- Email: [email protected]
Thermodynamic properties of pure fluids using the GEOS3C equation of state
Fluid Phase Equilibria 174 (2000) 51–68
Thermodynamic properties of pure fluids using the GEOS3C equation of state Dan Gean˘a∗ , Viorel Feroiu Department of Applied Physical Chemistry and Electrochemistry, University “Politehnica” Bucharest, Spl. Independen¸tei 313, 78126 Bucharest, Romania
Abstract Thermodynamic properties were predicted for two groups of pure fluids, along the saturation curve and in the single-phase region: IUPAC pure fluids (propylene, chlorine, oxygen and nitrogen) and halogenated hydrocarbons (R11, R12, R21, R22, R152a). A new form of equation GEOS, named GEOS3C, was used. A wide comparison with literature recommended data, as well as with results of other six cubic equations of state (Soave-Redlich-Kwong, Peng-Robinson, Stryjek-Vera, Schmidt-Wenzel, Freze et al. and Salim and Trebble) was made. The GEOS3C equation has three parameters estimated by matching three points on the saturation curve (vapor pressures and the corresponding liquid volumes). The three fixed temperatures are the triple point, the boiling point and the reduced temperature T r = 0.7. The GEOS3C equation leads to the best results in predicting vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties in the saturation range, but this do not lead necessarily to similar best predictions of all properties in the single-phase region. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Thermodynamic properties; Equation of state; Pure fluids
1. Introduction In the classical thermodynamic framework it is possible to obtain typical relationships for the calculation of enthalpy, entropy, fugacity and other thermodynamic properties. These properties can be related to operating variables of the processes, e.g. the temperature of a fluid in a heat exchanger. It is, therefore, important to establish relationships between the properties and independent variables like temperature or pressure. Such relationships for thermodynamic properties of fluids can be obtained using equations of states. In a previous work, Gean˘a and T˘an˘asescu [1] presented these functions for a general cubic equation of state called GEOS, and calculated the thermodynamic properties of methane. In order to predict accurate ∗ Corresponding author. Fax: 40-1-3129647. E-mail addresses: d [email protected] (D. Gean˘a), v [email protected] (V. Feroiu).
0378-3812/00/$20.00 © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 4 1 7 - 9
52
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
values for a wide range of fluid thermodynamic properties, an equation of state must first be able to properly represent the PVT behavior of pure substances. In this work, thermodynamic properties were predicted for two groups of pure fluids, along the saturation curve and in the single-phase region: IUPAC pure fluids (propylene, chlorine, oxygen and nitrogen) and halogenated hydrocarbons R11(CFCl3 ), R12(CF2 Cl2 ), R21(CHFCl2 ), R22(CHF2 Cl), R152a(CHF2 CH3 ). A new form of equation GEOS [2–6], named GEOS3C [7], was used. A wide comparison with IUPAC data [8], respectively, with Vargaftik data [9] for halogenated chlorofluorocarbons, hydrochlorofluorocarbons, and ASHRAE [10] for hydrofluorocarbons was made. A comparison with results of other six cubic equations of state is also presented: Soave-Redlich-Kwong (SRK) [11], Peng-Robinson (PR) [12], Stryjek-Vera (PRSV) [13], Schmidt-Wenzel (SW) [14], Freze et al. (C-1) [15] and Salim and Trebble (TBS) [16,17]. The three parameters of the GEOS3C equation were estimated by constraining the EOS to reproduce the experimental vapor pressure and liquid volume at three selected temperatures. The three fixed temperatures are the triple point, the boiling point and the reduced temperature T r = 0.7. The GEOS3C equation leads to the best results in predicting vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties at saturation, but this does not lead necessarily to similar best predictions of all properties in the single-phase region. 2. The GEOS3C equation of state The general cubic equation of state (GEOS) has the form P =
a(T ) RT − V − b (V − d)2 + c
(1)
The four parameters a, b, c, d for a pure component are expressed by a = ac β 2 (Tr );
ac = Ωa
R 2 Tc2 ; Pc
b = Ωb
RTc ; Pc
c = Ωc
R 2 Tc2 ; Pc2
d = Ωd
RTc Pc
(2)
A new temperature function is used: β(Tr ) = 1 + C1 y + C2 y 2 + C3 y 3 β(Tr ) = 1 + C1 y
for Tr ≤ 1
for Tr > 1
(3) (4)
y = 1 − Tr0.5
(5)
Setting four critical conditions [2–6], the expressions of the parameters Ω a , Ω b , Ω c , Ω d are Ωa = (1 − B)3 ;
Ωb = Zc − B
Ωc = (1 − B)2 (B − 0.25); B=
Ωd = Zc −
1 + C1 αc + C1
with α c as the Riedel’s criterion [2–6].
(6a) (1 − B) 2
(6b) (7)
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
53
Using experimental values of the critical constants and the acentric factor for the calculation of α c from the equation αc = 5.808 + 4.93ω
(8)
the C1 , C2 and C3 parameters were obtained by matching three points on the saturation curve (vapor pressures together with the corresponding liquid volumes). The three fixed temperatures are the triple point, the boiling point and the reduced temperature T r = 0.7. Thus, the EOS is forced to properly reproduce the region from the triple point to the critical point. The following features of the GEOS3C parameterization are remarkable: • It gives exactly the experimental critical point of any substance, and also the experimental critical compressibility factor Zc . • The prediction of liquid volume is improved, without translation, by the parameter C1 from the temperature function of the attractive term, involved in the B relation (7). • The parameters C1 , C2 and C3 , estimated from only three fixed temperatures, can predict with accuracy the saturation pressure curve from the triple point to the critical point. • The involving of the parameter C1 in the expressions of the a and b, leads to a “coupling” between the repulsive and attractive terms in a cubic equation of state. As pointed out previously [2–6], the relations (6a and b) are general forms for all the cubic equations of state with two, three and four parameters. In Appendix A, it is shown how the Eqs. (6a) and (6b) can be applied to obtain the parameters of several well known cubic forms (vdW, SRK, PR, SW, C-1, TBS). Our software used in calculations is based on the Eqs. (A.1)–(A.12) for the reduction of GEOS to different cubic EOS from the literature. This is the meaning of the statement “general cubic equation of state” used for GEOS. The temperature function (Eqs. (3)–(5)) as well as its first derivative are continuous in the critical point. The second derivative is not continuous, and this fact leads to a discontinuity in the isochoric heat capacity and other properties involving the second derivative, at the critical point. This discontinuity is generally of the order of some J/mole/K and it seems that do not lead to difficulties in the calculations, near the critical point. On the other side, it is known that the classical cubic EOS do not predict the correct divergent behavior of the isochoric heat capacity in the critical region. Recently, Teja and Sun [18] presented a method to transform an analytical EOS into a non-analytical equation which shows the correct divergence in the critical region. Although the GEOS3C equation with the temperature function (Eqs. (3)–(5)) shows a discontinuity in CV at the critical point, to obtain the correct divergence behavior a method as that suggested by Teja and Sun [18] should be applied.
3. Results and discussion 3.1. IUPAC pure fluids Seven equations of state SRK, PR, PRSV, SW, C-1, TBS and GEOS3C have been used to calculate thermodynamic properties for oxygen, chlorine nitrogen and propylene. The investigated PVT range covers single-phase (liquid or gas) and two-phase (liquid–vapor) regions. The calculations have been compared with data of IUPAC collections [8], as recommended by Deiters and De Reuck [19].
54
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Table 1 Values of C1 , C2 and C3 parameters for oxygen, chlorine, nitrogen and propylenea Component
C1
C2
C3
Oxygen Chlorine Nitrogen Propylene
0.2304 0.2241 0.1751 0.3959
0.0689 0.3454 0.1974 0.1251
0.2062 −0.0299 0.2250 0.1505
a
Critical data and acentric factors from IUPAC tables [8].
The values of the GEOS3C parameters C1 , C2 and C3 for oxygen, chlorine, nitrogen and propylene, estimated only by three points on the saturation curve corresponding to the triple point temperature, the boiling temperature and the reduced temperature T r = 0.7, are presented in Table 1. The following thermodynamic properties have been calculated: compressibility factor, Z; enthalpy, H; enthalpy of vaporization, 1v H; entropy, S; heat capacity at constant pressure, CP ; heat capacity at constant volume, CV ; adiabatic index, CP /CV ; speed of sound, WS ; fugacity coefficient, ϕ; Joule–Thomson coefficient, JT. The departure (residual) thermodynamic properties from the ideal gas at the same temperature and pressure are given for GEOS in Appendix B. The same relations were used for all EOSs under appropriate restrictions imposed for GEOS parameters (given in Appendix A). The ideal gas contribution to the thermodynamic properties was calculated using the heat capacity functions recommended in IUPAC tables [8]. The average absolute deviations for a property Y are relative (%): PN exp exp |(Y eos − Yi )/Yi | × 100 AAD% = i=1 i (9) N excepting the enthalpy and entropy where PN exp eos − Hi | i=1 |Hi AAD H (or S) = N
(10)
Table 2 Thermodynamic function deviations at saturation for propylenea EOS
SRK PR PRSVb SW C-1 TBSb GEOS3C GEOS3C (59P) a b
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
2.0 10.6 0.6 4.5 7.4 26.4 2.4 1.5
7.6 6.0 6.4 2.9 5.5 3.3 4.0 3.9
2.9 7.8 1.1 6.9 6.1 14.7 2.6 1.9
1.9 2.5 2.0 1.7 2.3 7.7 1.5 1.3
218.6 427.1 237.5 177.2 350.5 1546.0 279.5 220.2
134.5 129.0 161.2 121.7 129.4 169.7 136.7 138.3
0.8 2.0 0.8 0.7 1.7 6.3 1.2 1.0
2.4 3.2 3.0 2.7 3.0 4.5 2.7 2.6
Temperature range (K): 87.89–365.57; pressure range (MPa): 9.5 × 10−10 –4.66. Number of data points: 59. Temperature range (K): 140–365.57; pressure range (MPa): 10−4 –4.66.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
55
Table 3 Thermodynamic function deviations at saturation for propylenea EOS
SRK PR PRSV SW C-1 TBS GEOS3C GEOS3C (59P) a
AAD (%) CPL
CPV
(CP /CV )L
(CP /CV )V
ϕ
WSL
WSV
JTL
JTV
6.7 5.9 5.5 6.8 5.8 14.8 3.4 3.8
6.5 7.0 6.9 6.7 7.1 8.6 7.8 7.8
9.5 9.2 8.8 9.6 9.2 8.0 9.7 9.5
2.1 2.3 2.2 2.2 2.3 2.8 2.9 2.9
0.6 0.6 0.7 0.5 0.7 1.3 1.1 1.2
17.5 17.1 17.6 17.1 16.6 16.0 6.2 7.5
1.2 0.5 0.6 0.9 0.4 1.3 1.2 1.1
444 386 404 411 378 274 29.0 58.8
16.9 16.4 16.3 16.5 16.2 17.9 16.2 16.0
Temperature range (K): 198.8–363.56; pressure range (MPa): 0.025–4.5. Number of data points: 29.
