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Cubic crossover equation of state1
Fluid Phase Equilibria 147 Ž1998. 7–23
Cubic crossover equation of state S.B. Kiselev
1
)
Physical and Chemical Properties DiÕision, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303, USA Received 22 September 1997; accepted 5 December 1997
Abstract In this paper we develop a cubic crossover equation of state for pure fluids which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original classical cubic equation of state far away from the critical point. We use the modified Patel–Teja ŽPT. cubic equation of state as a starting point. A comparison is made with experimental data for pure CO 2 , water, and refrigerants R32 and R125 in the one- and two-phase regions. We show that the crossover Patel–Teja equation of state yields a much better representation of the thermodynamic properties of pure fluids, especially in the critical region and for vapor–liquid equilibrium, than the original PT equation of state. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Carbon dioxide; Critical state; Cubic equation of state; Vapor–liquid equilibria; R32; R125; Water
1. Introduction The development of a universal model for the description of the thermodynamic properties of fluids over a wide range of parameters of state including the ideal gas limit and the critical region has always been one of the most difficult tasks of the theory of fluids. The classical solution of this problem was first given by van der Waals w1x, who proposed a simple cubic equation of state Ž EOS. based on the ideal gas equation as a zeroth approximation, and included the effects of intermolecular interaction to a first approximation. In spite of its simplicity, the van der Waals EOS allows the qualitative prediction of vapor–liquid equilibrium and a critical point in real fluids, but the quantitative difference between theory and experiment is rather substantial. However, the approach )
Corresponding author. Institute for Oil and Gas Research of the Russian Academy of Sciences, Leninsky Prospect 63r2, Moscow 117917, Russian Federation. 1 Contribution of the National Institute of Standards and Technology, not subject to copyright in the United States, November 26, 1997. 0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 2 2 2 - 2
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S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
formulated by van der Waals was so fruitful that there have been many attempts to improve the representation of the liquid phase as well as the vapor–liquid equilibrium by modifying the van der Waals EOS. The first well-known attempts to improve the van der Waals EOS were made by Redlich and Kwong w2x, by Soave w3x, and by Peng and Robinson w4x. The equations of state proposed by these authors, and their different empirical and semiempirical modifications w5–10x, give a much better representation of the thermodynamic properties of fluids than the original van der Waals EOS, especially for vapor–liquid equilibrium ŽVLE.. However, all of these analytical equations of state fail to reproduce the non-analytical, singular behavior of fluids in the critical region, caused by long-scale fluctuations in density. The asymptotic singular critical behavior of the thermodynamic properties can be described in terms of scaling laws with universal critical exponents and universal scaling functions w11,12x. Theoretical crossover equations of state which incorporate the scaling laws asymptotically close to the critical point and are transformed into the regular classical expansion far away from the critical point have been developed by Chen et al. w13,14x, Jin et al. w15x and by Kiselev et al. w16–19x. Although these crossover equations of state give a very accurate representation of the thermodynamic properties of fluids in the wide region around the critical point, in the limit of zero density they do not reproduce the ideal gas equation of state and, therefore, they cannot be extrapolated to low densities. A phenomenological approach to the incorporation of scaling laws in a classical cubic EOS was first proposed by Fox w20,21x. The principal idea of this approach consists in the renormalization of the temperature and the density in the cubic EOS in such a way that in the asymptotic critical region they become non-analytic scaling functions of the dimensionless distance to the critical point. For these scaling functions some empirical correlations have been used w20,21x. A more rigorous theoretical method was proposed by van Pelt et al. w22x. This method has a theoretical foundation in the renormalization-group theory w13,23–25x and can be applied to the transformation of any classical equation of state, so that the equation incorporates a crossover from classical behavior far away from the critical point to a singular behavior in the vicinity of the critical point. It is the aim of this paper to develop, on the basis of the modern theory of critical phenomena, a cubic crossover equation of state for pure fluids which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original cubic equation of state far away from the critical point. We proceed as follows. In Section 2 we describe a general method for the transformation of a classical expression for the thermodynamic potential of a system into the crossover form. The original Patel–Teja equation of state is considered in Section 3. In Section 4 we formulate the crossover expression for the Helmholtz free energy for the Patel–Teja EOS. Comparisons with experimental data for pure CO 2 , water, R32 and R125 are discussed in Section 5. Our results are summarized in Section 6.
2. Theoretical background In the critical region, the van der Waals equation of state corresponds to the mean-field, or Landau theory of critical phenomena w26,27x. The Landau theory is based on the introduction of an order parameter Dh which is zero in the more symmetric Ždisordered. phase and nonzero in the less
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
9
symmetric Ž ordered. phase. The main assumption of the classical theory of critical phenomena is that the critical part of the free energy, D A of the system can be represented by a Taylor expansion in powers of the order parameter D A Ž t ,Dh . s
Ý Ý a i jt iDh j
Ž1.
is0 js1
where t s TrTc y 1 is the dimensionless deviation of the temperature T from the critical temperature Tc , and a i j are the system-dependent coefficients. In the critical region
Ž2.
that correspond to the Landau theory of critical phenomena w26x. The quantity h s Ž ED ArEDh .t , conjugated to the order parameter Dh , plays the role of the ordering field. From the condition h s 0 at t F 0 we have that the equilibrium value of the order parameter on the coexistence curve Dhcxs A
bL s
1 2
, and g L s 1
Ž3.
The Landau theory is valid only in the temperature region Gi <
a
D A Ž t ,Dh . s a12t Yy 2D 1 Dh 2 Y
2D1
Žg y2 b .
q a04 Dh 4 Y
D1
y K Žt 2 .
Ž4.
where Y denotes a crossover function to be specified below, while the kernel term K Žt 2 ., which provides the correct scaling behavior of the isochoric specific heat asymptotically close to the critical point, has the form K Žt 2 . s
1 2
a
a 20t 2 ž Yy D1 y 1 /
Ž5.
In these equations g s 1.24, b s 0.325, As 2 y g y 2 b s 0.110, and D1 s 0.51 are the current best estimates of the nonclassical critical exponents w11,12x. The crossover function Y modifies each term in Eq. Ž4. such that at Gi <
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
10
Mathematically, Eq. Ž4. is equivalent to the Landau expansion Ž 2. with the replacement of the dimensionless temperature t and the order parameter Dh by the renormalized values Žg y2 b .
a
t s t Yy 2D 1 , Dh s Dh Y
4D 1
Ž6.
and adding the kernel term K Žt 2 . . As one moves further from the critical point,
a
ž
D A Ž t ,Dh . s D A Ž t ,Dh . s D A t Yy 2D 1 Dh Y
4D 1
/
y K Žt 2 .
Ž7.
3. Classical equation of state The method described above does not depend on the particular type of classical EOS to be renormalized in the crossover form. In the present work we apply the crossover theory to the modified Patel–Teja ŽPT. equation of state w5,28x. The PT equation of state has the form Ps
RT
a
Ž8.
y Vyb
V ŽV qb. qcŽV yb.
where P is the pressure, V is the molar volume, and R is the universal gas constant. The parameters a, b and c of the PT equation of state are given by a s Va
2 R 2 T0c
P0c
a0 Ž T . , b s V b
RTO c P0c
, c s Vc
RT0 c P0c
Ž9.
where V a , V b , and V c are functions of the critical compressibility factor Z0c . The critical parameters T0c , V0c and P0c corresponding to the PT equation of state can be found from the condition EP
ž / EV
s 0, T0c
E2P
ž / EV
s 0,
2
T0c
P0cV0c RT0 c
s Z0c
Ž 10.
with the following relationships between dimensionless parameters V a , V b , and V c and the critical compressibility factor Z0c
V c s 1 y 3Z0c
Ž 11.
