Generalized Fundamental Equation of State for the

Generalized Fundamental Equation of State for the Normal Alkanes $$(\hbox {C}_{5}{-}\hbox {C}_{50})$$ ( C 5 - C 50 ) Igor Alexandrov, Anatoly Gerasimov & Boris Grigor’ev

International Journal of Thermophysics Journal of Thermophysical Properties and Thermophysics and Its Applications ISSN 0195-928X Volume 34 Number 10 Int J Thermophys (2013) 34:1865-1905 DOI 10.1007/s10765-013-1512-1

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Author's personal copy Int J Thermophys (2013) 34:1865–1905 DOI 10.1007/s10765-013-1512-1

Generalized Fundamental Equation of State for the Normal Alkanes (C5 −C50 ) Igor Alexandrov · Anatoly Gerasimov · Boris Grigor’ev

Received: 21 September 2012 / Accepted: 30 August 2013 / Published online: 29 September 2013 © Springer Science+Business Media New York 2013

Abstract Based on the extended three-parameter corresponding-states principle and the most reliable experimental data of n-alkanes, a generalized fundamental equation of state for technical calculations has been developed. This equation is in the form of the reduced Helmholtz free energy and takes the reduced density, reduced temperature, and acentric factor as variables. The proposed equation satisfies the critical conditions and Maxwell rule, shows correct behavior for the ideal curves and for the derivatives of the thermodynamic potentials, and allows the calculation of all thermodynamic properties including phase equilibrium of n-alkanes from n-pentane (C5 ) to n-pentacontane (C50 ) over a temperature range from the triple point to 700 K with pressures up to 100 MPa. The new equation differs from the previous generalized equations of other authors by a wider range of variation of the acentric factor ω = 0.25 to 1.8, and by more accurately predicting thermal properties. Keywords Acentric factor · Density · Equation of state · Heat capacity · Hydrocarbons · Normal alkanes · Saturated vapor pressure · Speed of sound 1 Introduction One of the fundamental tasks of thermodynamics and statistical physics is to find equations of state of matter. By the end of the twentieth century, over a hundred different

I. Alexandrov (B) · A. Gerasimov Kaliningrad State Technical University, Sovietsky prospekt 1, Kaliningrad 236022, Russia e-mail: [email protected] B. Grigor’ev Institute for Oil and Gas Problems, Russian Academy of Sciences, ul. Gubkina 3, Moscow 119333, Russia

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Int J Thermophys (2013) 34:1865–1905

equations of state were developed. However, most do not possess sufficient generality and universality, so the search to find more sophisticated equations of state continues. A complete overview of the contemporary equations of state is presented in the monograph of Span [1]. This monograph presents new empirical fundamental equations of state for the calculation of thermodynamic properties of fluids, in particular—the technically important hydrocarbons. These equations with relatively few coefficients have sufficiently high accuracy, numerical stability and good extrapolation behavior, because their form is determined by using a powerful optimization algorithm. These equations refute the generally accepted opinion that empirical equations of state can only be used in regions with reliable experimental data. However, even equations of state with optimized functional forms are limited by the fact that uncertainties are higher if there is only a small amount of experimental data, or the data are of low quality. In connection with this, there is a need for a generalized equation of state with numerical stability and, at the same time, has the ability to describe the thermodynamic properties of a wide range of technically important substances used in the chemical and petrochemical industries. A large multitude of models and equations of state have been developed either to fulfill the corresponding technical demands or to advance the scientific search for the physically correct description of the thermodynamic properties of fluids. Generalized equations of state for pure substances that are in use today may be subdivided into several groups. The first group is composed of the cubic equations of state that are direct descendants of the famous van der Waals equation of state. Classical representatives of this group are the well-known equations of state by Redlich and Kwong [2], Soave [3], and Peng and Robinson [4]. The work on cubic equations of state continues. A fairly complete overview can be found in the monograph of Brusilovsky [5]. However, the main disadvantage of these equations—the low accuracy of thermodynamic properties for a wide range of state parameters—cannot be overcome, as it is based on the structure of these equations. The second group includes equations of state that are derived on the framework of the statistical theory of associated fluids (SAFT) [6,7]. Despite the fact that they are the most modern of the equations, their capabilities, in view of their relatively simple structures, are close to the capabilities of the cubic equations of state. Because the purpose of this study was to develop a generalized equation of state, which is characterized by high precision in the calculation of thermodynamic properties, the model should use complex semi-empirical multiparameter equations. The most famous of these equations is the Benedict–Webb–Rubin equation of state [8]. Coefficients of the BWR equation have been determined by many authors and for many substances and their binary mixtures (a large bibliography is presented in [9]). Several generalized forms of this equation have also been published. The most successful generalization for hydrocarbons was made by Lee and Kessler [10] in the framework of the extended corresponding states principle in the formulation of Pitzer and Curl [11]. This generalization has been tested, and the main conclusions are as follows: the error in the description of the density is 1.0 % to 3.0 % for liquid and gaseous phases, excluding the critical region, and isobaric heat capacities are described with a similar error. The range of applicability of the equation, according to the authors [10],

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is 0.3 ≤ T /Tc ≤ 4.0 and p/ pc ≤ 10.0. Further investigations have shown that the range of pressure, without a significant increase in errors, can be increased up to 20 pc , and that the temperature can be reduced to 0.4Tc . These errors are observed for hydrocarbons with an acentric factor ω ≤ 0.8. The errors in density for more “heavy” hydrocarbons increase monotonically. A recent and promising development in the tradition of semi-empirical equations of state is the “BACKONE” family of equations of state. Among the works on these equations are those of Saager et al. [12], Saager and Fischer [13], Muller et al. [14], and Calero et al. [15]. However, this approach is far from complete and extensive in practical usage. Empirical approaches to generalize the description of thermodynamic properties of substances are usually based on more or less simple multiparameter equations of state, which are used in technical calculations. Platzer and Maurer [16] used a four-parameter correlation for corresponding states in order to generalize the coefficients of the Bender equation of state [17]. However, the results included the disadvantages of the basic equation, which are analyzed in detail in the work of Span [1]. A feature in the approach of Lee and Kessler [10] is that it can be applied to a variety of equations, regardless of their complexity. Many works are published on this subject that are not considered here. The most successful and modern modification of this approach is implemented in the work of Sun and Ely [29]. Using a four-parameter correlation for corresponding states, Sun and Ely obtained a generalized equation that can be applied to both nonpolar and polar substances. Thermodynamic properties are calculated with the use of three reference substances (propane, n-octane, and water) that are each composed of 14-term equations of state proposed by the same authors in [18]. In [29] the authors analyzed the quality of the description of the properties of 22 nonpolar, polar, and associated fluids. We have tested this equation for calculating properties of n-alkanes with carbon numbers n from 5 up to 50. For hydrocarbons with n > 8, the error in describing the density increases monotonically from 0.5 % to 1.0 % at n = 9, up to 3.0 % to 4.0 % at n = 10 to 15, up to 4.5 % to 5.0 % at n = 16 to 20, and up to 5.5 % to 8.0 % at n = 21 to 40. The deviation for saturated vapor pressure varies from ∼ 1.0 % for n-octane and up to 50 % to 60 % at n > 40. However, it should be kept in mind that the values of saturated vapor pressures of heavy hydrocarbons in the investigated temperature range are small and reliable experimental data are not available. The caloric properties isobaric heat capacity, isochoric heat capacity, and speed of sound are described with reasonable accuracy by deviations that do not exceed 1.0 % to 3.0 %. The monograph of Span [1], in addition to individual equations for the technically important substances, presents a generalized 10-term equation of state with an optimized functional form. The generalization is made within the framework of the extended three-parameter corresponding states principle. The acentric factor was chosen as the determining criterion of similarity. The equation is recommended for nonpolar substances, however, the high accuracy in the calculation of properties is insured only for those substances that are included in the optimization procedure. Critical properties are adjusted during the optimization; as a result they became adjustable

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Author's personal copy 1868 Table 1 Exponents of Eq. 3