The results of the calculations for propylene are summarized in Tables 2–4. The average absolute deviations (AAD), between calculated values by EOSs and IUPAC data are given. For each table the number of data points, the pressure and temperature ranges are indicated. The two-phase region properties have been calculated at temperatures from the triple point to the critical point. The notation GEOS3C in the tables indicates that the parameters C1 , C2 and C3 were estimated from vapor pressure and liquid volume at three fixed temperatures (triple point, boiling point and T r = 0.7). For propylene, a second set of parameters was used, obtained by correlating data at 59 temperatures on the saturation curve from the triple point to the critical point (notation GEOS3C (59P)). It may be observed that the results (deviations) obtained with the both sets of parameters are almost the same, and therefore, confirm the ability of the parameters obtained from only three experimental points to reproduce well the full saturation range. For all other substances only the results obtained with parameters estimated from three points on the saturation curve will be presented. Propylene is a severe test in predictions due to the temperature extension of the saturation range and the very low values of the vapor pressures towards the triple point (the order of magnitude is 10−10 MPa). Table 4 Thermodynamic function deviations in single-phase region for propylenea EOS
SRK PR PRSV SW C-1 TBS GEOS3C GEOS3C (59P) a b
AAD
AAD (%)
S (J/mol/K)
H (J/mol)
Z
CP
WS
CP /CV
ϕb
JT
1.7 2.5 2.1 1.9 2.3 6.5 2.8 2.8
816.5 473.2 322.8 606.4 464.2 2336.0 986.8 1057.0
4.0 4.1 4.2 2.2 3.9 5.7 3.8 4.0
3.6 5.4 4.4 4.3 5.2 14.3 3.8 3.5
19.1 17.5 17.6 18.2 17.6 39.8 30.2 32.3
5.7 5.3 5.4 5.5 5.3 4.1 5.9 5.8
2.5 4.2 4.1 1.3 3.7 26.5 4.1 4.0
63.3 91.1 75.2 90.4 90.7 76.3 50.3 45.5
Temperature range (K): 150–575; pressure range (MPa): 0.025–1000. Number of data points: 182. Pressure range (MPa): 0.025–100.
56
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
The following observations can be made on the basis of the results of the Tables 2 and 3: • The vapor pressures and saturated volumes are better predicted by GEOS3C, compared to the results obtained using other equations. Surprisingly, small are the deviations in vapor pressure predicted by SRK. • The low deviation in vapor pressure of the PRSV EOS (with an optimized parameter, k1 ) is explained by the restricted temperature range (140 K — Tc ), recommended by the authors [13]. • For the enthalpy of vaporization the GEOS3C gives the best results. For the CP , WSL and JTL the GEOS3C gives also the best results, but with larger deviations in the last two properties. • The difference in performance between the EOSs is less noticeable for the other properties. • The deviations given by TBS EOS with four optimized parameters, although in a restricted temperature range (140 K — Tc ), are very high (excepting VL ). This is probably explained by the fact that the optimized parameters of the TBS EOS have been obtained at temperatures higher than 140 K (not indicated by the authors [16]). In the single-phase region (Table 4) the PRSV, PR and C-1 EOSs give the best results for enthalpy. Notably larger are the deviations in WS , CP and ϕ for the TBS equation. Somewhat better results for CP and Joule–Thomson coefficient are obtained with GEOS3C. The pressure range was restricted at 100 MPa in the calculation of the fugacity coefficient deviations. At pressures higher than 200 MPa the fugacity coefficients can take values of 1012 –1015 and the deviations between the cubic EOS and the IUPAC data can become correspondingly of the magnitude order of 103 –104 %. Similar calculations have been done for chlorine, oxygen and nitrogen, with seven equations of state. The results obtained for chlorine are presented in Tables 5–7. In the saturation range (from the triple point to the critical point) GEOS3C gives the best results for vapor pressure, volumes, liquid enthalpy, enthalpy of vaporization and entropies. The deviations in fugacity coefficient, adiabatic index and sound velocity in vapor phase are larger for TBS and GEOS3C than for the other equations (Table 6). Somewhat better results predict the GEOS3C for Joule–Thomson coefficient and adiabatic index in the liquid phase. The parameter k1 was not available for chlorine and the results of the PRSV EOS are nearly the same as with PR EOS. In the single-phase region (Table 7) SRK and GEOS3C give the best results for enthalpy, entropy and fugacity coefficient, whereas SW EOS is the best for the compressibility factor. For the heat capacity CP the smallest deviations are obtained from the TBS and GEOS3C equations. Table 5 Thermodynamic function deviations at saturation for chlorinea EOS
SRK PR PRSV (k1 = 0) SW C-1 TBS GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
4.4 7.3 7.1 5.5 6.0 4.0 0.3
7.4 6.9 6.7 4.3 4.7 3.6 3.6
4.1 6.6 6.3 5.5 5.2 5.1 1.3
1.9 3.0 2.9 2.3 2.6 3.2 1.6
376.8 666.6 652.8 368.7 591.9 635.1 342.8
203.8 196.3 196.2 187.0 205.1 223.0 214.4
1.0 1.9 1.9 1.0 1.7 1.8 1.0
0.6 0.9 0.9 0.8 0.9 1.0 0.6
Temperature range (K): 172.17–416.95; pressure range (MPa): 1.38 × 10−3 –7.99. Number of data points: 52.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
57
Table 6 Thermodynamic function deviations at saturation for chlorinea EOS
AAD (%)
SRK PR PRSV (k1 = 0) SW C-1 TBS GEOS3C a
CPL
CPV
(CP /CV )L
(CP /CV )V
ϕ
WSL
WSV
JTL
JTV
19.0 18.0 18.0 21.1 18.2 10.7 13.2
6.1 7.5 7.5 6.2 7.4 10.6 9.8
12.3 12.0 12.0 12.5 12.1 11.6 10.9
3.5 4.2 4.2 3.9 4.2 9.3 8.5
0.9 0.7 0.7 0.7 0.4 3.0 2.5
18.1 19.3 19.3 17.3 14.6 23.6 19.5
1.1 1.5 1.5 1.3 1.2 5.6 4.8
108.0 94.6 94.4 100.0 75.1 50.4 27.3
10.0 9.4 9.4 9.3 9.5 9.6 9.4
Temperature range (K): 196.96–412.77; pressure range (MPa): 0.01–7.5. Number of data points: 11.
Table 7 Thermodynamic function deviations in single-phase region for chlorinea EOS
AAD
SRK PR PRSV (k1 = 0) SW C-1 TBS GEOS3C a
AAD (%)
S (J/mol/K)
H (J/mol)
Z
CP
WS
CP /CV
ϕ
JT
0.6 1.3 1.3 0.8 1.2 0.9 0.5
199.8 409.2 401.9 260.5 376.0 360.3 205.2
2.5 3.7 3.7 1.2 1.9 4.0 2.7
5.2 7.1 7.1 5.9 6.9 3.4 3.4
6.3 5.7 5.7 6.2 5.4 27.8 19.5
7.5 7.2 7.2 7.4 7.2 8.6 9.3
3.6 5.8 5.7 4.6 5.4 5.8 2.2
37.7 40.1 40.0 37.2 37.1 38.4 30.1
Temperature range (K): 180–900; pressure range (MPa): 0.01–25. Number of data points: 114.
The results obtained for oxygen are presented in Tables 8–10. In the saturation range (from the triple point to the critical point) C-1, TBS and GEOS3C equations give the best results for vapor pressure, volumes and enthalpy of vaporization. The deviations in fugacity coefficient are higher for GEOS3C than for the other equations, but the same equation is better in predicting CPL and JTL (Table 9). In the single-phase region (Table 10), PRSV and C-1 EOSs give the best results for enthalpy and entropy. The Table 8 Thermodynamic function deviations at saturation for oxygena EOS
SRK PR PRSV SW C-1 TBS GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
3.1 2.0 0.5 0.9 0.9 0.9 0.7
4.2 9.1 9.0 3.7 2.6 2.1 2.8
3.7 2.8 1.4 1.6 1.4 1.6 1.5
2.8 2.1 1.9 2.1 1.9 1.8 2.1
131.2 110.8 77.9 75.0 86.2 84.8 115.7
39.5 36.2 36.5 32.9 39.5 39.3 41.2
1.2 1.0 0.7 0.8 0.8 0.9 1.2
0.5 0.5 0.4 0.3 0.4 0.5 0.5
Temperature range (K): 54.36–154.59; pressure range (MPa): 1.5 × 10−4 –5.05. Number of data points: 53.
