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Equation of State for Supercooled Water at Pressures up to 400 Mpa

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Equation of State for Supercooled Water at Pressures up to 400 Mpa Equation of state for supercooled water at pressures up to 400 MPa Vincent Holten, Jan V. Sengers, and Mikhail A. Anisimova) Institute for Physical Science and Technology and Department of Chemical and Biomolecular Engineering, University of Maryland, College Park, Maryland 20742, USA (Dated: 9 September 2014) An equation of state is presented for the thermodynamic properties of cold and supercooled water. It is valid for temperatures from the homogeneous ice nucleation temperature up to 300 K and for pressures up to 400 MPa, and can be extrapolated up to 1000 MPa. The equation of state is compared with experimental data for the density, expansion coefficient, isothermal compressibility, speed of sound, and heat capacity. Estimates for the accuracy of the equation are given. The melting curve of ice I is calculated from the phase-equilibrium condition between the proposed equation and an existing equation of state for ice I. Key words: compressibility; density; equation of state; expansivity; heat capacity; speed of sound; supercooled water; thermodynamic properties Contents C. Tables 21 1. Introduction 1 References 21 2. Experimental Data 2 2.1. Density2 2.2. Density derivatives2 1. Introduction 2.3. Speed of sound2 2.4. Heat capacity4 Supercooled water has been of interest to science since it was 1 2.5. Values from IAPWS-955 first described by Fahrenheit in 1724. At atmospheric pres- 2.6. Adjustment of data5 sure, water can exist as a metastable liquid down to 235 K, 2.7. Values for extrapolation6 and supercooled water has been observed in clouds down to this temperature.2,3 Properties of supercooled water are 3. Equation of State 7 important for meteorological and climate models4,5 and for 3.1. Structure of the equation7 cryobiology.6,7 Furthermore, the thermodynamic properties of 3.2. Optimization method9 cold and supercooled water at high pressure are needed for the design of food processing.8 4. Comparison with Experimental Data 10 It is well known that several properties of supercooled wa- 4.1. Density 10 ter – such as the isobaric heat capacity, the expansion coeffi- 4.2. Expansivity 11 cient, and the isothermal compressibility – show anomalous 4.3. Isothermal compressibility 13 behavior; they increase or decrease rapidly with cooling. A 4.4. Speed of sound 14 liquid–liquid phase transition, terminated by a critical point, 4.5. Heat capacity 15 hidden below the homogeneous ice nucleation temperature 4.6. Extrapolation to 1000 MPa 15 has been proposed to explain this anomalous thermodynamic 4.7. Connection to IAPWS-95 16 behavior.9,10 4.8. Uncertainty estimates 17 Several equations of state for supercooled water have been 4.9. Ice I melting curve 17 published. Sato11 proposed an equation of state for water in 4.10. Vapor pressure 19 the liquid phase including the metastable state, valid up to 100 MPa. Jeffery and Austin12,13 developed an analytic equa- 5. Conclusion 19 tion of state of H2O that also covers the supercooled region. arXiv:1403.6777v2 [cond-mat.stat-mech] 8 Sep 2014 Kiselev and Ely14 made an early attempt to describe super- Acknowledgments 20 cooled water in terms of an equation of state incorporating critical behavior. Anisimov and coworkers15–18 also based A. Homogeneous Nucleation Curve 20 their equations of state on an assumed liquid–liquid critical B. Derivatives 20 point. Since the publication of these equations, new experi- mental data have become available that enable development of an equation of state with a significantly improved accuracy. The equation of state of this work was developed with the a)Author to whom correspondence should be addressed; electronic mail: following aims: [email protected] 1. It should represent the experimental data of liquid water 2 in the metastable region as well as possible. This work only al.,35 Asada et al.,37 and Kell and Whalley.30 To enable ex- considers supercooled water above the homogeneous nucle- trapolation of the equation above 400 MPa, density data from ation temperature. The equation does not cover the glassy state Grindley and Lind28 up to 800 MPa were included in the fit. of water (below 136 K at atmospheric pressure19). 