Int J Thermophys (2013) 34:1865–1905 k

dk

tk

lk

1

1

0.686

2

1

1.118

0 0

3

1

0.857

0

4

3

0.559

0

5

7

0.442

0

6

2

0.831

0

7

1

0.484

1

8

1

2.527

1 1

9

2

1.549

10

5

0.757

1

11

1

3.355

2

12

1

1.905

2

13

4

4.941

2

14

2

12.805

3

parameters that are defined by experimental data. This equation does not allow extrapolation in the value of the acentric factor ω. The equation should not be applied at ω > 0.4. Analysis has shown that in order to further improve the accuracy of the calculation of properties and phase behavior of complex hydrocarbon mixtures, only the equation of Sun and Ely [29] is recommended. However, we believe it is possible to increase the accuracy of the calculation of thermal properties using a corresponding-states principle based on the acentric factor and pseudo-critical properties, as is done in this work. 2 Generalized Fundamental Equation of State The purpose of this work is the development of a generalized fundamental equation of state for n-alkanes from C5 to C50 . The need to develop such an equation is not only to calculate the thermodynamic properties of heavy n-alkanes, but also to model the properties and phase behavior of complex hydrocarbon mixtures of undetermined composition (natural hydrocarbon mixtures at reservoir conditions, and the products of their processing). For these calculations the use of the cubic equation has inherent disadvantages. A more reliable prediction of the properties and phase behavior can be carried out with the use of multicomponent Helmholtz equations. In the case of mixtures of undefined composition, generalized equations are required. With the help of these equations properties of subfractions simulating a complex mixture can be calculated. Hydrocarbon gases (with carbon number n < 5) were not included in the generalization, because they do not exist in liquid hydrocarbon mixtures. The contribution of gas components in gas–condensate systems can be taken into account using individual equations of state that have been developed elsewhere.

123


1869

Table 2 Coefficients and exponents of Eq. 4 i

c1,i

c2,i

c3,i

c4,i

1

0.534 107 34 × 101

0.668 194 73 × 101

0.166 924 14 × 101

0.294 469 22 × 101

2

−0.227 781 89 × 101

−0.128 468 93 × 101

0.137 953 02 × 101

0.232 843 96 × 101

3

−0.387 854 99 × 101

−0.860 956 96 × 101

−0.267 078 21 × 101

0.279 601 14 × 101

4

−0.121 909 59 × 10−1

0.368 694 92

−0.206 272 85

0.637 314 70

5

0.929 421 59 × 10−3

0.807 310 74 × 10−1

−0.813 581 86 × 10−1

0.996 199 92

6

−0.166 312 29 × 10−1

−0.803 141 82 × 10−1

−0.353 437 19

0.118 709 29 × 101

7

−0.165 728 87 × 101

0.216 463 46 × 102

−0.160 189 67 × 102

0.103 751 03 × 101

8

0.126 426 06 × 101

0.216 458 43 × 101

−0.257 262 22 × 101

0.137 334 37 × 101

9

0.960 086 62 × 10−1

0.442 219 76 × 101

0.115 913 67 × 101

0.111 685 57 × 101

10

0.929 508 30 × 10−1

−0.574 638 93 × 10−1

0.444 196 82

0.763 904 20

−0.382 712 99

−0.204 297 13 × 101

0.117 514 52 × 101

0.148 290 49 × 101

12

0.349 360 66

0.640 556 42 × 101

−0.835 987 49 × 101

0.100 805 16 × 101

13

0.417 187 09 × 10−1

−0.902 876 49

0.230 698 11

0.133 204 74 × 101

14

−0.121 499 15 × 10−1

−0.154 742 03

0.232 330 99

0.120 624 11 × 101

11

For the generalization, an equation in the form of the reduced Helmholtz free energy is selected; a 0 (ρ, T ) + a r (ρ, T ) a(ρ, T ) = = α 0 (δ, τ ) + α r (δ, τ ) RT RT

(1)

where a (ρ, T ) is the Helmholtz free energy; α 0 (δ, τ ) is the reduced Helmholtz energy in the ideal-gas state; α r (δ, τ ) is the residual part of the reduced Helmholtz energy; δ = ρ/ρr is the reduced density; τ = Tr /T is the inverse reduced temperature; and ρr , Tr are reducing parameters. Thermodynamic properties can be calculated as derivatives of Eq. 1. The corresponding ratios are given in Appendix 1.

Table 3 Coefficients ai of Eq. 5 for Tc , bi of Eq. 6 for pc , and ci of Eq. 7 for ω

i

ai

1

1200

0.734 993 18

2

7.235 346 1

2.115 168 4

1.9087

3

−0.318 197 03

−0.362 293 42

0.005

4

0.436 006 96

5

−0.269 056 63

6

bi

ci

0.691 231 21 −0.220 560 59

6.6747

0.019 219 0.75

−2.889 041 6

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Table 4 Results of comparison of experimental data for thermodynamic properties of n-alkanes with the generalized fundamental equation of state, Eq. 3 Years References

Points Temperature and pressure range T (K)

p (MPa)

Average absolute deviation (AAD) (%) Liquid Gas

Critical Super -critical

0.581

4.577

C5 (n − pentane) p, ρ, T 1942 Sage and Lacey [30]

237

311–511 0.101–68.9

1952 Beattie et al. [31]

52

473–573 2.6–35.2

1953 Li and Canjar [32]

128

373–573 1.05–22.1

0.843

1980 Scaife and Lyons [33]

165

248–373 0.1–284

0.724

1.268

1985 Kratzke et al. [34]

119

238–573 0.747–60.6

0.440

1986 Grigor’ev et al. [35]

125

453–523 2.98–14.4

1.570

4.499

134

373–648 0.191–12

139

318–443 7.5–70.3

1.036

1997 Poehler and Kiran [37]

65

323–423 8.07–69.8

0.382

1998 Abdulagatov et al. [38]

15

647

5–37.5

2001 Tohidi et al. [39]

9

323

6.95–140

1.547

1928 Young [40]

28

243–309 0.005–0.101

1.105

1942 Sage and Lacey [30]

5

311–444 0.108–2.27

0.818

1945 Willingham et al. [41]

9

286–310 0.043–0.104

0.319

1951 Nicolini [42]

13

273–303 0.025–0.082

0.532

1951 Beattie et al. [43]

6

373–470 0.593–3.38

0.936

1964 Zanolini [44]

14

314–456 0.121–2.75

0.684 0.784

4.915

0.681 2.827

1.296 0.369

Saturated vapor pressure

1970 Douslin [45]

9

260–331 0.013–0.203

1974 Osborn and Douslin [46]

15

269–341 0.02–0.27

0.578

1975 Horner et al. [47]

34

265–297 0.017–0.064

0.483

1981 Hossenlopp and Scott [48]

9

260–331 0.013–0.203

0.816

1983 Olivares Fuentes et al. [49] 25

275–315 0.027–0.124

0.428

2002 Maia de Oliveira [50]

9

278–309 0.03–0.101

0.525

7

183–303

Saturated liquid density 0.427

1942 Carney [52]

6

243–293

0.100

1985 Kratzke et al. [34]

12

237–440

0.648

1995 Holcomb et al. [54]

29

250–409

0.494

1996 Mirskaya and Kamilov [55]

8

313–449

0.827

Saturated vapor density 1995 Holcomb et al. [54]

23

312–409

5.122

1998 Abdulagatov [56]

12

436–470

4.295

17

313–468 Saturated

Isobaric heat capacity

123

2.381 0.666

1990 Kurumov [23]

1971 Amirkhanov et al. [57]

8.751

14.97

1992 Kiran and Sen [36]

1930 Dornte and Smyth [51]

1.001 2.548

1.337


1871

Table 4 continued Years References


p (MPa)


40

298–523 0.019–0.203

2000 Gerasimov [24]

291

293–697 0.5–60



0.217 1.068

2.974

0.856

Saturated heat capacity 1940 Messerly and Kennedy [58]