2 V a s 3Z0c q 3 Ž 1 y 2 Z0c . V b q V b2 q V c
Ž 12.
and where V b is the smallest positive root of the cubic equation 2 3 V b3 q Ž 2 y 3Z0c . V b2 q 3Z0c V b y Z0c s0
Ž 13.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
11
The PT equation of state can be easily transformed into the Redlich–Kwong–Soave EOS Ž c s 0., and into the Peng–Robinson EOS Ž c s b ., by the special choice of the temperature dependent function a 0 Ž T . and the coefficient c. In the present work for a 0 Ž T . we chose a form proposed by Patel w28x a 0 Ž T . s 1 q c1Ž Tt y 1 . q c 2 Tt y 1 q c 3 Ž Tt N y 1 .
ž(
/
Ž 14.
where the coefficients c i Ž i s 1, 2, 3. and exponent N are the system dependent parameters, and Tt s TrT0c . The Helmholtz free energy and some relevant thermodynamic quantities related to the PT equation of state are presented in the Appendix A.
4. Crossover equation of state In order to apply the method described in Section 2 to the PT equation of state, one first rewrites the classical expression for the Helmholtz free energy in the dimensionless form A Ž T ,V . s
A Ž T ,V . RT
s D A Ž DT ,DV . y
V V0c
P0 Ž T . q m 0 Ž T .
Ž 15.
where AŽ T,V . is the Helmholtz free energy per mole, D A is the critical part of the Helmholtz free energy, m 0 Ž T . is the analytic function of temperature, DT s Tt y 1 is the dimensionless deviation of the temperature from the critical temperature T0c , P0 Ž T . s P Ž V0c , T . V0crRT is the dimensionless pressure at the critical isochore V s V0c , and DV s VrV0c y 1 is the order parameter in the PT equation of state Ž see Appendix A. . Secondly, we need to renormalize DT and DV in the critical part of the classical Helmholtz free energy D AŽ DT, DV . according to Eq. Ž6.. These equations are written in terms of the real critical parameters of the system. The critical temperature and pressure of the PT equation of state can be taken from available measurements or predictions, while the critical density is usually found as fitting parameters of the model, chosen to best describe the VLE surface far away from the critical point. Since any classical equation of state does not reproduce the thermodynamic surface of a fluid in the critical region, the critical density found by this method does not coincide with the real critical density of the system. In the present work, we have introduced in Eq. Ž 6. additional terms which take into account the difference between the classical critical temperature T0c and critical volume V0c , and the real critical parameters Tc and Vc : 2 Ž2y a .
a
t s t Yy 2D 1 q Ž 1 q t . Dtc Y
3D 1
Žg y2 b .
D h s Dh Y
4D 1
Ž 16. Ž2y a .
q Ž 1 q Dh. Dhc Y
2D1
Ž 17.
where the order parameter Dh s VrVc y 1, while Dtc s DTcrT0c and Dhc s DVcrV0c are the dimensionless shifts of the critical temperature and the critical volume, respectively.The crossover function Y in Eqs. Ž16. and Ž17. can be written in the parametric form w17,18x Y Ž q . s q 2rR Ž q .
D1
Ž 18.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
12
For the function RŽ q . , we adopted the same expressions as employed earlier by Kiselev in the parametric crossover model w19x
ž
RŽ q. s 1 q
2
q2 1qq
/
Ž 19.
The parametric variable q, which has a meaning of a renormalized measure of the distance from the critical point, can be found from the solution of the equation q2s
t Gi
q b 2LM
ž
Dh q d1t q d 2t 2 Gi
b
Ž1y2 b .
2
/
Y
D1
Ž q.
Ž 20.
b 2LM s Ž g y 2 b .rg Ž1 y 2 b . s 1.359 is the universal linear-model parameter w18,19x. The last two terms A t and A t 2 in the brackets at the right-hand side of Eq. Ž 20. correspond to a projection of the rectilinear diameter of the coexistence curve in the temperature–density variables rd s Ž rG q r L .r2 s rc Ž1 q dt . on the temperature–volume plane Vd s 1rrd ( Vc Ž1 q d1t q d 2t 2 .. In order to complete the transformation of the classical Helmholtz free energy into the crossover form, one must next replace the classical dimensionless temperature DT and the volume DV in Eq. Ž15. with the renormalized values t andDh , and add the kernel term K Žsee Eqs. Ž5. and Ž7.. . Finally, the crossover expression for the Helmholtz free energy can be written in the form A Ž T ,V . s D A Ž t ,Dh . y
V V0c
P0 Ž T . q m 0 Ž T . y K Ž t 2 .
Ž 21.
where, for the PT EOS, P0 Ž T . s
1 y b1
T0c V a a 0 Ž T .
Ž 22.
T Z0c b 2 b 3
The critical part D A is given by D A Ž t ,Dh . s yln
ž
Dh b1
/
q1 q
T0c V a a o Ž t . T
V
ln
ž
Dhrb 2 q 1 Dhrb 3 q 1
/
Dh q
y b1
T 0 c V a a 0 Ž t . Dh T
Z0c b 2 b 3
Ž 23. with the coefficients bi Ž i s 1,2,3. and V given in Appendix A. The crossover cubic equation of state can be obtained from the crossover expression Ž 21. by differentiation with respect to volume P Ž T ,V . s y
EA
ž / EV
RT s T
y V0 c
V0c ED A Vc
ž / EDh
q P0 Ž T . q T
V0c Vc
EK
ž / EDh
Ž 24. T
Eqs. Ž16. – Ž23. completely determine the crossover Helmholtz free energy for the PT equation of state. Asymptotically close to the critical point q < 1 Ž
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
13
Table 1 System-dependent constants for CO 2 , H 2 O, R32 and R125 CO 2
H 2O
R32
R125
Classical critical parameters P0c ŽMPa. 7.3830 T0c ŽK. 304.210 Z0c 0.31330
22.064 647.130 0.26950
5.7950 351.350 0.28195
3.6290 339.330 0.31136
Critical shifts DTc ŽK. DVc Žl moly1 .
6.5000=10y2 y9.9035=10y3
0 y2.0299=10y2
0 y3.2057=10y2
Classical PT EOS parameters c1 0.63199 c2 y2.69935 c3 0 N 0
0.60462 y2.56713 0 0
4.69539 y7.45749 y0.81188 2.0
y0.79912 0.32801 y3.86059=10y2 2.0
CrossoÕer parameters Gi 6.18200=10y2 d1 0 d2 y2.69935
1.02977=10y1 4.68835 y1.07896
5.95804=10y2 2.71799 y1.91253=10
6.39983=10y2 1.98093 y1.04839=10
1.5500=10y1 y1.3213=10y2
Fig. 1. PVT data w35,36x for carbon dioxide with predictions of the cubic crossover equation of state Žsolid curves. and the Patel–Teja equation Eq. Ž8. Ždashed curves..
14
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
values t ™ DT and Dh ™ DV, and Eq. Ž21. is transformed into the classical Helmholtz free energy density Ž15. for the original PT equation of state Ž 8.. In the limit of zero density V ™ `, q 4 Gi for all values of the temperature and the Ginzburg number, and Eq. Ž21. is transformed into the ideal gas expression A Ž T ,V . s yln Ž VrV0c . q A 0 Ž T . Ž 25. Ž . and Eq. 24 reproduces the ideal gas equation of state PV s RT Ž 26. The crossover cubic equation of state defined by Eqs. Ž 16. – Ž 21. , except for the parameters in the PT EOS, contains the following additional system-dependent parameters: critical shifts DTc and DVc , the Ginzburg number Gi, and the rectilinear diameter amplitudes d1 and d 2 . Since the critical parameters Tc and Vc of a pure fluid are usually known, the critical shifts can be easily determined by DTc s Tc y T0c and DVc s Vc y V0c Ž 27. if the classical critical parameters T0c and V0c are known. In practice, the simplest way is to set DTc s 0, and consider the shift of the critical volume, DVc , only. The parameters Gi, d1 and d 2 , similar to the coefficients c i Ž i s 1,2,3. in the PT EOS, can be represented as functions of the acentric factor V w29x; however, in the present work we find them from a fit of Eqs. Ž16. – Ž21. to experimental PVT data.
Fig. 2. VLE data w35x for carbon dioxide with predictions of the cubic crossover equation of state Žsolid curves. and the Patel–Teja equation Eq. Ž8. Ždashed curves..