2. The current reference for the thermodynamic properties 2.2. Density derivatives of water is the IAPWS-95 formulation.20,21 IAPWS-95 is, strictly speaking, valid only at temperatures above the melt- Several data sets exist for temperature and pressure deriva- ing curve. When extrapolated into the supercooled region, tives of the density r. The cubic expansion coefficient aP, IAPWS-95 also yields a good description of the data in the also known as expansivity, is defined as supercooled region that were available at the time the formu- lation was developed. For practical use, a new formulation for 1 ¶r aP = − ; (1) the thermodynamic properties of supercooled water should r ¶T smoothly connect with the IAPWS-95 formulation at higher P temperatures without significant discontinuities at the point of where T is the temperature and P is the pressure. The isother- switching. mal compressibility kT is defined as 3. The correlation should allow extrapolation up to 1000 MPa. There are only a few data in the supercooled region 1 ¶r = : (2) above 400 MPa, but smooth extrapolation up to 1000 MPa kT r ¶P T would be desirable. The data sets listed in Table2 were all included in the fit, with the exception of the compressibility data of Mishima.38 Mishima’s data were not included because they may be af- 2. Experimental Data fected by systematic errors of unknown size at low temper- atures. In previous work,17,18 expansivities reported by Hare Most of the experimental data that were considered in this and Sorensen34 were included in the fit. However, Hare and work have been reviewed before.17,20,22–25 In this section, we Sorensen did not measure the expansivity directly, but derived mainly discuss new data and data that were treated differently it from a fit to their density data. Because we already included than in our previous work.17,18,26 Hare and Sorensen’s density data in our fit, their expansiv- ity data were not used in the fit. Expansivity values from Ter Minassian et al.44 were calculated from their empirical corre- 2.1. Density lation. The accuracy of their correlation is not given; the rela- The experimental density data that were considered in this tive difference with expansivities calculated from IAPWS-95 work are listed in Table1 and shown in Fig.1(a). Additional is at most 3.2% in the range of 300 K to 380 K and 0 MPa to references to older data can be found in the articles of Tekácˇ 400 MPa. et al.50 and Wagner and Pruß.20 In a large part of the su- At points in the phase diagram where the expansivity is percooled region, the only available density data are those of zero, the density has a maximum with respect to temperature. Mishima.38 As a result, it is difficult to estimate the system- The temperature at which this occurs is usually referred to 45 atic error of these data at low temperatures. In a graph in his as the temperature of maximum density (TMD). Caldwell article,38 Mishima showed the random (type A) uncertainty measured the TMD for pressures up to 38 MPa, and these for each data point, which is 0.2% on average and at most measurements were included in the expansivity data set of the 0.5%. The systematic (type B) uncertainty can only be esti- fit as aP = 0 points. The recent TMD measurements of Hiro 51 mated above 253 K, in the region of overlap with density data et al. were not used, because they deviate systematically by of Kell and Whalley,30 Sotani et al.,35 and Asada et al.37 In about 1.5 K from more accurate data. this region, the densities of Mishima deviate systematically by up to 0.4% from these other data. Below 253 K, the systematic 2.3. Speed of sound uncertainty is unknown. As in earlier work,17 we adjusted the density values of Mishima, under the assumption that the sys- The experimental data on the speed of sound considered in tematic deviation at low temperatures, where it is not known, this article are given in Table3 and shown in Fig.2. Recent is the same as at higher temperatures, where it can be calcu- data that were not considered in previous work are the accu- lated. It was found that the adjusted data of Mishima do not rate measurements of the speed of sound by Lin and Trusler52 completely agree with the expansivity measurements of Ter down to of 253 K and from 1 MPa to 400 MPa. Although there Minassian et al.,44 which we consider to be more accurate. are few data points in the supercooled region, the accuracy Therefore, the adjusted data of Mishima were included in the of 0.03%–0.04% makes this an important data set. Lin and fit of the equation of state with a relatively low weight. Trusler also derived densities and isobaric heat capacities by The only experimental density data at atmospheric pressure integrating their speed-of-sound data. We have not considered that were included in the fit are those of Hare and Sorensen,34 these derived properties in the development of the equation of which are considered to be the best available. For pressures state in this work for the following reason. To enable integra- higher than atmospheric, we included data from Sotani et tion of the speed of sound, Lin and Trusler represented their 3 400 300 ) a P M ( e r u 200 s s e r P 100 (a) (b) 0 180 200 220 240 260 280 300 180 200 220 240 260 280 300 Temperature (K) Temperature (K) Adams (1931) Hare and Sorensen (1987) Speedy and Angell (1976) Ter Minassian et al.
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