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Application of the crossover lattice equation of state for fluid mixtures
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J. Chem. Thermodynamics 40 (2008) 741–749 www.elsevier.com/locate/jct
Application of the crossover lattice equation of state for fluid mixtures Yongjin Lee a, Moon Sam Shin a, Byoungjo Ha b, Hwayong Kim a,* a
School of Chemical and Biological Engineering and Institute of Chemical Processes, Seoul National University, Seoul 151-744, Republic of Korea b Eulji University, 212, Yangji-dong, Sujeong-Gu, Seongnam 461-713, Republic of Korea Received 28 December 2007; received in revised form 25 January 2008; accepted 25 January 2008 Available online 2 February 2008
Abstract In previous work, we developed the crossover lattice equation of state (xLF EOS) for pure fluids and the xLF EOS yielded the saturated vapour pressure and the density values with a much better accuracy than the classical LF EOS over a wide range. In this work, we extended xLF EOS to fluid mixtures. Classical composition-dependent mixing rules with only adjustable two binary interaction parameters same as the LF EOS are used. A comparison is made upon experimental data for fluids mixtures in the one- and two-phase regions. The xLF EOS shows more improved representations than the LF EOS, especially in the critical region. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Critical point; Crossover theory; Equation of state; Lattice fluid equation of state; Sanchez–Lacombe equation of state; Supercritical fluid
1. Introduction Accurate predictions of the thermodynamic properties and phase behaviour of fluid mixtures are very important for chemical process and product design in various industries. Over decades, significant efforts have been made for the development of better thermodynamic models. The most of these classical models gave very accurate predictions of the thermodynamics properties of fluids over a wide range, but these classical models did not calculate them well near the critical region due to mean field approximation. In the critical region, phase behaviour of fluids can be expressed exactly by the universal critical exponents and many models based on those exponents had been developed. But they did not reproduce the ideal gas equation of state in the limit of zero density. To connect both of the conceptually different theories, the crossover theory has been studied. A simple theory proposed and improved by Kiselev [1] gave successful representations of the thermodynamic properties of fluids over a wide range including the ideal gas limit and the critical region. In previous study
*
Corresponding author. Tel.: +82 2 888 7406; fax: +82 2 888 6695. E-mail address: [email protected] (H. Kim).
0021-9614/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2008.01.016
[2], the lattice fluid equation of state (LF EOS) was combined with the crossover theory developed by Kiselev [1] to obtain the crossover lattice fluid equation of state which incorporates the critical scaling laws valid asymptotically close to the critical point and reduces to the original classical equation of state far from the critical point. For various pure systems, the crossover lattice equation of state (xLF EOS) yielded the saturated data and the PVT data with a much better accuracy than the LF EOS. In this work, we continue a study begun in our previous paper about the development of the xLF EOS and extend the xLF EOS to fluid mixtures. In Section 2, we describe the crossover equation of state for pure components and mixture systems. Comparisons with experimental data are presented in Section 3; and conclusions are drawn in Section 4. 2. Theory In previous work, the Sanchez–Lacombe EOS [3,4] was chosen as a reference LF model and the crossover lattice equation of state was obtained. The LF EOS was derived from the configurational partition function according to statistical mechanics and was capable of describing thermodynamics properties of fluids in a wide range, except
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
near the critical region. The LF EOS was represented as follows: ~ Pe 1 1 q ~Þ 1 Z¼r ¼ r lnð1 q ; ð1Þ ~ r q ~ Te q Te where the reduced temperature, pressure, and density are defined by T RT P Pv q ~ ¼ : Te ¼ ¼ ; Pe ¼ ¼ ; q ð2Þ T e P q e The LF EOS has three molecular parameters; e*, v*, and r, or equivalently the scale factors T*, P*, and q*. In order to derive the crossover (xLF) EOS using the crossover theory, we followed the method developed by Kiselev [1]. Firstly, we need to recast the classical expression for Helmholtz energy into dimensionless form as follows: AðT ; vÞ ¼ DAðDT ; DvÞ þ Abg ðDT ; DvÞ;
ð3Þ
where b2 is the universal linear-model parameter and p2 the universal sine-model parameters, p2 = b2 = 1.359. The Gi is the Ginzburg number for the fluid of interest [7] and m0 = 1 is set in this study and v1, d1, and Gi are the system-dependent parameters. Finally, the crossover expression for the Helmholtz energy can be written in the form Þ DvP 0 ðT Þ þ Ar0 ðT Þ þ A0 ðT Þ ln voc ; AðT ; vÞ ¼ DAðs; u ð10Þ where Ar0 ðT Þ and P 0 ðT Þ are given by 1 1 Ar0 ðT Þ ¼ r þ ð~voc 1Þ ln 1 þ1 ; ~voc Te ~voc 1 1 1 P 0 ðT Þ ¼ r ~voc ln 1 1 : ~voc r Te ~voc
ð11Þ ð12Þ
The critical part DA is written as Þ ¼ Ar ðs; u Þ Ar ðs; 0Þ lnð P 0 ðs; 0Þ; DAðs; u u þ 1Þ þ u
where the critical part of the Helmholtz free energy
ð13Þ
DAðDT ; DvÞ ¼ Ar ðDT ;DvÞ Ar ðDT ;0Þ lnðDv þ 1Þ þ DvP 0 ðDT Þ; ð4Þ
and the background contribution are given by: Abg ðDT ; DvÞ ¼ DvP 0 ðT Þ þ Ar0 ðT Þ þ A0 ðT Þ ln voc :
