\rm r\*}(T\_{r\prime}\boldsymbol{\nu}\_r),\tag{12}$$

where *νr* = *<sup>ν</sup>*/*<sup>ν</sup>c* is the reduced volume, *s*r<sup>∗</sup> = (*s*r − *sc*)/*<sup>R</sup>* is a dimensionless residual entropy (that depends on the equation of state of the fluid), and *s*ig<sup>∗</sup> = *s*ig/*<sup>R</sup>* is the ideal gas contribution to the entropy:

$$\mathrm{s}^{\mathrm{ig}\*}\left(T\_{r\prime}\nu\_{r}\right) = \int\_{T\_{r\prime}0}^{T\_{r}} \frac{c\_{p}^{\mathrm{ig}\*}\left(T\_{r}\right) - 1}{T\_{r}} \mathrm{d}T\_{r} + \log\left(\frac{\nu\_{r}}{\nu\_{r\prime}0}\right) + s\_{0}^{\mathrm{ig}\*}\,. \tag{13}$$

In Equation (13), the subscript 0 refers to an arbitrary reference state and *c*ig<sup>∗</sup> *p* (*Tr*) = *c*ig*p* (*Tr*)/*R* is the dimensionless ideal gas isobaric heat capacity. Several approximate expressions for *c*ig<sup>∗</sup> *p* have been used in the literature. Common choices are polynomial expansions of the form

$$c\_p^{\text{ig}\*}(T\_r) = \sum\_{i=0,4} a\_i T\_{r\text{\textquotedblleft}r\text{\textquotedblright}}^i \tag{14}$$

and the well-known Aly-Lee equation [24] used by the DIPPR database [25]

$$c\_p^{\rm ig\star}(T\_r) = d\_0 + d\_1 \left[ \frac{d\_2 / T\_r}{\sinh(d\_2 / T\_r)} \right] + d\_3 \left[ \frac{d\_4 / T\_r}{\sinh(d\_4 / T\_r)} \right] \tag{15}$$

where the parameters *di* (*ai* in Equation (14)) are obtained by a fit to experimental data, and are, therefore, fluid-dependent. Furthermore, it is worth mentioning that, depending on the fluid, the RefProp 9.1 program [17] uses different approximate expressions for *c*ig*p* that provide an accurate fit to experimental data.

We find it convenient to split the ideal gas contribution to the entropy in two parts:

$$\mathbf{s}^{\text{ig}\*}\left(T\_{r\prime}\boldsymbol{\nu}\_{r}\right) = \mathbf{s}^{\text{ig}\rm O\*}\left(T\_{r}\right) + \mathbf{s}^{\text{ig}\rm 1}\left(T\_{r\prime}\boldsymbol{\nu}\_{r}\right),\tag{16}$$

where

$$\mathrm{s}^{\mathrm{ig0\*}}\left(T\_r\right) = -\int\_{T\_r}^{1} \frac{\mathrm{c}\_p^{\mathrm{ig\*}}\left(T\_r\right)}{T\_r} \mathrm{d}T\_{r\prime} \tag{17}$$

and

$$s^{\rm ig1\*}\left(T\_{r\prime}\upsilon\_{r}\right) = \log\left(\frac{T\_{r,0}\upsilon\_{r}}{T\_{r}\upsilon\_{r,0}}\right) + s\_{0}^{\rm ig\*} - s^{\rm ig0\*}\left(T\_{r,0}\right) \tag{18}$$

so that *s*ig0<sup>∗</sup> carries the temperature dependence of the dimensionless molar entropy due to *c*ig<sup>∗</sup> *p* , and *s*ig1<sup>∗</sup> is a function of the temperature and the molar volume. From Equations (12) and (16)–(18), one can write the following expressions for the entropies of the saturated vapor and liquid

$$s\_{\mathfrak{g}}^{\*}(T\_r) = s^{\mathrm{i}\mathfrak{g}\otimes \*}(T\_r) + s^{\mathrm{ex}\*}(T\_r, \upsilon\_{r,\mathfrak{g}}) \, , \tag{19}$$

$$s\_1^\*(T\_r) = \mathbf{s^{i\_0^\*0\*}}(T\_r) + \mathbf{s^{e\infty\*}}(T\_{r\prime} \boldsymbol{\nu}\_{r\prime 1}) \,\,\,\,\tag{20}$$

where *<sup>ν</sup>r*,g and *<sup>ν</sup>r*,<sup>l</sup> are, respectively, the reduced volumes of the saturated vapor and liquid. The excess entropy *<sup>s</sup>*ex<sup>∗</sup>(*Tr*, *<sup>ν</sup>r*) = *<sup>s</sup>*ig1∗(*Tr*, *<sup>ν</sup>r*) + *<sup>s</sup>*r<sup>∗</sup>(*Tr*, *<sup>ν</sup>r*), is a function of the reduced temperature and the reduced volume of the fluid. Equations (19) and (20) show that the entropies of the saturated liquid and vapor can be separated in two contributions. The first one, *<sup>s</sup>*ig0<sup>∗</sup>, is the same for the two saturated phases and only depends on *Tr* through an integral of *c*ig<sup>∗</sup> *p* /*Tr*. The second contribution *s*ex<sup>∗</sup> is different for each saturated phase and depends on the equation of state of the fluid.