18

148–286

1.686

1967 Messerly et al. [59]

25

149–303

1.590

1957 Richardson and Tait [60]

48

288–317 3.45–55.2

2.542

1990 Lainez et al. [61]

220

263–433 0.22–213

3.522

1940 Kelso and Felsing [62]

56

373–548 0.567–31.6

0.622

1963 Schaffenger [63]

48

371–583 0.064–0.233

1970 Oeder and Schneider [64]

30

193–273 10–100

0.593


169

248–373 0.1–253

0.634

Speed of sound

C6 (n-hexane) p, ρ, T 0.242

1981 Grigor’ev and Kurumov [65] 130

398–623 0.196–5.98

1988 Moriyoshi and Aono [66]

71

298–313 3.1–146

0.551

2.764 11.6

1992 Kiran and Sen [36]

76

313–448 0.21–65.8

0.578

1992 Susnar et al. [67]

51

294

0.1–34.6

0.314

1995 Sauermann et al. [68]

78

263–473 0.1–49.8

0.237

1998 Daridon et al. [69]

279

293–373 5–150

0.830


30

643–648 5–37.5

2.954

2.426

Saturated vapor pressure 1945 Willingham et al. [41]

16

286–343 0.012–0.104

0.536

1952 Brown [70]

19

310–342 0.032–0.101

0.246 0.322

1973 Li et al. [71]

9

301–336 0.023–0.085

1977 Mousa [72]

10

485–508 2.16–3.03

0.280

1978 Wieczorek and Stecki [73]

10

298–343 0.02–0.104

0.363

1980 Genco et al. [74]

22

383–507 0.314–3.03

0.588

1988 de Loos et al. [75]

5

472–507 1.77–3.02

0.402

1988 Wu and Sandler [76]

8

307–339 0.03–0.093

0.261

1992 Bich et al. [77]

39

295–351 0.017–0.134

0.534


27

308–443 0.03–1.09

0.499

1966 Schmidt et al. [78]

4

298–313

0.158

1967 Rozhnov [79]

4

303–413

0.147

1970 Findenegg [80]

26

252–333

0.242

1980 Aicart et al. [81]

4

298–333

0.148

1980 Dymond and Young [82]

13

238–393

0.834

Saturated liquid density

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p (MPa)


1995 Yu and Tsai [83]

5

293–313

0.188


12

263–428

0.420


10

343–503

0.377

2002 Garcia et al. [84]

5

278–318

0.277

12

473–508


Saturated vapor density 1998 Abdulagatov et al. [38]

4.240 13.14

Isobaric heat capacity 1931 Huffman et al. [85]

5

189–328

0.936

1946 Douslin and Huffman [86]

24

180–301

1.382

1947 Waddington and Douslin [87]

11

334–469 0.031–0.064

2000 Gerasimov [24]

114

293–624 0.5–60

0.384 0.638

0.791

Saturated heat capacity 1931 Huffman et al. [85]

5

93.4–293

1.258

2002 Paramo et al. [88]

15

278–348

0.689

36

509–673

1967 Boelhouwer [89]

40

253–333 20–140

1991 Verveiko et al. [90]

16

293–323 0.1–600

3.291


275

293–373 0.1–150

2.340

2001 Khasanshin and Shchemelev [91] 20

298–433 0.1–49.1

4.096

2001 Ball and Trusler [92]

298–373 0.1–101

3.101

Isochoric heat capacity 1998 Abdulagatov et al. [38]

14.38

0.927

Speed of sound

82

2.681

C7 (n-heptane) p, ρ, T 1937 Smith et al. [93]

46

303–523 0.72–35.6

0.407

1955 Nichols et al. [94]

278

278–511 1.38–69.1

0.321


64

273–393 0.1–117.7

0.417

1964 Doolittle [96]

60

303–573 5–500

1.263


232

248–373 0.1–200.1

0.319

1982 Golik et al. [97]

33

323–453 0.1–196.1

0.847

1982 Zawisza and Vejrosta [98]

59

423–573 0.2–5.45

1.196 2.900

1985 Muringer et al. [99]

215

198–311 0.1–263.4

0.697

1989 Toscani et al. [100]

96

298–373 0.1–100

0.287

1990 Kurumov [23]

316

188–623 0.1–150

0.533 1.666

1992 Susnar et al. [67]

51

294.3

0.153

1940 Smith [101]

16

313–403

0.178

1945 Willingham et al. [41]

40

298–372

0.419

0.1–34.6


123

3.678

7.467

2.128


1873



p (MPa)


1949 Forziati et al. [102]

20

299–372

1952 Brown [70]

8

313–371

0.237

1990 Kurumov [23]

8

398–540

0.405


0.433

2000 Weber [103]

59

335–503

0.329

2005 Ewing and Ochoa [104]

43

372– 537

0.364

Saturated liquid density 1967 Rozhnov [79]

4

303–413

0.463

1976 Christopher et al. [105]

12

298–353

0.309

1987 Stephan and Hildwein [106]

70

182–535

0.844

4.603

1990 Kurumov [23]

20

188–539

0.399

5.043

1995 Yu and Tsai [83] 2003 Kahl et al. [107]

5 12

293–313 278–333

0.303 0.328

1987 Stephan and Hildwein [106]

74

182–540

2.571 11.32

1990 Kurumov [23]

9

423–540

1.420

1976 San Jose et al. [108]

26

413–513 1.0–2.0

1.613

1980 Kalinowska et al. [109]

96

185–301 Saturated

0.808

1994 Zabransky and Ruzicka [110] 32

182–480 Saturated

0.518

2000 Gerasimov [24]

293–623 0.5–60

1.100

Saturated vapor density 9.485


276

1.608 11.91

1.311

3.562

0.782

Saturated heat capacity 9

278–318

0.319

2000 Becker and Aufderhaar [112] 8

1947 Osborne and Ginnings [111]

302–337

0.101


15

288–348

0.225

1998 Abdulagatov [56]

349

374–673 0.665–6.1

1.535

2002 Polikhronidi et al. [113]

33

324–422 5.7–6.6

3.700

Isochoric heat capacity 5.341

Speed of sound 1953 Kling et al. [114]

23

293–373 0.1–49

3.259


60

253–453 Sat. – 140

2.221


68

313–453 0.1–196

2.758

1985 Muringer et al. [99]

113

186–311 0.1–263.4

1.408

C8 (n-octane) p, ρ, T 1942 Felsing and Watson [115]

89

373–548 0.507–30.4

0.773


51

303–393 0.098–118

0.348

1963 Schaffenger [63]

46

359–575 0.062–0.193

1985 Dymond et al. [116]

47

298–373 0.1–540

0.522 1.233

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p (MPa)


1991 Banipal et al. [117]

66

318–373 0.1–10

0.375

1991 Tanaka et al. [118]

19

298–348 0.1–151

0.454

1996 Goodwin et al. [119] Saturated vapor pressure

14

298–303 0.675–32.6 0.770

1928 Young [40] 1942 Felsing and Watson [115]

14 7

299–557 398–548

0.354 0.993 0.170

1986 Badalyan et al. [120]

18

423–568

1988 Gierycz et al. [121]

3

373–398

0.133

1996 Dejoz et al. [122]

39

291–409

0.293

2003 Ewing and Sanchez Ochoa [123] 60

323–563

0.296


Saturated liquid density 1967 Das and Kuloor [124]

28

300–560

0.500

1971 Chappelow et al. [125]

31

245–395

0.512


15

443–568

0.903

1962 Connolly and Kandalic [126]

10

463–553

1.882

1967 Das and Kuloor [124]

28

299–568

2.053

1971 Amirkhanov et al. [57]

16

403–567

2.072


60

318–373 0.1–10

0.328


28

385–523

2000 Gerasimov [24]

173

220–630

0.86


6

283–308

0.453

1954 Finke et al. [127]

18

222–298

0.348

399

453–693

1.257 2.917

Saturated vapor density


0.22 0.8

2.0

Saturated heat capacity

Isochoric heat capacity 1998 Abdulagatov [56] Speed of sound 1967 Boelhouwer [89]