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
15
Fig. 3. PVT and VLE data for water with predictions of the cubic crossover equation of state Žsolid curves. and the Patel–Teja equation Eq. Ž8. Ždashed curves.. Empty circles represent experimental data obtained by Rivkin et al. w38–41x, and filled circles correspond to the data generated with the fundamental equation of state of Pruss and Wagner w42x.
Fig. 4. PVT data w44x Žempty circles., w46x Žempty squares., and w43x Žfilled triangles., for R32 with predictions of the cubic crossover equation of state Žsolid curves..
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S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
5. Comparison with experimental data It is well-known that cubic equations of state give poorer predictions of the densities and pressures of liquids than of gases, especially for polar components and strongly hydrogen bonded fluids, like water w30,31x. Water is essentially a nonclassical system. Even such multiparameter analytical equations as the equation of Benedict–Webb–Rubin w32x and the equation of Lee–Kesler w33x Ž which can be successfully applied for the description of the thermodynamic properties of other fluids. fail in their application to water even far away from the critical point Žsee for example Ref. w31x.. For this reason, water and other polar fluids are good examples for testing the predictive capability of the model. We consider here the volumetric properties of carbon dioxide and water in the one- and two-phase regions and compare the cubic crossover equation of state with experimental data. Refrigerant mixtures such as R125q R32 are of interest as candidate to replace ozone-depleting refrigerants. Most models that are currently used for predicting the thermophysical properties of this mixture cannot accurately represent properties in and around the critical region. The parametric crossover model developed recently by Kiselev and Huber w34x gives a good representation of the thermodynamic surface for this system in and beyond the critical region, but fails at low densities and temperatures. Therefore, we also applied our crossover model to represent the thermodynamic properties of refrigerants R32 and R125.
Fig. 5. PVT data w48x Žempty circles. and w51x Žfilled squares., for R125 with predictions of the cubic crossover equation of state Žsolid curves..
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
17
For carbon dioxide and water, PT equations of state already exist, and for the classical critical parameters P0c , T0c and Z0c , for the coefficients c1, c 2 , c 3 , and the exponent N in Eq. Ž14. for carbon dioxide and water we adopt the same values as obtained by Patel w28x. The classical critical parameters for CO 2 and water reported by Patel w28x differ from the experimental values reported by Duschek et al. w35x Gilgen et al. w36x for CO 2 , and to the critical parameters adopted by Levelt Sengers et al. w37x for water; therefore, for CO 2 and water we use both shifts, DTc and DVc , in Eqs. Ž16. and Ž17.. The Ginzburg number Gi and the rectilinear-diameter amplitudes d1 and d 2 in Eq. Ž20. have been found from a fit of our crossover model to the experimental PVT data obtained by Duschek et al. w35x Gilgen et al. w36x for CO 2 , and to the experimental PVT data obtained by Rivkin et al. w38–41x for water. Additional PVT data along separate isochores at temperatures T ) 750 K and VLE data for water were generated with the fundamental equation of state developed by Pruss and Wagner w42x. The values of all system dependent constants for CO 2 and water are listed in Table 1. A comparison of the cubic crossover equation of state with experimental data and with the original PT equation of state is shown in Figs. 1 and 2 for carbon dioxide, and in Fig. 3 for water. As one can see, the crossover cubic equation of state gives a much better representation of thermodynamic properties of carbon dioxide and water than the original PT equation of state, especially in the critical region. The average deviation of pressures calculated with the crossover cubic EOS from the experimental data in the density range 0.5 rc F r F 1.5 rc is about 0.5% for all isotherms shown in Figs. 1 and 3. The crossover equation clearly yields a major improvement in the representation of the
Fig. 6. VLE data for R32 obtained by Holcomb et al. w52x, by Malbrunot et al. w46x, by Defibaugh et al. w44x, by Higashi w53x, by Magee w45x and by Malbrunot et al. w46x for R32, with predictions of the crossover cubic equation of state Žsolid curves.. The dashed curves calculated with the parametric crossover model w34x ŽCREQS97..
18
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
liquid densities along the saturated curve and the PVT data at high densities Ž r G rc . in the supercritical region where the original PT EOS yields up to 15% deviations for the saturated liquid densities Ž see Fig. 2. , and up to 20–25% for pressures Ž see Figs. 1 and 3.. At low densities r < rc and in the limit of zero densities. Both equations yield the same results in the entire temperature range and solid and dashed curves in Figs. 1 and 3 practically coincide. For R32 and R125, we adopt for P0c and T0c the same values of the critical parameters as used in the parametric crossover model by Kiselev and Huber w34x. The classical critical compressibility factor Z0c , the shift of the critical volume DVc , and all other system-dependent parameters in the PT equation of state Ž8. and in the CUCREQ were found from a fit of Eqs. Ž 16. – Ž 23. Eqs. Ž 24. – Ž 26. to experimental PVT data. For R32, the PVT data were taken from Refs. w43–46x and for R125 we used Refs. w47–51x. We also used in the fit few additional experimental VLE data points from Refs. w52–54x. The values of all system-dependent constants for R32 and R125 are presented in Table 1. Comparisons of the crossover cubic equation of state with experimental PVT data in the one-phase region are shown in Figs. 4 and 5. A reasonably good agreement between the crossover cubic equation of state ŽCUCREQ. and experimental data is observed. The average deviation of the calculated pressures from the experimental data in the range of densities 0 F r F 1.8 rc for all isotherms is about 1–2%, which is approximately 2–3 times less than the accuracy achieved for pure
Fig. 7. VLE data for R125 obtained by Magee w45x, by Boyes and Weber w50x, by Ye et al. w49x, by Duarte-Garza et al. w47x, by Defibaugh and Morrison w48x and by Kuwubara et al. w54x, with predictions of the crossover cubic equation of state Žsolid curves.. The dashed curves calculated with the parametric crossover model w34x ŽCREQS97..
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
19
R32 and R125 with the parametric crossover model ŽCREQS97. w34x which was optimized for the critical region only. At densities r G 1.8 rc the deviations between the calculated and experimental values of pressures increase and reach 10–20% at near critical isotherms at densities r ( 2 rc . As we mentioned above, a simple two- or three-parameter cubic equation of state, in principle, cannot describe the PVT surface of dense fluids. Since far away from the critical point, at densities r G 2 rc , the crossover PT equation of state and the original Eq. Ž8. practically coincide, in order to improve a description of the PVT surface of dense fluids with the crossover equation of state, one needs to improve the original PT equation of state. Comparisons of the cubic crossover equation of state with VLE data and with the parametric crossover equation of state obtained by Kiselev and Huber w34x for R32 and R125 are shown in Figs. 6 and 7. These figures show that in the critical region at 0.95Tc F T F Tc , the cubic crossover EOS and the parametric crossover model give essentially identical results. However, unlike the parametric crossover model w34x which is restricted to temperatures T G 0.8Tc , the cubic crossover EOS yields a good representation of vapor pressures and saturated liquid and vapor densities over the entire range of temperatures from the critical temperature down to the triple point.
6. Discussion In this paper we develop a simple method for incorporating of scaling laws in an analytical equation of state. Unlike the crossover approach proposed by van Pelt et al. w22x, the crossover
Fig. 8. The isochoric specific heat Cy of water at the critical isochore as a function of temperature. The solid curve represents values calculated with the cubic crossover equation of state, the dashed curve calculated with the Patel–Teja equation Eq. Ž8., the symbols indicate values generated with the parametric crossover model w19x.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
20
function in our method does not depend on the particular type of classical EOS to be renormalized and can be applied to any analytical equation of state. In the present paper, we use the modified Patel–Teja cubic equation of state w28x as an example. The crossover cubic equation of state reproduces the non-analytical, singular behavior asymptotically close to the critical point and is transformed into the original Patel–Teja equation of state far from the critical point. In the limit of zero density, the crossover equation is transformed into the ideal gas equation of state. Application of this equation to PVT and VLE data for carbon dioxide, water, and refrigerants R32 and R125 shows that the crossover cubic equation yields a satisfactory representation of the thermodynamic surface of pure fluids in a temperature range from the triple point to T F 2Tc and a density range r F 2 rc . Since in the present paper we consider the volumetric properties only, the coefficient a 20 in the kernel term, which determines the asymptotic critical behavior of the isochoric specific heat Cy Žsee Eq. Ž5.. , was set equal zero. However, it is easy to show that with the kernel term, the crossover cubic equation of state is able to reproduce a singular behavior of the isochoric specific heat in the critical region. The results of our calculations of Cy for water with a 20 s 9.21 are shown in Fig. 8. As one can see, the crossover cubic equation of state does reproduce an asymptotic singular behavior of the isochoric specific heat at the critical isochore unlike the analytical cubic equation of state. In order to extend the range of validity of our crossover equation to a wide range of densities, a better analytical equation of state has to be chosen w55x. In the present paper we restrict the application to the cubic equations only because of their widespread using for engineering’s calculations of the phase behavior in binary- and multi-component fluid mixtures. The method described here can be also applied to mixture calculations. Further research toward this goal is in the progress and the results will be presented in the next publication.