58
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Table 9 Thermodynamic function deviations at saturation for oxygena EOS
SRK PR PRSV SW C-1 TBS GEOS3C a
AAD (%) CPL
CPV
(CP /CV )L
(CP /CV )V
ϕ
WSL
WSV
JTL
JTV
10.7 7.3 7.4 11.6 7.3 6.7 6.1
10.9 11.7 11.7 10.9 11.7 12.1 12.9
12.4 11.8 11.4 12.3 11.7 11.5 13.5
3.2 3.2 2.9 3.2 3.1 2.8 2.7
0.3 0.9 0.9 0.3 0.6 0.9 1.6
18.1 17.3 17.4 18.0 10.3 7.3 12.3
2.8 1.9 1.8 3.1 2.2 2.0 2.0
70.9 66.8 61.1 61.1 39.6 25.1 15.8
12.4 12.2 12.1 12.2 12.4 12.1 12.0
Temperature range (K): 78.65–154.36; pressure range (MPa): 0.025–5. Number of data points: 29.
fugacity coefficient and the compressibility factor are best predicted by C-1 EOS. SRK EOS gives the smallest deviation in CP . Similar results were obtained for nitrogen, both at saturation and in the single-phase region (not presented here). Examples of calculated properties are presented in the Figs. 1–5. Fig. 1 shows the enthalpy-pressure diagram of propylene. Points figure IUPAC data, while the curves represent calculations performed by GEOS3C for saturation region and five isotherms. The entropy–pressure diagram of propylene calculated by GEOS3C is presented in Fig. 2. The agreement with IUPAC data is acceptable at pressures under 100 MPa, but larger errors of calculation should be, however, counted in the high-pressure range (P > 100 MPa). The CP predictions from the GEOS3C and PR equations for chlorine on the saturation curve are presented in Fig. 3. Both perform similarly, excepting the liquid region where GEOS3C gives better predictions. The Joule–Thomson coefficient predictions from the GEOS3C equation for chlorine on the saturation curve and on the single-phase isotherms are presented in Fig. 4. The dependence of pressure of the GEOS3C fugacity coefficient of oxygen at different temperatures is presented in Fig. 5. A satisfactory agreement with IUPAC data may be remarked, confirmed also by the AAD% given in Table 10. Table 10 Thermodynamic function deviations in single-phase region for oxygena EOS
SRK PR PRSV SW C-1 TBS GEOS3C a
AAD
AAD (%)
S (J/mol/K)
H (J/mol)
Z
CP
WS
CP /CV
ϕ
JT
0.5 0.5 0.3 0.4 0.4 0.5 0.6
58.8 61.6 39.1 51.3 38.5 43.8 70.9
0.8 5.5 5.6 1.0 0.8 1.2 1.9
1.6 3.1 2.6 2.5 2.9 2.5 3.7
5.3 5.9 6.2 5.4 5.9 9.5 17.5
3.6 2.8 3.1 3.6 2.8 3.2 4.7
2.5 3.8 3.7 1.5 1.1 1.8 2.9
15.2 20.1 19.0 16.9 17.9 18.6 19.2
Temperature range (K): 60–300; pressure range (MPa): 0.025–80. Number of data points: 151.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
59
Fig. 1. Enthalpy-pressure diagram of propylene. Points: IUPAC data [8].
3.2. Halogenated chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs) and hydrofluorocarbons (HFCs) The second group of pure fluids studied in this work is that of halogenated chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs) and hydrofluorocarbons (HFCs). The knowledge of thermodynamic properties of such compounds is of importance in relation to their use as refrigerants.
Fig. 2. Entropy-pressure diagram of propylene. Points: IUPAC data [8].
60
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Fig. 3. CP prediction for chlorine on the saturation curve. Points: IUPAC data [8].
Five equations of state have been used to calculate thermodynamic properties for halogenated hydrocarbons, because parameters were not available for PRSV and TBS EOSs. The investigated PVT range covers single-phase (liquid or gas) and two-phase (liquid–vapor) regions. The calculations have been compared with recommended data of Vargaftik [9] and ASHRAE [10]. Ideal gas heat capacities were taken from Reid et al. [20].
Fig. 4. Joule–Thomson coefficient prediction for chlorine with GEOS3C. Points: IUPAC data [8].
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
61
Fig. 5. Fugacity coefficient prediction for oxygen. Points: IUPAC data [8].
The values of the parameters C1 , C2 and C3 of the GEOS3C equation for two CFCs (R11, R12), two HCFCs (R21, R22) and a HFC (R152a) are presented in Table 11. The results of the calculations for R11 and R21 are summarized in Tables 12–16. Average absolute deviations (AAD) calculated with Eqs. (9) and (10) between EOS and literature data [9] are presented. For each table, number of data points taken into calculation, pressure and temperature ranges are indicated. The two-phase region properties have been calculated at temperatures ranging between the lowest available value and the critical point [9]. The notation GEOS3C in the tables indicates that the parameters C1 , C2 and C3 were obtained by using only data at three fixed temperatures on the saturation curve (the lowest available temperature, the boiling point and T r = 0.7). For R11 the best results are obtained at saturation by the GEOS3C and SW equations. Notably are the small deviations in liquid enthalpy and entropy of the SRK EOS. The enthalpies and entropies on the isobars are predicted with similar accuracy from all equations as shown in Table 13. Surprising is the small deviation in volume given by SRK EOS. Table 11 Values of C1 , C2 and C3 parameters for several CFCs, HCFCs and HFCsa Component
C1
C2
C3
R11 (CFCl3 ) R12 (CF2 Cl2 ) R21 (CHFCl2 ) R22 (CHF2 Cl) R152a (CHF2 CH3 )
0.3350 0.3114 0.3019 0.2800 0.2554
0.3268 0.3306 0.4976 0.4589 0.9813
0.1358 0.1854 −0.1360 0.1038 −1.1251
a
Critical data from Vargaftik [9] and ASHRAE [10], acentric factors from Reid et al. [12].
62
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Table 12 Thermodynamic function deviations at saturation for R11a EOS
SRK PR SW C-1 GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
1.2 1.9 0.6 1.5 0.5
6.3 5.9 1.4 9.5 2.3
1.6 2.1 1.0 1.9 0.9
0.7 1.3 0.8 0.9 0.6
81.0 505.6 296.6 325.9 307.3
190.2 188.3 189.5 190.4 187.1
0.5 2.0 1.4 1.6 1.3
1.0 0.8 0.9 1.0 0.9
Temperature range (K): 213.15–413.15; pressure range (bar): 0.01–17.9. Number of data points: 62.
Table 13 Thermodynamic function deviations in single-phase region for R11a EOS
AAD V (%)
SRK PR SW C-1 GEOS3C a
AAD
0.2 0.5 0.4 0.6 0.8
H (J/mole)
S (J/mole/K)
247.8 232.7 230.7 229.3 225.3
1.4 1.4 1.4 1.4 1.4
Temperature range (K): 243.15–523.15; pressure range (bar): 0.03–25. Number of data points: 436.
For R21 the best results are obtained at saturation by the GEOS3C and SW equations. On the isobars the volume predictions from the GEOS3C are slightly worse than the results of the SW equation, but the predicted values of enthalpies and entropies are better. For the refrigerant R152a at saturation the best results are those predicted by GEOS3C excepting the vapor volumes and the enthalpy of vaporization predicted slightly better by the C-1 EOS. The comparison is made with ASHRAE [10] recommended data, from the triple point to the critical temperature.
Table 14 Thermodynamic function deviations at saturation for R21a EOS
SRK PR SW C-1 GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
0.8 1.6 0.9 1.3 0.8
12.8 4.7 4.2 7.7 3.7
2.3 2.7 2.1 2.8 2.5
3.5 3.6 3.2 3.3 2.8
1489.2 1235.0 1421.4 1382.5 1181.4
1242.3 1248.9 1258.0 1254.7 1220.0
4.3 3.3 3.9 3.7 3.2
3.5 3.5 3.5 3.4 3.2
Temperature range (K): 213.15–451.4; pressure range (bar): 0.025–51.8. Number of data points: 119.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
63
Table 15 Thermodynamic function deviations in single-phase region for R21a EOS
AAD V (%)
SRK PR SW C-1 GEOS3C a
AAD
7.5 3.1 1.6 6.2 3.7
H (J/mole)
S (J/mole/K)
1001.0 834.5 928.0 955.9 783.4
2.9 2.2 2.5 2.4 2.1
Temperature range (K): 273.15–473.15; pressure range (bar): 1–200. Number of data points: 300.
Table 16 Thermodynamic function deviations at saturation for R152aa EOS
SRK PR SW C-1 GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
3.5 3.1 2.8 1.8 0.4
24.8 10.4 13.3 4.4 4.6
2.6 2.2 2.6 1.5 2.1
1.7 1.3 1.4 1.6 2.2
821.7 674.7 765.5 810.2 753.3
647.6 650.7 649.2 654.9 634.0
3.5 2.9 3.2 3.2 2.7
3.0 2.9 2.9 2.8 2.4
Temperature range (K): 154.41–386.26; pressure range (bar): 8 × 10−4 –45.1. Number of data points: 67.
Fig. 6. Temperature–enthalpy dependence (saturation curve) for R21. Points: Vargaftik data [9].