64

253–393 20–140

1.686

1985 Takagi and Teranishi [128]

3

298

2.454

1997 Ding et al. [129]

142

293–363 5–90

1.804

2001 Khasanshin et al. [91]

46

303–433 0.1–49.1

2.496

0.1–100

C9 (n-nonane) p, ρ, T 1953 Carmichael et al. [130]

86

311–511 0.167–69.2 0.488


52

303–393 0.098–118 0.292

1964 Doolittle [96]

66

303–573 5–500

1.104


188

248–373 0.1–301

0.561

123

3.647

0.472


1875



p (MPa)


1991

Kurumov [23]

434

223–623 0.089–130 0.818

1991

Banipal et al. [117]

72

313–373 0.1–10


1.757 23.54

1.92

0.460

Saturated vapor pressure 1931

White and Rose [131]

4

424

0.185

1945

Willingham et al. [41]

20

343–425

0.277

1949

Forziati et al. [102]

20

343–425

0.283

1953 1986

Carmichael et al. [130] Paul et al. [132]

13 6

311–511 333–424

0.478 0.330

1991

Kurumov [23]

16

473–594

2.622

2001

Ortega et al. [133]

46

403–439

0.626

0.934

Saturated liquid density 1930

Dornte and Smyth [51]

11

223–423

0.678

1953

Carmichael et al. [130]

13

311–511

0.740

1978

Grindley and Lind [134]

13

303–423

0.176

1986

Plebanski et al. [135]

10

300–390

0.441

1991

Amirkhanov [136]

27

273–594

0.502

6.169

1991

Kurumov [23]

23

223–594

0.438

1.69

6

498–594

Saturated vapor density 1991

Kurumov [23]

4.405

8.467

1.284

9.973

2.37

2.602

7.78

1.109

Isobaric heat capacity 1930

Parks et al. [137]

8

225–299

1.304

1991


72

318–373 0.1–10

0.649

1997

Tovar et al. [138]

3

288–308

0.245

2000

Gerasimov [24]

146

324–625 0.5–60

1.669


Huffman et al. [85]

8

228–298

0.951

1947

Osborne and Ginnings [111] 9

278–318

0.335

1954

Finke et al. [127]

22

225–314

0.442

226

425–693 0–19.8

1.379

Isochoric heat capacity 1991

Amirkhanov [136]

Speed of sound 1953

Kling et al. [114]

24

293–373 0.098–49

1.872

1967

Boelhouwer [89]

89

253–413 10–140

1.310

1981

Kiryakov [139]

126

303–393 0.098–196 0.908

2005

Lago et al. [140]

66

293–393 0.109–100 1.374

55

294–394 1.72–24.1

C10 (n-decane) p, ρ, T 1940

Sage et al. [141]

0.191

123


Int J Thermophys (2013) 34:1865–1905



p (MPa)


1942 Reamer et al. [142]

217

311–511 1.38–68.9

0.497

1970 Snyder and Winnick [143]

116

298–358 0.101–656

0.595


77

248–373 0.1–294

0.638

1983 Gehrig and Lentz [144]

395

298–673 0–302

0.715 11.07


72

313–373 0.1–10

0.267


9.589 2.577

1991 Kurumov [23]

263

248–623 0.1–103

1.173 0.995 5.452

2004 Caudwell et al. [145]

34

298–373 0.1–192

0.572

2005 Zuniga-Moreno et al. [146]

148

313–363 1.01–25.1

0.375

1945 Willingham et al. [41] 1989 Chirico et al. [147]

19 33

368–448 0.008–0.104 268–490 0–0.27

0.310 0.350 0.322 1.471

1.825


1991 Kurumov [23]

12

498–618 0.317–2.15

1994 Morgan and Kobayashi [148]

16

323–588 0.001–1.4

0.436

1996 Dejoz et al. [122]

37

315–458 0.001–0.133

0.721

1930 Bingham and Fornwalt [149]

16

273–373

0.643


8

283–393

0.626

1991 Amirkhanov [136]

24

274–618

0.678

1991 Kurumov [23]

25

248–598

1.089

1995 Yu and Tsai [83]

5

293–313

0.667

1991 Amirkhanov [136]

11

578–618

2.985 1.752

1991 Kurumov [23]

5

498–598

3.443


4.49

Saturated vapor density

Isobaric heat capacity 1986 Gates et al. [151]

4

298–368 Saturated

0.526


72

318–373 0.1–10

0.296

2000 Gerasimov [24]

135

293–635 0.5–60

1.455 1.505

2002 Peleteiro et al. [152]

5

280–318 Saturated

0.311

2.572


6

251–298

0.852


9

278–318

0.403

1954 Finke et al. [127]

17

247–319

0.352


15

278–348

0.260

122

453–693 1.68–8.08

1.775 6.803 8.605 0.940

1994 Aminabhavi and Gopalakrishna [153] 3

298–318

2.056

2001 Khasanshin and Shchemelev [91]

298–433 0.1–49.1

1.166

Isochoric heat capacity 1991 Amirkhanov [136] Speed of sound

123

40


1877



2002 Casas et al. [154]

3

p (MPa)

288–308



1.936

C11 (n-undecane) p, ρ, T 303–573 5–500

1.074

1980 Landau and Wuerflinger [155] 147

1964 Doolittle [96]

70

258–313 10–300

0.639

1991 Kurumov [23]

253–398 0.099–111

0.560

77

1.126

Saturated vapor pressure 1955 Camin and Rossini [156]

20

378–470 0.006–0.105

0.572

1991 Kurumov [23]

19

248–623 0–1.58

0.268

2005 Calculation on [27]

7

360–480

0.208

2010 Calculation on Cs [157]

6

248–340

0.319

Saturated liquid density 1930 Bingham and Fornwalt [149]

8

273–373

0.930

1930 Dornte and Smyth [51]

11

263–463

0.799 0.432

1946 Vogel [158]

4

293–359

1964 Doolittle [96]

6

303–573

0.522


5

288–308

0.606

1991 Kurumov [23]

19

248–623

0.981

19

248–623

61

250–625 0.1–60

0.916

65

250–460 0.5–60

3.131

Saturated vapor density 1991 Kurumov [23]

0.932

Isobaric heat capacity 2000 Gerasimov [24]

0.460

Isochoric heat capacity 2000 Gerasimov [24] Saturated heat capacity 1931 Huffman et al. [85]

5

259–298

0.779

1954 Finke et al. [127]

12

252–299

0.479

Speed of sound 1962 Golik and Ivanova [160]

6

293–343

1.180

1985 Melikhov [161]

126

303–393 0.101–203

0.663

2000 Plantier et al. [162]

9

293–373

1.422

2002 Casas et al. [154]

3

288–308

1.141

78

311–408 0.1–689

0.270 0.354

C12 (n-dodecane) p, ρ, T 1958 Cutler et al. [163] 1960 Boelhouwer [95]

28

303–393 0–118

1970 Snyder and Winnick [143]

96

298–358 0.101–417

0.497

268–313 10–250

0.494

1980 Landau and Wuerflinger [155] 88

123


Int J Thermophys (2013) 34:1865–1905



p (MPa)


1991 Kurumov [23]

156

273–623 0.093–124

0.683 0.255

1991 Tanaka et al. [118]

21

298–348 0.1–151

0.359

2004 Caudwell et al. [145]

64

298–473 0.1–192

0.565


Saturated vapor pressure 1945 Willingham et al. [41]

20

400–491 0.006–0.104

0.396

1985 Gierycz et al. [164]

13

378–418 0.003–0.013

0.491


13

353–588 0.001–0.671

0.566

1991 Kurumov [23]

7

523–632 0.213–1.33

1.947

1996 Dejoz et al. [122]

38

344–502 0–0.134

0.788

Saturated liquid density 1930 Dornte and Smyth [51]

12

263–483


9

283–393

0.882 0.784

1991 Kurumov [23]

22

289–623

0.932


5

278–318

0.761

Isobaric heat capacity 2000 Gerasimov [24]