ð5Þ
In equations (3)–(5), DT = T/Toc 1, Dv = v/voc 1 are dimensionless distances from the classical temperature Toc and molar volume voc, respectively. P 0 ðT Þ ¼ P ðT ; voc Þ=RT is the dimensionless pressure, Ar0 ðT Þ ¼ Ar ðT ; voc Þ is the dimensionless residual part of the Helmholtz energy along the critical isochore v = voc. Then the classical dimensional temperature DT and Dv in the singular or critical term is replaced with renormalized values. The s ¼ sY
2Da
¼ uY u
1
þ ð1 þ sÞDsc Y
ðc2bÞ 4D1
2ð2aÞ 3D1
þ ð1 þ uÞDvc Y
ð6Þ
;
2ð2aÞ 2D1
;
r
r
Þ, A ðs; 0Þ, and P 0 ðs; 0Þ are given by where A ðs; u " 1 Þ ¼ r Ar ðs; u þ e Þ T ð1 þ sÞ~voc ð1 þ u 1 Þ 1Þ ln 1 ð~voc ð1 þ u þ1 ; Þ ~voc ð1 þ u " Ar ðs; 0Þ ¼ r
1 1 þ ð~voc 1Þ ln 1 ~voc Te ð1 þ sÞ~voc
ð14Þ #
þ1 ;
ð15Þ # 1 1 1 P 0 ðs; 0Þ ¼ r ~voc ln 1 : 1 ~voc r Te ð1 þ sÞ~voc "
ð7Þ
where a = 0.11, b = 0.325, c = 2 2b a = 1.24 and D1 = 0.51 are universal non-classical critical exponent [5]. The s = (T/Tc) 1 is a dimensionless deviation of the temperature from the real critical temperature Tc, u = (v/vc) 1 is a dimensionless deviation of the molar volume from the real critical molar volume vc, and Dsc = (Tc/ Toc) 1, Duc = (vc/voc) 1. The crossover function Y can be written in the parametric form 2D1 q Y ðqÞ ¼ ; ð8Þ 1þq where q is a renormalized distance to the critical point and can be found from the solution of the crossover sine-model (SM) [6] and s p2 s q2 1 2 1 2 ¼ Gi q Gi 4b 2 ð12bÞ 2 Dg½1 þ v1 expð10DgÞ þ d 1 s Y D1 ; ð9Þ b b m0 Gi
ð16Þ TABLE 1 System-dependent parameters for the xLF EOS Components
Classical parameters *
Carbon dioxide Methane Ethane Propane n-Butane i-Butane n-Pentane n-Hexane R123 R124 R125 R143a R134a R32 DME
*
*
Crossover parameters
T /K
P/ MPa
q/ (g cm3)
Gi
v1
d1
266.25
617.82
1.345
0.0884
0.0034
3.232
205.35 300.50 331.30 380.76 364.18 415.67 434.76 404.30 335.75 286.05 291.90 313.17 298.93 350.62
234.15 308.72 313.97 304.53 284.18 285.77 293.27 337.85 349.13 368.27 389.03 403.06 583.26 433.98
0.449 0.584 0.608 0.654 0.641 0.664 0.686 1.661 1.647 1.730 1.254 1.450 1.167 0.760
0.0309 0.0884 0.0490 0.0425 0.0441 0.0447 0.0296 0.0940 0.0890 0.0273 0.1585 0.0487 0.2093 0.0801
0.0032 0.0049 0.0035 0.0041 0.0039 0.0038 0.0041 0.0049 0.0042 0.0042 0.0026 0.0249 0.0012 0.0029
3.084 2.945 2.943 3.113 2.960 2.639 3.603 2.989 3.039 4.207 4.219 3.040 3.215 3.041
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
The xLF EOS can be obtained by differentiation of equation (10) with respect to volume Þ oA RT voc oDAðs; u P ðT ; vÞ ¼ ¼ þ P 0 ðT Þ : ov T voc ou vc T
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In order to obtain an xLF model for mixtures, mixing rules should be expressed by a ‘‘field” variable rather than
ð17Þ
TABLE 2 Binary parameters of the LF and xLF model regressed from the experimental phase-equilibrium data Systems
LF
xLF lij
kij
kij
lij
One-phase region CO2(1) + ethane(2) x1 = 0.252 x1 = 0.492 x1 = 0.739
0.123 0.080 0.190
0.087 0.133 0.255
0.076 0.085 0.109
0.036 0.068 0.047
R32(1) + R134a(2) x1 = 0.330 x1 = 0.500
0.152 0.113
0.525 0.453
0.009 0.010
0.015 0.016
R125(1) + R134a(2) 0.172 x1 = 0.500
0.305
0.011
0.015
R125(1) + R143a(2) 0.018 x1 = 0.500
0.163
0.003
0.013
CO2(1) + n-butane(2) T = 310.93 K 0.025
0.006
0.204
0.122
CO2(1) + i-butane(2) T = 310.93 K 0.001
0.041
0.128
0.085
CO2(1) + n-pentane(2) T = 323 K 0.041 T = 348 K 0.073
0.140 0.253
0.260 0.140
0.339 0.268
Two-phase region Ethane(1) + n-hexane(2) T = 298.15 K 0.032 0.011 T = 338.71 K 0.067 0.034
0.010 0.101
0.168 0.298
CO2(1) + R123(2) T = 313.15 K T = 323.15 K T = 333.15 K
0.018 0.015 0.016
0.006 0.039 0.033
0.037 0.050 0.048
0.015 0.033 0.022
Propane(1) + n-butane(2) T = 343.17 K 0.004 T = 373.15 K 0.004 T = 393.15 K 0.004
0.003 0.016 0.011
0.015 0.017 0.013
0.142 0.083 0.065
0.035 0.016 0.023 0.022
0.419 0.453 0.379 0.361
0.025 0.026 0.033 0.069
0.282 0.291 0.041 0.051
R143a(1) + n-butane(2) T = 333.15 K 0.051 T = 343.15 K 0.053 T = 353.15 K 0.048 T = 363.15 K 0.051
0.079 0.074 0.078 0.062
0.016 0.031 0.018 0.019
0.239 0.157 0.194 0.179
0.053 0.071 0.101
0.009 0.001 0.006
0.096 0.078 0.078
CO2(1) + ethane(2) T = 230 K T = 270 K T = 293.15 K T = 298.15 K
CO2(1) + R124(2) T = 313.15 K T = 323.15 K T = 333.15 K
0.031 0.029 0.039
FIGURE 1. Plot of pressure against temperature [15] for (CO2 + ethane) with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves) (a) 0.252 CO2 + 0.748 ethane; q = d, 6.52 mol dm3; s, 7.21 mol dm3; ., 8.23 mol dm3; M, 8.95 mol dm3; j, 9.67 mol dm3, (b) (0.492 CO2 + 0.508 ethane); d, 7.05 mol dm3; s, 7.51 mol dm3; ., 7.77 mol dm3; M, 8.0 mol dm3; j, 8.43 mol dm3, (c) (0.739 CO2 + 0.261 ethane); d, 7.31 mol dm3; s, 8.26 mol dm3; ., 9.13 mol dm3; M 9.98 mol dm3; j, 11.85 mol dm3.
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
the ‘‘density” variable [8–10]. So, mixing rules are given in terms of the chemical potential of a mixture l and the equation of state becomes a function of temperature, density and the chemical potential of a mixture. The critical parameters are found from the following conditions, 3 oP oP P ðT c ; vc ; lc Þ ¼ P c ; ¼ 0; ¼ 0: ov T c ;vc ;lc ov3 T c ;vc ;lc ð18Þ However, this approach following the principle of the critical point universality makes the model very complex and the calculation progress time-consuming. As shown by Kiselev and co-workers [11,12], if one does not have an interest in reproducing all scaling laws near the critical point of a binary mixture, the ‘‘density” variable can be used. So for the original LF model parameter e* and v*, the following mixing and combining rules are used [13,14].
