Relation between the Isentropic Index and the GrÃ¼neisen Parameter

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Relation between the Isentropic Index and the Grü neisen Parameter for Saturated Liquids Kandadai Srinivasan,†,‡ Santiago Velasco,§,⊥ and Juan A. White*,§,⊥ †

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India School of Mechanical and Chemical Engineering, University of Western Australia, Crawley 6009, Australia § Departamento de Fisica Aplicada and ⊥IUFFyM, Universidad de Salamanca, 37008 Salamanca, Spain ‡

S Supporting Information *

ABSTRACT: We investigate the isentropic index along the saturated vapor line as a correlating parameter with quantities both in the saturated liquid phase and the saturated vapor phase. The relation is established via temperatures such as Thgmax and T* where the saturated vapor enthalpy and the product of saturation temperature and saturated liquid density attain a maximum, respectively. We obtain that the saturated vapor isentropic index is correlated with these temperatures but also with the saturated liquid Grüneisen parameters at Thgmax and T*.

1. INTRODUCTION Thermodynamic properties of pure fluids along the liquid− vapor coexistence curve have been a fascinating field of research as they play an important role in the choice of working fluids for engineering applications in refrigeration, heat pumping, or power cycles.1 The most widely investigated properties are the vapor pressure2−4 and the heats of vaporization,5,6 while densities are also given prominence.7,8 In a recent paper,9 we have analyzed the relation between the saturated vapor phase and the isentropic index k = ∂ ln p/∂ ln ρ|s. One of the findings of ref 9 is the existence of a linear relation between the minimum in the saturated vapor isentropic index kgmin, which is a slope property, and the reduced saturated vapor density ρg/ρc, which is an equilibrium property, at the temperature where that minimum occurs. Srinivasan10 identified a linear relation between the mean molecular length parameter and the ratio of temperatures at which the saturated vapor enthalpy has a maximum (Thgmax) to the temperature at which the product of saturation temperature and saturated liquid density attains a maximum (T*) based on the analysis of a limited number of refrigerants and cryogenic fluids. When extended to the entire range of liquids covered by the NIST program RefProp 9.0, Srinivasan et al.9 observed that the linear relation is no longer valid because Thgmax tends to the critical temperature Tc for high critical molar volume fluids and hence Thgmax/T* flattens out deviating from linearity. We note that most of the previous efforts were oriented toward correlating either the saturated liquid phase thermodynamic properties or the saturated vapor properties. In this context it is worth recalling that T* determines a characteristic point of the saturated liquid line which can be connected with critical properties. In particular, T* is linearly related to Tc and the saturated liquid density at this point shows a linear correlation with the critical density. On the other hand, although Thgmax is determined from the saturated vapor enthalpy, it bears a good quadratic relation with T* (ref 9) and, as mentioned above, the ratio Thgmax/T* is related to the critical temperature thus providing a link between © 2014 American Chemical Society

saturated vapor, saturated liquid, and critical properties. In this work we shall deepen into relationships of this kind. In an article describing the slopes of thermodynamic properties on various coordinate systems involving pressure− volume−temperature and entropy (p−v−T and s), Srinivasan et al.11 showed that most of them can be reduced to a form where the Grüneisen parameter γ = ∂ ln T/∂ ln ρ|s appears repeatedly. This dimensionless parameter is also referred to as Grüneisen’s gamma/constant and is generally used as a characterizing property of solids. There have been some studies to apply significance of this parameter to fluids. For example, Mizushima12 applied the dislocation model of liquid structure to determine Grüneisen’s constant of molecular liquids. While this parameter is known to be reasonably constant over a wide temperature range for solids, Knopoff and Shapiro13 showed that for water and mercury it is not so. It was shown to increase with decreasing specific volume contrary to what happens with solids. Further, they observed differences even between these two fluids and attributed them to the higher compressibility of mercury than that of water. The work of Arp et al.14 is a major application of the Grüneisen parameter for fluids in hydrodynamics. They listed a number of variants in which this parameter can be expressed. They found it to be in the range of 0.2−2, which is somewhat less than that of solids. They highlighted the peculiarities of water, which shows negative values for the Grüneisen parameter because of the negative value of the coefficient of expansion at temperatures 1 (σc−2 > 5.4 nm−2 or vc < 0.15 m3 kmol−1). It also emerges that the oddball fluids are water, heavy water, and methanol. If these fluids are not included, the linear regression yields the following empirical correlation Thgmax /T * = −1.3124 − 0.2317kgmin (2)