7. List of symbols a ai j A b c di Gi K N P q R RŽ q . T Tr V Y Z
coefficient of cubic equation of state coefficients of Landau expansion Helmholtz free energy pert mole coefficient of cubic equation of state coefficient of cubic equation of state coefficients of rectilinear diameter Ginzburg number kernel term exponent pressure argument of crossover function gas constant crossover function temperature dimensionless temperature molar volume crossover function compressibility factor
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
21
Greek letters a critical exponent b critical exponent g critical exponent D difference D1 critical exponent Dh order parameter t reduced temperature difference Vi dimensionless coefficients of cubic equation of state r molar density Subscripts c critical L Landau theory 0 classical Acknowledgements The author is indebted to D.G. Friend, J.C. Rainwater and A.S. Teja for valuable discussions which stimulated this work. The author also would like to thank the Physical and Chemical Properties Division, National Institute of Standards and Technology for the opportunity to work as a Guest Researcher at NIST during the course of this research. Appendix A. Helmholtz free energy for the PT equation of state The PT equation of state RT a P Ž T ,V . s y Vyb V ŽV q b. q cŽV yb.
Ž A.1 .
can be written in the dimensionless form P Ž T ,V . s
PV0c
1 s
RT
y DV q b 1
T0c V a
a0 Ž T .
T Z0c Ž DV q b 2 .Ž DV q b 3 .
Ž A.2 .
where DT s TrT0c y 1, DV s VrV0c y 1, and the coefficients bi Ž i s 1,2,3. and V are expressed through the dimensionless parameters V a , V b , V c , and the critical compressibility factor Z0c : Vb b1 s 1 y Ž A.3 . Z0c b2 s 1 q b3 s 1 q
V b q Vc y V 2 Z0c
V b q Vc q V 2 Z0c
Ž A.4 . Ž A.5 .
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
22
(
V s V b2 q 6 V b V c q V c2
Ž A.6 .
At the critical isochore V s V0c , and Eq. ŽA.2. reads P0 Ž T . s
1 y b1
T0c V a a 0 Ž T .
Ž A.7 .
T Z0c b 2 b 3
The Helmholtz free energy A Ž T ,V .
s yHPdVq A 0 Ž T . s yRT ln Ž DV q b 1 . q RT0c
Va a0 Ž T . V
ln
ž
DV q b 2 DV q b 3
/
Ž A.8 .
q A0 ŽT .
where A 0 Ž T . corresponds to the temperature dependent function of the ideal-gas part of the Helmholtz free energy. The dimensionless Helmholtz free energy can be represented in the form A Ž T ,V . s
A Ž T ,V . RT
s D A Ž DT ,DV . y
V V0c
P0 Ž T . q m 0 Ž T .
Ž A.9 .
where D AŽ DT, DV . corresponds to the critical part of the Helmholtz free energy, and m 0 Ž T . is an analytical function of the temperature. The functions D A and m 0 are found from the condition D A Ž DT ,DV s 0 . s 0, and
ED A
ž / EDV
s0
Ž A.10 .
DT Ž DVs0 .
With the Helmholtz free energy as given by Eq. ŽA.8. we have D A Ž DT ,DV . s yln
ž
DV b1
/
q1 q
T0c V a a 0 Ž DT .
V
T
ln
ž
DVrb 2 q 1 DVrb 3 q 1
/
DV q
y b1
T0c V a a 0 Ž DT . DV T
Z0c b 2 b 3 Ž A.11 .
and
m 0 Ž T . s P0 Ž T . y ln b 1 q
T0c V a a 0 Ž T . T
V
ln
b2
ž / b3
q A0 ŽT .
Ž A.12 .
References w1x J.D. Van der Waals, Kohnstam, Lehrbuch der Thermostatik, 2er Teil, Verlag von Johann Ambrosius Barth, Leipzig, 1927. w2x O. Redlich, J.N.S. Kwong, Chem. Rev. 44 Ž1949. 233. w3x G. Soave, Chem. Eng. Sci. 27 Ž1972. 1197. w4x D.Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 Ž1976. 59. w5x N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 Ž1982. 463. w6x A. Peneloux, E. Rauzy, R. Freze, Fluid Phase Equilibria 8 Ž1982. 7. w7x R. Solimando, M. Rogalski, E. Neau, A. Peneloux, Fluid Phase Equilibria 106 Ž1995. 59. w8x E. Behar, R. Simonet, E. Rauzy, Fluid Phase Equilibria 21 Ž1985. 237.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x w37x w38x w39x w40x w41x w42x w43x w44x w45x w46x w47x w48x w49x w50x w51x w52x w53x w54x w55x
23
A.S. Lawal, E.T. Van der Laan, R.K.M. Thambynayagam, Soc. Pet. Eng. of AIME, Pap. 14269, 1985. P.M. Mathias, T. Naheiri, E.M. Oh, Fluid Phase Equilibria 47 Ž1989. 77. J.V. Sengers, J.M.H. Levelt Sengers, Annu. Rev. Phys. Chem. 37 Ž1986. 189. M.A. Anisimov, S.B. Kiselev, Sov. Tech. Rev. B Therm. Phys. 3 Ž2. Ž1992., Harwood Academic, Chur-Melbourne. Z.Y. Chen, P.C. Albright, J.V. Sengers, Phys. Rev. A 41 Ž1990. 3161. Z.Y. Chen, A. Abbaci, S. Tang, J.V. Sengers, Phys. Rev. A 42 Ž1990. 4470. G.X. Jin, S. Tang, J.V. Sengers, Phys. Rev. E 47 Ž1993. 388. S.B. Kiselev, High Temp. 28 Ž1990. 42. S.B. Kiselev, I.G. Kostyukova, A.A. Povodyrev, Int. J. Thermophys. 12 Ž1991. 877. S.B. Kiselev, J.V. Sengers, Int. J. Thermophys. 14 Ž1993. 1. S.B. Kiselev, Fluid Phase Equilibria 128 Ž1997. 1. J.F. Fox, J. Stat. Phys. 21 Ž1979. 243. J.F. Fox, Fluid Phase Equilibria 14 Ž1983. 45. A. van Pelt, G.X. Jin, J.V. Sengers, Int. J. Thermophys. 15 Ž1994. 687. J.F. Nicoll, J.K. Bhattachafjee, Phys. Rev. B 23 Ž1981. 389. J.F. Nicoll, P.C. Albright, Phys. Rev. B 31 Ž1985. 4576. J.F. Nicoll, P.C. Albright, Phys. Rev. B 34 Ž1986. 1991. L.D. Landau, E.M. Lifshitz, Statistical Physics, 3rd edn., Pergamon, New York, 1980. A.Z. Patashinskii, V.L. Pokrovskii, Fluctuation Theory of Phase Transitions, Pergamon, New York, 1979. N.C. Patel, Int. J. Thermophys. 17 Ž1996. 673. S.B. Kiselev, J.C. Rainwater, M.L. Huber, Fluid Phase Equilibria, in press. R.C. Reid, J.M. Prausnitz, T.K. Sherwood, The Properties of Gases and Liquids, McGraw-Hill, New York, 1977. S.M. Walas, Phase Equilibria in Chemical Engineering, Butterworth, Boston, 1985. M. Benedict, G.B. Webb, L.C. Rubin, Chem. Eng. Prog. 47 Ž1951. 419. B.I. Lee, M.G. Kesler, AIChE J. 21 Ž1975. 510. S.B. Kiselev, M.L. Huber, Int. J. Refrigeration, in press. W. Duschek, R. Kleinrahm, W. Wagner, J. Chem. Thermodynamics 22 Ž1990. 827–841. R. Gilgen, R. Kleinrahm, W. Wagner, J. Chem. Thermodynamics 24 Ž1992. 1243. J.M.H. Levelt Sengers, J. Straub, K. Watanabe, P.G. Hill, J. Phys. Chem. Ref. Data 14 Ž1985. 193. S.L. Rivkin, T.S. Akhundov, Teploenergetika 9 Ž1. Ž1962. 57. S.L. Rivkin, T.S. Akhundov, Teploenergetika 10 Ž9. Ž1963. 66. S.L. Rivkin, G.V. Troyanovskaya, Teploenergetika 11 Ž10. Ž1964. 72. S.L. Rivkin, T.S. Akhundov, E.A. Kremenevskaya, N.N. Asadullaeva, Teploenergetika 13 Ž4. Ž1966. 59. A. Pruss, W. Wagner, Physical chemistry of aqueous system: meeting the needs of industry, in: H.J. White, J.V. Sengers, D.B. Neumann, J.C. Bellows ŽEds.., Begell House, N.Y.-Wallingford, 1995, pp. 66–77. J.C. Holste, H.A. Duarte-Garza, M.A. Villamanan-Olfos, Paper No. 93-WArHT-60, ASME Winter Annual Meeting, New Orleans, December, 1993. D.R. Defibaugh, G. Morrison, L.A. Weber, J. Chem. Eng. Data 39 Ž1994. 333. J.W. Magee, Int. J. Thermophys. 17 Ž1996. 803. P.F. Malbrunot, P.A. Meunier, G.M. Scatena, W.H. Mears, K.P. Murphy, J.V. Sinka, J. Chem. Eng. Data 13 Ž1968. 16. H.A. Duarte-Garza, C.E. Stouffer, K.R. Hall, J.C. Holste, K.N. Marsh, B.E. Gammon, J. Chem. Eng. Data Ž1997. submitted. D.R. Defibaugh, G. Morrison, Fluid Phase Equilibria 80 Ž1992. 157. F. Ye, H. Sato, K. Watanabe, J. Chem. Eng. Data 40 Ž1995. 148. S.J. Boyes, L.A. Weber, J. Chem. Thermodynamics 27 Ž1995. 163. L.C. Wilson, W.V. Wilding, G.M. Wilson, R.L. Rowley, V.M. Felix, T. Chisolm-Carter, Fluid Phase Equilibria 80 Ž1992. 167. C.D. Holcomb, V.G. Niesen, L.J. van Poolen, S.L. Outcalt, Fluid Phase Equilibria 91 Ž1993. 145. Y. Higashi, Int. J. Refrigeration 17 Ž1994. 524. S. Kuwubara, H. Aoyama, H. Sato, K. Watanabe, J. Chem. Eng. Data 40 Ž1995. 112. S.B. Kiselev, D.G. Friend, submitted for publication.