The behavior of the ideal gas contribution *s*ig0∗(*Tr*) defined in Equation (17) is shown in Figure 1 in a *T*r − *s*∗ diagram for: (a) ammonia; (b) benzene; and (c) methyl palmitate. In this figure, one can see that the main source for the inclination of the *T*r − *s*∗ saturation boundary is the ideal gas contribution *<sup>s</sup>*ig0∗(*Tr*). This implies that the dry or wet character of a fluid is mainly driven by the slope of *<sup>s</sup>*ig0∗(*Tr*). This fact was previously observed by Groniewsky and coworkers [14,15], but using the contribution to the molar entropy due to the isochoric molar heat capacity of the ideal gas *c*ig*v* instead of the isobaric one, *c*ig*p* .

We also show in Figure 1 the line of constant quality *s*∗*q*¯ (with *q*¯ = 0.385) obtained from Equation (4) using RefProp 9.1 results [17] for *s*∗l and *s*∗g, and the result *<sup>s</sup>*<sup>∗</sup>*q*¯,A1 of approximation A1 for the modified rectilinear diameter relation in Equation (5) with the parameter *b*A1 given by Equation (11). To calculate *b*A1, the following values were used [23] (see also Table A1): *ω* = 0.256, *T*M*r* = 0.8162, and *ξ*<sup>∗</sup>M = −8.6109 for ammonia; *ω* = 0.211, *T*M*r* = 0.8252, and *ξ*<sup>∗</sup>M = 6.1558 for benzene; and *ω* = 0.91, *T*M*r* = 0.7388, and *ξ*<sup>∗</sup>M = 86.6279 for methyl palmitate. We obtained *b*A1 = −4.4225 for ammonia, *b*A1 = −18.5302 for benzene, and *b*A1 = −110.949 for methyl palmitate.

In Figure 1a, we consider ammonia as an example of wet fluid. As one can observe, in this case, the ideal gas contribution *s*ig0∗(*Tr*) shows a noticeable deviation from both the line of constant quality *s*∗*q*¯ obtained from RefProp 9.1 and the approximate (straight line) result *<sup>s</sup>*<sup>∗</sup>*q*¯,A1. The best agreemen<sup>t</sup> for ammonia was obtained for *Tr* ≈ 0.8. Much better agreemen<sup>t</sup> was obtained for benzene (Figure 1b), especially in the range 0.7 < *Tr* < 0.99. The case of methyl palmitate presented in Figure 1c shows a good agreemen<sup>t</sup> between *s*ig0∗(*Tr*) and *s*∗*q*¯, with larger deviations with *<sup>s</sup>*<sup>∗</sup>*q*¯,A1. We note that methyl palmitate is a "very dry" fluid, with a large value of *ξ*<sup>∗</sup>M and approximation A1 is known to give large deviations for these fluids [16]. In Figure 1d, we plot the results for *<sup>s</sup>*ig0<sup>∗</sup>, *s*∗*q*¯, and *<sup>s</sup>*<sup>∗</sup>*q*¯,A1 in the same scale, showing that the main discrepancies take place between *s*∗*q*¯ and *<sup>s</sup>*<sup>∗</sup>*q*¯,A1for methyl palmitate.

From the results presented in Figure 1, which are also valid for the remaining fluids of RefProp 9.1 (not shown), we can conclude that the ideal gas contribution *s*ig0<sup>∗</sup> is close to the line of constant quality *s*∗*q*¯ with *q*¯ = 0.385 but some differences do arise, especially for *Tr* < 0.7. Therefore, it is not advisable to replace *s*∗*q*¯ with *s*ig0<sup>∗</sup> in Equations (6) and (7) in order to obtain a new approximation for the entropy of the saturation boundary. As shown in the next section, it is more convenient to consider the approximate expressions of Equations (8) and (9) but using information from *s*ig0<sup>∗</sup> instead of Equation (11).

**Figure 1.** *T*r-*s*<sup>∗</sup> diagram for: (**a**) ammonia; (**b**) benzene; and (**c**) methyl palmitate. The solid red line corresponds to the ideal gas contribution to the entropy *s*ig0<sup>∗</sup> defined in Equation (17), the dotted black line is the constant quality entropy *s*∗*q*¯ obtained from Equation (4) with *q*¯ = 0.385, and the dashed blue line shows *<sup>s</sup>*<sup>∗</sup>*q*¯,A1 = *b*(1 − *<sup>T</sup>*r) with *b* obtained from Equation (11) (approximation A1). The solid black line represents the liquid–vapor saturation boundary. (**d**) A comparison of the results for *<sup>s</sup>*ig0<sup>∗</sup>, *s*∗*q*¯, and *<sup>s</sup>*<sup>∗</sup>*q*¯,A1presented in (**<sup>a</sup>**–**<sup>c</sup>**). All data were obtained using RefProp 9.1 [17].