35

334–597 1–60

0.967

2000 Bessieres et al. [165]

77

313–373 0.1–100

0.997


4

275–298

0.871

1954 Finke et al. [127]

11

267–317

0.340

2000 Gerasimov [24]

11

267–317

0.340

55

324–377

7.546 0.464

Isochoric heat capacity 2002 Polikhronidi et al. [113] Speed of sound 1967 Boelhouwer [89]

85

273–473 10–140

1985 Melikhov [161]

191

303–433 0.1–589

1.200

303–433 0.1–49.1

0.253

2001 Khasanshin and Shchemelev [91] 30 C13 (n-tridecane) p, ρ, T 1964 Doolittle [96]

70

303–573 5–500

1.118


31

323–453 0.1–196

0.644

1991 Kurumov [23]

71

270–630 0.1–80

0.869

Saturated vapor pressure 1947 Fenske et al. [166]

1

380

1955 Camin and Rossini [156]

14

412–509 0.006–0.103

0.369

1991 Kurumov [23]

43

268–670 0–1.57

0.841

1996 Viton et al. [167]

33

274–467 0–0.035

0.869

2010 Calculation on Cs [157]

8

268–340

0.760

123

0.001

0.594 1.14


1879



p (MPa)



Saturated liquid density 1946 Schiessler et al. [168]

5

273–372

0.822

1946 Vogel [158]

4

293–359

0.638

1991 Kurumov [23]

43

268–670

1.442

1998 Wu et al. [169]

4

293–313

0.911


5

280–318

0.974

42

268–670

2000 Gerasimov [24]

36

321–607

1.129


5

280–318

0.777

8

272–306

0.512


64

313–453 0.1–196

0.302


9

293–373 Saturated 0.099


276

293–373 0.1–150

0.431


48

303–433 0.1–49.1

0.287

0.810

13.5

Saturated vapor density 2011 Gerasimov et al. [171]

1.239


Saturated heat capacity 1954 Finke et al. [127] Speed of sound

C14 (n-tetradecane) p, ρ, T 1970 Snyder and Winnick [143]

117

298–358 0.1–367

1978 Gouel [172]

90

298–393 5.16–40.63 0.599

1987 Holzapfel et al. [173]

12

293

0.1–10

1.060

1995 Gawronska et al. [174]

8

323–364 7.9–24

0.687


11

427–527

0.303

1987 Kneisl and Zondlo [175]

24

404–524

0.932


16

373–588

0.512

1996 Viton et al. [167]

34

284–467

1.251

Saturated liquid density 1882 Krafft [176]

13

277–293

1941 Calingaert et al. [177]

4

293–373

0.868

1946 Vogel [158]

4

293–359

0.851

1965 Roshchupkin [178]

11

288–523

0.550

298–323

0.935

1994 Aminabhavi and Gopalakrishna [153] 9

0.947

Speed of sound 2000 Daridon and Lagourette [179]

251

293–373 0.1–150

0.480


9

293–373

0.311

123


Int J Thermophys (2013) 34:1865–1905



p (MPa)


2001 Pardo et al. [180]

4

288–308

0.226


48

303–433 0.1–49.1

0.425

C15 (n-pentadecane) p, ρ, T 1958 Cutler et al. [163]

69

293–383 0.1–551

0.303


142

293–408 0.1–149

0.403


10

442–543

0.283

1996 Viton et al. [167]

20

293–467

1.753

Saturated liquid density 1946 Schiessler et al. [168]

4

293–372

0.981

1946 Vogel [158]

4

293–359

0.849

1955 Landa et al. [182]

4

293–298

0.476

1958 Cutler et al. [163]

6

310–408

0.843

1978 Diaz Pena and Tardajos [183]

4

298–333

1.013

Speed of sound 2000 Plantier et al. [162]

10

293–383

0.617


54

303–433 0.1–49.1

0.551

2000 Daridon and Lagourette [179]

170

293–383 0.101–150 0.635

1964 Doolittle [96] 1970 Snyder and Winnick [143]

60 93

323–573 5–500 298–358 0.1–290

1978 Gouel [172]

75

314–392 5.17–40.6 0.734

C-16 (n-hexadecane) p, ρ, T 0.851 0.709

1979 Dymond et al. [184]

27

298–373 0.1–450


72

318–373 0.1–10

0.518 0.959

1998 Chang et al. [185]

21

333–413 0.1–30

0.575


16

463–559

0.232

1987 Mills et al. [186]

11

388–560

0.991

1994 Morgan and Kobayashi [148] 20

393–583

0.101

1996 Viton et al. [167]

303–467

2.705

24

Saturated liquid density 1946 Vogel [158]

4

293–360

0.956

1964 Doolittle [96]

6

323–573

0.223


11

298–393

0.976

1986 Plebanski et al. [135]

11

299–489

1.033

123



1881



p (MPa)



Speed of sound 1967 Boelhouwer [89]

74

293–473 10–140

0.502


9

303–383

0.846

2001 Ball and Trusler [92]

65

298–373 0.1–101

0.588

2001 Khasanshin and Shchemelev [91] 29

303–433 0.1–49.1

0.644

0.680

C17 (n-heptadecane) p, ρ, T 1964 Doolittle [96]

66

323–573 0–500


124

303–383 0.1–149.5 0.570

Saturated vapor pressure 1981 Grenier-Loustalot, et al. [187]

5

357–434

5.055

1996 Viton et al. [167]

24

313–467

1.145


5

295–372

1.260

1941 Calingaert et al. [177]

4

293–373

1.167

1947 Schiessler [188]

5

293–372

1.254


9

303–383

0.888

2000 Daridon and Lagourette [179]

151

303–383 0.101–150 0.852

53

333-408 0.1-551

Speed of sound

C18 (n-octadecane) p, ρ, T 1958 Cutler et al. [163]

0.292

Saturated vapor pressure 1955 Myers and Fenske [189]

23

375–567

3.408


17

413–588

0.605

1996 Viton et al. [167]

16

333–467

2.207

Saturated liquid density 1950 Buckland and Seyer [190]

6

333–553

1954 Gray and Smith [191]

10

313–521

0.561 1.293

1958 Cutler et al. [163]

10

333–408

0.964

8

313–383

0.155


16

423–588

1.093

1996 Viton et al. [167]

15

334–467

1.594

4

305–372

Speed of sound 2000 Plantier et al. [162] C19 (n-nonadecane) Saturated vapor pressure


1.309

123


Int J Thermophys (2013) 34:1865–1905



p (MPa)


1896

Eykman [192]

2

307–355

1.211

1924

McKinney [193]

1

313

0.026

1986

Chu et al. [194]

8

308–343

1.300

8

313–383

0.840

50

373–573 5–500

0.738

Speed of sound 2000

Plantier et al. [162]

C20 (n-icosane) p, ρ, T 1964

Doolittle [96]


Sasse et al. [195]

21

363–467

2.538

1989

Chirico et al. [147]

29

388–625

0.224

1994

Morgan and Kobayashi [148] 32

433–583

0.385


Eykman [192]

5

311–409

1.400

1964

Doolittle [96]

5

373–573

1.369

1988

Rodden et al. [196]

5

374–534

0.454

Speed of sound 2000

Plantier et al. [162]

8

0.859

C21 (n-geneicosane) Saturated vapor pressure 1948

Mazee [197]

8

440–468

2.990

1981

Grenier-Loustalot et al. [187] 4

379–434

2.175


Schmidt [198]

4

316–333

1.37

1981


379–434

1988

Sasse et al. [195]

16

353–462

5.158

1994


453–573

0.523

C22 (n-docosane) Saturated vapor pressure 3.839


Schmidt [198]

4

319–333

1.416

7

470–492

2.211

C23 (n-tricosane) Saturated vapor pressure 1948

Mazee [197]

C24 (n-tetracosane) Saturated vapor pressure 1971

Meyer and Stec [199]

6

498–550

0.960

1988

Sasse et al. [195]

12

373–462

4.107

123



1883



1994


p (MPa)