FIGURE 2. Plot of pressure against temperature [15] for (R32 + R134a) with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves) (a) (0.33R32 + 0.67R134a); q = d, 1.12 mol dm3; s, 2.33 mol dm3; ., 4.65 mol dm3; M, 7.21 mol dm3; j, 8.45 mol dm3; h, 9.19 mol dm3; , 10.67 mol dm3; e, 11.80 mol dm3; N, 12.75 mol dm3; O, 13.58, (b) (0.50R32 + 0.50R134a); q = d, 1.10 mol dm3; s, 2.12 mol dm3; ., 3.82 mol dm3; M, 6.25 mol dm3; j, 8.17 mol dm3; h, 9.18 mol dm3; , 11.65 mol dm3; e, 13.21 mol dm3; N, 14.40 mol dm3.
e ¼ v ¼
XX XX
1=2
/i /j eij ;
eij ¼ ðeii ejj Þ
/i /j vij ;
1 vij ¼ ðvii þ vjj Þð1 lij Þ: 2
ð1 k ij Þ;
ð19Þ ð20Þ
The simple-linear mixing rules are used for crossover parameters: X/ 1 i ¼ ; ð21Þ GiðxÞ Gii X v1 ðxÞ ¼ /i v1;i ; ð22Þ X d 1 ðxÞ ¼ /i d 1;i : ð23Þ Because these mixing rules based on the composition, the real critical parameters for mixtures are not obtained. Experimental critical lines for mixtures are available for some simple systems. Thus, the real critical point is ob-
FIGURE 3. Plot of pressure against temperature [16] for (R125 + R134a) and R143a with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves) (a) (0.50R125 + 0.50R134a); q = d, 1.66 mol dm3; s, 3.25 mol dm3; ., 5.32 mol dm3; M, 6.61 mol dm3; j, 7.35 mol dm3; h, 7.66 mol dm3; , 8.22 mol dm3; e, =8.80 mol dm3; N, 10.02 mol dm3; O, 10.96 mol dm3; , 11.72 mol dm3; , 12.36 (b) (0.50R125 + 0.50R143a); q = d, 0.883 mol dm3; s, 2.20 mol dm3; ., 4.17 mol dm3; M, 5.95 mol dm3; j, 7.02 mol dm3; h, 7.72 mol dm3; , 8.96 mol dm3; e, 10.07 mol dm3; N, 10.81 mol dm3; O, 11.50 mol dm3; , 12.09 mol dm3; , 12.55 mol dm3.
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
tained indirectly by using the pseudo critical parameters and the mixture critical shifts. The pseudo critical parameters are given by the following simple expressions that are derived from the LF model [3]: e ðxÞ 1 2r1 ðxÞ pffiffiffiffiffiffiffiffiffiffi 2 ; R 1 þ r1 ðxÞ 2e ðxÞ 1 1 1 pffiffiffiffiffiffiffiffiffiffi þ P oc ðxÞ ¼ r r ðxÞlg 1 þ ðxÞ ; 1 1 pffiffiffiffiffiffiffiffiffiffi v ðxÞ 1 þ r1 ðxÞ 2 r1 ðxÞ 2 pffiffiffiffiffiffiffiffiffiffi 1 1 pffiffiffiffiffiffiffiffiffiffi þ r1 ðxÞ ; Z oc ðxÞ ¼ 1 þ r1 ðxÞ r1 ðxÞ lg 1 þ r1 ðxÞ 2 RZ oc ðxÞT oc ðxÞ voc ðxÞ ¼ : P oc ðxÞ ð24Þ
T oc ðxÞ ¼
745
The mixture critical shifts Dsc, Duc, which are used to correct the difference between the pseudo critical parameters and the real critical point, are given by following mixing rules: X Dsc ¼ DsðiÞ c xi ; i
Duc ¼
X
DuðiÞ c xi :
ð25Þ
i
The real critical parameters can be estimated as follows: T c ¼ T oc ð1 þ Dsc Þ; vc ¼ voc ð1 þ Duc Þ:
ð26Þ
Finally, the fugacity of component of i in mixture for xLF model was derived as follows:
These analytical expressions for the pseudo critical parameters makes calculation processes using the xLF model simpler than the crossover SAFT and the cubic EOS that need the numerical process to obtain pseudo critical parameters.
FIGURE 4. Plot of compressibility factor against pressure [17] for (a) (CO2 + xn-butane) at T = 310.93 K; s, 0.0291 n-butane; h, 0.0507 n-butane and (b) (CO2 + xi-butane) system at T = 310.93 K; O, 0.0289 i-butane; s, 0.0507 i-butane with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves).
FIGURE 5. Plot of pressure against density [18] for the (CO2 + n-pentane) (a) at T = 323 K and (b) T = 348 K with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); q = d, 0.88; s, 0.80; ., 0.72; M, 0.62; j, 0.50; h, 0.35; , 0.20; e, 0.10.
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
oDAðs; u; xk Þ DvP 0 ðT ; xk Þ onk oP 0 ðT ; xk Þ oAres ðT ; xk Þ þ A0 ðT ; xk Þ þ n 0 þ nDv onk onk 1 oDv lnðDv þ 1Þ þ n ln Z: Dv þ 1 onk
ln wk ¼ DAðs; u; xk Þ þ n
ð27Þ
3. Comparison with experimental data To demonstrate the accuracy of the xLF equation of state for the prediction of mixture fluid behaviour, we present results for PVT and VLE phase behaviour of the various systems (n-alkane and refrigerant + carbon dioxide), (n-alkane + n-alkane), and (refrigerant + refrigerant) binary mixtures. System-dependent parameters for pure components are presented in table 1 and the binary interaction parameters for the xLF EOS of mixture systems are listed in table 2. We first present results for single-phase PVT behaviour of fluid mixtures. The interaction parameters were obtained from a fit of the xLF model to the experimental data in the one-phase region. A comparison of the calculated values of pressure along isochores with experimental data is presented in figures 1 to 3. The xLF model shows good agreement for all mixtures over a wide range of temperatures, pressures and densities. In figure 4, the xLF model is compared with the LF model for experimental compressibility factors of the two binaries, (CO2 + n-C4H10) and (CO2 + i-C4H10) at T = 310.93 K. Calculated results with the LF model show considerable disagreement and the LF model yields the two-phase results in the one-phase region near the critical
FIGURE 6. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [19,20] of the (ethane + n-hexane) system with predictions of the xLF EOS (solid curve) and the LF EOS (dotted curve); T = s, 298.15 K; M, 338.71 K.
point. However, the xLF model represents the compressibility factor of fluid mixtures well, especially rapid changes near the critical point. Figure 5 shows comparisons between the calculated results using two models and experimental P, q, T data for the (carbon dioxide + n-pentane) mixture. The xLF model reproduces the experimental results better than the LF model in a wide range. The xLF model also was used to predict the two-phase VLE behaviour of fluid mixtures. Comparisons of the xLF model predictions with the experimental results are presented in figures 6 to 11. The xLF model shows a great improvement in prediction of the thermodynamic properties in the critical region.
FIGURE 7. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [21] of the (CO2 + R123) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 313.15 K; h, 323.15 K; e, 333.15 K.
FIGURE 8. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [22,23] of the (propane + n-butane) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = M, 343.17 K; s, 373.15 K; e, 393.15 K.
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
FIGURE 9. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [24,25] of the (CO2 + ethane) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 230 K; h, 270 K; e, 293.15 K; M, 298.15 K.