#### **4. A New Approximation for the** *T***-***<sup>s</sup>* **Saturation Curve**

Differentiating Equation (19) with respect to *Tr*, we obtain the following expression for the slope of the entropy of the saturated vapor

$$\frac{\mathrm{d}s\_{\mathrm{g}}^{\*}}{\mathrm{d}T\_{r}} = \frac{c\_{p}^{\mathrm{ig}\*}(T\_{r})}{T\_{r}} - \frac{\psi^{\*}(T\_{r})}{T\_{r}},\tag{21}$$

where 
$$\psi^\*(T\_l) = -T\_l \frac{\mathrm{d}s^{\mathrm{ex}\*} (T\_{r\_l} \nu\_{r, \mathrm{g}}(T\_l))}{\mathrm{d}T\_r} \,. \tag{22}$$

We note that, as pointed out by Garrido et al. [12], according to Equation (21), a fluid behaves as dry for reduced temperatures such that

$$
\omega\_p^{\rm ig^\*} (T\_r) > \Psi^\*(T\_r) \,. \tag{23}
$$

A rigorous approach for obtaining *ψ*∗ from an explicit equation of state (Eos) model has been developed by Garrido et al. [12]. This implies that *ψ*∗ should be correlated with the acentric factor *ω* of the fluid. In Figure 2, we plot *ψ*∗ vs. *ω* for the 121 fluids of RefProp 9.1 [17] at a reduced temperature *Tr* = *T*M*r* = 0.81, which is taken as reference. Instead of considering a given Eos model, the values of *ψ*∗ were obtained from Equation (21) and RefProp 9.1 data for *c*ig*p* and <sup>d</sup>*s*<sup>∗</sup>g/d*Tr*. As one can observe, there is a rather good correlation between *ψ*∗ and *ω*. A quadratic fit excluding two oddball fluids (ethanol and methanol, black dots in Figure 2) yields

$$
\psi^\*(\omega, \overline{T}\_{\text{Mr}}) = 8.7872 + 8.7191\omega - 1.9704\omega^2 \tag{24}
$$

with a coefficient of determination *R*<sup>2</sup> = 0.9838. *Energies* **2019**, *12*, 3266

Equating Equations (10) and (21) and taking as reference the reduced temperature *T*M*r* = 0.81, we obtain

$$b\_{\rm A3} = -\frac{c\_p^{\rm ig^\ast}(T\_{\rm Mr}) + \delta(\omega)}{T\_{\rm Mr}},\tag{25}$$

where the label A3 has been chosen to compare with approximations A1 and A2 in [16], and, using Equations (3) and (24),

$$\delta(\omega) = -\psi^\*(\omega, T\_{\rm Mr}) + (1 - \bar{\eta})K(\omega)\frac{(1 - 0.62T\_{\rm Mr})}{\overline{T}\_{\rm Mr}(1 - \overline{T}\_{\rm Mr})^{0.62}}$$

$$= -1.0901 + 2.3893\omega + 2.6119\omega^2. \tag{26}$$

**Figure 2.** *ψ*∗ vs. *ω* for the 121 fluids of RefProp 9.1 [17] at a reduced temperature *T*M*r* = 0.81. The dashed blue line is the result of a quadratic fit to the data (red circles) excluding ethanol and methanol (solid black circles).

Finally, inserting Equation (25) into Equations (8) and (9), we obtain

$$s\_{\overline{g}\mathcal{A}3}^{\*}(T\_r) = -\frac{c\_p^{\text{ig}\ast}(\overline{T}\_{\text{Mr}}) + \delta(\omega)}{\overline{T}\_{\text{Mr}}}(1 - T\_r) + (1 - \overline{q})K(\omega)\frac{(1 - T\_r)^{0.38}}{T\_r},\tag{27}$$

$$s\_{\rm l,A3}^{\*}(T\_r) = -\frac{c\_p^{\rm lq^\circ}(T\_{\rm Mr}) + \delta(\omega)}{T\_{\rm Mr}}(1 - T\_r) - \bar{q}K(\omega)\frac{(1 - T\_r)^{0.38}}{T\_r},\tag{28}$$

which are the results of our new approximation A3 for the entropies of the saturated vapor and liquid. Taking into account that the new approximation is also based in the modified rectilinear diameter relation in Equation (5), we expect that the results of A3 should apply in the range 0.6 < *Tr* < 0.99. We would like to recall here that Equations (27) and (28) only require the knowledge of the acentric factor of the fluid, its critical temperature, and the ideal gas molar isobaric heat capacity at a reduced temperature *T*M*r* = 0.81. These parameters are easily accessible for a large amount of fluids using well known databases such as DIPPR [25].

In the next section, we analyze the results of Equations (27) and (28) by comparing with RefProp data and with the results of approximation A1 derived in [16].