453–588


0.217


Wakefield and Marsh [200]

3

318–338

1.565

28

300–600 0.1–80

1.291

8

350–700

8

350–700

0.702

38

330–700

1.985

C25 (n-pentacosane) p, ρ, T 1999

Calculation from [26]



0.466





C26 (n-hexacosane) Saturated vapor pressure 1928

Young [40]

3

532–556

7.530

1981


379–434

7.231

C28 (n-octacosane) p, ρ, T 1964

Doolittle [96]

6

373–573 0–0.003

0.784


Mazee [197]

8

493–521

6.259

1989

Chirico et al. [147]

13

453–575

1.120

1994


483–588

1.878


Krafft [176]

3

335

1.883

1964

Doolittle [96]

6

323–573

0.784

1988

Rodden et al. [196]

5

372–533

0.583

C30 (n-triacontane) p, ρ, T 1964

Doolittle [96]

50

373–573 5–500

0.624

1999


28

300–600 0.1–80

1.642


Mazee [197]

7

531–547

1.361

2005


8

350–700

0.620


Mazee [197]

2

343–363

1.194

1964

Doolittle [96]

5

373–573

0.525

2006


8

350–700

0.997

123


Int J Thermophys (2013) 34:1865–1905



p (MPa)


Saturated heat capacity 1999 Calculation from [26] 37

340–700

2.339

C31 (n-gentriacontane) Saturated vapor pressure 1948 Mazee [197]

7

517–553

2.853

C35 (n-pentatriacontane) p, ρ, T 1999 Calculation from [26] 28

300–600 0.1–80

2.012


556–570

4.188

2005 Calculation from [26] 8

1948 Mazee [197]

350–700

0.850

Saturated liquid density 2006 Calculation from [28] 8

350–700

1.338

350–700

2.558

Saturated heat capacity 1999 Calculation from [26] 36 C36 (n-hexatriacontane) p, ρ, T 5

373–573 1×10−5 –1×10−4 0.709

7

550–567

1939 Waterman et al. [201]

2

355–373

1.206

1964 Doolittle [96]

5

373–573

0.709

1968 API Proj. 42 [157]

3

353–372

1.248

40

423–573 5–500

0.846


300–600 0.1–80

2.476

1964 Doolittle [96] Saturated vapor pressure 1948 Mazee [197]

5.949


C40 (n-tetracontane) p, ρ, T 1964 Doolittle [96] Saturated vapor pressure 2005 Calculation from [27] 7

400–700

1.250

Saturated liquid density 1964 Doolittle [96]

4

423–573

0.989


400–700

1.713

360–700

2.604

Saturated heat capacity 1999 Calculation from [26] 35 C45 (n-pentatetracontane) Saturated vapor pressure 2005 Calculation from [27] 7

123

400–700

1.651



1885

Table 4 continued Years

References

Points

Temperature and pressure range

Average absolute deviation (AAD) (%)

T (K)

Liquid

p (MPa)

Gas

Critical

Super -critical



7

400–700

2.193

35

360–700

1.845

7

400–700

7

400–700

2.712

34

370–700

2.140



C50 (n-pentacontane) Saturated vapor pressure 2005


2.367



Saturated heat capacity Calculation from [26]

AAD, %

1999

Fig. 1 Comparisons of average absolute deviations for p, ρ, T data calculated from Eq. 3

The ideal-gas reduced Helmholtz energy, in dimensionless form, can be represented by h0τ s0 δτ0 τ − α (δ, τ ) = 0 − 0 − 1 + ln RTc R δ0 τ R

τ

0

τ0

c0p

1 dτ + τ2 R

τ τ0

c0p τ

dτ

(2)

where δ0 = ρ0 /ρc and τ0 = Tc /T0 . T0 and p0 are arbitrary constants, and ρ0 is the ideal gas density at T0 and p0 (ρ0 = p0 /(T0 R)). h 00 is the ideal-gas enthalpy at the reference state, and s00 is the ideal-gas entropy at the reference state. To describe the residual part of the reduced Helmholtz energy, an optimized functional form developed by Sun and Ely [18] has been used. This form of the equation can be further extended to a broader class of substances:

123


AAD, %

1886

AAD, %

Fig. 2 Comparisons of average absolute deviations for saturated vapor pressures calculated from Eq. 3

AAD, %

Fig. 3 Comparisons of average absolute deviations for saturated liquid densities calculated from Eq. 3

Fig. 4 Comparisons of average absolute deviations for isobaric heat capacity calculated from Eq. 3

α r (δ, τ ) =

6 k=1

N k δ d k τ tk +

14

Nk δ dk τ tk exp −δlk

(3)

k=7

where τ = Tr /T, δ = ρ/ρr , and Tr , ρr are reducing parameters. The exponents of Eq. 3 are given in Table 1. The temperature exponents of Eq. 3 were optimized using a nonlinear procedure which is mentioned below.

123


AAD, %

Int J Thermophys (2013) 34:1865–1905

Fig. 5 Comparisons of average absolute deviations for speed of sound calculated from Eq. 3

The generalization was performed within the framework of the theory of thermodynamic similarity with one defining criterion of similarity—the acentric factor. The coefficients of the generalized equations of state can be defined by the following relation: Nk = c1,i + c2,i ω + c3,i ωc4,i

(4)

where ω is the acentric factor. The coefficients c j,i of Eq. 4 are given in Table 2. A nonlinear optimization procedure was used to develop Eq. 3. The same fitting procedure was used in [19]. Optimization of the coefficients and exponents c j,i of Eq. 4 and temperature exponents of Eq. 3 took place simultaneously in a nonlinear form. During the fitting procedure, various constraints were used, which were imposed in the form of inequalities on the thermodynamic surface. The main types of constraints are: critical conditions, controlling the ideal curves, and controlling various derivatives of thermodynamic quantities. The fitting procedure used experimental p, ρ, T data, saturated vapor pressure, saturated heat capacity, isobaric heat capacity, and speed of sound. The experimental data that were used in the regression are marked in bold in Table 4. The acentric factor and critical properties are represented as functions of the carbon number. To calculate the acentric factor, the critical temperature and pressure were calculated with the relations proposed in [19], Tc = a1 − exp(a2 + a3 n a4 + γ a5 /n 5 ), K pc = b1 − exp[b2 + b3 n b4 + b5 /n + ζ b6 /(n + 1)4 ], MPa

(5) (6)

ω = c1 − exp(c2 n c3 − c4 n c5 )

(7)

where n is the number of carbon atoms in a molecule, γ = 1 for even carbon numbers, γ = 0 for odd carbon numbers, ζ = 0 for even carbon numbers, and ζ = 1 for odd carbon numbers. Coefficients of Eqs. 5–7 are taken from [19] and given in Table 3. To calculate the critical density, the authors used the ratio proposed in [20]: ρc = M W/(61.7193 + 38.3386n 1.16536 ), g · mL−1

(8)

where MW is the molecular weight.