747
FIGURE 11. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [21] of the (CO2 + R124) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 313.15 K; h, 323.15 K; e, 333.15 K.
FIGURE 10. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [26] of the (R143a + n-butane) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 333.15 K; h, 343.15 K; e, 353.15 K; M, 363.15 K.
Figures 12 and 13 present q–x1 and q–x1 plots of (CO2 + R124) and (R143a + n-butane) binary mixtures. Calculated results of the xLF model show that this model can represent a drastic change of densities near the critical point, but the LF model does not show this behaviour. A q value means a renormalized distance to the critical point and this value determines a degree of singular effects on the behaviour of fluids and the fluid mixture. As represented in a q–x1 plot, the q value of the vapour and liquid phase has a large difference in the state far from the critical point. As the fluid approaches the critical point, a difference between two values gradually get smaller and finally two values become equal at the critical point. This effect makes the xLF model predict the fluid properties well near the critical point.
FIGURE 12. Plot of (a) density against mole fraction x1; solid curve, xLF EOS (353.15 K); dotted curve, LF EOS (353.15 K); long dashed curve, xLF EOS (363.15 K); dash-dot-dot curve, LF EOS (363.15 K) and (b) q against mole fraction x1; solid curve, liquid q (353.15 K); dotted curve, vapour q (353.15 K); long dashed curve, liquid q (363.15 K); dash-dot-dot curve, vapour q (363.15 K) for (R143a + n-butane).
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749 TABLE 3 The deviation of the LF and xLF models Systems
LF
xLF
AADP
AADY
AADP
AADY
One-phase region CO2(1) + ethane(2) x1 = 0.252 x1 = 0.492 x1 = 0.739
7.872 4.710 3.074
0.241 0.313 0.232
R32(1) + R134a(2) x1 = 0.330 x1 = 0.500
17.149 15.732
2.847 2.391
R125(1) + R134a(2) x1 = 0.500 21.612
2.585
R125(1) + R143a(2) 13.298 x1 = 0.500
2.294
CO2(1) + n-butane(2) T = 310.93 K 9.554
0.587
CO2(1) + i-butane(2) T = 310.93 K 8.707
0.845
CO2(1) + n-pentane(2) T = 323 K 40.391 T = 348 K 42.013
10.771 9.883 Two-phase region
FIGURE 13. Plots of (a) density against mole fraction x1; solid curve, xLF EOS (313.15 K); long dashed curve, xLF EOS (323.15 K); short dashed curve, xLF EOS (333.15 K); dotted curve, LF EOS (313.15 K); dash-dot curve, LF EOS (323.15 K); dash-dot-dot curve, LF EOS (333.15 K) and (b) q against mole fraction x1 for (CO2 + R124); solid curve, liquid q (313.15 K); dotted curve, vapour q (313.15 K); short dashed curve, liquid q (323.15 K); dash-dot-dot curve, vapour q (323.15 K); long dashed curve, liquid q (333.15 K); dash-dot curve, vapour q (333.15 K).
The deviation of results calculated with the xLF EOS and the LF EOS models are listed in table 3. The average absolute deviation (AAD) for the various systems is improved by the crossover treatment. 4. Conclusion The xLF EOS was applied to fluid mixtures using the classical method of phase equilibria calculation. This approach neglects the critical exponents along the critical line but retains them at the critical points in the limits of pure components. This approach makes calculation processes simple and eliminates the need for the critical lines in the calculations. As presented in the results for various mixtures, the xLF model represented the PVT behaviour well to high pressures and showed improvements for the VLE calculation near the critical point using the two binary interaction parameters that were used for the LF model.
Ethane(1) + n-hexane(2) T = 298.15 K 3.652 T = 338.71 K 2.153
1.833 1.546
0.319 0.694
0.518 0.293
CO2(1) + R123(2) T = 313.15 K T = 323.15 K T = 333.15 K
0.844 1.817 1.487
1.271 1.197 1.663
0.762 1.019 1.093
0.311 0.301 0.210
Propane(1) + n-butane(2) T = 343.17 K 1.014 T = 373.15 K 0.825 T = 393.15 K 0.974
0.976 0.935 1.541
0.538 0.357 0.141
0.433 0.480 0.262
CO2(1) + ethane(2) T = 230 K T = 270 K T = 293.15 K T = 298.15 K
0.622 0.462 2.075 2.319
0.456 0.561 2.782 2.972
0.601 0.731 0.890 1.013
0.399 0.501 1.101 0.939
R143a(1) + n-butane(2) T = 333.15 K 0.411 T = 343.15 K 0.465 T = 353.15 K 1.477 T = 363.15 K 1.909
0.625 0.732 0.905 0.979
0.424 0.569 0.555 0.296
0.768 0.748 0.498 0.126
1.039 3.590 0.615 2.308 3.587 1.155 0.714 1.207 0.385
P AADP ð%Þ ¼ ð1=nÞ Pni ð P_ i;exp P_ i;cal Þ= P _ i;exp 100. n ð_y i;exp y_ i;cal Þ 100. AADY ð%Þ ¼ ð1=nÞ
0.413 0.694 0.733
CO2(1) + R124(2) T = 313.15 K T = 323.15 K T = 333.15 K
i
Acknowledgement This work was supported by the BK21 project of the Ministry of Education of Korea and by the National Laboratory (NRL) Program.
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
S.B. Kiselev, Fluid Phase Equilibr. 147 (1998) 7–23. M.S. Shin, Y. Lee, H. Kim, J. Chem. Thermodyn. 40 (2008) 174–179. I.C. Sanchez, R.H. Lacombe, J. Phys. Chem. 80 (1976) 2352–2362. P.C. Albright, J.V. Sengers, J.F. Nicoll, M. Leykoo, Int. J. Thermophys. 7 (1986) 75–85. K. Gauter, R.A. Heideman, Ind. Eng. Chem. Res. 39 (2000) 1115–1117. S.B. Kiselev, J.F. Ely, Fluid Phase Equilibr. 119 (2003) 8645–8662. M.A. Anisimov, S.B. Kiselev, J.V. Sengers, S. Tang, Physica A 188 (1992) 487–525. M. Fisher, Phys. Rev. 176 (1968) 257–271. R.B. Griffiths, J.C. Wheeler, Phys. Rev. A 2 (1970) 1047–1064. W.F. Saam, Phys. Rev. A. 2 (1970) 1461–1466. S.B. Kiselev, J.F. Ely, J. Chem. Phys. 119 (2003) 8645–8662. L. Sun, H. Zhao, C. Kiselev, S.B. McCabe, J. Phys. Chem. B 109 (2005) 9047–9058. H. Orbey, C.P. Bokis, C.C. Chen, Ind. Eng. Chem. Res. 37 (1998) 4481–4491. B.L. West, D. Bush, N.H. Brandley, M.F. Vincent, S.G. Kazarian, C.A. Eckert, Ind. Eng. Chem. Res. 37 (1998) 3305–3311.