Cubic crossover equation of state S.B. Kiselev
1
)
Physical and Chemical Properties DiÕision, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303, USA Received 22 September 1997; accepted 5 December 1997
Abstract In this paper we develop a cubic crossover equation of state for pure fluids which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original classical cubic equation of state far away from the critical point. We use the modified Patel–Teja ŽPT. cubic equation of state as a starting point. A comparison is made with experimental data for pure CO 2 , water, and refrigerants R32 and R125 in the one- and two-phase regions. We show that the crossover Patel–Teja equation of state yields a much better representation of the thermodynamic properties of pure fluids, especially in the critical region and for vapor–liquid equilibrium, than the original PT equation of state. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Carbon dioxide; Critical state; Cubic equation of state; Vapor–liquid equilibria; R32; R125; Water
1. Introduction The development of a universal model for the description of the thermodynamic properties of fluids over a wide range of parameters of state including the ideal gas limit and the critical region has always been one of the most difficult tasks of the theory of fluids. The classical solution of this problem was first given by van der Waals w1x, who proposed a simple cubic equation of state Ž EOS. based on the ideal gas equation as a zeroth approximation, and included the effects of intermolecular interaction to a first approximation. In spite of its simplicity, the van der Waals EOS allows the qualitative prediction of vapor–liquid equilibrium and a critical point in real fluids, but the quantitative difference between theory and experiment is rather substantial. However, the approach )
Corresponding author. Institute for Oil and Gas Research of the Russian Academy of Sciences, Leninsky Prospect 63r2, Moscow 117917, Russian Federation. 1 Contribution of the National Institute of Standards and Technology, not subject to copyright in the United States, November 26, 1997. 0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 8 . 0 0 2 2 2 - 2
8
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
formulated by van der Waals was so fruitful that there have been many attempts to improve the representation of the liquid phase as well as the vapor–liquid equilibrium by modifying the van der Waals EOS. The first well-known attempts to improve the van der Waals EOS were made by Redlich and Kwong w2x, by Soave w3x, and by Peng and Robinson w4x. The equations of state proposed by these authors, and their different empirical and semiempirical modifications w5–10x, give a much better representation of the thermodynamic properties of fluids than the original van der Waals EOS, especially for vapor–liquid equilibrium ŽVLE.. However, all of these analytical equations of state fail to reproduce the non-analytical, singular behavior of fluids in the critical region, caused by long-scale fluctuations in density. The asymptotic singular critical behavior of the thermodynamic properties can be described in terms of scaling laws with universal critical exponents and universal scaling functions w11,12x. Theoretical crossover equations of state which incorporate the scaling laws asymptotically close to the critical point and are transformed into the regular classical expansion far away from the critical point have been developed by Chen et al. w13,14x, Jin et al. w15x and by Kiselev et al. w16–19x. Although these crossover equations of state give a very accurate representation of the thermodynamic properties of fluids in the wide region around the critical point, in the limit of zero density they do not reproduce the ideal gas equation of state and, therefore, they cannot be extrapolated to low densities. A phenomenological approach to the incorporation of scaling laws in a classical cubic EOS was first proposed by Fox w20,21x. The principal idea of this approach consists in the renormalization of the temperature and the density in the cubic EOS in such a way that in the asymptotic critical region they become non-analytic scaling functions of the dimensionless distance to the critical point. For these scaling functions some empirical correlations have been used w20,21x. A more rigorous theoretical method was proposed by van Pelt et al. w22x. This method has a theoretical foundation in the renormalization-group theory w13,23–25x and can be applied to the transformation of any classical equation of state, so that the equation incorporates a crossover from classical behavior far away from the critical point to a singular behavior in the vicinity of the critical point. It is the aim of this paper to develop, on the basis of the modern theory of critical phenomena, a cubic crossover equation of state for pure fluids which incorporates the scaling laws asymptotically close to the critical point and is transformed into the original cubic equation of state far away from the critical point. We proceed as follows. In Section 2 we describe a general method for the transformation of a classical expression for the thermodynamic potential of a system into the crossover form. The original Patel–Teja equation of state is considered in Section 3. In Section 4 we formulate the crossover expression for the Helmholtz free energy for the Patel–Teja EOS. Comparisons with experimental data for pure CO 2 , water, R32 and R125 are discussed in Section 5. Our results are summarized in Section 6.