64
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
An example of calculated property is presented in Fig. 6. Enthalpy predictions of GEOS3C and SRK equations for R21 are compared with data of Vargaftik [9] in the saturation range. All other thermodynamic properties are also predicted, similarly to IUPAC fluids, but are not reproduced in this work, having not available data for comparison. On the basis of the studied CFC, HCFC and HFC substances, it is possible to conclude that GEOS3C and SW equations predict accurate thermodynamic properties for this group of chemical compounds.
4. Conclusions The thermodynamic functions were predicted by GEOS3C and other six equations of state on a wide PVT range, including the entire saturation region. The GEOS3C equation, has three parameters estimated by constraining the EOS to reproduce the experimental vapor pressure and liquid volume at three selected temperatures. For the substances considered in this study, the GEOS3C equation gives the best results in vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties in the saturation range, but this do not lead necessarily to similar best predictions of all properties in the single-phase region. At very high pressures, property predictions are unsatisfactory. Although the original SRK and PR equations give generally poor predictions of vapor pressure and saturated liquid volume, they lead to surprisingly good results for some other properties (enthalpy and entropy). On the basis of the studied CFC, HCFC and HFC substances it appears that GEOS3C and SW equations predict better the thermodynamic properties for this group of chemical compounds. An inconsistency given by all equations is that an infinite value for isochoric heat capacity is not predicted at the critical point. Although the second derivative of the temperature function of GEOS3C presents a discontinuity at critical temperature, to obtain the correct divergence behavior of the isochoric heat capacity a crossover method should be applied. List of symbols a, b, c, d AAD B C1 , C2 and C3 CV , CP E F H JT M P, PS R S T U
parameters in GEOS absolute average deviation dimensionless parameter in GEOS, defined by Eq. (7) parameters in GEOS3C temperature function isochoric and isobaric heat capacities expression based on GEOS, defined by Eqs. (A.10)–(A.12) Helmholtz function enthalpy Joule–Thomson coefficient molar mass pressure, saturation pressure universal gas constant entropy temperature internal energy
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
V, VL , VV Ws Y Z
65
molar volume, liquid volume, vapor volume speed of sound thermodynamic function (general notation) compressibility factor
Greek symbols αc β ϕ ω Ω a, Ω b, Ω c, Ω d
Riedel’s criterium (parameter in GEOS) reduced temperature function in GEOS fugacity coefficient acentric factor parameters of GEOS
Subscripts c critical property r reduced property
Acknowledgements The authors are grateful to National Council for Scientific Research of Romania, for financial support. Appendix A A.1. Reduction of GEOS to other cubic EOSs To obtain the parameters of the vdW EOS from the Eqs. (6a) and (6b), we set [2–4] the following restrictions: Ωc = (1 − B)2 (B − 0.25) = 0;
Ωd = Zc − 0.5(1 − B) = 0
(A.1)
It results: B (vdW) = 0.25, Zc (vdW) = 0.375, Ω a (vdW) = (1 − B)3 = 27/64, Ω b (vdW) = Z c − B = 1/8. Similarly, to obtain the parameters of the SRK EOS from the Eqs. (6a) and (6b) we set [2–4] the following restrictions: Ωc = −(Ωb /2)2 and Ωd = −Ωb /2. It follows (Zc − B)2 (Zc − B) ; Ωd = Zc − 0.5(1 − B) = − 4 2 It results: Zc (SRK) = 1/3, and the relation for B (SRK) 1 1 − 3B 2 B = 0.25 − 36 1 − B Ωc = (1 − B)2 (B − 0.25) = −
(A.2)
(A.3)
Solving iteratively this equation gives B (SRK) = 0.2467, and correspondingly Ω a (SRK) = (1 − B)3 = 0.42748 and Ω b (SRK) = Z c − B = 0.08664.
66
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
For PR EOS we set the restrictions [2–4]: Ωc = −2(Ωb )2 and Ωd = −Ωb . It results 1 B = 0.25 − 8
1 − 3B 1−B
2 ;
Zc =
1+B 4
(A.4)
giving B (PR) = 0.2296 and Zc (PR) = 0.3074. The SW general form results from the conditions u = −2
Ωd ; Ωb
w=
(Ωc + Ωd2 ) Ωb2
(A.5)
The equations for B (SW) and Zc (SW) are B = 0.25 +
u+w+1 1 − (u + 2)2 4
1 − 3B 1−B
2 ;
Zc = B +
1 − 3B u+2
(A.6)
The particular relations for SW EOS (u + w = 1), Yu and Lu (u − w = 3) [21], and Iwai et al. (u = −w) [22] are readily obtained. The C-1 EOS results from the following conditions: Ωa1/3 = (1 − B) = ΩC1 = 0.77; Ωc = − Ωd =
B = 0.23;
Ωb = Zc − B = Zc − 0.23
(Ωr1 − Ωr2 )2 = (1 − B)2 (B − 0.25) 4
(Ωr1 + Ωr2 ) (1 − B) = Zc − 2 2
(A.7) (A.8) (A.9)
Similarly, for the EOS of Sugie et al. [23], B is obtained from the optimized value of Ω ac , and Zc from the Ω b1 value, with the relations (A.7). The TBS EOS results from the following conditions: Ωc = (1 − B)2 (B − 0.25) = −Bc Cc − Ω d = Zc −
(Bc + Cc )2 2 − Dc2
(1 − B) (Bc + Cc ) =− 2 2
giving, for optimized Zc (TBS) and Dc (TBS) parameters, the equation of the B (TBS): " # (1 − B) 2 1 − Dc2 B = 0.25 + (Zc − B)(3Zc − 1) − Zc − (1 − B)2 2
(A.10)
(A.11)
(A.12)
with the corresponding value for Ω a , Ω b , Ω c , Ω d from Eqs. (6a) and (6b). All equations for B can be solved iteratively, starting with a zero value of B in the right hand term.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
67
Appendix B B.1. Departure (residual) functions for GEOS The following equations for the departure (residual) functions from ideal gas state at the same temperature and pressure were used with the GEOS equation (or GEOS3C) [1]: ∂a R 1U = − a − T E (B.1) ∂T ∂a R E + RT(Z − 1) (B.2) 1H = − a − T ∂T V ∂a + R ln Z + E V −b ∂T V 1F R = RT ln − RT ln Z − aE V −b V 1GR = RT ln + RT(Z − 1) − RT ln Z − aE V −b V a ln ϕ = ln + Z − 1 − ln Z − E V −b RT 0,5 CP V 2 ∂P WS = − CV M ∂V T ∂(1U R ) ∂ 2a R 1CV = =T E ∂T ∂T 2 V 1S R = −R ln
1CPR with
=
1CVR
(∂P /∂T )2V −T −R (∂P /∂V )T
√ 1 V − d + −c E= √ ln √ ; for c < 0 2 −c V − d − −c √ c 1 ; for c > 0 E = √ arctg V −d c
(B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9)
(B.10) (B.11)
1 ; for c = 0 (B.12) V −d The above expressions of the residual departure functions based on GEOS equation may be also used for other cubic equations, which can be converted to the GEOS form. The values of the thermodynamic functions are calculated from E=
Y = Y ∗ + 1Y R ∗
where Y is the ideal gas contribution.
(B.13)
68
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
D. Gean˘a, R. T˘an˘asescu, Rev. Chim. (Bucharest) 40 (1989) 682–688. D. Gean˘a, Rev. Chim. (Bucharest) 37 (1986) 303–309. D. Gean˘a, Rev. Chim. (Bucharest) 37 (1986) 951–959. D. Gean˘a, Rev. Chim. (Bucharest) 38 (1987) 975–979. V. Feroiu, D. Gean˘a, Fluid Phase Equilibria 55 (1990) 263–270. D. Gean˘a, V. Feroiu, Fluid Phase Equilibria 77 (1992) 121–132. D. Gean˘a, V. Feroiu, in: Proceedings of the National Conference on Chemistry and Chemical Engineering, Bucharest, Romania, October 1995, Vol. 3, pp. 133–139. IUPAC Thermodynamic Tables of Fluid State, Vol. 6: Nitrogen, 1979, Vol. 7: Propylene, 1980, Vol. 8: Chlorine, 1985, Vol. 9: Oxygen, 1987, IUPAC Thermodynamic Tables Project Center, Imperial College, London. N.B. Vargaftik, Tables on the thermophysical properties of liquids and gases, Second Edition, Wiley, New York, 1975. ASHRAE Handbook, Fundamentals, SI Edition, Ashrae, Atlanta, 1997. G. Soave, Chem. Eng. Sci. 27 (1972) 1197–1203. D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 (1986) 323–333. G. Schmidt, H. Wenzel, Chem. Eng. Sci. 35 (1980) 1503–1512. R. Freze, J.-L. Chevalier, A. Peneloux, E. Rauzy, Fluid Phase Equilibria 15 (1983) 33–66. P.H. Salim, M.A. Trebble, Fluid Phase Equilibria 65 (1991) 59–70. P.H. Salim, M.A. Trebble, Fluid Phase Equilibria 65 (1991) 41–57. A.S. Teja, T. Sun, Fluid Phase Equilibria 161 (1999) 63–76. U.K. Deiters, K.M. De Reuck, Fluid Phase Equilibria 161 (1999) 205–219. R. Reid, J.M. Prausnitz, B. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1986. J.-M. Yu, B.C.-Y. Lu, Fluid Phase Equilibria 34 (1987) 1–19. Y. Iwai, M.R. Margerum, B.C.-Y. Lu, Fluid Phase Equilibria 42 (1988) 21–41. H. Sugie, Y. Iwahori, B.C.-Y. Lu, Fluid Phase Equilibria 50 (1989) 1–20.