with a coefficient of determination (R2) of 0.880. In Figure 1 the symbols represent the results for the fluids included in the NIST RefProp 9.1 program.21 We note that, for the sake of comparison with Figure 2, different symbols are used for methanol, water, and heavy water. We would like to remark about the different behavior obtained for fluids with σc−2 < 5.4 nm−2 (vc > 0.15 m3 kmol−1) and those with σc−2 > 5.4 nm−2. In the first case a fairly good linear behavior is obtained; whereas in the second case, the data are scattered to a

with a coefficient of determination of 0.968. The large deviation observed for methanol from the linear behavior in Figure 1 and especially in Figure 2 could be due to the equation of state used in RefProp 9.0 and 9.1 for this fluid. Problems arising with this equation of state where also found in the analysis of the Riedel function presented in ref 3. Two remarks should be made when comparing Figures 1 and 2. First we note that there is a gap in both figures around σc−2 ∼ 5.4 nm−2 (kgmin ∼1). This implies that there is also a gap in the

Thgmax /T * = −0.0317σc−2/nm−2 + 1.2927

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dx.doi.org/10.1021/ie500578v | Ind. Eng. Chem. Res. 2014, 53, 6866−6870

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critical molar volume among the pure fluids included in RefProp 9.1 between 0.146 m3 kmol−1 (ethane) and 0.159 m3 kmol−1 (R161). Second, while kgmin is linearly correlated with Thgmax/T* for kgmin > 1 in Figure 2, this is not the case for σc−2 > 5.4 nm−2 in Figure 1. The linear behavior observed in Figure 2 and given by eq 2 is one of the main results of the present paper. From this result kgmin emerges as a relevant parameter in the analysis of saturated fluid properties as shown below. Through eq 2 we have been able to correlate two saturated vapor phase properties (kgmin and Thgmax) with a saturated liquid phase property (T*). Other linear correlations involving these and related properties are listed in Table 1. The subscript f is Table 1. Empirical Correlations between Various Properties y

x

slope

intercept

R2

T* Thgmax kgThgmax

Tkgmin Tkgmin kgmin

0.7624 0.9954 0.9973

30.2263 −12.885 0.0426

0.993 0.987 0.995

Figure 4. Reduced temperature dependence of saturated Grüneisen parameters: (legend) full lines liquid phase, broken lines vapor phase, □ water (vc = 0.056 m3 kmol−1), ○ oxygen (vc = 0.073 m3 kmol−1), Δ ethane (vc = 0.146 m3 kmol−1), × propyne (vc = 0.164 m3 kmol−1), ∗ toluene (vc = 0.316 m3 kmol−1).

used for saturated liquid phase and g for saturated vapor phase. We note that the saturated vapor isentropic index at Thgmax (kgThgmax) shows an excellent linear correlation with kgmin while its value at T* (kgT*) has a very different relation with kgmin, as shown in Figure 3. A quadratic seems to describe their mutual dependence as given below, with no oddball fluids:

with a coefficient of determination of 0.992.

phase remains independent of temperature except close to the critical temperature. In the liquid phase, its variation can be classified as decreasing slope (for most cryogenic fluids), linear (most inorganic fluids except water), or increasing slope (e.g.: all refrigerants, high molecular weight organic fluids). Among the liquids represented in Figure 4, propyne has a very steep increase in liquid Grüneisen parameter, which is somewhat odd. For the sake of completeness Figure 5 shows the Grüneisen

Figure 3. Relation between kgT* and kgmin: (legend) ○ all fluids in RefProp 9.1.

Figure 5. Reduced pressure dependence of saturated Grüneisen parameters: (legend) full lines liquid phase, broken lines vapor phase, □ water (vc = 0.056 m3 kmol−1), ○ oxygen (vc = 0.073 m3 kmol−1), Δ ethane (vc = 0.146 m3 kmol−1), × propyne (vc = 0.164 m3 kmol−1), ∗ toluene (vc = 0.316 m3 kmol−1).