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[15] L.A. Weber, Int. J. Thermophys. 13 (1992) 1011–1032. [16] J.W. Magee, W.M. Haynes, Int. J. Thermophys. 21 (2000) 113– 150. [17] T. Tsuji, S. Honda, T. Kiaki, M. Hongo, J. Supercrit. Fluids 13 (1998) 15–21. [18] E. Kiran, H. Pohler, Y. Xiong, J. Chem. Eng. Data 41 (1996) 158– 172. [19] E.J. Zais, I.H. Silberberg, J. Chem. Eng. Data 15 (1970) 253–256. [20] K. Ohgaki, F. Sano, T. Katayama, J. Chem. Eng. Data 21 (1976) 55– 58. [21] K. Jeong, J. Lim, G. Lee, Y. Lee, H. Kim, Fluid Phase Equilibr. 251 (2007) 63–67. [22] P. Beranek, I. Wichterle, Fluid Phase Equilibr. 6 (1981) 279–282. [23] W.B. Kay, J. Chem. Eng. Data 15 (1970) 46–52. [24] T.S. Brown, A.J. Kidney, E.D. Sloan, Fluid Phase Equilibr. 40 (1988) 169–184. [25] K. Ohgaki, T. Katayama, Fluid Phase Equilibr. 1 (1977) 27–32. [26] J. Lim, G. Lee, Y. Lee, H. Kim, J. Chem. Eng. Data 52 (2007) 391– 394.
JCT 08-3
J. Chem. Thermodynamics 40 (2008) 741–749 www.elsevier.com/locate/jct
Application of the crossover lattice equation of state for fluid mixtures Yongjin Lee a, Moon Sam Shin a, Byoungjo Ha b, Hwayong Kim a,* a
School of Chemical and Biological Engineering and Institute of Chemical Processes, Seoul National University, Seoul 151-744, Republic of Korea b Eulji University, 212, Yangji-dong, Sujeong-Gu, Seongnam 461-713, Republic of Korea Received 28 December 2007; received in revised form 25 January 2008; accepted 25 January 2008 Available online 2 February 2008
Abstract In previous work, we developed the crossover lattice equation of state (xLF EOS) for pure fluids and the xLF EOS yielded the saturated vapour pressure and the density values with a much better accuracy than the classical LF EOS over a wide range. In this work, we extended xLF EOS to fluid mixtures. Classical composition-dependent mixing rules with only adjustable two binary interaction parameters same as the LF EOS are used. A comparison is made upon experimental data for fluids mixtures in the one- and two-phase regions. The xLF EOS shows more improved representations than the LF EOS, especially in the critical region. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Critical point; Crossover theory; Equation of state; Lattice fluid equation of state; Sanchez–Lacombe equation of state; Supercritical fluid
1. Introduction Accurate predictions of the thermodynamic properties and phase behaviour of fluid mixtures are very important for chemical process and product design in various industries. Over decades, significant efforts have been made for the development of better thermodynamic models. The most of these classical models gave very accurate predictions of the thermodynamics properties of fluids over a wide range, but these classical models did not calculate them well near the critical region due to mean field approximation. In the critical region, phase behaviour of fluids can be expressed exactly by the universal critical exponents and many models based on those exponents had been developed. But they did not reproduce the ideal gas equation of state in the limit of zero density. To connect both of the conceptually different theories, the crossover theory has been studied. A simple theory proposed and improved by Kiselev [1] gave successful representations of the thermodynamic properties of fluids over a wide range including the ideal gas limit and the critical region. In previous study
*
Corresponding author. Tel.: +82 2 888 7406; fax: +82 2 888 6695. E-mail address: [email protected] (H. Kim).
0021-9614/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2008.01.016
[2], the lattice fluid equation of state (LF EOS) was combined with the crossover theory developed by Kiselev [1] to obtain the crossover lattice fluid equation of state which incorporates the critical scaling laws valid asymptotically close to the critical point and reduces to the original classical equation of state far from the critical point. For various pure systems, the crossover lattice equation of state (xLF EOS) yielded the saturated data and the PVT data with a much better accuracy than the LF EOS. In this work, we continue a study begun in our previous paper about the development of the xLF EOS and extend the xLF EOS to fluid mixtures. In Section 2, we describe the crossover equation of state for pure components and mixture systems. Comparisons with experimental data are presented in Section 3; and conclusions are drawn in Section 4. 2. Theory In previous work, the Sanchez–Lacombe EOS [3,4] was chosen as a reference LF model and the crossover lattice equation of state was obtained. The LF EOS was derived from the configurational partition function according to statistical mechanics and was capable of describing thermodynamics properties of fluids in a wide range, except
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
near the critical region. The LF EOS was represented as follows: ~ Pe 1 1 q ~Þ 1 Z¼r ¼ r lnð1 q ; ð1Þ ~ r q ~ Te q Te where the reduced temperature, pressure, and density are defined by T RT P Pv q ~ ¼ : Te ¼ ¼ ; Pe ¼ ¼ ; q ð2Þ T e P q e The LF EOS has three molecular parameters; e*, v*, and r, or equivalently the scale factors T*, P*, and q*. In order to derive the crossover (xLF) EOS using the crossover theory, we followed the method developed by Kiselev [1]. Firstly, we need to recast the classical expression for Helmholtz energy into dimensionless form as follows: AðT ; vÞ ¼ DAðDT ; DvÞ þ Abg ðDT ; DvÞ;
ð3Þ