Thermodynamic properties of pure fluids using the GEOS3C equation of state Dan Gean˘a∗ , Viorel Feroiu Department of Applied Physical Chemistry and Electrochemistry, University “Politehnica” Bucharest, Spl. Independen¸tei 313, 78126 Bucharest, Romania
Abstract Thermodynamic properties were predicted for two groups of pure fluids, along the saturation curve and in the single-phase region: IUPAC pure fluids (propylene, chlorine, oxygen and nitrogen) and halogenated hydrocarbons (R11, R12, R21, R22, R152a). A new form of equation GEOS, named GEOS3C, was used. A wide comparison with literature recommended data, as well as with results of other six cubic equations of state (Soave-Redlich-Kwong, Peng-Robinson, Stryjek-Vera, Schmidt-Wenzel, Freze et al. and Salim and Trebble) was made. The GEOS3C equation has three parameters estimated by matching three points on the saturation curve (vapor pressures and the corresponding liquid volumes). The three fixed temperatures are the triple point, the boiling point and the reduced temperature T r = 0.7. The GEOS3C equation leads to the best results in predicting vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties in the saturation range, but this do not lead necessarily to similar best predictions of all properties in the single-phase region. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Thermodynamic properties; Equation of state; Pure fluids
1. Introduction In the classical thermodynamic framework it is possible to obtain typical relationships for the calculation of enthalpy, entropy, fugacity and other thermodynamic properties. These properties can be related to operating variables of the processes, e.g. the temperature of a fluid in a heat exchanger. It is, therefore, important to establish relationships between the properties and independent variables like temperature or pressure. Such relationships for thermodynamic properties of fluids can be obtained using equations of states. In a previous work, Gean˘a and T˘an˘asescu [1] presented these functions for a general cubic equation of state called GEOS, and calculated the thermodynamic properties of methane. In order to predict accurate ∗ Corresponding author. Fax: 40-1-3129647. E-mail addresses: d [email protected] (D. Gean˘a), v [email protected] (V. Feroiu).
0378-3812/00/$20.00 © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 0 ) 0 0 4 1 7 - 9
52
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
values for a wide range of fluid thermodynamic properties, an equation of state must first be able to properly represent the PVT behavior of pure substances. In this work, thermodynamic properties were predicted for two groups of pure fluids, along the saturation curve and in the single-phase region: IUPAC pure fluids (propylene, chlorine, oxygen and nitrogen) and halogenated hydrocarbons R11(CFCl3 ), R12(CF2 Cl2 ), R21(CHFCl2 ), R22(CHF2 Cl), R152a(CHF2 CH3 ). A new form of equation GEOS [2–6], named GEOS3C [7], was used. A wide comparison with IUPAC data [8], respectively, with Vargaftik data [9] for halogenated chlorofluorocarbons, hydrochlorofluorocarbons, and ASHRAE [10] for hydrofluorocarbons was made. A comparison with results of other six cubic equations of state is also presented: Soave-Redlich-Kwong (SRK) [11], Peng-Robinson (PR) [12], Stryjek-Vera (PRSV) [13], Schmidt-Wenzel (SW) [14], Freze et al. (C-1) [15] and Salim and Trebble (TBS) [16,17]. The three parameters of the GEOS3C equation were estimated by constraining the EOS to reproduce the experimental vapor pressure and liquid volume at three selected temperatures. The three fixed temperatures are the triple point, the boiling point and the reduced temperature T r = 0.7. The GEOS3C equation leads to the best results in predicting vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties at saturation, but this does not lead necessarily to similar best predictions of all properties in the single-phase region. 2. The GEOS3C equation of state The general cubic equation of state (GEOS) has the form P =
a(T ) RT − V − b (V − d)2 + c
(1)
The four parameters a, b, c, d for a pure component are expressed by a = ac β 2 (Tr );
ac = Ωa
R 2 Tc2 ; Pc
b = Ωb
RTc ; Pc
c = Ωc
R 2 Tc2 ; Pc2
d = Ωd
RTc Pc
(2)
A new temperature function is used: β(Tr ) = 1 + C1 y + C2 y 2 + C3 y 3 β(Tr ) = 1 + C1 y
for Tr ≤ 1
for Tr > 1
(3) (4)
y = 1 − Tr0.5
(5)
Setting four critical conditions [2–6], the expressions of the parameters Ω a , Ω b , Ω c , Ω d are Ωa = (1 − B)3 ;
Ωb = Zc − B
Ωc = (1 − B)2 (B − 0.25); B=
Ωd = Zc −
1 + C1 αc + C1
with α c as the Riedel’s criterion [2–6].
(6a) (1 − B) 2
(6b) (7)
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
53
Using experimental values of the critical constants and the acentric factor for the calculation of α c from the equation αc = 5.808 + 4.93ω
(8)
the C1 , C2 and C3 parameters were obtained by matching three points on the saturation curve (vapor pressures together with the corresponding liquid volumes). The three fixed temperatures are the triple point, the boiling point and the reduced temperature T r = 0.7. Thus, the EOS is forced to properly reproduce the region from the triple point to the critical point. The following features of the GEOS3C parameterization are remarkable: • It gives exactly the experimental critical point of any substance, and also the experimental critical compressibility factor Zc . • The prediction of liquid volume is improved, without translation, by the parameter C1 from the temperature function of the attractive term, involved in the B relation (7). • The parameters C1 , C2 and C3 , estimated from only three fixed temperatures, can predict with accuracy the saturation pressure curve from the triple point to the critical point. • The involving of the parameter C1 in the expressions of the a and b, leads to a “coupling” between the repulsive and attractive terms in a cubic equation of state. As pointed out previously [2–6], the relations (6a and b) are general forms for all the cubic equations of state with two, three and four parameters. In Appendix A, it is shown how the Eqs. (6a) and (6b) can be applied to obtain the parameters of several well known cubic forms (vdW, SRK, PR, SW, C-1, TBS). Our software used in calculations is based on the Eqs. (A.1)–(A.12) for the reduction of GEOS to different cubic EOS from the literature. This is the meaning of the statement “general cubic equation of state” used for GEOS. The temperature function (Eqs. (3)–(5)) as well as its first derivative are continuous in the critical point. The second derivative is not continuous, and this fact leads to a discontinuity in the isochoric heat capacity and other properties involving the second derivative, at the critical point. This discontinuity is generally of the order of some J/mole/K and it seems that do not lead to difficulties in the calculations, near the critical point. On the other side, it is known that the classical cubic EOS do not predict the correct divergent behavior of the isochoric heat capacity in the critical region. Recently, Teja and Sun [18] presented a method to transform an analytical EOS into a non-analytical equation which shows the correct divergence in the critical region. Although the GEOS3C equation with the temperature function (Eqs. (3)–(5)) shows a discontinuity in CV at the critical point, to obtain the correct divergence behavior a method as that suggested by Teja and Sun [18] should be applied.
3. Results and discussion 3.1. IUPAC pure fluids Seven equations of state SRK, PR, PRSV, SW, C-1, TBS and GEOS3C have been used to calculate thermodynamic properties for oxygen, chlorine nitrogen and propylene. The investigated PVT range covers single-phase (liquid or gas) and two-phase (liquid–vapor) regions. The calculations have been compared with data of IUPAC collections [8], as recommended by Deiters and De Reuck [19].
54
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Table 1 Values of C1 , C2 and C3 parameters for oxygen, chlorine, nitrogen and propylenea Component
C1
C2
C3
Oxygen Chlorine Nitrogen Propylene
0.2304 0.2241 0.1751 0.3959
0.0689 0.3454 0.1974 0.1251
0.2062 −0.0299 0.2250 0.1505
a
Critical data and acentric factors from IUPAC tables [8].
The values of the GEOS3C parameters C1 , C2 and C3 for oxygen, chlorine, nitrogen and propylene, estimated only by three points on the saturation curve corresponding to the triple point temperature, the boiling temperature and the reduced temperature T r = 0.7, are presented in Table 1. The following thermodynamic properties have been calculated: compressibility factor, Z; enthalpy, H; enthalpy of vaporization, 1v H; entropy, S; heat capacity at constant pressure, CP ; heat capacity at constant volume, CV ; adiabatic index, CP /CV ; speed of sound, WS ; fugacity coefficient, ϕ; Joule–Thomson coefficient, JT. The departure (residual) thermodynamic properties from the ideal gas at the same temperature and pressure are given for GEOS in Appendix B. The same relations were used for all EOSs under appropriate restrictions imposed for GEOS parameters (given in Appendix A). The ideal gas contribution to the thermodynamic properties was calculated using the heat capacity functions recommended in IUPAC tables [8]. The average absolute deviations for a property Y are relative (%): PN exp exp |(Y eos − Yi )/Yi | × 100 AAD% = i=1 i (9) N excepting the enthalpy and entropy where PN exp eos − Hi | i=1 |Hi AAD H (or S) = N
(10)
Table 2 Thermodynamic function deviations at saturation for propylenea EOS
SRK PR PRSVb SW C-1 TBSb GEOS3C GEOS3C (59P) a b
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
2.0 10.6 0.6 4.5 7.4 26.4 2.4 1.5
7.6 6.0 6.4 2.9 5.5 3.3 4.0 3.9
2.9 7.8 1.1 6.9 6.1 14.7 2.6 1.9
1.9 2.5 2.0 1.7 2.3 7.7 1.5 1.3
218.6 427.1 237.5 177.2 350.5 1546.0 279.5 220.2
134.5 129.0 161.2 121.7 129.4 169.7 136.7 138.3
0.8 2.0 0.8 0.7 1.7 6.3 1.2 1.0
2.4 3.2 3.0 2.7 3.0 4.5 2.7 2.6
Temperature range (K): 87.89–365.57; pressure range (MPa): 9.5 × 10−10 –4.66. Number of data points: 59. Temperature range (K): 140–365.57; pressure range (MPa): 10−4 –4.66.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
55
Table 3 Thermodynamic function deviations at saturation for propylenea EOS
SRK PR PRSV SW C-1 TBS GEOS3C GEOS3C (59P) a
AAD (%) CPL
CPV
(CP /CV )L
(CP /CV )V
ϕ
WSL
WSV
JTL
JTV
6.7 5.9 5.5 6.8 5.8 14.8 3.4 3.8
6.5 7.0 6.9 6.7 7.1 8.6 7.8 7.8
9.5 9.2 8.8 9.6 9.2 8.0 9.7 9.5
2.1 2.3 2.2 2.2 2.3 2.8 2.9 2.9
0.6 0.6 0.7 0.5 0.7 1.3 1.1 1.2
17.5 17.1 17.6 17.1 16.6 16.0 6.2 7.5
1.2 0.5 0.6 0.9 0.4 1.3 1.2 1.1
444 386 404 411 378 274 29.0 58.8
16.9 16.4 16.3 16.5 16.2 17.9 16.2 16.0
Temperature range (K): 198.8–363.56; pressure range (MPa): 0.025–4.5. Number of data points: 29.