2. Theoretical background In the critical region, the van der Waals equation of state corresponds to the mean-field, or Landau theory of critical phenomena w26,27x. The Landau theory is based on the introduction of an order parameter Dh which is zero in the more symmetric Ždisordered. phase and nonzero in the less
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
9
symmetric Ž ordered. phase. The main assumption of the classical theory of critical phenomena is that the critical part of the free energy, D A of the system can be represented by a Taylor expansion in powers of the order parameter D A Ž t ,Dh . s
Ý Ý a i jt iDh j
Ž1.
is0 js1
where t s TrTc y 1 is the dimensionless deviation of the temperature T from the critical temperature Tc , and a i j are the system-dependent coefficients. In the critical region
Ž2.
that correspond to the Landau theory of critical phenomena w26x. The quantity h s Ž ED ArEDh .t , conjugated to the order parameter Dh , plays the role of the ordering field. From the condition h s 0 at t F 0 we have that the equilibrium value of the order parameter on the coexistence curve Dhcxs A
bL s
1 2
, and g L s 1
Ž3.
The Landau theory is valid only in the temperature region Gi <
a
D A Ž t ,Dh . s a12t Yy 2D 1 Dh 2 Y
2D1
Žg y2 b .
q a04 Dh 4 Y
D1
y K Žt 2 .
Ž4.
where Y denotes a crossover function to be specified below, while the kernel term K Žt 2 ., which provides the correct scaling behavior of the isochoric specific heat asymptotically close to the critical point, has the form K Žt 2 . s
1 2
a
a 20t 2 ž Yy D1 y 1 /
Ž5.
In these equations g s 1.24, b s 0.325, As 2 y g y 2 b s 0.110, and D1 s 0.51 are the current best estimates of the nonclassical critical exponents w11,12x. The crossover function Y modifies each term in Eq. Ž4. such that at Gi <
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
10
Mathematically, Eq. Ž4. is equivalent to the Landau expansion Ž 2. with the replacement of the dimensionless temperature t and the order parameter Dh by the renormalized values Žg y2 b .
a
t s t Yy 2D 1 , Dh s Dh Y
4D 1
Ž6.
and adding the kernel term K Žt 2 . . As one moves further from the critical point,
a
ž
D A Ž t ,Dh . s D A Ž t ,Dh . s D A t Yy 2D 1 Dh Y
4D 1
/
y K Žt 2 .
Ž7.
3. Classical equation of state The method described above does not depend on the particular type of classical EOS to be renormalized in the crossover form. In the present work we apply the crossover theory to the modified Patel–Teja ŽPT. equation of state w5,28x. The PT equation of state has the form Ps
RT
a
Ž8.
y Vyb
V ŽV qb. qcŽV yb.
where P is the pressure, V is the molar volume, and R is the universal gas constant. The parameters a, b and c of the PT equation of state are given by a s Va
2 R 2 T0c
P0c
a0 Ž T . , b s V b
RTO c P0c
, c s Vc
RT0 c P0c
Ž9.
where V a , V b , and V c are functions of the critical compressibility factor Z0c . The critical parameters T0c , V0c and P0c corresponding to the PT equation of state can be found from the condition EP
ž / EV
s 0, T0c
E2P
ž / EV
s 0,
2
T0c
P0cV0c RT0 c
s Z0c
Ž 10.
with the following relationships between dimensionless parameters V a , V b , and V c and the critical compressibility factor Z0c
V c s 1 y 3Z0c
Ž 11.
2 V a s 3Z0c q 3 Ž 1 y 2 Z0c . V b q V b2 q V c
Ž 12.
and where V b is the smallest positive root of the cubic equation 2 3 V b3 q Ž 2 y 3Z0c . V b2 q 3Z0c V b y Z0c s0
Ž 13.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
11
The PT equation of state can be easily transformed into the Redlich–Kwong–Soave EOS Ž c s 0., and into the Peng–Robinson EOS Ž c s b ., by the special choice of the temperature dependent function a 0 Ž T . and the coefficient c. In the present work for a 0 Ž T . we chose a form proposed by Patel w28x a 0 Ž T . s 1 q c1Ž Tt y 1 . q c 2 Tt y 1 q c 3 Ž Tt N y 1 .
ž(
/
Ž 14.
where the coefficients c i Ž i s 1, 2, 3. and exponent N are the system dependent parameters, and Tt s TrT0c . The Helmholtz free energy and some relevant thermodynamic quantities related to the PT equation of state are presented in the Appendix A.
4. Crossover equation of state In order to apply the method described in Section 2 to the PT equation of state, one first rewrites the classical expression for the Helmholtz free energy in the dimensionless form A Ž T ,V . s
A Ž T ,V . RT
s D A Ž DT ,DV . y
V V0c
P0 Ž T . q m 0 Ž T .
Ž 15.
where AŽ T,V . is the Helmholtz free energy per mole, D A is the critical part of the Helmholtz free energy, m 0 Ž T . is the analytic function of temperature, DT s Tt y 1 is the dimensionless deviation of the temperature from the critical temperature T0c , P0 Ž T . s P Ž V0c , T . V0crRT is the dimensionless pressure at the critical isochore V s V0c , and DV s VrV0c y 1 is the order parameter in the PT equation of state Ž see Appendix A. . Secondly, we need to renormalize DT and DV in the critical part of the classical Helmholtz free energy D AŽ DT, DV . according to Eq. Ž6.. These equations are written in terms of the real critical parameters of the system. The critical temperature and pressure of the PT equation of state can be taken from available measurements or predictions, while the critical density is usually found as fitting parameters of the model, chosen to best describe the VLE surface far away from the critical point. Since any classical equation of state does not reproduce the thermodynamic surface of a fluid in the critical region, the critical density found by this method does not coincide with the real critical density of the system. In the present work, we have introduced in Eq. Ž 6. additional terms which take into account the difference between the classical critical temperature T0c and critical volume V0c , and the real critical parameters Tc and Vc : 2 Ž2y a .
a
t s t Yy 2D 1 q Ž 1 q t . Dtc Y
3D 1
Žg y2 b .
D h s Dh Y
4D 1
Ž 16. Ž2y a .
q Ž 1 q Dh. Dhc Y
2D1
Ž 17.
where the order parameter Dh s VrVc y 1, while Dtc s DTcrT0c and Dhc s DVcrV0c are the dimensionless shifts of the critical temperature and the critical volume, respectively.The crossover function Y in Eqs. Ž16. and Ž17. can be written in the parametric form w17,18x Y Ž q . s q 2rR Ž q .
D1
Ž 18.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
12
For the function RŽ q . , we adopted the same expressions as employed earlier by Kiselev in the parametric crossover model w19x
ž
RŽ q. s 1 q
2
q2 1qq
/
Ž 19.
The parametric variable q, which has a meaning of a renormalized measure of the distance from the critical point, can be found from the solution of the equation q2s
t Gi
q b 2LM
ž
Dh q d1t q d 2t 2 Gi
b
Ž1y2 b .
2
/
Y
D1
Ž q.
Ž 20.
b 2LM s Ž g y 2 b .rg Ž1 y 2 b . s 1.359 is the universal linear-model parameter w18,19x. The last two terms A t and A t 2 in the brackets at the right-hand side of Eq. Ž 20. correspond to a projection of the rectilinear diameter of the coexistence curve in the temperature–density variables rd s Ž rG q r L .r2 s rc Ž1 q dt . on the temperature–volume plane Vd s 1rrd ( Vc Ž1 q d1t q d 2t 2 .. In order to complete the transformation of the classical Helmholtz free energy into the crossover form, one must next replace the classical dimensionless temperature DT and the volume DV in Eq. Ž15. with the renormalized values t andDh , and add the kernel term K Žsee Eqs. Ž5. and Ž7.. . Finally, the crossover expression for the Helmholtz free energy can be written in the form A Ž T ,V . s D A Ž t ,Dh . y
V V0c
P0 Ž T . q m 0 Ž T . y K Ž t 2 .
Ž 21.
where, for the PT EOS, P0 Ž T . s
1 y b1
T0c V a a 0 Ž T .
Ž 22.