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The required relation for the ideal gas part, α 0 (τ, δ), can easily be obtained from an integrable equation for the heat capacity of the ideal gas, c0p (T ), which is known rather accurately for many technically important fluids. For n-alkanes from n-pentane to n-tridecane, individual equations for c0p were used that are presented in the literature [1,53]. For the heavier n-alkanes, the values of c0p were calculated by the group contribution method described in [22]. The work described here focuses on the description of the residual part of the reduced Helmholtz energy, α r (τ, δ). 3 Results and Discussion The most reliable experimental data were selected in order to develop the generalized Helmholtz Eq. 3. For n-alkanes C5 −C13 , an array of data was included in the fitting procedure consisting largely of p, ρ, T and vapor–pressure data [23] and c p , p, T data [24] obtained in the 1980s in the laboratory of the Grozny Petroleum Institute. The uncertainty of the measurements [23] in the liquid-phase density was estimated to be 0.05 % to 0.1 %, and in the gas-phase density—0.1 % to 0.23 %. The uncertainty of the measurements of c p , p, T data [24] was estimated to be 0.3 % to 0.5 %. For nalkanes C13 −C40 the p, ρ, T data of Doolittle [96] were used. The uncertainty of the density measurements was estimated to be 0.1 % to 0.2 %. In addition, different data of other authors were used if the accuracy of measurement conforms to the following conditions: saturated vapor pressure—the uncertainty of the measurements do not exceed 0.1 % at temperatures above the normal boiling point; liquid-phase density: 0.1 %; gas phase density: 0.25 %; heat capacity: 1.0 %; and speed of sound: 0.2 %. For n-alkanes C5 −C18 the lack of experimental data on the saturated vapor pressures near the triple point was filled by calculated data using methods given in [25]. For heavier n-alkanes C40 −C50 , calculated caloric and p, ρ, T data [26] were used, as well as calculated data of saturated vapor pressure and saturated liquid density defined by [27,28]. The inclusion of calculated data in the fitting procedure and the use of different constraints increased the stability of Eq. 3 and improved the extrapolation behavior. Table 4 presents a detailed comparison with the most representative experimental data of the thermodynamic properties of n-alkanes and Figs. 1, 2, 3, 4, and 5 present the overall characteristics of the uncertainties for each substance. In addition, in Appendix 2 for n-alkanes C7 −C12 , Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21 present the characteristics of the uncertainties for the thermodynamic properties calculated using the most reliable modern equations of Span [1], Lemmon and Huber [21], and Lemmon and Span [202]. In Fig. 1, comparisons of the predictions for single-phase densities are shown. The density of the liquid phase is described with an uncertainty of 0.5 % to 0.8 %. The uncertainty in the gas phase is slightly higher, and for all available data, the uncertainties vary from 1 % to 1.8 %. For the supercritical region the uncertainties do not exceed 2 %, excluding n-hexane and n-heptane. We compared the same data sets with the equation of Sun and Ely [29]. In general, the uncertainty of the description of the thermal properties is slightly worse than the equation of [29] for n-pentane, n-hexane, and n-heptane, and significantly better for heavy hydrocarbons with carbon number n > 10.

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Figures 2 and 3 present the comparison of the saturation boundary. For the majority of n-alkanes, the uncertainty in the vapor pressure does not exceed 1.5 %, and only for the last two (n-pentatetracontane and n-pentacontane), the uncertainties are slightly higher. A similar situation is observed for saturated liquid densities: for most substances, the uncertainties do not exceed 1.5 % and only for the last two n-alkanes (n-pentatetracontane and n-pentacontane), the uncertainty is higher than 2 %. Figure 4 shows comparisons of the isobaric heat capacity data. Reliable experimental data on the isobaric heat capacity are available only up to n-tridecane. The isobaric heat capacity of the liquid phase is described with uncertainties of 1 %, excluding n-nonane. In the gas phase, uncertainties are slightly higher up to 1.5 %, excluding npentane with uncertainties greater than 2.5 %. Uncertainties in the supercritical region do not exceed 1 % to 1.5 %, excluding n-nonane and n-decane with uncertainties of 2.5 %. Figure 5 shows comparisons of the speed-of-sound data. Uncertainties for the majority of substances in the description of the speed of sound do not exceed 1.5 %, and increases to 2 % to 3.5 % for the lighter n-alkanes starting with n-octane Table 4 presents the results of comparisons with the most representative literature data. Selecting data for Table 4 was carried out using information about the purity of the sample, errors of the experimental measurements, and range of the experimental measurements. The statistics used to evaluate the equation are based on the percent deviation for any property, X , X = 100 ×

X data − X calc X data

,%

(9)

Using this definition, the average absolute deviation (AAD) in Table 4 is defined as A AD =

n 1 | X i | , % n

(10)

i=1

where n is the number of data points. A critical region for Table 4 was defined as 0.7ρc ≤ ρ ≤ 1.3ρc and 0.98Tc ≤ T ≤ 1.02Tc . 4 Conclusion The information on the thermodynamic properties is widely needed in engineering design. One of the reliable sources of the thermodynamic properties is the equation of state. The reliable experimental data are not readily available for most substances, and it is not always possible to develop an individual equation. In this case, the predictive methods such as the corresponding-states principle are good alternatives. In this work, we developed a generalized fundamental equation of state based on the extended three-parameter corresponding-states principle. The proposed equation has enough precision to calculate thermodynamic properties and phase behavior of n-alkanes (from n-pentane to n-pentacontane) over the temperature range from the triple point to 700 K and at pressures up to 100 MPa.

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Acknowledgments The authors would like to thank Dr. Eric W. Lemmon at the National Institute of Standards and Technology for his help in collecting experimental data and for his valuable comments. The authors are grateful to the Russian Foundation for Basic Research (RFBR) for financial support under Grant No. 09-08-00683a.

Appendix 1: Definitions of Common Thermodynamic Properties and their Relation to the reduced Helmholtz Energy The functions used for calculating pressure ( p), compressibility factor (Z ), internal energy (u), enthalpy (h), entropy (s), Gibbs energy (g), isochoric heat capacity (cv ), isobaric heat capacity (c p ), Joule–Thomson coefficient (μ), and the speed of sound (w) from Eq. 1 are given below.

r ∂α = ρ RT 1 + δ ∂δ τ T r p ∂α Z = =1+δ ρ RT ∂δ τ 0 r a + Ts ∂α u ∂α = =τ + RT RT ∂τ δ ∂τ δ r 0 r h ∂α u + pv ∂α ∂α +δ = =τ + +1 RT RT ∂τ δ ∂τ δ ∂δ τ 0 r 1 ∂a ∂α ∂α s − α0 − αr =− =τ + R R ∂T ρ ∂τ δ ∂τ δ r h − Ts g ∂α = = 1 + α0 + αr + δ RT RT ∂δ τ 2 0 2 r ∂ α 1 ∂u ∂ α cv 2 = = −τ + R R ∂T ρ ∂τ 2 δ ∂τ 2 δ

r 2 r 2 ∂α ∂ α 1 + δ − δτ ∂δ τ ∂δ∂τ cp 1 ∂h cv = +

= r R R ∂T p R 2 ∂ 2 αr 1 + 2δ ∂α 2 ∂δ τ + δ ∂δ p = ρ2

μRρ = Rρ

∂T ∂p

∂a ∂ρ

(11) (12) (13) (14) (15) (16) (17)

(18)

τ

h

r 2 r ∂ α 2 ∂ 2 αr − δ ∂α + δτ ∂δ∂τ ∂δ τ + δ ∂δ 2 τ = 2 0 2 r 2 2 r r r ∂ α ∂ α 2 ∂ 2 αr 1 + δ ∂α 1 + 2δ ∂α − τ2 + ∂∂τα2 ∂δ τ − δτ ∂δ∂τ ∂δ τ + δ ∂τ 2 ∂δ 2 δ

w2 M RT

=

∂α r

M ∂p = 1 + 2δ + δ2 RT ∂ρ s ∂δ τ

r 2 r 2 ∂ α 1 + δ ∂α − δτ ∂δ ∂δ∂τ

2 0 τ 2 r − τ 2 ∂∂τα2 + ∂∂τα2 δ

123

δ

δ

τ

(19)

∂ 2 αr ∂δ 2

τ

(20)


1891

The fugacity coefficient and second and third virial coefficients are given in the following equations: ϕ = exp Z − 1 − ln (Z ) + α r r 1 ∂α B (T ) = lim δ→0 ρc ∂δ τ 2 r 1 ∂ α C (T ) = lim δ→0 ρc2 ∂δ 2 τ

(21) (22) (23)

Other derived properties, given below, include the first derivative of pressure with respect to density at constant temperature (∂ p/∂ρ)T , the second derivative of pressure with respect to density at constant temperature (∂ 2 p/∂ρ 2 )T , and the first derivative of pressure with respect to temperature at constant density (∂ p/∂ T )ρ . 2 r r ∂ α ∂α 2 = RT 1 + 2δ +δ ∂δ ∂δ 2 τ T τ 2 2 r 3 r r ∂ p ∂ α ∂ α ∂α RT 2 3 2δ = + 4δ + δ 2 2 ∂ρ T ρ ∂δ τ ∂δ τ ∂δ 3 τ r 2 r ∂p ∂α ∂ α = Rρ 1 + δ − δτ ∂T ρ ∂δ τ ∂δ∂τ