The results of the calculations for propylene are summarized in Tables 2–4. The average absolute deviations (AAD), between calculated values by EOSs and IUPAC data are given. For each table the number of data points, the pressure and temperature ranges are indicated. The two-phase region properties have been calculated at temperatures from the triple point to the critical point. The notation GEOS3C in the tables indicates that the parameters C1 , C2 and C3 were estimated from vapor pressure and liquid volume at three fixed temperatures (triple point, boiling point and T r = 0.7). For propylene, a second set of parameters was used, obtained by correlating data at 59 temperatures on the saturation curve from the triple point to the critical point (notation GEOS3C (59P)). It may be observed that the results (deviations) obtained with the both sets of parameters are almost the same, and therefore, confirm the ability of the parameters obtained from only three experimental points to reproduce well the full saturation range. For all other substances only the results obtained with parameters estimated from three points on the saturation curve will be presented. Propylene is a severe test in predictions due to the temperature extension of the saturation range and the very low values of the vapor pressures towards the triple point (the order of magnitude is 10−10 MPa). Table 4 Thermodynamic function deviations in single-phase region for propylenea EOS
SRK PR PRSV SW C-1 TBS GEOS3C GEOS3C (59P) a b
AAD
AAD (%)
S (J/mol/K)
H (J/mol)
Z
CP
WS
CP /CV
ϕb
JT
1.7 2.5 2.1 1.9 2.3 6.5 2.8 2.8
816.5 473.2 322.8 606.4 464.2 2336.0 986.8 1057.0
4.0 4.1 4.2 2.2 3.9 5.7 3.8 4.0
3.6 5.4 4.4 4.3 5.2 14.3 3.8 3.5
19.1 17.5 17.6 18.2 17.6 39.8 30.2 32.3
5.7 5.3 5.4 5.5 5.3 4.1 5.9 5.8
2.5 4.2 4.1 1.3 3.7 26.5 4.1 4.0
63.3 91.1 75.2 90.4 90.7 76.3 50.3 45.5
Temperature range (K): 150–575; pressure range (MPa): 0.025–1000. Number of data points: 182. Pressure range (MPa): 0.025–100.
56
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
The following observations can be made on the basis of the results of the Tables 2 and 3: • The vapor pressures and saturated volumes are better predicted by GEOS3C, compared to the results obtained using other equations. Surprisingly, small are the deviations in vapor pressure predicted by SRK. • The low deviation in vapor pressure of the PRSV EOS (with an optimized parameter, k1 ) is explained by the restricted temperature range (140 K — Tc ), recommended by the authors [13]. • For the enthalpy of vaporization the GEOS3C gives the best results. For the CP , WSL and JTL the GEOS3C gives also the best results, but with larger deviations in the last two properties. • The difference in performance between the EOSs is less noticeable for the other properties. • The deviations given by TBS EOS with four optimized parameters, although in a restricted temperature range (140 K — Tc ), are very high (excepting VL ). This is probably explained by the fact that the optimized parameters of the TBS EOS have been obtained at temperatures higher than 140 K (not indicated by the authors [16]). In the single-phase region (Table 4) the PRSV, PR and C-1 EOSs give the best results for enthalpy. Notably larger are the deviations in WS , CP and ϕ for the TBS equation. Somewhat better results for CP and Joule–Thomson coefficient are obtained with GEOS3C. The pressure range was restricted at 100 MPa in the calculation of the fugacity coefficient deviations. At pressures higher than 200 MPa the fugacity coefficients can take values of 1012 –1015 and the deviations between the cubic EOS and the IUPAC data can become correspondingly of the magnitude order of 103 –104 %. Similar calculations have been done for chlorine, oxygen and nitrogen, with seven equations of state. The results obtained for chlorine are presented in Tables 5–7. In the saturation range (from the triple point to the critical point) GEOS3C gives the best results for vapor pressure, volumes, liquid enthalpy, enthalpy of vaporization and entropies. The deviations in fugacity coefficient, adiabatic index and sound velocity in vapor phase are larger for TBS and GEOS3C than for the other equations (Table 6). Somewhat better results predict the GEOS3C for Joule–Thomson coefficient and adiabatic index in the liquid phase. The parameter k1 was not available for chlorine and the results of the PRSV EOS are nearly the same as with PR EOS. In the single-phase region (Table 7) SRK and GEOS3C give the best results for enthalpy, entropy and fugacity coefficient, whereas SW EOS is the best for the compressibility factor. For the heat capacity CP the smallest deviations are obtained from the TBS and GEOS3C equations. Table 5 Thermodynamic function deviations at saturation for chlorinea EOS
SRK PR PRSV (k1 = 0) SW C-1 TBS GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
4.4 7.3 7.1 5.5 6.0 4.0 0.3
7.4 6.9 6.7 4.3 4.7 3.6 3.6
4.1 6.6 6.3 5.5 5.2 5.1 1.3
1.9 3.0 2.9 2.3 2.6 3.2 1.6
376.8 666.6 652.8 368.7 591.9 635.1 342.8
203.8 196.3 196.2 187.0 205.1 223.0 214.4
1.0 1.9 1.9 1.0 1.7 1.8 1.0
0.6 0.9 0.9 0.8 0.9 1.0 0.6
Temperature range (K): 172.17–416.95; pressure range (MPa): 1.38 × 10−3 –7.99. Number of data points: 52.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
57
Table 6 Thermodynamic function deviations at saturation for chlorinea EOS
AAD (%)
SRK PR PRSV (k1 = 0) SW C-1 TBS GEOS3C a
CPL
CPV
(CP /CV )L
(CP /CV )V
ϕ
WSL
WSV
JTL
JTV
19.0 18.0 18.0 21.1 18.2 10.7 13.2
6.1 7.5 7.5 6.2 7.4 10.6 9.8
12.3 12.0 12.0 12.5 12.1 11.6 10.9
3.5 4.2 4.2 3.9 4.2 9.3 8.5
0.9 0.7 0.7 0.7 0.4 3.0 2.5
18.1 19.3 19.3 17.3 14.6 23.6 19.5
1.1 1.5 1.5 1.3 1.2 5.6 4.8
108.0 94.6 94.4 100.0 75.1 50.4 27.3
10.0 9.4 9.4 9.3 9.5 9.6 9.4
Temperature range (K): 196.96–412.77; pressure range (MPa): 0.01–7.5. Number of data points: 11.
Table 7 Thermodynamic function deviations in single-phase region for chlorinea EOS
AAD
SRK PR PRSV (k1 = 0) SW C-1 TBS GEOS3C a
AAD (%)
S (J/mol/K)
H (J/mol)
Z
CP
WS
CP /CV
ϕ
JT
0.6 1.3 1.3 0.8 1.2 0.9 0.5
199.8 409.2 401.9 260.5 376.0 360.3 205.2
2.5 3.7 3.7 1.2 1.9 4.0 2.7
5.2 7.1 7.1 5.9 6.9 3.4 3.4
6.3 5.7 5.7 6.2 5.4 27.8 19.5
7.5 7.2 7.2 7.4 7.2 8.6 9.3
3.6 5.8 5.7 4.6 5.4 5.8 2.2
37.7 40.1 40.0 37.2 37.1 38.4 30.1
Temperature range (K): 180–900; pressure range (MPa): 0.01–25. Number of data points: 114.
The results obtained for oxygen are presented in Tables 8–10. In the saturation range (from the triple point to the critical point) C-1, TBS and GEOS3C equations give the best results for vapor pressure, volumes and enthalpy of vaporization. The deviations in fugacity coefficient are higher for GEOS3C than for the other equations, but the same equation is better in predicting CPL and JTL (Table 9). In the single-phase region (Table 10), PRSV and C-1 EOSs give the best results for enthalpy and entropy. The Table 8 Thermodynamic function deviations at saturation for oxygena EOS
SRK PR PRSV SW C-1 TBS GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
3.1 2.0 0.5 0.9 0.9 0.9 0.7
4.2 9.1 9.0 3.7 2.6 2.1 2.8
3.7 2.8 1.4 1.6 1.4 1.6 1.5
2.8 2.1 1.9 2.1 1.9 1.8 2.1
131.2 110.8 77.9 75.0 86.2 84.8 115.7
39.5 36.2 36.5 32.9 39.5 39.3 41.2
1.2 1.0 0.7 0.8 0.8 0.9 1.2
0.5 0.5 0.4 0.3 0.4 0.5 0.5
Temperature range (K): 54.36–154.59; pressure range (MPa): 1.5 × 10−4 –5.05. Number of data points: 53.