T Z0c b 2 b 3
The critical part D A is given by D A Ž t ,Dh . s yln
ž
Dh b1
/
q1 q
T0c V a a o Ž t . T
V
ln
ž
Dhrb 2 q 1 Dhrb 3 q 1
/
Dh q
y b1
T 0 c V a a 0 Ž t . Dh T
Z0c b 2 b 3
Ž 23. with the coefficients bi Ž i s 1,2,3. and V given in Appendix A. The crossover cubic equation of state can be obtained from the crossover expression Ž 21. by differentiation with respect to volume P Ž T ,V . s y
EA
ž / EV
RT s T
y V0 c
V0c ED A Vc
ž / EDh
q P0 Ž T . q T
V0c Vc
EK
ž / EDh
Ž 24. T
Eqs. Ž16. – Ž23. completely determine the crossover Helmholtz free energy for the PT equation of state. Asymptotically close to the critical point q < 1 Ž
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
13
Table 1 System-dependent constants for CO 2 , H 2 O, R32 and R125 CO 2
H 2O
R32
R125
Classical critical parameters P0c ŽMPa. 7.3830 T0c ŽK. 304.210 Z0c 0.31330
22.064 647.130 0.26950
5.7950 351.350 0.28195
3.6290 339.330 0.31136
Critical shifts DTc ŽK. DVc Žl moly1 .
6.5000=10y2 y9.9035=10y3
0 y2.0299=10y2
0 y3.2057=10y2
Classical PT EOS parameters c1 0.63199 c2 y2.69935 c3 0 N 0
0.60462 y2.56713 0 0
4.69539 y7.45749 y0.81188 2.0
y0.79912 0.32801 y3.86059=10y2 2.0
CrossoÕer parameters Gi 6.18200=10y2 d1 0 d2 y2.69935
1.02977=10y1 4.68835 y1.07896
5.95804=10y2 2.71799 y1.91253=10
6.39983=10y2 1.98093 y1.04839=10
1.5500=10y1 y1.3213=10y2
Fig. 1. PVT data w35,36x for carbon dioxide with predictions of the cubic crossover equation of state Žsolid curves. and the Patel–Teja equation Eq. Ž8. Ždashed curves..
14
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
values t ™ DT and Dh ™ DV, and Eq. Ž21. is transformed into the classical Helmholtz free energy density Ž15. for the original PT equation of state Ž 8.. In the limit of zero density V ™ `, q 4 Gi for all values of the temperature and the Ginzburg number, and Eq. Ž21. is transformed into the ideal gas expression A Ž T ,V . s yln Ž VrV0c . q A 0 Ž T . Ž 25. Ž . and Eq. 24 reproduces the ideal gas equation of state PV s RT Ž 26. The crossover cubic equation of state defined by Eqs. Ž 16. – Ž 21. , except for the parameters in the PT EOS, contains the following additional system-dependent parameters: critical shifts DTc and DVc , the Ginzburg number Gi, and the rectilinear diameter amplitudes d1 and d 2 . Since the critical parameters Tc and Vc of a pure fluid are usually known, the critical shifts can be easily determined by DTc s Tc y T0c and DVc s Vc y V0c Ž 27. if the classical critical parameters T0c and V0c are known. In practice, the simplest way is to set DTc s 0, and consider the shift of the critical volume, DVc , only. The parameters Gi, d1 and d 2 , similar to the coefficients c i Ž i s 1,2,3. in the PT EOS, can be represented as functions of the acentric factor V w29x; however, in the present work we find them from a fit of Eqs. Ž16. – Ž21. to experimental PVT data.
Fig. 2. VLE data w35x for carbon dioxide with predictions of the cubic crossover equation of state Žsolid curves. and the Patel–Teja equation Eq. Ž8. Ždashed curves..
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
15
Fig. 3. PVT and VLE data for water with predictions of the cubic crossover equation of state Žsolid curves. and the Patel–Teja equation Eq. Ž8. Ždashed curves.. Empty circles represent experimental data obtained by Rivkin et al. w38–41x, and filled circles correspond to the data generated with the fundamental equation of state of Pruss and Wagner w42x.
Fig. 4. PVT data w44x Žempty circles., w46x Žempty squares., and w43x Žfilled triangles., for R32 with predictions of the cubic crossover equation of state Žsolid curves..
16
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
5. Comparison with experimental data It is well-known that cubic equations of state give poorer predictions of the densities and pressures of liquids than of gases, especially for polar components and strongly hydrogen bonded fluids, like water w30,31x. Water is essentially a nonclassical system. Even such multiparameter analytical equations as the equation of Benedict–Webb–Rubin w32x and the equation of Lee–Kesler w33x Ž which can be successfully applied for the description of the thermodynamic properties of other fluids. fail in their application to water even far away from the critical point Žsee for example Ref. w31x.. For this reason, water and other polar fluids are good examples for testing the predictive capability of the model. We consider here the volumetric properties of carbon dioxide and water in the one- and two-phase regions and compare the cubic crossover equation of state with experimental data. Refrigerant mixtures such as R125q R32 are of interest as candidate to replace ozone-depleting refrigerants. Most models that are currently used for predicting the thermophysical properties of this mixture cannot accurately represent properties in and around the critical region. The parametric crossover model developed recently by Kiselev and Huber w34x gives a good representation of the thermodynamic surface for this system in and beyond the critical region, but fails at low densities and temperatures. Therefore, we also applied our crossover model to represent the thermodynamic properties of refrigerants R32 and R125.
Fig. 5. PVT data w48x Žempty circles. and w51x Žfilled squares., for R125 with predictions of the cubic crossover equation of state Žsolid curves..
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
17
For carbon dioxide and water, PT equations of state already exist, and for the classical critical parameters P0c , T0c and Z0c , for the coefficients c1, c 2 , c 3 , and the exponent N in Eq. Ž14. for carbon dioxide and water we adopt the same values as obtained by Patel w28x. The classical critical parameters for CO 2 and water reported by Patel w28x differ from the experimental values reported by Duschek et al. w35x Gilgen et al. w36x for CO 2 , and to the critical parameters adopted by Levelt Sengers et al. w37x for water; therefore, for CO 2 and water we use both shifts, DTc and DVc , in Eqs. Ž16. and Ž17.. The Ginzburg number Gi and the rectilinear-diameter amplitudes d1 and d 2 in Eq. Ž20. have been found from a fit of our crossover model to the experimental PVT data obtained by Duschek et al. w35x Gilgen et al. w36x for CO 2 , and to the experimental PVT data obtained by Rivkin et al. w38–41x for water. Additional PVT data along separate isochores at temperatures T ) 750 K and VLE data for water were generated with the fundamental equation of state developed by Pruss and Wagner w42x. The values of all system dependent constants for CO 2 and water are listed in Table 1. A comparison of the cubic crossover equation of state with experimental data and with the original PT equation of state is shown in Figs. 1 and 2 for carbon dioxide, and in Fig. 3 for water. As one can see, the crossover cubic equation of state gives a much better representation of thermodynamic properties of carbon dioxide and water than the original PT equation of state, especially in the critical region. The average deviation of pressures calculated with the crossover cubic EOS from the experimental data in the density range 0.5 rc F r F 1.5 rc is about 0.5% for all isotherms shown in Figs. 1 and 3. The crossover equation clearly yields a major improvement in the representation of the
Fig. 6. VLE data for R32 obtained by Holcomb et al. w52x, by Malbrunot et al. w46x, by Defibaugh et al. w44x, by Higashi w53x, by Magee w45x and by Malbrunot et al. w46x for R32, with predictions of the crossover cubic equation of state Žsolid curves.. The dashed curves calculated with the parametric crossover model w34x ŽCREQS97..