∂p ∂ρ

(24) (25) (26)

The derivatives of the residual Helmholtz energy are given in the following equations. δ δ2

δ3

∂ 2 αr = ∂δ 2

k=1 6

(27)

k=7

Nk δ dk τ tk [dk (dk − 1)] +

k=1

14

N k δ d k τ tk

k=7

dk − lk δlk dk − 1 − lk δlk − lk2 δlk exp −δlk

(28)

6 14 ∂ 3αr d k tk = N δ τ − 1) − 2)] + Nk δ dk τ tk exp −δlk [d (d (d k k k k 3 ∂δ k=1 k=7

dk (dk − 1) (dk − 2) + lk δlk −2 + 6dk − 3dk2 − 3dk lk +3lk − lk2 + 3lk2 δ 2lk [dk − 1 + lk ] − lk3 δ 3lk

τ τ2

6 14

∂α r = dk − lk δlk Nk δ dk τ tk dk + Nk δ dk τ tk exp −δlk ∂δ

∂α r = N k δ d k τ tk t k + Nk δ dk τ tk exp −δlk tk ∂τ

∂ 2 αr = ∂τ 2

6

14

k=1 6

k=7

k=1

Nk δ dk τ tk [tk (tk −1)]+

14

(29) (30)

Nk δ dk τ tk exp −δlk [tk (tk −1)]

(31)

k=7

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τδ δτ 2

6 14 ∂ 2 αr = tk dk −lk δlk Nk δ dk τ tk [dk tk ]+ Nk δ dk τ tk exp −δlk ∂τ ∂δ

∂ 3αr = ∂δ∂τ 2

k=1 6

(32)

k=7

Nk δ dk τ tk [dk tk (tk − 1)]

k=1 14

+

tk (tk − 1) dk − lk δlk Nk δ dk τ tk exp −δlk

(33)

k=7

Appendix 2: Comparison of Thermodynamic Properties of Selected n-alkanes (C7 , C8 , C10 , C12 ) calculated by Individual Equations of State with the Generalized Fundamental Equation of State, Eq. 3 200-300 K

AAD, %

300-400 K

400-500 K

500-650 K

Pressure, MPa Boelhouwer [96] G olik et al. [98] Nichols et al. [95] Scaife and Lyons [33] Susnar et al. [67] Zawisza et al. [99]

Doolittle [97] Mur inger et al. [100] Kur umov [23] Smith et al. [94] Toscani et al. [101] Span [1]

Fig. 6 Comparisons of average absolute deviations for p, ρ, T data calculated from Eq. 3 for n-heptane

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AAD, %

Int J Thermophys (2013) 34:1865–1905

Temperature, K Brown [70] Kur umov [23] Smith [102] Willingham et al. [41]

Forziati et al. [103] Ewing and O choa [105] Weber [104] Span [1]

AAD, %

Fig. 7 Comparisons of average absolute deviations for vapor pressures calculated from Eq. 3 for n-heptane

Temperature, K Christopher et al. [106] Kur umov [23] Stephan et al. [107] Span [1]

Kahl et al. [108] Rozhnov [79] Yu and Tsai [83]

Fig. 8 Comparisons of average absolute deviations for saturated liquid densities calculated from Eq. 3 for n-heptane

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Int J Thermophys (2013) 34:1865–1905 0.1-1 MPa

1-5 MPa

AAD, %

5-10 MPa

10-30 MPa

50-60 MPa

Temperature, K Kalinowska et al. [110] San Jose et al. [109] Span [1]

Gerasimov [24] Zabr ansky et al. [111]

Fig. 9 Comparisons of average absolute deviations for isobaric heat capacity calculated from Eq. 3 for n-heptane

123


1895 300 K

AAD, %

300-400 K

400-500 K

Pressure, MPa Tanaka et al. [119] Banipal et al. [118] Felsing et al. [116] Schaffenger [63]

G oodwin et al. [120] Boelhouwer [96] Dymond et al. [117] Span [1]

AAD, %

Fig. 10 Comparisons of average absolute deviations for p, ρ, T data calculated from Eq. 3 for n-octane

Temperature, K Badalyan et al. [121] F elsing et al. [116] Young [40] Span [1]

Dejoz et al. [123] G ier ycz et al. [122] Ewing et al. [124]

Fig. 11 Comparisons of average absolute deviations for vapor pressures calculated from Eq. 3 for n-octane

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AAD, %

Fig. 12 Comparisons of average absolute deviations for saturated liquid densities calculated from Eq. 3 for n-octane

Temperature, K Abdulagatov et al. [38] Das and Kuloor [125]

Chappelow et al. [126] Span [1]

Fig. 13 Comparisons of average absolute deviations for isobaric heat capacity calculated from Eq. 3 for n-octane

0.1-1 MPa

1-5 MPa

AAD, %

10-20 MPa

30 MPa

50-60 MPa

100 MPa

Temperature, K Banipal et al. [118] Hossenlopp, Scott [48]

123

Gerasimov [24] Span [1]


1897 250-300 K

AAD, %

300-450 K

450-550 K

550-600 K

Pressure, MPa Kur umov [23] Caudwell et al. [146] Reamer et al. [143] Scaife and Lyons [33] Zuniga-Moreno et al. [147]

Banipal et al. [118] G ehr ig and Lentz [145] Sage et al. [142] Snyder et al. [144] Lemmon and Span [203]

AAD, %

Fig. 14 Comparisons of average absolute deviations for p, ρ, T data calculated from Eq. 3 for n-decane

Temperature, K Chir ico et al. [148] Morgan et al. [149] Willingham et al. [41]

Dejoz et al. [123] Kur umov [23] Lemmon and Span [203]

Fig. 15 Comparisons of average absolute deviations for vapor pressures calculated from Eq. 3 for n-decane

123


AAD, %

1898

Temperature, K Amirkhanov et al. [137] Dymond and Young [151] Yu and Tsai [83]

Bingham et al. [150] Kurumov [23] Lemmon and Span [203]

Fig. 16 Comparisons of average absolute deviations for saturated liquid densities calculated from Eq. 3 for n-decane 0.1-0.5 MPa

1-10 MPa

AAD, %

10-30 MPa

50-60 MPa

100 MPa

Temperature, K G er asimov [24]


Lemmon and Span [203]

Fig. 17 Comparisons of average absolute deviations for isobaric heat capacity calculated from Eq. 3 for n-decane

123


1899 270-330 K

AAD, %

350-420 K

450-520 K

550-570 K

Pressure, MPa Kur umov [23] Tanaka et al. [119] Cutler et al. [164] Snyder et al. [144]

Caudwell et al. [146] Boelhouwer [96] Landau et al. [156] Lemmon and Huber [21]

AAD, %

Fig. 18 Comparisons of average absolute deviations for p, ρ, T data calculated from Eq. 3 for n-dodecane

Temperature, K Dejoz et al. [123] Mor gan et al. [149] Willingham et al. [41]

G ier ycz et al. [165] Kur umov [23] Lemmon and Huber [21]

Fig. 19 Comparisons of average absolute deviations for vapor pressures calculated from Eq. 3 for ndodecane

123


AAD, %

1900

Temperature, K Dornte and Smyth [51] G ar cia et al. [84] Lemmon and Huber [21]

Dymond and Young [82] Kur umov [23]

Fig. 20 Comparisons of average absolute deviations for saturated liquid densities calculated from Eq. 3 for n-dodecane 0.1 MPa

10 MPa

AAD, %

20 MPa

30-50 MPa

60-100 MPa

Temperature, K Bessier es et al. [166] Lemmon and Huber [21]

G er asimov [24]

Fig. 21 Comparisons of average absolute deviations for isobaric heat capacity calculated from Eq. 3 for n-dodecane

123


1901

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