58
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Table 9 Thermodynamic function deviations at saturation for oxygena EOS
SRK PR PRSV SW C-1 TBS GEOS3C a
AAD (%) CPL
CPV
(CP /CV )L
(CP /CV )V
ϕ
WSL
WSV
JTL
JTV
10.7 7.3 7.4 11.6 7.3 6.7 6.1
10.9 11.7 11.7 10.9 11.7 12.1 12.9
12.4 11.8 11.4 12.3 11.7 11.5 13.5
3.2 3.2 2.9 3.2 3.1 2.8 2.7
0.3 0.9 0.9 0.3 0.6 0.9 1.6
18.1 17.3 17.4 18.0 10.3 7.3 12.3
2.8 1.9 1.8 3.1 2.2 2.0 2.0
70.9 66.8 61.1 61.1 39.6 25.1 15.8
12.4 12.2 12.1 12.2 12.4 12.1 12.0
Temperature range (K): 78.65–154.36; pressure range (MPa): 0.025–5. Number of data points: 29.
fugacity coefficient and the compressibility factor are best predicted by C-1 EOS. SRK EOS gives the smallest deviation in CP . Similar results were obtained for nitrogen, both at saturation and in the single-phase region (not presented here). Examples of calculated properties are presented in the Figs. 1–5. Fig. 1 shows the enthalpy-pressure diagram of propylene. Points figure IUPAC data, while the curves represent calculations performed by GEOS3C for saturation region and five isotherms. The entropy–pressure diagram of propylene calculated by GEOS3C is presented in Fig. 2. The agreement with IUPAC data is acceptable at pressures under 100 MPa, but larger errors of calculation should be, however, counted in the high-pressure range (P > 100 MPa). The CP predictions from the GEOS3C and PR equations for chlorine on the saturation curve are presented in Fig. 3. Both perform similarly, excepting the liquid region where GEOS3C gives better predictions. The Joule–Thomson coefficient predictions from the GEOS3C equation for chlorine on the saturation curve and on the single-phase isotherms are presented in Fig. 4. The dependence of pressure of the GEOS3C fugacity coefficient of oxygen at different temperatures is presented in Fig. 5. A satisfactory agreement with IUPAC data may be remarked, confirmed also by the AAD% given in Table 10. Table 10 Thermodynamic function deviations in single-phase region for oxygena EOS
SRK PR PRSV SW C-1 TBS GEOS3C a
AAD
AAD (%)
S (J/mol/K)
H (J/mol)
Z
CP
WS
CP /CV
ϕ
JT
0.5 0.5 0.3 0.4 0.4 0.5 0.6
58.8 61.6 39.1 51.3 38.5 43.8 70.9
0.8 5.5 5.6 1.0 0.8 1.2 1.9
1.6 3.1 2.6 2.5 2.9 2.5 3.7
5.3 5.9 6.2 5.4 5.9 9.5 17.5
3.6 2.8 3.1 3.6 2.8 3.2 4.7
2.5 3.8 3.7 1.5 1.1 1.8 2.9
15.2 20.1 19.0 16.9 17.9 18.6 19.2
Temperature range (K): 60–300; pressure range (MPa): 0.025–80. Number of data points: 151.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
59
Fig. 1. Enthalpy-pressure diagram of propylene. Points: IUPAC data [8].
3.2. Halogenated chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs) and hydrofluorocarbons (HFCs) The second group of pure fluids studied in this work is that of halogenated chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs) and hydrofluorocarbons (HFCs). The knowledge of thermodynamic properties of such compounds is of importance in relation to their use as refrigerants.
Fig. 2. Entropy-pressure diagram of propylene. Points: IUPAC data [8].
60
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Fig. 3. CP prediction for chlorine on the saturation curve. Points: IUPAC data [8].
Five equations of state have been used to calculate thermodynamic properties for halogenated hydrocarbons, because parameters were not available for PRSV and TBS EOSs. The investigated PVT range covers single-phase (liquid or gas) and two-phase (liquid–vapor) regions. The calculations have been compared with recommended data of Vargaftik [9] and ASHRAE [10]. Ideal gas heat capacities were taken from Reid et al. [20].
Fig. 4. Joule–Thomson coefficient prediction for chlorine with GEOS3C. Points: IUPAC data [8].
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
61
Fig. 5. Fugacity coefficient prediction for oxygen. Points: IUPAC data [8].
The values of the parameters C1 , C2 and C3 of the GEOS3C equation for two CFCs (R11, R12), two HCFCs (R21, R22) and a HFC (R152a) are presented in Table 11. The results of the calculations for R11 and R21 are summarized in Tables 12–16. Average absolute deviations (AAD) calculated with Eqs. (9) and (10) between EOS and literature data [9] are presented. For each table, number of data points taken into calculation, pressure and temperature ranges are indicated. The two-phase region properties have been calculated at temperatures ranging between the lowest available value and the critical point [9]. The notation GEOS3C in the tables indicates that the parameters C1 , C2 and C3 were obtained by using only data at three fixed temperatures on the saturation curve (the lowest available temperature, the boiling point and T r = 0.7). For R11 the best results are obtained at saturation by the GEOS3C and SW equations. Notably are the small deviations in liquid enthalpy and entropy of the SRK EOS. The enthalpies and entropies on the isobars are predicted with similar accuracy from all equations as shown in Table 13. Surprising is the small deviation in volume given by SRK EOS. Table 11 Values of C1 , C2 and C3 parameters for several CFCs, HCFCs and HFCsa Component
C1
C2
C3
R11 (CFCl3 ) R12 (CF2 Cl2 ) R21 (CHFCl2 ) R22 (CHF2 Cl) R152a (CHF2 CH3 )
0.3350 0.3114 0.3019 0.2800 0.2554
0.3268 0.3306 0.4976 0.4589 0.9813
0.1358 0.1854 −0.1360 0.1038 −1.1251
a
Critical data from Vargaftik [9] and ASHRAE [10], acentric factors from Reid et al. [12].
62
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
Table 12 Thermodynamic function deviations at saturation for R11a EOS
SRK PR SW C-1 GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
1.2 1.9 0.6 1.5 0.5
6.3 5.9 1.4 9.5 2.3
1.6 2.1 1.0 1.9 0.9
0.7 1.3 0.8 0.9 0.6
81.0 505.6 296.6 325.9 307.3
190.2 188.3 189.5 190.4 187.1
0.5 2.0 1.4 1.6 1.3
1.0 0.8 0.9 1.0 0.9
Temperature range (K): 213.15–413.15; pressure range (bar): 0.01–17.9. Number of data points: 62.
Table 13 Thermodynamic function deviations in single-phase region for R11a EOS
AAD V (%)
SRK PR SW C-1 GEOS3C a
AAD
0.2 0.5 0.4 0.6 0.8
H (J/mole)
S (J/mole/K)
247.8 232.7 230.7 229.3 225.3
1.4 1.4 1.4 1.4 1.4
Temperature range (K): 243.15–523.15; pressure range (bar): 0.03–25. Number of data points: 436.
For R21 the best results are obtained at saturation by the GEOS3C and SW equations. On the isobars the volume predictions from the GEOS3C are slightly worse than the results of the SW equation, but the predicted values of enthalpies and entropies are better. For the refrigerant R152a at saturation the best results are those predicted by GEOS3C excepting the vapor volumes and the enthalpy of vaporization predicted slightly better by the C-1 EOS. The comparison is made with ASHRAE [10] recommended data, from the triple point to the critical temperature.
Table 14 Thermodynamic function deviations at saturation for R21a EOS
SRK PR SW C-1 GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
0.8 1.6 0.9 1.3 0.8
12.8 4.7 4.2 7.7 3.7
2.3 2.7 2.1 2.8 2.5
3.5 3.6 3.2 3.3 2.8
1489.2 1235.0 1421.4 1382.5 1181.4
1242.3 1248.9 1258.0 1254.7 1220.0
4.3 3.3 3.9 3.7 3.2
3.5 3.5 3.5 3.4 3.2
Temperature range (K): 213.15–451.4; pressure range (bar): 0.025–51.8. Number of data points: 119.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
63
Table 15 Thermodynamic function deviations in single-phase region for R21a EOS
AAD V (%)
SRK PR SW C-1 GEOS3C a
AAD
7.5 3.1 1.6 6.2 3.7
H (J/mole)
S (J/mole/K)
1001.0 834.5 928.0 955.9 783.4
2.9 2.2 2.5 2.4 2.1
Temperature range (K): 273.15–473.15; pressure range (bar): 1–200. Number of data points: 300.
Table 16 Thermodynamic function deviations at saturation for R152aa EOS
SRK PR SW C-1 GEOS3C a
AAD (%)
AAD
PS
VL
VV
1v H
HL (J/mol)
HV (J/mol)
SL (J/mol/K)
SV (J/mol/K)
3.5 3.1 2.8 1.8 0.4
24.8 10.4 13.3 4.4 4.6
2.6 2.2 2.6 1.5 2.1
1.7 1.3 1.4 1.6 2.2
821.7 674.7 765.5 810.2 753.3
647.6 650.7 649.2 654.9 634.0
3.5 2.9 3.2 3.2 2.7
3.0 2.9 2.9 2.8 2.4
Temperature range (K): 154.41–386.26; pressure range (bar): 8 × 10−4 –45.1. Number of data points: 67.
Fig. 6. Temperature–enthalpy dependence (saturation curve) for R21. Points: Vargaftik data [9].
64
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
An example of calculated property is presented in Fig. 6. Enthalpy predictions of GEOS3C and SRK equations for R21 are compared with data of Vargaftik [9] in the saturation range. All other thermodynamic properties are also predicted, similarly to IUPAC fluids, but are not reproduced in this work, having not available data for comparison. On the basis of the studied CFC, HCFC and HFC substances, it is possible to conclude that GEOS3C and SW equations predict accurate thermodynamic properties for this group of chemical compounds.