18
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
liquid densities along the saturated curve and the PVT data at high densities Ž r G rc . in the supercritical region where the original PT EOS yields up to 15% deviations for the saturated liquid densities Ž see Fig. 2. , and up to 20–25% for pressures Ž see Figs. 1 and 3.. At low densities r < rc and in the limit of zero densities. Both equations yield the same results in the entire temperature range and solid and dashed curves in Figs. 1 and 3 practically coincide. For R32 and R125, we adopt for P0c and T0c the same values of the critical parameters as used in the parametric crossover model by Kiselev and Huber w34x. The classical critical compressibility factor Z0c , the shift of the critical volume DVc , and all other system-dependent parameters in the PT equation of state Ž8. and in the CUCREQ were found from a fit of Eqs. Ž 16. – Ž 23. Eqs. Ž 24. – Ž 26. to experimental PVT data. For R32, the PVT data were taken from Refs. w43–46x and for R125 we used Refs. w47–51x. We also used in the fit few additional experimental VLE data points from Refs. w52–54x. The values of all system-dependent constants for R32 and R125 are presented in Table 1. Comparisons of the crossover cubic equation of state with experimental PVT data in the one-phase region are shown in Figs. 4 and 5. A reasonably good agreement between the crossover cubic equation of state ŽCUCREQ. and experimental data is observed. The average deviation of the calculated pressures from the experimental data in the range of densities 0 F r F 1.8 rc for all isotherms is about 1–2%, which is approximately 2–3 times less than the accuracy achieved for pure
Fig. 7. VLE data for R125 obtained by Magee w45x, by Boyes and Weber w50x, by Ye et al. w49x, by Duarte-Garza et al. w47x, by Defibaugh and Morrison w48x and by Kuwubara et al. w54x, with predictions of the crossover cubic equation of state Žsolid curves.. The dashed curves calculated with the parametric crossover model w34x ŽCREQS97..
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
19
R32 and R125 with the parametric crossover model ŽCREQS97. w34x which was optimized for the critical region only. At densities r G 1.8 rc the deviations between the calculated and experimental values of pressures increase and reach 10–20% at near critical isotherms at densities r ( 2 rc . As we mentioned above, a simple two- or three-parameter cubic equation of state, in principle, cannot describe the PVT surface of dense fluids. Since far away from the critical point, at densities r G 2 rc , the crossover PT equation of state and the original Eq. Ž8. practically coincide, in order to improve a description of the PVT surface of dense fluids with the crossover equation of state, one needs to improve the original PT equation of state. Comparisons of the cubic crossover equation of state with VLE data and with the parametric crossover equation of state obtained by Kiselev and Huber w34x for R32 and R125 are shown in Figs. 6 and 7. These figures show that in the critical region at 0.95Tc F T F Tc , the cubic crossover EOS and the parametric crossover model give essentially identical results. However, unlike the parametric crossover model w34x which is restricted to temperatures T G 0.8Tc , the cubic crossover EOS yields a good representation of vapor pressures and saturated liquid and vapor densities over the entire range of temperatures from the critical temperature down to the triple point.
6. Discussion In this paper we develop a simple method for incorporating of scaling laws in an analytical equation of state. Unlike the crossover approach proposed by van Pelt et al. w22x, the crossover
Fig. 8. The isochoric specific heat Cy of water at the critical isochore as a function of temperature. The solid curve represents values calculated with the cubic crossover equation of state, the dashed curve calculated with the Patel–Teja equation Eq. Ž8., the symbols indicate values generated with the parametric crossover model w19x.
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
20
function in our method does not depend on the particular type of classical EOS to be renormalized and can be applied to any analytical equation of state. In the present paper, we use the modified Patel–Teja cubic equation of state w28x as an example. The crossover cubic equation of state reproduces the non-analytical, singular behavior asymptotically close to the critical point and is transformed into the original Patel–Teja equation of state far from the critical point. In the limit of zero density, the crossover equation is transformed into the ideal gas equation of state. Application of this equation to PVT and VLE data for carbon dioxide, water, and refrigerants R32 and R125 shows that the crossover cubic equation yields a satisfactory representation of the thermodynamic surface of pure fluids in a temperature range from the triple point to T F 2Tc and a density range r F 2 rc . Since in the present paper we consider the volumetric properties only, the coefficient a 20 in the kernel term, which determines the asymptotic critical behavior of the isochoric specific heat Cy Žsee Eq. Ž5.. , was set equal zero. However, it is easy to show that with the kernel term, the crossover cubic equation of state is able to reproduce a singular behavior of the isochoric specific heat in the critical region. The results of our calculations of Cy for water with a 20 s 9.21 are shown in Fig. 8. As one can see, the crossover cubic equation of state does reproduce an asymptotic singular behavior of the isochoric specific heat at the critical isochore unlike the analytical cubic equation of state. In order to extend the range of validity of our crossover equation to a wide range of densities, a better analytical equation of state has to be chosen w55x. In the present paper we restrict the application to the cubic equations only because of their widespread using for engineering’s calculations of the phase behavior in binary- and multi-component fluid mixtures. The method described here can be also applied to mixture calculations. Further research toward this goal is in the progress and the results will be presented in the next publication.
7. List of symbols a ai j A b c di Gi K N P q R RŽ q . T Tr V Y Z
coefficient of cubic equation of state coefficients of Landau expansion Helmholtz free energy pert mole coefficient of cubic equation of state coefficient of cubic equation of state coefficients of rectilinear diameter Ginzburg number kernel term exponent pressure argument of crossover function gas constant crossover function temperature dimensionless temperature molar volume crossover function compressibility factor
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
21
Greek letters a critical exponent b critical exponent g critical exponent D difference D1 critical exponent Dh order parameter t reduced temperature difference Vi dimensionless coefficients of cubic equation of state r molar density Subscripts c critical L Landau theory 0 classical Acknowledgements The author is indebted to D.G. Friend, J.C. Rainwater and A.S. Teja for valuable discussions which stimulated this work. The author also would like to thank the Physical and Chemical Properties Division, National Institute of Standards and Technology for the opportunity to work as a Guest Researcher at NIST during the course of this research. Appendix A. Helmholtz free energy for the PT equation of state The PT equation of state RT a P Ž T ,V . s y Vyb V ŽV q b. q cŽV yb.
Ž A.1 .
can be written in the dimensionless form P Ž T ,V . s
PV0c
1 s
RT
y DV q b 1
T0c V a
a0 Ž T .
T Z0c Ž DV q b 2 .Ž DV q b 3 .
Ž A.2 .
where DT s TrT0c y 1, DV s VrV0c y 1, and the coefficients bi Ž i s 1,2,3. and V are expressed through the dimensionless parameters V a , V b , V c , and the critical compressibility factor Z0c : Vb b1 s 1 y Ž A.3 . Z0c b2 s 1 q b3 s 1 q
V b q Vc y V 2 Z0c
V b q Vc q V 2 Z0c
Ž A.4 . Ž A.5 .
S.B. KiseleÕr Fluid Phase Equilibria 147 (1998) 7–23
22
(
V s V b2 q 6 V b V c q V c2
Ž A.6 .
At the critical isochore V s V0c , and Eq. ŽA.2. reads P0 Ž T . s
1 y b1
T0c V a a 0 Ž T .
Ž A.7 .
T Z0c b 2 b 3
The Helmholtz free energy A Ž T ,V .
s yHPdVq A 0 Ž T . s yRT ln Ž DV q b 1 . q RT0c
Va a0 Ž T . V
ln
ž
DV q b 2 DV q b 3
/
Ž A.8 .
q A0 ŽT .
where A 0 Ž T . corresponds to the temperature dependent function of the ideal-gas part of the Helmholtz free energy. The dimensionless Helmholtz free energy can be represented in the form A Ž T ,V . s
A Ž T ,V . RT
s D A Ž DT ,DV . y
V V0c
P0 Ž T . q m 0 Ž T .
Ž A.9 .
where D AŽ DT, DV . corresponds to the critical part of the Helmholtz free energy, and m 0 Ž T . is an analytical function of the temperature. The functions D A and m 0 are found from the condition D A Ž DT ,DV s 0 . s 0, and
ED A
ž / EDV
s0
Ž A.10 .
DT Ž DVs0 .
With the Helmholtz free energy as given by Eq. ŽA.8. we have D A Ž DT ,DV . s yln
ž
DV b1
/
q1 q
T0c V a a 0 Ž DT .
V
T
ln
ž
DVrb 2 q 1 DVrb 3 q 1
/
DV q
y b1
T0c V a a 0 Ž DT . DV T
Z0c b 2 b 3 Ž A.11 .
and
m 0 Ž T . s P0 Ž T . y ln b 1 q
T0c V a a 0 Ž T . T
V
ln
b2
ž / b3
q A0 ŽT .
Ž A.12 .
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