where b2 is the universal linear-model parameter and p2 the universal sine-model parameters, p2 = b2 = 1.359. The Gi is the Ginzburg number for the fluid of interest [7] and m0 = 1 is set in this study and v1, d1, and Gi are the system-dependent parameters. Finally, the crossover expression for the Helmholtz energy can be written in the form Þ DvP 0 ðT Þ þ Ar0 ðT Þ þ A0 ðT Þ ln voc ; AðT ; vÞ ¼ DAðs; u ð10Þ where Ar0 ðT Þ and P 0 ðT Þ are given by 1 1 Ar0 ðT Þ ¼ r þ ð~voc 1Þ ln 1 þ1 ; ~voc Te ~voc 1 1 1 P 0 ðT Þ ¼ r ~voc ln 1 1 : ~voc r Te ~voc
ð11Þ ð12Þ
The critical part DA is written as Þ ¼ Ar ðs; u Þ Ar ðs; 0Þ lnð P 0 ðs; 0Þ; DAðs; u u þ 1Þ þ u
where the critical part of the Helmholtz free energy
ð13Þ
DAðDT ; DvÞ ¼ Ar ðDT ;DvÞ Ar ðDT ;0Þ lnðDv þ 1Þ þ DvP 0 ðDT Þ; ð4Þ
and the background contribution are given by: Abg ðDT ; DvÞ ¼ DvP 0 ðT Þ þ Ar0 ðT Þ þ A0 ðT Þ ln voc :
ð5Þ
In equations (3)–(5), DT = T/Toc 1, Dv = v/voc 1 are dimensionless distances from the classical temperature Toc and molar volume voc, respectively. P 0 ðT Þ ¼ P ðT ; voc Þ=RT is the dimensionless pressure, Ar0 ðT Þ ¼ Ar ðT ; voc Þ is the dimensionless residual part of the Helmholtz energy along the critical isochore v = voc. Then the classical dimensional temperature DT and Dv in the singular or critical term is replaced with renormalized values. The s ¼ sY
2Da
¼ uY u
1
þ ð1 þ sÞDsc Y
ðc2bÞ 4D1
2ð2aÞ 3D1
þ ð1 þ uÞDvc Y
ð6Þ
;
2ð2aÞ 2D1
;
r
r
Þ, A ðs; 0Þ, and P 0 ðs; 0Þ are given by where A ðs; u " 1 Þ ¼ r Ar ðs; u þ e Þ T ð1 þ sÞ~voc ð1 þ u 1 Þ 1Þ ln 1 ð~voc ð1 þ u þ1 ; Þ ~voc ð1 þ u " Ar ðs; 0Þ ¼ r
1 1 þ ð~voc 1Þ ln 1 ~voc Te ð1 þ sÞ~voc
ð14Þ #
þ1 ;
ð15Þ # 1 1 1 P 0 ðs; 0Þ ¼ r ~voc ln 1 : 1 ~voc r Te ð1 þ sÞ~voc "
ð7Þ
where a = 0.11, b = 0.325, c = 2 2b a = 1.24 and D1 = 0.51 are universal non-classical critical exponent [5]. The s = (T/Tc) 1 is a dimensionless deviation of the temperature from the real critical temperature Tc, u = (v/vc) 1 is a dimensionless deviation of the molar volume from the real critical molar volume vc, and Dsc = (Tc/ Toc) 1, Duc = (vc/voc) 1. The crossover function Y can be written in the parametric form 2D1 q Y ðqÞ ¼ ; ð8Þ 1þq where q is a renormalized distance to the critical point and can be found from the solution of the crossover sine-model (SM) [6] and s p2 s q2 1 2 1 2 ¼ Gi q Gi 4b 2 ð12bÞ 2 Dg½1 þ v1 expð10DgÞ þ d 1 s Y D1 ; ð9Þ b b m0 Gi
ð16Þ TABLE 1 System-dependent parameters for the xLF EOS Components
Classical parameters *
Carbon dioxide Methane Ethane Propane n-Butane i-Butane n-Pentane n-Hexane R123 R124 R125 R143a R134a R32 DME
*
*
Crossover parameters
T /K
P/ MPa
q/ (g cm3)
Gi
v1
d1
266.25
617.82
1.345
0.0884
0.0034
3.232
205.35 300.50 331.30 380.76 364.18 415.67 434.76 404.30 335.75 286.05 291.90 313.17 298.93 350.62
234.15 308.72 313.97 304.53 284.18 285.77 293.27 337.85 349.13 368.27 389.03 403.06 583.26 433.98
0.449 0.584 0.608 0.654 0.641 0.664 0.686 1.661 1.647 1.730 1.254 1.450 1.167 0.760
0.0309 0.0884 0.0490 0.0425 0.0441 0.0447 0.0296 0.0940 0.0890 0.0273 0.1585 0.0487 0.2093 0.0801
0.0032 0.0049 0.0035 0.0041 0.0039 0.0038 0.0041 0.0049 0.0042 0.0042 0.0026 0.0249 0.0012 0.0029
3.084 2.945 2.943 3.113 2.960 2.639 3.603 2.989 3.039 4.207 4.219 3.040 3.215 3.041
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
The xLF EOS can be obtained by differentiation of equation (10) with respect to volume Þ oA RT voc oDAðs; u P ðT ; vÞ ¼ ¼ þ P 0 ðT Þ : ov T voc ou vc T
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In order to obtain an xLF model for mixtures, mixing rules should be expressed by a ‘‘field” variable rather than
ð17Þ
TABLE 2 Binary parameters of the LF and xLF model regressed from the experimental phase-equilibrium data Systems
LF
xLF lij
kij
kij
lij
One-phase region CO2(1) + ethane(2) x1 = 0.252 x1 = 0.492 x1 = 0.739
0.123 0.080 0.190
0.087 0.133 0.255
0.076 0.085 0.109
0.036 0.068 0.047
R32(1) + R134a(2) x1 = 0.330 x1 = 0.500
0.152 0.113
0.525 0.453
0.009 0.010
0.015 0.016
R125(1) + R134a(2) 0.172 x1 = 0.500
0.305
0.011
0.015
R125(1) + R143a(2) 0.018 x1 = 0.500
0.163
0.003
0.013
CO2(1) + n-butane(2) T = 310.93 K 0.025
0.006
0.204
0.122
CO2(1) + i-butane(2) T = 310.93 K 0.001
0.041
0.128
0.085
CO2(1) + n-pentane(2) T = 323 K 0.041 T = 348 K 0.073
0.140 0.253
0.260 0.140
0.339 0.268
Two-phase region Ethane(1) + n-hexane(2) T = 298.15 K 0.032 0.011 T = 338.71 K 0.067 0.034
0.010 0.101
0.168 0.298
CO2(1) + R123(2) T = 313.15 K T = 323.15 K T = 333.15 K
0.018 0.015 0.016
0.006 0.039 0.033
0.037 0.050 0.048
0.015 0.033 0.022
Propane(1) + n-butane(2) T = 343.17 K 0.004 T = 373.15 K 0.004 T = 393.15 K 0.004
0.003 0.016 0.011
0.015 0.017 0.013
0.142 0.083 0.065
0.035 0.016 0.023 0.022
0.419 0.453 0.379 0.361
0.025 0.026 0.033 0.069
0.282 0.291 0.041 0.051
R143a(1) + n-butane(2) T = 333.15 K 0.051 T = 343.15 K 0.053 T = 353.15 K 0.048 T = 363.15 K 0.051
0.079 0.074 0.078 0.062
0.016 0.031 0.018 0.019
0.239 0.157 0.194 0.179
0.053 0.071 0.101
0.009 0.001 0.006
0.096 0.078 0.078
CO2(1) + ethane(2) T = 230 K T = 270 K T = 293.15 K T = 298.15 K
CO2(1) + R124(2) T = 313.15 K T = 323.15 K T = 333.15 K
0.031 0.029 0.039
FIGURE 1. Plot of pressure against temperature [15] for (CO2 + ethane) with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves) (a) 0.252 CO2 + 0.748 ethane; q = d, 6.52 mol dm3; s, 7.21 mol dm3; ., 8.23 mol dm3; M, 8.95 mol dm3; j, 9.67 mol dm3, (b) (0.492 CO2 + 0.508 ethane); d, 7.05 mol dm3; s, 7.51 mol dm3; ., 7.77 mol dm3; M, 8.0 mol dm3; j, 8.43 mol dm3, (c) (0.739 CO2 + 0.261 ethane); d, 7.31 mol dm3; s, 8.26 mol dm3; ., 9.13 mol dm3; M 9.98 mol dm3; j, 11.85 mol dm3.