4. Conclusions The thermodynamic functions were predicted by GEOS3C and other six equations of state on a wide PVT range, including the entire saturation region. The GEOS3C equation, has three parameters estimated by constraining the EOS to reproduce the experimental vapor pressure and liquid volume at three selected temperatures. For the substances considered in this study, the GEOS3C equation gives the best results in vapor pressure and volumes at saturation. It also gives reasonable deviations for the other thermodynamic properties in the saturation range, but this do not lead necessarily to similar best predictions of all properties in the single-phase region. At very high pressures, property predictions are unsatisfactory. Although the original SRK and PR equations give generally poor predictions of vapor pressure and saturated liquid volume, they lead to surprisingly good results for some other properties (enthalpy and entropy). On the basis of the studied CFC, HCFC and HFC substances it appears that GEOS3C and SW equations predict better the thermodynamic properties for this group of chemical compounds. An inconsistency given by all equations is that an infinite value for isochoric heat capacity is not predicted at the critical point. Although the second derivative of the temperature function of GEOS3C presents a discontinuity at critical temperature, to obtain the correct divergence behavior of the isochoric heat capacity a crossover method should be applied. List of symbols a, b, c, d AAD B C1 , C2 and C3 CV , CP E F H JT M P, PS R S T U
parameters in GEOS absolute average deviation dimensionless parameter in GEOS, defined by Eq. (7) parameters in GEOS3C temperature function isochoric and isobaric heat capacities expression based on GEOS, defined by Eqs. (A.10)–(A.12) Helmholtz function enthalpy Joule–Thomson coefficient molar mass pressure, saturation pressure universal gas constant entropy temperature internal energy
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
V, VL , VV Ws Y Z
65
molar volume, liquid volume, vapor volume speed of sound thermodynamic function (general notation) compressibility factor
Greek symbols αc β ϕ ω Ω a, Ω b, Ω c, Ω d
Riedel’s criterium (parameter in GEOS) reduced temperature function in GEOS fugacity coefficient acentric factor parameters of GEOS
Subscripts c critical property r reduced property
Acknowledgements The authors are grateful to National Council for Scientific Research of Romania, for financial support. Appendix A A.1. Reduction of GEOS to other cubic EOSs To obtain the parameters of the vdW EOS from the Eqs. (6a) and (6b), we set [2–4] the following restrictions: Ωc = (1 − B)2 (B − 0.25) = 0;
Ωd = Zc − 0.5(1 − B) = 0
(A.1)
It results: B (vdW) = 0.25, Zc (vdW) = 0.375, Ω a (vdW) = (1 − B)3 = 27/64, Ω b (vdW) = Z c − B = 1/8. Similarly, to obtain the parameters of the SRK EOS from the Eqs. (6a) and (6b) we set [2–4] the following restrictions: Ωc = −(Ωb /2)2 and Ωd = −Ωb /2. It follows (Zc − B)2 (Zc − B) ; Ωd = Zc − 0.5(1 − B) = − 4 2 It results: Zc (SRK) = 1/3, and the relation for B (SRK) 1 1 − 3B 2 B = 0.25 − 36 1 − B Ωc = (1 − B)2 (B − 0.25) = −
(A.2)
(A.3)
Solving iteratively this equation gives B (SRK) = 0.2467, and correspondingly Ω a (SRK) = (1 − B)3 = 0.42748 and Ω b (SRK) = Z c − B = 0.08664.
66
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
For PR EOS we set the restrictions [2–4]: Ωc = −2(Ωb )2 and Ωd = −Ωb . It results 1 B = 0.25 − 8
1 − 3B 1−B
2 ;
Zc =
1+B 4
(A.4)
giving B (PR) = 0.2296 and Zc (PR) = 0.3074. The SW general form results from the conditions u = −2
Ωd ; Ωb
w=
(Ωc + Ωd2 ) Ωb2
(A.5)
The equations for B (SW) and Zc (SW) are B = 0.25 +
u+w+1 1 − (u + 2)2 4
1 − 3B 1−B
2 ;
Zc = B +
1 − 3B u+2
(A.6)
The particular relations for SW EOS (u + w = 1), Yu and Lu (u − w = 3) [21], and Iwai et al. (u = −w) [22] are readily obtained. The C-1 EOS results from the following conditions: Ωa1/3 = (1 − B) = ΩC1 = 0.77; Ωc = − Ωd =
B = 0.23;
Ωb = Zc − B = Zc − 0.23
(Ωr1 − Ωr2 )2 = (1 − B)2 (B − 0.25) 4
(Ωr1 + Ωr2 ) (1 − B) = Zc − 2 2
(A.7) (A.8) (A.9)
Similarly, for the EOS of Sugie et al. [23], B is obtained from the optimized value of Ω ac , and Zc from the Ω b1 value, with the relations (A.7). The TBS EOS results from the following conditions: Ωc = (1 − B)2 (B − 0.25) = −Bc Cc − Ω d = Zc −
(Bc + Cc )2 2 − Dc2
(1 − B) (Bc + Cc ) =− 2 2
giving, for optimized Zc (TBS) and Dc (TBS) parameters, the equation of the B (TBS): " # (1 − B) 2 1 − Dc2 B = 0.25 + (Zc − B)(3Zc − 1) − Zc − (1 − B)2 2
(A.10)
(A.11)
(A.12)
with the corresponding value for Ω a , Ω b , Ω c , Ω d from Eqs. (6a) and (6b). All equations for B can be solved iteratively, starting with a zero value of B in the right hand term.
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
67
Appendix B B.1. Departure (residual) functions for GEOS The following equations for the departure (residual) functions from ideal gas state at the same temperature and pressure were used with the GEOS equation (or GEOS3C) [1]: ∂a R 1U = − a − T E (B.1) ∂T ∂a R E + RT(Z − 1) (B.2) 1H = − a − T ∂T V ∂a + R ln Z + E V −b ∂T V 1F R = RT ln − RT ln Z − aE V −b V 1GR = RT ln + RT(Z − 1) − RT ln Z − aE V −b V a ln ϕ = ln + Z − 1 − ln Z − E V −b RT 0,5 CP V 2 ∂P WS = − CV M ∂V T ∂(1U R ) ∂ 2a R 1CV = =T E ∂T ∂T 2 V 1S R = −R ln
1CPR with
=
1CVR
(∂P /∂T )2V −T −R (∂P /∂V )T
√ 1 V − d + −c E= √ ln √ ; for c < 0 2 −c V − d − −c √ c 1 ; for c > 0 E = √ arctg V −d c
(B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9)
(B.10) (B.11)
1 ; for c = 0 (B.12) V −d The above expressions of the residual departure functions based on GEOS equation may be also used for other cubic equations, which can be converted to the GEOS form. The values of the thermodynamic functions are calculated from E=
Y = Y ∗ + 1Y R ∗
where Y is the ideal gas contribution.
(B.13)
68
D. Gean˘a, V. Feroiu / Fluid Phase Equilibria 174 (2000) 51–68
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
D. Gean˘a, R. T˘an˘asescu, Rev. Chim. (Bucharest) 40 (1989) 682–688. D. Gean˘a, Rev. Chim. (Bucharest) 37 (1986) 303–309. D. Gean˘a, Rev. Chim. (Bucharest) 37 (1986) 951–959. D. Gean˘a, Rev. Chim. (Bucharest) 38 (1987) 975–979. V. Feroiu, D. Gean˘a, Fluid Phase Equilibria 55 (1990) 263–270. D. Gean˘a, V. Feroiu, Fluid Phase Equilibria 77 (1992) 121–132. D. Gean˘a, V. Feroiu, in: Proceedings of the National Conference on Chemistry and Chemical Engineering, Bucharest, Romania, October 1995, Vol. 3, pp. 133–139. IUPAC Thermodynamic Tables of Fluid State, Vol. 6: Nitrogen, 1979, Vol. 7: Propylene, 1980, Vol. 8: Chlorine, 1985, Vol. 9: Oxygen, 1987, IUPAC Thermodynamic Tables Project Center, Imperial College, London. N.B. Vargaftik, Tables on the thermophysical properties of liquids and gases, Second Edition, Wiley, New York, 1975. ASHRAE Handbook, Fundamentals, SI Edition, Ashrae, Atlanta, 1997. G. Soave, Chem. Eng. Sci. 27 (1972) 1197–1203. D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. R. Stryjek, J.H. Vera, Can. J. Chem. Eng. 64 (1986) 323–333. G. Schmidt, H. Wenzel, Chem. Eng. Sci. 35 (1980) 1503–1512. R. Freze, J.-L. Chevalier, A. Peneloux, E. Rauzy, Fluid Phase Equilibria 15 (1983) 33–66. P.H. Salim, M.A. Trebble, Fluid Phase Equilibria 65 (1991) 59–70. P.H. Salim, M.A. Trebble, Fluid Phase Equilibria 65 (1991) 41–57. A.S. Teja, T. Sun, Fluid Phase Equilibria 161 (1999) 63–76. U.K. Deiters, K.M. De Reuck, Fluid Phase Equilibria 161 (1999) 205–219. R. Reid, J.M. Prausnitz, B. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1986. J.-M. Yu, B.C.-Y. Lu, Fluid Phase Equilibria 34 (1987) 1–19. Y. Iwai, M.R. Margerum, B.C.-Y. Lu, Fluid Phase Equilibria 42 (1988) 21–41. H. Sugie, Y. Iwahori, B.C.-Y. Lu, Fluid Phase Equilibria 50 (1989) 1–20.