4. SATURATED FLUID PHASE GRÜ NEISEN PARAMETER Figure 4 shows some representative saturated fluid phase Grüneisen parameters covering fluids with critical molar volumes well below 0.15 m3 kmol−1, around it and well above it. Water has a peculiar behavior due to its negative coefficient of expansion close to the triple point at which the Grüneisen parameters would become negative. Consequently, it will show a maximum at a particular temperature which is at about 0.75Tc. Broadly, the Grüneisen parameter in the vapor

parameter for the same fluids as Figure 4 but now in terms of the reduced pressure, with a similar behavior (note that the maximum for water occurs at a reduced pressure near 0.1). A comparison of Grüneisen parameter at various saturation temperatures namely Tkgmin, T*, and Thgmax will be the theme of further discussion. Figure 6 shows a relation between the saturated vapor isentropic indices at Thgmax and its minimum value with the Grüneisen parameters of the saturated liquid phase at the same temperature. This indicates that it is possible to correlate the slope properties of vapor and liquid phases at the phase

kgT * = 0.4126kgmin 2 − 0.2252kgmin + 0.9167

(3)

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seen in Figure 8. Again methanol and ethanol cease to comply with the general trend, with propyne joining the group of

Figure 6. Dependence of the saturated liquid phase Grüneisen parameter with the isentropic index of the saturated vapor phase: (legend) × kgmin, + kThgmax; (oddball fluids) ○ methanol, Δ ethanol, □ water, ◇ heavy water; open symbols at Tkgmin, filled symbols at Thgmax.

Figure 8. Variation of the ratio of liquid phase Grüneisen parameter ratios at Thgmax and T* with kgmin: (legend) ○ all fluids except ● methanol, ∗ ethanol, and × propyne.

boundaries. Water and alcohol specimens again do not follow the general trend. If they are not included in the regression, the following empirical relation can be obtained combining both variants: γfT hgmax(or γfTk gmin) = 0.0215exp[3.0786kgT hgmax(or kgmin)]

noncomplying alcohols. Surprisingly, both specimens of water comply reasonably well. The empirical correlation between the properties shown in Figure 8 is given below:

(4)

with a coefficient of determination of 0.948. While there is a good correlation between vapor phase isentropic indices and Grüneisen parameters at Tkgmin and Thgmax, properties at T* are not that well correlated as shown in Figure 7, where we plot the saturated liquid Grüneisen

γfT hmax /γfT * = 0.7245kgmin + 0.0224

(5)

with a coefficient of determination of 0.967.

5. SUMMARY In this work we have shown the feasibility to correlate some of the isentropic slope properties at the boundaries of fluid phases. The Grüneisen parameter and the isentropic index have been chosen as representative properties to achieve the pursued objectives. Three saturation states have been chosen in the study: (i) the state at which the vapor phase isentropic index is minimum, (ii) the state where the vapor phase enthalpy is maximum, and (iii) the state where the product of liquid density by temperature attains its maximum value. A number of closely linear correlations have been observed. These empirical relations may be useful in refining the equations of state and further strengthening the corresponding states principles of fluids.

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ASSOCIATED CONTENT

S Supporting Information *

Table listing data used in the figures, obtained from the fluids available in the NIST program RefProp 9.1.21 This material is available free of charge via the Internet at http://pubs.acs.org/.

Figure 7. Variation of saturated liquid Grüneisen parameter and saturated vapor isentropic index at T*: (legend) ○ all fluids in RefProp 9.1 except ● alcohols and waters.

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AUTHOR INFORMATION

Corresponding Author

parameters against the saturated vapor isentropic index at T*. We note that for fluids with vc > ∼0.47 m3 kmol−1, kgT* seems to increase although γfT* monotonically decreases. Like in most results presented along this work, waters and alcohols arise as the oddball fluids with a behavior far away from the general trend. However, the ratios of saturated liquid phase Grüneisen parameters at Thgmax and T* correlate quite well with kgmin as

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.V. and J.A.W. thank the Ministerio de Educación y Ciencia of Spain for financial support under Grant FIS2009-07557. 6869



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REFERENCES

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Relation between the Isentropic Index and the GrÃ¼neisen Parameter

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