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
the ‘‘density” variable [8–10]. So, mixing rules are given in terms of the chemical potential of a mixture l and the equation of state becomes a function of temperature, density and the chemical potential of a mixture. The critical parameters are found from the following conditions, 3 oP oP P ðT c ; vc ; lc Þ ¼ P c ; ¼ 0; ¼ 0: ov T c ;vc ;lc ov3 T c ;vc ;lc ð18Þ However, this approach following the principle of the critical point universality makes the model very complex and the calculation progress time-consuming. As shown by Kiselev and co-workers [11,12], if one does not have an interest in reproducing all scaling laws near the critical point of a binary mixture, the ‘‘density” variable can be used. So for the original LF model parameter e* and v*, the following mixing and combining rules are used [13,14].
FIGURE 2. Plot of pressure against temperature [15] for (R32 + R134a) with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves) (a) (0.33R32 + 0.67R134a); q = d, 1.12 mol dm3; s, 2.33 mol dm3; ., 4.65 mol dm3; M, 7.21 mol dm3; j, 8.45 mol dm3; h, 9.19 mol dm3; , 10.67 mol dm3; e, 11.80 mol dm3; N, 12.75 mol dm3; O, 13.58, (b) (0.50R32 + 0.50R134a); q = d, 1.10 mol dm3; s, 2.12 mol dm3; ., 3.82 mol dm3; M, 6.25 mol dm3; j, 8.17 mol dm3; h, 9.18 mol dm3; , 11.65 mol dm3; e, 13.21 mol dm3; N, 14.40 mol dm3.
e ¼ v ¼
XX XX
1=2
/i /j eij ;
eij ¼ ðeii ejj Þ
/i /j vij ;
1 vij ¼ ðvii þ vjj Þð1 lij Þ: 2
ð1 k ij Þ;
ð19Þ ð20Þ
The simple-linear mixing rules are used for crossover parameters: X/ 1 i ¼ ; ð21Þ GiðxÞ Gii X v1 ðxÞ ¼ /i v1;i ; ð22Þ X d 1 ðxÞ ¼ /i d 1;i : ð23Þ Because these mixing rules based on the composition, the real critical parameters for mixtures are not obtained. Experimental critical lines for mixtures are available for some simple systems. Thus, the real critical point is ob-
FIGURE 3. Plot of pressure against temperature [16] for (R125 + R134a) and R143a with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves) (a) (0.50R125 + 0.50R134a); q = d, 1.66 mol dm3; s, 3.25 mol dm3; ., 5.32 mol dm3; M, 6.61 mol dm3; j, 7.35 mol dm3; h, 7.66 mol dm3; , 8.22 mol dm3; e, =8.80 mol dm3; N, 10.02 mol dm3; O, 10.96 mol dm3; , 11.72 mol dm3; , 12.36 (b) (0.50R125 + 0.50R143a); q = d, 0.883 mol dm3; s, 2.20 mol dm3; ., 4.17 mol dm3; M, 5.95 mol dm3; j, 7.02 mol dm3; h, 7.72 mol dm3; , 8.96 mol dm3; e, 10.07 mol dm3; N, 10.81 mol dm3; O, 11.50 mol dm3; , 12.09 mol dm3; , 12.55 mol dm3.
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
tained indirectly by using the pseudo critical parameters and the mixture critical shifts. The pseudo critical parameters are given by the following simple expressions that are derived from the LF model [3]: e ðxÞ 1 2r1 ðxÞ pffiffiffiffiffiffiffiffiffiffi 2 ; R 1 þ r1 ðxÞ 2e ðxÞ 1 1 1 pffiffiffiffiffiffiffiffiffiffi þ P oc ðxÞ ¼ r r ðxÞlg 1 þ ðxÞ ; 1 1 pffiffiffiffiffiffiffiffiffiffi v ðxÞ 1 þ r1 ðxÞ 2 r1 ðxÞ 2 pffiffiffiffiffiffiffiffiffiffi 1 1 pffiffiffiffiffiffiffiffiffiffi þ r1 ðxÞ ; Z oc ðxÞ ¼ 1 þ r1 ðxÞ r1 ðxÞ lg 1 þ r1 ðxÞ 2 RZ oc ðxÞT oc ðxÞ voc ðxÞ ¼ : P oc ðxÞ ð24Þ
T oc ðxÞ ¼
745
The mixture critical shifts Dsc, Duc, which are used to correct the difference between the pseudo critical parameters and the real critical point, are given by following mixing rules: X Dsc ¼ DsðiÞ c xi ; i
Duc ¼
X
DuðiÞ c xi :
ð25Þ
i
The real critical parameters can be estimated as follows: T c ¼ T oc ð1 þ Dsc Þ; vc ¼ voc ð1 þ Duc Þ:
ð26Þ
Finally, the fugacity of component of i in mixture for xLF model was derived as follows:
These analytical expressions for the pseudo critical parameters makes calculation processes using the xLF model simpler than the crossover SAFT and the cubic EOS that need the numerical process to obtain pseudo critical parameters.
FIGURE 4. Plot of compressibility factor against pressure [17] for (a) (CO2 + xn-butane) at T = 310.93 K; s, 0.0291 n-butane; h, 0.0507 n-butane and (b) (CO2 + xi-butane) system at T = 310.93 K; O, 0.0289 i-butane; s, 0.0507 i-butane with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves).
FIGURE 5. Plot of pressure against density [18] for the (CO2 + n-pentane) (a) at T = 323 K and (b) T = 348 K with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); q = d, 0.88; s, 0.80; ., 0.72; M, 0.62; j, 0.50; h, 0.35; , 0.20; e, 0.10.
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Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
oDAðs; u; xk Þ DvP 0 ðT ; xk Þ onk oP 0 ðT ; xk Þ oAres ðT ; xk Þ þ A0 ðT ; xk Þ þ n 0 þ nDv onk onk 1 oDv lnðDv þ 1Þ þ n ln Z: Dv þ 1 onk
ln wk ¼ DAðs; u; xk Þ þ n
ð27Þ
3. Comparison with experimental data To demonstrate the accuracy of the xLF equation of state for the prediction of mixture fluid behaviour, we present results for PVT and VLE phase behaviour of the various systems (n-alkane and refrigerant + carbon dioxide), (n-alkane + n-alkane), and (refrigerant + refrigerant) binary mixtures. System-dependent parameters for pure components are presented in table 1 and the binary interaction parameters for the xLF EOS of mixture systems are listed in table 2. We first present results for single-phase PVT behaviour of fluid mixtures. The interaction parameters were obtained from a fit of the xLF model to the experimental data in the one-phase region. A comparison of the calculated values of pressure along isochores with experimental data is presented in figures 1 to 3. The xLF model shows good agreement for all mixtures over a wide range of temperatures, pressures and densities. In figure 4, the xLF model is compared with the LF model for experimental compressibility factors of the two binaries, (CO2 + n-C4H10) and (CO2 + i-C4H10) at T = 310.93 K. Calculated results with the LF model show considerable disagreement and the LF model yields the two-phase results in the one-phase region near the critical
FIGURE 6. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [19,20] of the (ethane + n-hexane) system with predictions of the xLF EOS (solid curve) and the LF EOS (dotted curve); T = s, 298.15 K; M, 338.71 K.
point. However, the xLF model represents the compressibility factor of fluid mixtures well, especially rapid changes near the critical point. Figure 5 shows comparisons between the calculated results using two models and experimental P, q, T data for the (carbon dioxide + n-pentane) mixture. The xLF model reproduces the experimental results better than the LF model in a wide range. The xLF model also was used to predict the two-phase VLE behaviour of fluid mixtures. Comparisons of the xLF model predictions with the experimental results are presented in figures 6 to 11. The xLF model shows a great improvement in prediction of the thermodynamic properties in the critical region.
FIGURE 7. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [21] of the (CO2 + R123) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 313.15 K; h, 323.15 K; e, 333.15 K.
FIGURE 8. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [22,23] of the (propane + n-butane) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = M, 343.17 K; s, 373.15 K; e, 393.15 K.
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
FIGURE 9. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [24,25] of the (CO2 + ethane) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 230 K; h, 270 K; e, 293.15 K; M, 298.15 K.
747
FIGURE 11. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [21] of the (CO2 + R124) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 313.15 K; h, 323.15 K; e, 333.15 K.
FIGURE 10. Plot of pressure against mole fraction to illustrate the (vapour + liquid) equilibrium [26] of the (R143a + n-butane) system with predictions of the xLF EOS (solid curves) and the LF EOS (dotted curves); T = s, 333.15 K; h, 343.15 K; e, 353.15 K; M, 363.15 K.
Figures 12 and 13 present q–x1 and q–x1 plots of (CO2 + R124) and (R143a + n-butane) binary mixtures. Calculated results of the xLF model show that this model can represent a drastic change of densities near the critical point, but the LF model does not show this behaviour. A q value means a renormalized distance to the critical point and this value determines a degree of singular effects on the behaviour of fluids and the fluid mixture. As represented in a q–x1 plot, the q value of the vapour and liquid phase has a large difference in the state far from the critical point. As the fluid approaches the critical point, a difference between two values gradually get smaller and finally two values become equal at the critical point. This effect makes the xLF model predict the fluid properties well near the critical point.
FIGURE 12. Plot of (a) density against mole fraction x1; solid curve, xLF EOS (353.15 K); dotted curve, LF EOS (353.15 K); long dashed curve, xLF EOS (363.15 K); dash-dot-dot curve, LF EOS (363.15 K) and (b) q against mole fraction x1; solid curve, liquid q (353.15 K); dotted curve, vapour q (353.15 K); long dashed curve, liquid q (363.15 K); dash-dot-dot curve, vapour q (363.15 K) for (R143a + n-butane).
748
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749 TABLE 3 The deviation of the LF and xLF models Systems
LF
xLF
AADP
AADY
AADP
AADY
One-phase region CO2(1) + ethane(2) x1 = 0.252 x1 = 0.492 x1 = 0.739
7.872 4.710 3.074
0.241 0.313 0.232
R32(1) + R134a(2) x1 = 0.330 x1 = 0.500
17.149 15.732
2.847 2.391
R125(1) + R134a(2) x1 = 0.500 21.612
2.585
R125(1) + R143a(2) 13.298 x1 = 0.500
2.294
CO2(1) + n-butane(2) T = 310.93 K 9.554
0.587
CO2(1) + i-butane(2) T = 310.93 K 8.707
0.845
CO2(1) + n-pentane(2) T = 323 K 40.391 T = 348 K 42.013
10.771 9.883 Two-phase region
FIGURE 13. Plots of (a) density against mole fraction x1; solid curve, xLF EOS (313.15 K); long dashed curve, xLF EOS (323.15 K); short dashed curve, xLF EOS (333.15 K); dotted curve, LF EOS (313.15 K); dash-dot curve, LF EOS (323.15 K); dash-dot-dot curve, LF EOS (333.15 K) and (b) q against mole fraction x1 for (CO2 + R124); solid curve, liquid q (313.15 K); dotted curve, vapour q (313.15 K); short dashed curve, liquid q (323.15 K); dash-dot-dot curve, vapour q (323.15 K); long dashed curve, liquid q (333.15 K); dash-dot curve, vapour q (333.15 K).
The deviation of results calculated with the xLF EOS and the LF EOS models are listed in table 3. The average absolute deviation (AAD) for the various systems is improved by the crossover treatment. 4. Conclusion The xLF EOS was applied to fluid mixtures using the classical method of phase equilibria calculation. This approach neglects the critical exponents along the critical line but retains them at the critical points in the limits of pure components. This approach makes calculation processes simple and eliminates the need for the critical lines in the calculations. As presented in the results for various mixtures, the xLF model represented the PVT behaviour well to high pressures and showed improvements for the VLE calculation near the critical point using the two binary interaction parameters that were used for the LF model.
Ethane(1) + n-hexane(2) T = 298.15 K 3.652 T = 338.71 K 2.153
1.833 1.546
0.319 0.694
0.518 0.293
CO2(1) + R123(2) T = 313.15 K T = 323.15 K T = 333.15 K
0.844 1.817 1.487
1.271 1.197 1.663
0.762 1.019 1.093
0.311 0.301 0.210
Propane(1) + n-butane(2) T = 343.17 K 1.014 T = 373.15 K 0.825 T = 393.15 K 0.974
0.976 0.935 1.541
0.538 0.357 0.141
0.433 0.480 0.262
CO2(1) + ethane(2) T = 230 K T = 270 K T = 293.15 K T = 298.15 K
0.622 0.462 2.075 2.319
0.456 0.561 2.782 2.972
0.601 0.731 0.890 1.013
0.399 0.501 1.101 0.939
R143a(1) + n-butane(2) T = 333.15 K 0.411 T = 343.15 K 0.465 T = 353.15 K 1.477 T = 363.15 K 1.909
0.625 0.732 0.905 0.979
0.424 0.569 0.555 0.296
0.768 0.748 0.498 0.126
1.039 3.590 0.615 2.308 3.587 1.155 0.714 1.207 0.385
P AADP ð%Þ ¼ ð1=nÞ Pni ð P_ i;exp P_ i;cal Þ= P _ i;exp 100. n ð_y i;exp y_ i;cal Þ 100. AADY ð%Þ ¼ ð1=nÞ
0.413 0.694 0.733
CO2(1) + R124(2) T = 313.15 K T = 323.15 K T = 333.15 K
i
Acknowledgement This work was supported by the BK21 project of the Ministry of Education of Korea and by the National Laboratory (NRL) Program.
Y. Lee et al. / J. Chem. Thermodynamics 40 (2008) 741–749
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JCT 08-3